c109 Module Questions and more

For analyzing graphs, consider these three levels of graph comprehension (Friel, Curcio, & Bright's, 2001):

1. Reading the data (literal), 2. Reading between the data (making comparisons 3. Reading beyond the data (making inferences).

Tiered Lesson

1. The degree of assistance. 2. How structured the task is 3. The complexity of the task given. 4. The complexity ofprocess.

Student D says, "In 100 spins, the blue will appear 20 times.

1/5 chance to occur Correct! Student D simply indicates that the probability will be 20/100, which simplifies to ⅕.

Module 5 part 2

Teaching Diverse Students

Avoid conflicting objectives in introductory lessons.

Is your lesson about what it means to measure area or about understanding square centimeter

3. The complexity of the task given.

This can include making a task more concrete or more abstract or including more difficult problems or applications.

Virtual number lines

Tool Correct! Virtual number lines are a tool to explore mathematical operations, integers, and more.

1Multistep Word Problems.

Two-step word problems appear for the first time in the Common Core State Standards (NGA Center & CCSSO, 2010) when second-graders are expected to solve two-step addition and subtraction word problems. Then two-step word problems in all four operations are part of the third-grade standards and multistep problems in all four operations with whole numbers are expected starting in fourth grade, including problems with remainders that must be interpreted. To start, be sure children can analyze the structure of one-step problems in the way that we have discussed.

Unit 10 Quiz

Unit 10 Quiz

Unit 10: Mathematical Representation

Unit 10: Mathematical Representation

Unit 6 Practice Quiz - Assessment

Unit 6 Practice Quiz - Assessment

Unit 7 Practice Quiz - Tools for Math

Unit 7 Practice Quiz - Tools for Math

Unit 7: Tools for Mathematics

Unit 7: Tools for Mathematics

Unit 8 Practice Quiz—Interdisciplinary Learning Experiences as Context for Math Instruction

Unit 8 Practice Quiz—Interdisciplinary Learning Experiences as Context for Math Instruction

Unit 8

Unit 8: Interdisciplinary Learning Experiences as Context for Math Instruction

The use of bar diagrams (also called strip or tape dia-grams)

as semiconcrete visual representations is a central fixture many international curricula and are increasingly found in U.S. classrooms. As with other tools, these diagrams support students' mathematical thinking by generating "meaning-making space" (Murata, 2008, p. 399) and are a precursor to the use of number lines

Unit 6:

Assessment

8) Think-pair-share

Formative Assessment Correct! Think-pair-share is a type of peer assessment where students share their ideas with a classmate to compare their thinking.

Software packages like spreadsheets

Non-Tool Correct! Entire software packages are not considered tools.

A teacher wants to show students their test scores without revealing their names. Which two graph types could be used? A) Stem-and-leaf plot B) Line graph C) Line plot D) Histogram E) Circle graph F) Bar graph

A) Stem-and-leaf plot C) Line plot Correct! A line plot and stem-and-leaf plot can be used. In a line plot the teacher could provide a numeric scale that shows the range of all test scores (72-100, for example) and place an X above each score received by a class member. A stem-and-leaf plot could also be used to display test scores without student names. Scores can be grouped by tens, and the digit in the tens place is in the stem column, with the digits on the right being placed in the leaves column.

Operational Symbol

+, -. / X addition, subtraction, division, and mulitplication symbols

Find the similarities and differences of a wide variety of shapes. A)Level 0: Visualization B)Level 1: Analysis C)Level 2: Informal Deduction

B) Level 1: Analysis Correct! By finding similarities and differences between shapes, students are operating at Level 1 if they are discovering properties of these shapes and putting them in certain classes.

A manager wants to show employee salaries at a company without revealing employee names. Which type of graph should be used to display this data? A)Histogram B)Line graph C)Stem-and-leaf plot D)Bar graph

C)Stem-and-leaf plot

Community Assets:

Common backgrounds and experiences that students bring from the community where they live, such as resources, local landmarks, community events, and practices that a teacher can draw upon to support learning.

Permit students to use technology when appropriate.

For example, except for tests of computa-tional skills, calculators can allow students to focus on what you really want to test.

Researchers suggest three strands of algebraic reasoning, all infusing the central notions of generalization and symbolization (Blanton, 2008; Kaput, 2008). We use these themes to organize this chapter as illustrated here (section headers in parenthesis)

1. The study of structures in the number system, including those arising in arithmetic. (Con-necting Number and Algebra and Properties of the Operations) 2. The study of patterns, relations, and functions (Patterns and Functions) 3. The process of mathematical modeling (Meaningful Use of Symbols and Mathematical Modeling)

Unit 5:

Differentiated Mathematics Instruction

Communitive Property

States that numbers are free to move around in addition and multiplication and the results will always be the same

40-13= 33

Student subtracted

The before phase

is important when using learning centers, as it is still important to elicit prior knowledge, ensure the task is understood, and establish clear expectations. It may include modeling what happens at each center.

There are four basic categories for adapting mathematics content for gifted mathematics students:

(1) acceleration and pacing, (2) enrichment (depth), (3) sophistication (complexity), and (4) creativity (Johnsen,

Say Decimal Fractions Correctly

Make sure you are reading and saying decimals in ways that support students' understand-ing and links to fraction numeration. Always say "five and two-tenths" instead of "five point two." Using the point terminology results in a disconnect to the fractional part that exists in every decimal. This practice of attending to precision in language provides your students with opportunities to hear connections between decimals and fractions, so that when they hear "two-tenths," they think of both 0.2 and 2/10. Research shows that labeling decimals by place value is particularly useful with supporting students' understanding of decimal comparisons when the role of a zero had to be considered

1Part-Part-Whole Problems.

Part-part-whole problems also known as put together and take apart problems in the Common Core State Standards (NGA Center & CCSSO, 2010), involve two parts that are conceptually or mentally combined into one collection or whole, as in Figure 8.1(c). These problems are different from change problems in that there is no action of physically joining or separating the quantities. In these situations, either the miss-ing whole (total unknown), one of the missing parts (one addend unknown), or both parts (two addends unknown) must be found.

After students solve a problem, you ask that they explain how they arrived at their solution

Reasoning Correct! Remember that allowing students to explain their solutions is a way for them to demonstrate their reasoning skills as well as justifying and communicating their solutions. You will also want to make sure that the student's reasoning progresses in a logical order.

1/5 A teacher is demonstrating probability with a spinner divided into six equal sections, each in a different color. After 3 spins, the outcomes are yellow, blue, yellow. Students are asked the probability of landing on blue next.

Select the student reasoning with the response.

Students were asked to compute the area of the shaded region of the following triangle. Identify whether each student's reasoning is based on a conception or misconception

Students were asked to compute the area of the shaded region of the following triangle. Identify whether each student's reasoning is based on a conception or misconception

6) End of term tests

Summative Assessment Correct! An end of term test is separate from state assessments since they come from you and will also be a big part of the students' grade. These tests will also take in material from all the units you have covere

Avoid interjecting clues or teaching.

The temptation to help is sometimes overwhelming. Watch and listen. Your goal is to use the interview not to teach but to find out where the student is in terms of conceptual understanding and procedural fluency.

1Contextual Problems.

There is more to think about than simply giving students problems to solve. In contrast to the rather straightforward and brief problems given in the previous section, you also need to set more complex problems in meaningful contexts.

Addition

When the parts of a set are known, addition is used to name the whole in terms of the parts. This simple definition of addition serves both action situations (join and separate) and static or no-action situations (part-part-whole)

A line plot

is used to show the frequency of data on a number line and is great to use with children in the early grades. Example: You can have your students gather data about the weight

Which of the following represents a misconception elementary students have about probability? A) The probability of rolling a 6 goes down every time a 6 is rolled. B) Probability can be expressed as a fraction, decimal, or percent. C) When performing a probability experiment, with replacement, the theoretical probability will not change. D) When calculating probability, the total number of possible outcomes should be the denominator. .

A) The probability of rolling a 6 goes down every time a 6 is rolled. Correct! In the above dice example, it would be common for young students to think that because a 1 or a 6 had already been rolled, then it is less likely that those numbers will come up again

All of the following are true about probability except __________. A) probability changes based on the number of trials B) probability can be determined in two ways: theoretically and experimentally C)experimental probability gets closer to theoretical probability with a larger data set D) probability of an event is a measure of the likelihood of an event occurring

A) probability changes based on the number of trials Correct! Theoretical probability is the likeliness of an event happening based on all possible outcomes. The theoretical probability will not change based on the number of trials. For example, the theoretical probability of rolling a 4 on a six-sided die is 1/6. The probability will not change as more trials are conducted. The theoretical probability of rolling a 4 will remain 1/6.

The mean of a data set is 12. How can the mean decrease? A)By adding two sixes to the data set B)There is not enough information. C)By adding a 12 to the data set D)By adding a 14 to the data set

A)By adding two sixes to the data set

1) Select all the planning strategies for all learners. A) Use different learning objectives for lower- and higher-level learners B) Include opportunities for collaboration and classroom discussion C) Use graphic organizers to help students organize work D) Do drill exercises to help students learn new concepts E) Choose a step-by-step method for all students

B) Include opportunities for collaboration and classroom discussion C) Use graphic organizers to help students organize work

Determine the mean, median, mode, and range for the following data set: 18, 18, 46, 2. A) Mean: 21; median: 16; mode: 18; range: 16 B) Mean: 21; median: 18; mode: 18; range: 44 C) Mean: 21; median: 18; mode: 44; range: 18 D) Mean: 21; median: 16; mode: 18; range: 16 .

B) Mean: 21; median: 18; mode: 18; range: 44 Correct! To determine the mean, add up all the values and divide by the number of addends. To find the median, arrange the numbers in numerical order and locate the number in the middle. To calculate the range, subtract the lowest value from the greatest value. To find the mode, determine which numbers appear the most

Complete the following sentence. An event in a probability experiment __________. A) is always the same as the sample space B) is a subset of the sample space C)must be part of a two-stage or multistage experiment D) must be part of a single-stage experiment

B) is a subset of the sample space Correct! A subset of an event is considered to have happened. In probability theory, the sample space of an experiment is the set of all possible outcomes or results of that experiment. If, for example, you teach your students about sample space by rolling a single die, then the sample space is the numbers 1 to 6.

When a student employs multiple representations, what have they demonstrated? A) Ability to recall facts with automaticity B) Instrumental understanding C) Relational understanding D) Recursive thinking

C) Relational understanding

Complete the following sentence. An event in a probability experiment __________. A) must be part of a single-stage experiment B) must be part of a two-stage or multistage experiment C) is a subset of the sample space D) is always the same as the sample space

C) is a subset of the sample space Correct! A subset of an event is considered to have happened. In probability theory, the sample space of an experiment is the set of all possible outcomes or results of that experiment. If, for example, you teach your students about sample space by rolling a single die, then the sample space is the numbers 1 to 6.

Classroom Questions Students want to learn about each other, their families and pets, measures such as arm span or time to get to school, their likes and dislikes, and so on. The easiest questions are those that can be answered by each class member contributing one piece of data. Here are a few ideas:

• Favorites: TV shows, games, movies, ice cream, video games, sports teams, music (when there are lots of possibilities, start by restricting the number of choices) • Numbers: Number of pets or siblings, hours watching TV or hours of sleep, bedtime, time spent on the computer • Measures: Height, arm span, area of foot, long-jump distance, shadow length, seconds to run around the track, minutes spent traveling to sch

Unit 5 Summary 3

In a constructivist classroom, teachers should avoid providing too much support. The goal is for all students to actually meet the standard or objective, but if teachers take over too much of the problem-solving process for students, the productive struggle is eliminated. While teachers want all students to meet the learning objective, they can provide reasonable accommodations such as manipulatives to help solve problems, or in limited cases, a step-by-step procedure for struggling students.

Unit 3

Instructional Strategies

MODULE 8 TOOLS FOR MATHEMATICS

MODULE 8 TOOLS FOR MATHEMATICS

3The best approach to improving estimation skills is to have students do a lot of estimating. Keep the following tips in mind:

5. Do not promote a "winning" estimate. This emphasis on a contest discourages estimation and promotes only seeking the "exact" answer. 6. Encourage students to give a range of estimates that they believe includes the actual measure. For example, the door is between 7 and 8 feet tall. This focus on reasonable minimum and maximum values, not only is a practical real-life approach but also helps focus on the approximate nature of estimation.

1)Select all instructional strategies that can be used to effectively hold small-group and classroom discussions. A) Discuss the correct answer and the procedure only one student used B) Have students share multiple methods about how to solve the problem C) Encourage students to think-pair-share and compare solutions and strategies D) Trade and grade papers so student can see how others solved the problem

B and C B) Have students share multiple methods about how to solve the problem C) Encourage students to think-pair-share and compare solutions and strategies

If you want to teach the concept of shape transformations (i.e., rotations, reflections, and dilations), which two tools or technology resources could you use? A) A compass B) Geometric modeling software C) Centimeter grid paper D) Graphic organizers

B) Geometric modeling software C) Centimeter grid paper Correct! Grid paper can be used to help students draw shapes, see relationships between the transformation, and teach concepts such as area. Geometry software programs will allow students to create and manipulate geometric shapes.

A student wants to construct a graph to show the population growth in a city over 10 years. Which type of graph would best show this data? A)Stem-and-leaf plot B)Line graph C)Bar graph D)Histogram

B)Line graph

When teaching a unit on geometry and symmetry, which tool or manipulative can be used to enhance learning outcomes? A) Base-ten blocks B) Calculator C) Dynamic geometry programs D)Cuisenaire rods

C) Dynamic geometry programs

Which of the following statements is true of the van Hiele model for geometric reasoning? A) Proof is the ultimate goal. B) Levels are age-dependent. C) Levels are sequential. D) Geometric communication is vital.

C) Levels are sequential. Correct! The van Hiele model is a five-level hierarchy of ways to understand spatial ideas. Each level describes the thinking processes used in geometric contexts. These levels are sequential and are not age-dependent.

A teacher is planning a lesson on mean and poses the following data set to students: 6, 8, 7, 3. A student responds and says the mean is 6. The teacher then changes the data set to: 6, 6, 8, 7, 3. How will the mean change? A) The mean will increase. B) The mean will decrease. C) The mean will not change. D)The mean will increase by 1 value.

C) The mean will not change. Correct! Practice this problem using a calculator and a whiteboard and check the work. Try calculating the answers with and without adding the extra 6. What happens? Now try it again with another set of numbers. Adding the mean to a set of numbers will not affect the mean of the entire set.

Combination Problems.

Combination problems involve counting the number of possible pairings that can be made between two or more sets (things or events). This structure is more complex and therefore not a good introductory point for multiplication, but it is important that you recognize it as another category of multiplicative problem structures. Students often start by using the model shown in Figure 8.10(e) where one set is the row (pants) and the other the column (jackets) in a matrix format. Counting how many combinations of two or more things or events are possible is important in determining probabilities—a seventh grade standard

2) What is an example of a peer-assisted teaching strategy? A) A student works with the teacher to solve math-related questions. B) A teacher explains how to solve a complex math problem. C) A student uses a manipulative to solve fraction equations. D) An older student partners with a younger student to explain a math problem.

D) An older student partners with a younger student to explain a math problem. Correct! This question is asking for an example of a peer-assisted learning strategy, which is not a best practice. Generally, it is not an effective strategy to pair a higher-level student with a lower-level student. However, this question is asking for an example of when a peer could help another peer. You may teach fifth grade and once a month go to the first-grade classroom to help them with some math concepts. Teachers should not do this daily, but once a month would be appropriate.

The mean of a data set is 12. Determine how the mean can increase. A) There is not enough information. B) By adding a 12 to the dataset C) By adding two sixes to the data set D) By adding a 14 to the data set

D) By adding a 14 to the data set Correct! The mean will increase when the additional data has a mean that is greater than the original mean. Here, the original mean is 12 and by adding the 14 this will shift the distribution, causing the mean to increase.

Which instructional strategy can the teacher use that will make use of constructivist teaching while also providing sufficient scaffolding and modeling for students by making known which steps they need to take and when? A) Think-alouds B) CSA sequence C) Peer-assisted learning D) Explicit strategy instruction

D) Explicit strategy instruction Correct! Explicit strategy instruction requires that you, the teacher, are carefully guiding students through the problem-solving process, making known thinking strategies that are involved in figuring out a specific problem. Modeling and scaffolding is an important part of this strategy as it will help students to see the approach that needs to be taken. This approach is constructivist in the sense that the teacher should be building on the students' prior knowledge and experience to help make sense and meaning of the problem.

You are teaching a lesson on adding fractions and want to relate this concept outside of the math curriculum. How could this be taught? A) Students use a fraction model to help solve problems. B) Students could complete a homework task on adding fractions. C) Students could complete an adding fractions quiz. D) Students use a recipe with 1/2 cup of flour, 1/4 cup of milk, and 1/3 cup of butter to determine the total amount of ingredients.

D) Students use a recipe with 1/2 cup of flour, 1/4 cup of milk, and 1/3 cup of butter to determine the total amount of ingredients. Correct! Relating curriculum to outside events helps contextualize the mathematical concepts. When you attempt to make a mathematical connection outside of the classroom, you are trying to relate the math concepts to real life. In this example, students are making a connection to a recipe and using math to determine a new portion size.

Unit 2

National, State, and Local Mathematics Standards

2) A drill is a form of ______that is best used when students are familiar with the process

Review Correct! Drill will help when conceptual understanding is already established.

Relational symbol

equal sign = means balances

In the after phase

you may decide to focus on one particular center, or you may begin with the center that was the least challenging for students and progress to the one that was most challenging. Or, you may not discuss what was done at each center, but instead ask students to talk about what they learned about the number 6, for example

Given the following table, which representation is correct? x 1 2 3 4 y 4 7 10 13 (a.)y = 3x (b.) illustrated graph (c.)The value of y is three times the value of x. (d.) illustrated graph

(d.) illustrated graph Correct! The table represents a linear function and the slope of the line is constant. To graph the points, apply each value as a coordinate point. For example, for the coordinate point (1, 4), go over one and up four on the grid.

Some common misconceptions in geometry:

*Confusing area, perimeter, surface area, and/or volume measurements/calculations *Triangle area calculation errors; do not understand the relationship between the area of a triangle and a quadrilateral that share the same base and height *Misunderstanding shared characteristics of quadrilaterals (i.e., a square is a rectangle but a rectangle is not always a square)

Some common number sense misconceptions:

*Confusing place value when writing numbers, especially decimal place value *Making place-value errors during mathematical calculations (i.e., 25 + 3 = 55) *Errors in fraction models *When using a traditional algorithm for multiplying multi-digit numbers, students do not use place value appropriately when regrouping. *Students believe that they should always get a smaller answer in a subtraction or division problem. Students also believe that they should always get a larger answer in an addition or multiplication problem. *Students often think that the divisor must be less than the dividend and that is not true. *Students can often choose the wrong operation when working on word problems or math expressions/equations. *Students think there is no relationship between multiplication and division.

When exploring with the virtual tools, consider these questions:

*Is this tool easy to drag around and manipulate? *Does this tool have any customizable features? *Is the tool user-friendly? *Does the tool support independent practice, or does the tool require a high level of teacher support? *Does the tool have any automatic features (snapping together, counting, etc.)?

Five Adaptations in Planning for All Learners The following five adaptions are not a process to follow but represent options that you can choose from as you meet the needs of your diverse range of students.

*Make accommodations and modifications *Differentiating instruction *Learning centers *Tiered lessons *Flexible grouping

As an example, consider what is important for a student to know about a fraction such as 6/8? At what point do they know enough that they can claim they "understand" fractions? It is more com-plicated than it might first appear. Here is a partial list of what they might know or be able to do

*Read the fraction *Identify the 6 and 8 as the numerator and denominator, respectively. *Recognize it is equivalent to 3/4. *Know it is more than 1/2 (recognize relative size) *Draw a region that is shaded in a way to show 6/8 *Locate 6/8 on a number line. *Illustrate 6/8 of a set of 48 pennies or counters. *Know that there are infinitely many equivalencies to6/8 *Recognize 6/8 as a rationale number *Realize 6/8 might also be describing a ratio (girls to boys, for example *Be able to represent 6/8 as a decimal fraction.

Select all misconceptions about place value. -A student writes 9 × 5 = 14. -A student writes 10 - 5 = 2. -A student writes three tenths as 0.03. -A student writes six-hundredths as 0.06. -A student concludes that 304 = 30 + 4.

-A student writes three tenths as 0.03. -A student concludes that 304 = 30 + 4. Correct! The student who writes three tenths as 0.03 has a misconception about decimal place value. Three tenths should be written as 0.3. Concluding that 304=30+4 is also a misconception because the student made a conclusive incorrect statement. The student in this example has a place-value misconception that hindered their ability to write the expanded form of a number.

Array and Area Problems The array is a model for an equal-group situation (Figure 8.10[c]). It is shown as a rectangular grouping, with the first factor (the number of groups) representing the number of rows and the second factor (the number of items in each equal group) representing the equal number found in each row (number of columns). This structure is the U.S. convention for what each factor represents. CCSS-M groups arrays with area rather than with the equal-group problems because arrays can be thought of as a logical lead-in to the row-and-column structure of an area problem

. But remember an array can be modeled with circular counters or any items as you see in the sample problems in Figure 8.10(c) (also using dots [Matney & Daugherty, 2013]). Yet, if you begin to use the small square tiles for the array and move the tiles tightly together, the result can be recorded on grid paper and connected more easily to the area problem structure. Using the discrete items first prepares students for the more sophisticated use of continuous units with measuring area

B. Ace of diamonds and ace of hearts have already been drawn and will not be drawn again.

0 Response for: undefined Correct! In response B, this student is indicating that the events that have already occurred will not occur again and therefore provides the answer 0 when asked what the theoretical probability is of drawing an ace. This is a common misconception among small children, that chance has a memory and probability is changed based on what has already happened.

Facts can be learned by rote memorization. This knowledge is still constructed, but it is not 1)_____ to other knowledge.

1) connected Correct! Once again, knowledge must be connected to prior knowledge in order to take on meaning and to be more easily retrieved from your memory.

to measure something, one must perform three steps:

1. Decide on the attribute to be measured. 2. Select a unit that has that attribute. 3. Compare the units—by filling, covering, matching, or using some other method—with the attribute of the object being measured. The number of units required to match the object is the measure

1Begin measurement activities with students making an estimate. Just as for computational estimation, students need specific strategies to estimate measures. Here are four strategies:

1. Develop benchmarks or referents. Students who have acquired mental benchmarks or reference points for measurements (for single units and also for useful multiples of standard units) and practice using them in class activities are much better estimators than students who have not learned to use benchmarks (Joram & Gabriele, 2016). Students must pay attention to the size of the unit to estimate well (Towers & Hunter, 2010). Referents should be things that students can easily visualize. One example is the height of a child (see Figure 18.3).

The National Council of Teachers of Mathematics' (NCTM's) position statement on Early Childhood Learning emphasizes that all children need an early foundation of challenging mathematics (2013). This document provides the following research-based recommendations:

1. Enhance children's natural interest in mathematics and assist them in using mathematics to make sense of their world. 2. Build on children's experience and knowledge using familiar contexts. 3. Base mathematics curriculum and teaching practices on a solid understanding of both mathematics and child development. 4. Use formal and informal experiences to strengthen children's problem solving and rea-soning processes. 5. Provide opportunities for children to explain their thinking about mathematical ideas. 6. Assess children's mathematical knowledge, skills, and strategies through a variety of for-mative assessment approache

1The best approach to improving estimation skills is to have students do a lot of estimating. Keep the following tips in mind:

1. Explicitly teach each strategy. After learning and practicing each strategy, students can choose from the options the one that works best in a particular situation. 2. Discuss how different students made their estimates. These conversations will confirm that there is no single right way to estimate while reminding students of other useful approaches

Teaching standard units of measure can be organized around three broad goals:

1. Familiarity with the unit. Students should have a basic idea of the size of commonly used units and what they measure. Knowing approximately how much 1 liter of water is or being able to estimate a shelf as 5 feet long is as important as measuring either accurately. 2. Ability to select an appropriate unit. Students should know both what is a reasonable mea-surement unit in a given situation and the level of precision required. Would you measure your lawn to purchase grass seed with the same precision as you would use in measuring a window to buy a pane of glass? Students need practice in selecting appropriate standard units and judging the level of precision. 3. Knowledge of relationships between units. Students should know the relationships that are commonly used, such as those between inches, feet, and yards or milliliters and liter

2Multistep Word Problems

1. Give students a one-step problem and have them solve it. Before discussing the answer, have the students use the answer to the first problem to create a second problem. The rest of the class can then be asked to solve the second problem. Here is an example: Given problem: It took 3 hours for the Morgan family to drive the 195 miles to Washington, D.C. What was their average speed? Second problem: The Morgan children remember crossing the river at about 10:30 a.m., or 2 hours after they left home. About how many miles from home is the river?

The following considerations can help maximize the value of your tests

1. The following considerations can help maximize the value of your tests 2. Include opportunities for explanations. 3. Use open-ended questions. 4. Permit students to use technology when appropriate.

1Here are four arguments against presenting the key word approach:

1. The key word strategy sends a terribly wrong message about doing mathematics. The most important approach to solving any contextual problem is to analyze it and make sense of it using all the words. The key word approach encourages children to ignore the meaning and structure of the problem. Mathematics is about reasoning and making sense of situations. Sense-making strategies always work!

Students struggle to understand and interpret histograms

1. Understanding the distinction between bar graphs and histograms 2. Finding the center of the distribution (students focus on the heights of the bars (y-axis) instead of the distribution (x-axis). 3. Interpreting a flatter histogram to mean there is less vari-ability in the dat

1) Instrumental Understanding

2) Having students memorize the multiplication table to be able to do their work fast Correct! This can be a helpful and even necessary activity, but make sure that students are able to use this for other activities that foster relational understanding.

3Multistep Word Problems

2. Make a hidden question. Repeat the approach above by giving groups different one-step problems. Have them solve the first problem and write a second problem. Then should write a single combined problem that leaves out the question from the first problem. That question from the first problem is the hidden question, as in this example: Given problem: Toby bought three dozen eggs for 89 cents a dozen. How much was the total cost? Second problem: How much change did Toby get back from $5? Hidden-question problem: Toby bought three dozen eggs for 89 cents a dozen. How much change did Toby get back from $5?

2Begin measurement activities with students making an estimate. Just as for computational estimation, students need specific strategies to estimate measures. Here are four strategies:

2. Use "chunking." Chunking involves subdividing a measurement into components to better estimate the amount. Figure 18.3 shows an example of chunking using windows, bulletin boards, and spaces between as chunks making the estimation easier. The weight of a stack of books is easier to estimate if some referent is given for the weight of an "average" book. But, if the wall length to be estimated has no useful chunks, it can be mentally subdivided in half and then in fourths or even eighths by repeated subdivisions until a manageable length is found. Length, volume, area, and surface area measurements all lend themselves to the estimation strategy of chunking.

2Here are ways to explore area of circles that build con-ceptual understanding for the formula:

2. Use radius squares. Draw a circle and draw a radius. Set the radius as the unit. Use the radius unit to make squares—we will call them radius squares. Form a grid that is 2 * 2 radius squares, and draw a circle inside. By observation, you see that the circle inscribed in this grid has an area of less than 4 radius squares (because much of the grid is outside the circle). Students need two identical copies of this picture—one to cut out the radius squares to see how many it takes to cover the circle in the other picture. Students can estimate that the number of radius squares needed is about 3.1 or 3.2. This technique reinforces the geometric idea that r 2 is a square with a side of

2The best approach to improving estimation skills is to have students do a lot of estimating. Keep the following tips in mind:

3. Create a list of benchmarks. Record and post on a class chart suggested benchmarks for common measures. 4. Accept a range ofestimates. Think in relative terms about what is a good estimate. Within 10 per-cent for length is reasonable. Even 30 percent "off" may be reasonable for weights or volumes.

3Here are ways to explore area of circles that build con-ceptual understanding for the formula:

3. Draw inscribed and superscribed squares. This approach is similar to the method that Archi-medes used to approximate π. Students draw a circle on grid paper with a given radius (or the circle can be drawn beforehand on a handout). They draw a square inside the circle and outside the circle (Figure 18.18). Find the areas of the two squares, and average them to find the area of the circle. Archimedes started using polygons that got closer to the shape of the circle (see the Archimedes Circles Activity Page for students to see how the estimates become closer). See NCTM Illuminations for full lessons in which this approach is used

4 Multistep Word Problems

3. Have other groups identify the hidden question. Because all students are working on a similar task (be sure to mix the operations), they will be more likely to understand what is meant by a hidden question. Pose standard two-step problems, and have the students identify and answer the hidden question. Consider this problem: The Marsal Company bought 275 widgets wholesale for $3.69 each. In the first month, the com-pany sold 205 widgets at $4.99 each. How much did the company make or lose on the widgets? Do you think the Marsal Company should continue to sell widgets?

3Begin measurement activities with students making an estimate. Just as for computational estimation, students need specific strategies to estimate measures. Here are four strategies:

3. Iterate units. For length, area, and volume, it is sometimes easy to mark off single units mentally (visualizing) or physically. You might use your hands or make marks to keep track as you go. If you know, for example, that your stride is about 3/4 meter, you can walk off a length and then multiply to get an estimate. Hand and finger widths are also useful

For any simulation, the following steps can serve as a useful guide. Here the steps are explained for use with Activity 21.17 1. Identify key components and assumptions of the problem. The key component in the water problem is the condition of a pump. Each pump is either working or not working. In this problem, the assumption is that the probability that a pump is working is 1/2. 2. Select a random device for the key components. Any random device can be selected that has outcomes with the same probability as those of the key component—in this case, the pumps. Here a simple choice might be tossing a coin, with heads representing a working pump. 3. Define a trial. A trial consists of simulating a series of key components until the situation has been completely modeled one time. In this problem, a trial could consist of tossing a coin five times, each toss representing a different pump (heads for pump is working and tails for pump is not working).

4. Conduct a large number of trials and record the information. For this problem, it would be useful to record the number of heads and tails in groups of five because each set of five is one trial and represents all of the pumps. 5. Use the data to draw conclusions. There are four possible paths for the water, each flowing through two of the five pumps. As they are numbered in the drawing, if any one of the pairs 1-2, 5-2, 5-3, and 4-3 is open, it makes no difference whether the other pumps are working. By counting the number of trials in which at least one of these four pairs of coins both come up heads, we can estimate the probability of water flowing. To answer the second question, the number of tails (pumps not working) per trial can be averaged. Here are a few more examples of problems for which a simulation can be used to gather empirical data.

4Here are ways to explore area of circles that build con-ceptual understanding for the formula:

4. Cut to make a parallelogram. Cut a circle apart into sec-tors and rearrange them to look like a parallelogram. For example, students can cut from 3 to 12 sectors from a circle and build them into what looks like parallelogram. Recall that in the angle investigation, students made a wax paper circle. This same circle can be remade and cut to form the pieces to explore area of a circle. You may need to help them notice that the smaller the size of the sectors used, the closer the arrangement gets to a rectangle. Figure 18.19 presents a common development of the area formula A = πr^2.

What are two appropriate uses for a calculator? A)Calculators can be used to improve student attitude and motivation. B)Calculators can be used to explore patterns and basic facts. C)Calculators can be used to teach concepts. D)Calculators can be used to practice computational skills.

A and B Correct! Research reveals that students who frequently use calculators have better attitudes toward subjects of math. There is also evidence that students are more motivated when their anxiety is reduced; therefore, supporting students during problem-solving activities is important. Students can also use a calculator to practice exploring patterns. For example, students may use a calculator to practice finding multiples of 7.

4. Select two learning theories that describe effective math learning. A) Students learn new ideas through collaboration with their teacher and peers. B) New math concepts should be taught in isolation. C) Students learn by rearranging previously learned concepts and making connections. D) Students should be presented with step-by-step procedures. E) Students learn best by being presented formulas and practice problems.

A and C A) Students learn new ideas through collaboration with their teacher and peers. C) Students learn by rearranging previously learned concepts and making connections.

Line Graph-

A line graph is used to track the relation between two pieces of data

To create an optimal environment for doing mathematics, the teacher should do which of the following? A) Create a spirit of inquiry, trust, and expectations. B) Demonstrate mathematical procedures. C) Focus on posing problems. D) Test ideas and conjectures and tell students the results.

A) Create a spirit of inquiry, trust, and expectations. Correct! In a constructivist math class, the teacher should act as a facilitator and ask guiding questions. This process takes time and the teacher must use class time effectively to build a classroom culture of respect and trust. Once this is established, students will feel comfortable talking about errors and learning from mistakes by a process called strategic competency.

7) Select all statements that describe challenges that students from diverse groups may face. A) Special education students may require extended time to complete a task. B) Students from diverse backgrounds may solve problems differently. C) Lower-level students benefit from different learning goals. D) Students with learning disabilities also struggle with organization.

A) Special education students may require extended time to complete a task. B) Students from diverse backgrounds may solve problems differently.

Acceleration and Pacing

Acceleration recognizes that your students may already understand the mathematics content that you plan to teach. Some teachers use "curriculum compacting" (Reis & Renzulli, 2005) to give a short overview of the content and assess students' ability to respond to mathematics tasks that would demonstrate their proficiency. Allowing students to increase the pace of their own learning can give access to curriculum different from grade level content while demanding more independent study. But, moving students to higher mathematics (by moving them up a grade, for example) will not alone succeed in engaging them if the learning remains at a slow pace. Research reveals that when gifted students are accelerated through the curriculum they become more likely to explore STEM

Flexible Grouping

Allowing students to collaborate on tasks provides support and challenges, increasing their chance to communicate about mathematics and build understanding Groups can be selected based on the students' academic abilities, language needs, social dynamics, and behavior. Avoid ability grouping! As opposed to differentiation, ability grouping means that groups are formed and those needing more support in the low group are meeting different (lesser) learning goals than students in the high group. Although this may be well-intentioned, it only puts the students in the low group further behind, increasing the gap between more and less dependent students and significantly damaging students' self-esteem

Object Graphs.

An object graph uses the actual objects being graphed. Examples include types of shoes, favorite apple, energy bar wrappers, and books. Each item can be placed in a square or on a floor tile so that comparisons and counts are easily made. Notice that an object graph is a small step from sorting. If real objects are sorted into groups, those groups can be lined up for comparison—an object graph!

shared responsibility

At the start of the year, it is important to do team building activities and to set the standard that all members contribute to the group understanding.

A teacher is demonstrating probability and rolls a die two times, revealing a 1 and then a 6. A group of students is then asked to determine the probability of rolling a 6 on the next roll. Because a 1 and 6 already appeared, the next roll will be either a 2, 3, 4, or 5. The probability of rolling a 6 on the next roll would be 1 out of 6. Because a 1 and 6 have already been rolled, a 6 will appear every other time the die is rolled. In all 220 outcomes, a 6 will appear 45 times.

Because a 1 and 6 already appeared, the next roll will be either a 2, 3, 4, or 5. 0% chance of occuring The probability of rolling a 6 on the next roll would be 1 out of 6. Correct Answer Because a 1 and 6 have already been rolled, a 6 will appear every other time the die is rolled. 1/2 chance to occur In all 220 outcomes, a 6 will appear 45 times. 9/44 chance to occur

performance indicators They should help focus students on the objectives.

Being aware of students' common misconceptions can help you create your performance indicators because you will know which weak spots you should help students overcome.

3)When teaching the concept of long division, what can a teacher do to encourage and promote student communication about their math thinking? Select all the methods that should be used. A) Allow students to practice long division using drill and practice methods B) Allow students to use a calculator to solve long division problems C) Have students draw a picture when solving long division problems D) Have students create their own word problem that would require long division

C and D C) Have students draw a picture when solving long division problems D) Have students create their own word problem that would require long division

A fifth-grade teacher is planning a unit of study on equivalent fractions. Select all tools or manipulatives that can enhance learning outcomes. A) Centimeter grid paper B) Spinner C) Fraction tiles D) Pattern blocks E) Compass

C) Fraction tiles D) Pattern blocks

Which two tools/manipulatives can be used to teach a lesson on fractions? A) Dice B) Graph paper C) Pattern blocks D) A protractor E) Fraction bars

C) Pattern blocks E) Fraction bars Correct! Pattern blocks provide a hands-on way to explore and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions, and finding divisors. Fraction bars are a great way for students to visualize relationships between fractional parts of the same whole. Students can compare and order fractions, see equivalent fractions, explore common denominators, and do basic operations with fractions.

Quantity,

Children explore quantity before they can count. They can identify which cup is bigger or which plate of blueberries has more berries. Soon they need to attach an amount to the quan-tities to explore them in greater depth. When you look at an number of objects, sometimes you are able to just "see" how many are there, particularly for a small group of items. For example, when you roll a die and immediately know that it is five without counting the dots, that ability to "just see it" is called subitizing

Productive Disposition.

Collecting data on students' ability to persevere, as well as their confidence and belief in their own mathematical abilities, is also important. This evidence is most often obtained with observation, students' self-reports, interviews, and writing. Informa-tion on perseverance and willingness to attempt problems is available to you every day when you use a problem-solving approach.

1) The student likely has a ________.. issue behind the error.

Conceptual Correct! When there is conceptual misunderstanding, this means that students do not understand the relationship of basic elements and how they function together. Conceptual knowledge will therefore help with procedural knowledge. In this case there is something that students do not conceptually understand about how numbers higher than three are changed when multiplied.

1Counting On and Counting Back

Counting on is the ability to start counting from a given number other than one and it is consid-ered a landmark on a child's path to number sense. For example, if a child is given a group of five cubes and then given two more and asked how many in all, if the child does not recount the five cubes but just uses the counting sequence and states six, seven, they are counting on. Counting on from a particular number and counting back are often difficult skills. In particular, for English learners, counting back is more difficult (try counting back in a second language you have learned).

A math teacher wants to work with a social studies teacher on integrating the content. How can the teacher apply a math connection? A) By exploring the how ratios, fractions, and percents can be expressed in a circle graph. B) By demonstrating the complexity of creating various graphs. C) By allowing students to use manipulatives when solving problems. D) By allowing students to create a timeline of the Native American cultures.

D) By allowing students to create a timeline of the Native American cultures. Correct! Integrating math content with other core areas can help students gain a better conceptual development of certain topics. Here the teacher is trying to show students that math can be used when building and constructing timelines.

A teacher is planning a lesson on transformation and displays the above graphic. A student suggests the triangle underwent a slide. At what level of van Hiele's would this response represent? A) Level A B) Level B C) Level 1 D) Level 0

D) Level 0 Correct! Slides, flips, and turns are the first transformations that Level 0 thinkers grasp. The important thing to note is that the student recognizes that it is the same shape that underwent a transformation.

Here are suggestions, each leading to a different approach to finding the area of a trapezoid and a Grid of Trapezoids Activity Sheet • Make a parallelogram inside the given trapezoid using three of the sides. • Make a parallelogram using three sides that surround the trapezoid. base = base 1 + base 2 A = height × (base 1 + base 2) Two congruent trapezoids always make a parallelogram with the same height and a base equal to the sum of the bases in the trapezoid. Therefore, A = × height × (base 1 + base 2) 1 2

FIGURE 18.16 Two congruent trapezoids always form a parallelogram. • Draw a diagonal forming two triangles. • Draw a line through the midpoints of the nonparallel sides. The length of that line is the average of the lengths of the two parallel sides. • Draw a rectangle inside the trapezoid, leaving two trian-gles, then put those two triangles together. • Use transformational geometry. • Enclose the trapezoid in a larger shape (Manizade & Mason, 2014

FORMATIVE ASSESSMENT Notes. Children who count successfully orally may not have attached meaning to their counts. Here is a diagnostic interview for assessing a child's understanding of cardinality and if they are using counting as a tool. Show a card with five to nine large dots in a row so that they can be easily counted. Ask the child to count the dots. If the count is accurate, ask, "How many dots are on the card?" Early on, children may need to count again, but a child who is beginning to grasp the meaning of counting will not need to recount. Now ask the child, "Please give me the same number of counters as there are dots on the card." Here is a sequence of indicators to watch for, listed in orde

FORMATIVE ASSESSMENT Notes. Does the child not count but instead take out counters and make a similar pattern? Will the child recount the dots on the card? Does the child place the counters in a one-to-one correspon-dence with the dots? Does the child just remember the number of dots and retrieve the correct number of counters? Can the child show that there are the same number of counters as dots? As the child shows competence with patterned sets, move to using random dot patterns.

True or False Theoretical probability changes based on the number of trials. True False

False Correct! Theoretical probability is the likeliness of an event happening based on all possible outcomes. The theoretical probability will not change based on the number of trials. For example, the theoretical probability of rolling a 4 on a six-sided dice is 1/6. There are six possible outcomes and only one 4 on the dice. They will not change as more trials are conducted. The theoretical probability of rolling a 4 will remain 1/6.

The Commutative Property for Addition The commutative property (sometimes known as the order property) for addition means you can change the order of the addends and it does not change the answer (expected of first graders in CCSS-M). Although the commutative property may seem obvious to us (simply reverse the two piles of counters on the part-part-whole mat), it may not be apparent to children. Because this property is essential in problem solving (counting on from the larger number), mastery of basic facts (if you know 3 + 9, you also know 9 + 3), and mental mathematics, there is value in spending time helping children construct the relation-ship (Baroody, Wilkins, & Tiilikainen, 2003).

First-graders do not need to be able to name the property as much as they need to understand and visualize the property, know why it applies to addition but not subtraction, and apply it. But, always name the property accurately and never use a "nickname" that will later confuse the student (and subsequent teachers), such as saying the "ring around the Rosie" or "flip flop" property for the commutative property. Those arbitrary names are confusing as students progress (Karp, Bush, & Dougherty, 2016). Use the precise terminology

3The Relations Core: More Than, Less Than, and Equal To

For all three relationships (more/greater than, less/less than, and same/equal to), children should construct sets using counters as well as make comparisons or choices between two given sets. Conduct the following activities in a spirit of inquiry with requests for children's explana-tions, such as "Can you show me how you know this group has l

Focus on the attribute being measured.

For example, when discussing how to measure the area of an irregular shape, units such as square tiles or circular counters may be sug-gested. Each unit covers area, and each will give a different result. The discussion can focus on what it means to measure area.

Change Problems—Join and Add To.

For the action of joining, there are three quantities involved: an initial or start amount, a change amount (the part being added or joined), and the resulting amount (the total amount after a change takes place). In Figure 8.1(a), this action is illustrated by the change being "added to" the start amount. Provide students with the Join Story Activity Page where they work with counters and model the problem on the story situation graphic organizer.

5) End of class quizzes

Formative Assessment Correct! You can always give short quizzes to check your students' understanding, thus helping you know how to adjust your instruction for the next day.

Cardinality

Fosnot and Dolk (2001) state that an understanding of cardinality and its connection to count-ing is not a simple task for 4-year-olds. Children will learn how to count (matching counting words with objects) before they understand that the last count word stated in a count indicates the amount of the set (how many you have in all) or the set's cardinality as shown in Figure 7.4. Children who make this connection are said to have the cardinality principle, which is a refine-ment of their early ideas about quantity and is an expectation for kindergartners (NGA Center & CCSSO, 2010). Most, but certainly not all, children by age 41 & Dolk, 2001)

Unit Summary In this unit you learned about encouraging students to collaborate and share ideas with other students. Students should not work in rows and in isolation but rather they should learn with peers. Teachers should encourage students to compare answers and share different methods and strategies. This topic is also about critical thinking and how teachers can ask open-ended questions to promote thinking. A teacher's responses should be broad and encourage students to explain and elaborate on their thinking or justify their answers or responses.

Furthermore, a teacher can also incorporate critical thinking by restating what the student just said in some format of a question. This is essentially repeating what the student just said, but asking a question for further clarification. Lastly, when teachers think about encouraging students to use precise academic language, they can use math word walls in the classroom or lead students through think-alouds while modeling math language. Students should be encouraged to use academic language while reflecting in a math journal or by representing their thinking in various ways (e.g., draw a picture or use a manipulative).

2) Instrumental Understanding

Having students complete a 10-question quiz at the end of the unit. Correct! Simply providing a quiz would be instrumental. Consider, however, turning this into relational understanding by having students explain their answers to each other.

Pie charts, like bar graphs, typically display categorical data. They also typically show per-centages and, as such, are perceived as too advanced for young students.

However, circle graphs can be set up to indicate the number of data points from a total set, without calculating per-centages. Also, an understanding of percentages is not required when using computer software to create the graph.

Learning Check

Imagine you are teaching multiplication facts to your third-grade class. Although most of the students are proficient at addition, many are struggling to understand how to multiply numbers higher than 3. If you were to use drill, you would have students repeat a series of exercises over and over that would essentially help them to memorize certain equations. However, this is not the best way to approach this situation. The following are statements about drill and are reasons why you would not want to use drill in this particular situation.

Subtraction

In a part-part-whole model, when the whole and one of the parts are known, subtraction can be used to name the missing part. If you start with a whole set of 8 and remove a set of 3, the 2 sets that you know are the sets of 8 and 3. The expression 8 - 3, read "eight minus three," names the set of 5 that remains (note that we didn't say "take away").

Words and phrases such as and, or, at least, and no more than may cause students some confusion and therefore require explicit attention. Of special note is the word or because its meaning in everyday usage is generally not the same as its strict logical meaning in mathematics.

In mathematics, or includes the case of both. For example, in the tack-cup toss experiment, the event of "tack (or cup) landing up" includes tack (only) up, cup (only) up, and both tack and cup u

Comparison Problems.

In multiplicative comparison problems, there are really two different sets or groups, as there were with comparison situations for addition and subtrac-tion. In additive situations, the comparison is an amount or quantity difference between the two groups. In multiplicative situations, the comparison is based on one group being a particular multiple of the other (a reference set). With multiplication comparison, there are three possibilities for the unknown: : the product, the group size, and the number of groups

Teaching is never a one-size-fits-all activity. In addition to EL students, you will also have students who are culturally and ethnically diverse, reluctant learners or those with low motivation, and gifted students. Following are some teaching strategies that can help provide appropriate accommodations for these students.

It may be that some of these strategies apply to other groups as well, but try to match the strategy with the group that it would help the most. (Hint: each group has two strategies connected with it, even though there are certainly more that are not listed.)

Numeral Writing and Recognition

Kindergartners are expected to write numbers from 0 to 20 (NGA Center & CCSSO, 2010). Helping children read and write the 10 single-digit numerals is similar to teaching them to read and write letters of the alphabet. Neither has anything to do with number concepts. Numeral writing can be engaging. For example, ask children to trace over pages of numerals, make numerals from clay, trace them in shaving cream on their desks, press the numeral on a calculator, write them on the interactive whiteboard or in the air, and so on.

The Van Hiele model is a five-level hierarchy of ways of understanding spatial ideas. Each level describes the thinking processes used in geometric contexts. Level 0 (Visualization)—The objects of thought at level 0 are shapes and what they look like. The properties of a figure are not perceived. At this level, students make decisions based on perception, not reasoning. Level 1 (Analysis)—The objects of thought at level 1 are classes of shapes rather than individual shapes. Students see figures as collections of properties. They can recognize and name properties of geometric figures, but they do not see relationships between these properties.

Level 2 (Informal Deduction)—The objects of thought at level 2 are the properties of shapes. At this level, students can create meaningful definitions and give informal arguments to justify their reasoning. Level 3 (Deduction)—The objects of thought at level 3 are relationships between properties of geometric shapes. At this level, students should be able to construct proofs such as those typically found in a high school geometry class. Level 4 (Rigor)—The objects of thought at level 4 are deductive axiomatic systems for geometry. See pages 403-407 for more information.

MODULE 10NUMBER SENSE AND PERFORMING OPERATIONS

MODULE 10NUMBER SENSE AND PERFORMING OPERATIONS

MODULE 11ALGEBRAIC THINKING

MODULE 11ALGEBRAIC THINKING

MODULE 12GEOMETRY AND MEASUREMENT

MODULE 12GEOMETRY AND MEASUREMENT

MODULE 13 PROBABILITY AND STATISTICS

MODULE 13 PROBABILITY AND STATISTICS

Open Questions

Many questions in textbooks are closed, meaning there is one answer, and often only one way to get there. Such a task cannot meet the needs of range of learners in the classroom. Open questions are broad-based questions that invite meaningful responses from students at many developmental levels

Model-Based Problems.

Many students will use counters, bar diagrams, or number lines to solve story problems. These models are thinking tools that help them understand what is happening in the problem and keep track of the numbers and steps in solving the problem. Problems can also be posed using models when there is no context involved

Your students who are English learners (EL) will likely come to your class with several challenges that make it more difficult for them to succeed without appropriate accommodations. Fortunately there are several ways to support these students.

Match the teaching technique that will help solve the given difficulty.

Distributive Property

Multiplication combined with addition or subtraction

Unit 2 Practice Quiz

National, State, and Local Mathematics Standards

Individual whiteboards

Non-Tool Correct! Individual whiteboards are an instructional resource rather than a tool to explore a math concept.

Student response systems

Non-Tool Correct! Student response systems are an instructional technology tool used in the classroom, but they do not help students explore a math concept.

Virtual math games and quizzes

Non-Tool Correct! Virtual math games and quizzes are instructional resources for practicing and assessing learning, but they are not tools used to explore a math concept.

Vocabulary flash cards

Non-Tool Correct! Vocabulary flash cards are an instructional resource, but they are not considered a tool without having some sort of a picture or drawing that can be used to help explore a math concept.

This is a way to gather less visible information over a longer period of time.

Observations Correct! Observations can take a long time and will require that you take individual notes on students. These notes will help you know how to adjust your instruction and provide feedback to both students and parents.

Order of operation's PEMDAS (Purple Elephants Marching Down A Street)

P is parenthesis E is Exponets M is multiplication D is division A is addition S is subtraction

2) EL students feel isolated from other students.

Pair them with another student. Correct! Even if EL students are with someone who speaks their native language, this is still an opportunity for them to collaborate in a nonthreatening situation. It is also important that these students not be placed in groups larger than two, as this makes it more likely for that student to be excluded.

Unit 6 Summary During this unit you learned about math assessments. When creating assessments for the elementary math classroom, it is important to make sure each assessment aligns with the objective. First, determine the objective. Then find the task or assessment that aligns with the objective. Furthermore, it is helpful to be familiar with observational assessment, which requires a teacher to observe students at work while taking narrative/anecdotal style notes to measure student performance rather than participation

Participation refers to whether or not the student is participating (raising their hand). Teachers want to assess performance or achievement using an observational assessment. Moreover, a rubric is used to score a student's work and not grade or find a percent. A rubric is a predetermined framework that will assign a score or evaluate, not a grade, average, or percent.

There are three types of assets: To create an environment that is conducive to learning, consider how you can use what you know about your students to design and deliver learning experiences that align with their motivations, interests, and abilities.

Personal Assets: Cultural Assets: Community Assets:

Pie charts (sometimes called pie graphs) and circle graphs mean the same thing; the term circle graph may be more common in curriculum and the term pie chart is used with graphing tools such as electronic spreadsheets.

Pie charts, like bar graphs, typ-ically display categorical data. They also typically show per-centages and, as such, are perceived as too advanced for young students. However, circle graphs can be set up to indicate the number of data points from a total set, without calculating per-centages. Also, an understanding of percentages is not required when using computer software to create the graph.

4) A more effective teaching strategy is to offer various _____. that are _________tasks.

Practices, Problem-based Correct! Problem-based tasks are the most effect because they cannot only offer variety but will also help to draw on students' prior knowledge and experiences to help make sense of new material. Correct! Practices should be seen as providing students with a variety of tasks or experiences that are designed to build conceptual knowledge.

1Introducing Symbolism

Preschoolers initially have little need for the symbols +, -, and = as they listen and respond orally to addition and subtraction situations, however, by kinder-garten and first grade these symbolic conventions are required. Therefore, whenever young children are engaged in solving story problems, introduce symbols as a way to record what they did as they share their thinking. Say, "You had the whole number of 12 in your problem, and the number 8 was one of the parts of 12. You found out that the unknown part was 4. Here is a way we write that: 12 - 8 = 4." The minus sign should be read as "minus" or "subtract" but not as "take away." The addition sign is easier because it is typically a substitute for "and." Some children may describe a counting-up strategy to find the solution of 4, so also record the equivalent equation as 8 + 4 = 12.

Giving the students a problem to solve amd telling them to find three different ways to solve it.

Problem-Solving Correct! In addition to asking students to find multiple ways of solving a problem, you can assess their problem-solving skills by also seeing how they approach the problem. In other words, do they try to understand it completely before they begin?

2) Encourage students to think of success as a result of effort rather than a lack of inherent ability to do math.

Reluctant Learners Correct! Many reluctant or low-motivation learners tend to think that they cannot do math. However, research shows that our success in math or any other subject is due in large part to the amount of effort invested. Do your best to encourage traits of resilience and effort. (For more information on this you can explore the work of Carol Dweck.)

Select which activity is appropriate for helping students at the given van Hiele level of geometric thinking.

Select which activity is appropriate for helping students at the given van Hiele level of geometric thinking.

Associated Property

States that can be grouped together in different combinations and still produce the same results in addition and multiplication.

Stem-and-Leaf Plots.

Stem-and-leaf plots (sometimes called stem plots) display numeric data as a list, grouped within ranges of data. By way of example, consider the National League baseball teams total wins for the 2016 season:

The Zero Property.

Story problems involving zero and using zeros in the three-addend sums (e.g., 4 + 0 + 3) are good opportunities to help children understand zero as an identity element in addition or subtraction believe that 6 + 0 must be more than 6 because "adding makes numbers bigger," or that 12 - 0 must be 11 because "subtracting makes numbers smaller." Instead of making meaningless rules about adding and subtracting zero, create opportunities for discussing adding and subtracting zero using contextual situation

Students are asked to calculate the area of this irregular shape and provide the following answers. Match their answers with their thinking. The figure is shape with 3 sides measuring 5in, 6in, and 9in. The 4th side has a partial length of 3in then dips down 3in with the next section measurement being unknown. The next segment goes up 2in and a final length 4in.

Student calculated the perimeter. 34 inches Student gave correct answer. 44 inches squared Student forgot to subtract one small rectangular portion in the middle. 48 inches squared Student calculated rectangular shapes as if they were triangular. 22 inches squared

26+4 = 66

Student may not understand place value if they added 4 and 2 or they stacked the numbers up they added the numbers incorrectly

Reasoning for question 7 Correct! Students will enter your classroom with various skills and diverse backgrounds. Teachers should embrace the different approaches students use when solving problems. Because the United States has people from all over the world, you will better serve the needs of these students by valuing their culture and language and not trying to force them into local culture and language. Valuing a person's background is more than a belief statement; it is a set of intentional actions that communicate to the student, "I want to know about you, I want to see math as part of your life, and I expect that you can do high-level math."

Students with disabilities have very specific difficulties with perceptual or cognitive processing. Students with disabilities can benefit from a variety of accommodations that include extended time and the ability to complete fewer problems on a task. Students with mild disabilities may also benefit from explicit instruction, which is a structured, teacher-led instructional approach that targets a specific strategy.

Tasks

Tasks refer to products, including problem-based tasks, writing, and students' self-assessments. Good assessment tasks for either instructional or formative assessment purposes should permit every student in the class, regardless of mathematical ability, to demonstrate his or her knowledge, skill, or understanding

The difference between equity and equality in teaching may seem trivial, but the distinction is an important one.

Teaching equally (which is not the aim in teaching) provides the same requirements for all students while offering identical instruction. *Teaching with equity, which is the goal of differentiated instruction, is to maintain high expectations in teaching while making reasonable accommodations for students who need them. But it is important to see these accommodations as a way to offer added support for students, not to make things easier for them.

Individual accountability

That means that while the group is working together on a product, individuals must be able to explain the process, the con-tent, and the product.

When asking questions in your classroom, consider the following:

The "level" of the question: Type of knowledge that is targeted: Pattern of questioning: Who is thinking of the answer: How you respond to an answer:

Provide a good rationale for using standard units.

The need for a standard unit has more meaning when your students have measured the same objects with their own collections of nonstandard units and arrived at different and sometimes confusing answers.

2Part-Part-Whole Problems.

There is no meaningful distinction between the two parts in a part-part-whole situation, so there is no need to have a different problem for each part as the unknown. The third situation in which the whole or total is known and the two parts are unknown creates opportunities to think about all the possible decompositions of the whole as instead of having one answer, this situation usually produces a set of correct answers (Caldwell, Kobett & Karp, 2014; Champagne, Schoen, & Riddell, 2014). This struc-ture links directly to the idea that numbers are embedded in other numbers. For example, students can break apart 7 into 5 and 2, where each of the addends (or parts) is embedded in the 7 (whole). See the Part-Part-Whole Story Activity Page for the corresponding graphic organizer.

26 + 4 = 66 40 - 13 = 33 69 x 2 = 148

These equations are clearly false, but they are examples of common mistakes that some of your students will make. This video will go over such mistakes, why students make them, and how you can help them figure out the correct way to think through some math problems.

Subtraction as Think-Addition.

Thinking about sub-traction as "think-addition" rather than "take-away" is signif-icant for mastering subtraction facts. Because the tiles for the remaining part or unknown addend are left hidden under the cover for Activity 8.2, when children do such activities they are encouraged to think: "What goes with the part I see to make the whole?" For example, if the total or whole number of tiles is 9, and 6 tiles are removed from under the cover, the child can think in terms of "6 and what amount add to 9?" or "What goes with 6 to equal 9?" The mental activity is "think-addition" instead of "count what's left." Later, when working on subtrac-tion facts, a subtraction fact such as 9 - 6 = □ should trigger the same thought pattern: "6 and what equals 9?"

Writing helps students reflect on, explain, and defend their thought process and answers.

This is a reason to have students write. The act of writing will help students pause and reflect on their problem-solving process, which can help develop their metacognitive skills.

Base-ten blocks

Tool Correct! Base-ten blocks are an excellent tool to help students explore and learn about place value and mathematical operations.

Rulers and protractors

Tool Correct! Rulers and protractors are measurement tools that can be used to explore length and angle measurements respectively.

Virtual fraction circles

Tool Correct! Virtual fraction circles are a tool to explore fractions.

True or False According to the Van Hiele model for geometric reasoning, levels are sequential. True False

True

True or False In a linear relationship, all the points on a graph lie on a straight line. True False

True Correct! Linear functions are a subset of growing patterns and functions, which can be linear or nonlinear. But because linearity is a major focus of middle school math, and because growing patterns in elementary school tend to be linear situations, it is important to understand linear functions. Linearity can be established by looking at other representations. If you make a table for the coordinate points and the graph forms a straight line, the graph is a linear function.

Attribute: Volume/Capacity

Units: cubes, balls, cups of water How many units will fill the bucket?

Attribute: Length

Units: rods, toothpicks, straws, string How many units are as tall as the bucket? How much string is needed to go around the bucket?

Contextual Problems.

When teaching multiplication and division, it is essential to use interesting contextual problems instead of more sterile story problems (or "naked numbers")—and yes, assessments focus on these contextual problems! Consider the following problem

Unit 5 Summary 4

When working with diverse students, teachers should meet students' individual readiness and provide accommodations as needed. Additionally, gifted students will need to be challenged in math class to reduce any potential boredom. Teachers may not consciously seek to stereotype students by gender; however, the gender-based biases of our society may affect teacher-student interactions. Lastly, a calculator can be an accommodation when students are using calculators and still meeting the full objective

3) EL students have difficulty understanding instructions and objectives.

Write and State the learning objective clearly and avoid using unnecessary words. Correct! It is always a good practice to state your learning objectives at the beginning of your lesson so that students know exactly what they are expected to learn. It is also important to be clear and concise in your instructions and use gestures when you are explaining something.

Process

can also be differentiated in various ways. Tomlinson and McTighe (2006) suggest that in thinking about process, teachers think about selecting strategies that build on students' readiness, interests, and learning preferences. In addition, the process should help students learn effective strategies and reflect on which strategies work best for them. Open questions and tiered lessons are two ways to differentiate the process.

Content

can be differentiated in many ways, including resources or manipulatives used, mathematics vocabulary developed, examples and nonexamples used to develop a concept, and teacher-directed groups used to provide foundations for a new concept

A histogram

can be thought of as a hybrid of a line graph and a bar graph where the data is represented in terms of its frequency. A common misconception about histograms is that students cannot calculate the arithmetic mean and other measures of central tendency; however, a student can determine the modal interval (tallest bar on the graph). Another misconception is that you can determine individual data points or values from a histogram. The histogram only displays frequency of ranges of data, not exact values.

A Venn diagram

can be used to compare, contrast, and show the relation between groups of things. Example: Have students use it to see the relation between numbers.

A stem-and-leaf plot

can be used to see the shape of data (e.g., bell-shaped curve). The stem is the tens place and Sthe leaf is the ones digit. Example: You can use it to show your students their quiz scores

A tool

is any object, picture, or drawing that can be used to explore a concept" One type of a tool is a manipulative. A tool does not "illustrate" a concept. The tool is used to visualize a mathematical con-cept and only your mind can impose the mathematical relationship on the object

Drill

refers to repetitive exercises designed to replicate a procedure or algorithm Most textbooks include sets of exercises with every lesson. There is a seemingly endless amount of downloaded worksheets that focus on drill. What has all of this drill accomplished over the years? It has worked against developing mathematical practices, and created genera-tions of people who don't remember the skills they learned, do not like mathematics, and do not pursue professions that involve mathematic

Consider these suggestions as you implement diagnostic interviews:

• Avoid revealing whether a student's answer is right or wrong. • Wait silently for the student to answer. • Avoid interjecting clues or teaching • Avoid interjecting clues or teaching

1There are four important principles of iterating units of length, whether they are nonstandard or standard (Dietiker, Gonulates, Figueras, & Smith, 2010). Units must be:

• Equal in length or you cannot iterate them by counting. • Aligned with the length being measured or a different quantity is measured. • Placed without gaps or a part of the length is not measured. • Placed without overlaps or the length has portions that are measured more than one time. Students can begin to measure length using a variety of nonstandard units, including these: • Giant footprints: Cut out about 20 copies of a large footprint about 1 1/2 - to 2-feet long from poster board. .

3) Class discussions

Formative Assessment Correct! Class discussions are a great way to see your students thinking at work and whether they are learning the intended material. During and after these discussions you will know what needs to be retaught.

bar diagram (see Figure 8.5) .

visually connects to the part-part-whole diagram students have been using since kindergarten

C. There are 4 aces in the deck.

1/13 Correct! In response C the student simplified 4/52 to 1/13.

A. Out of 52 cards in the deck, aces have already been drawn twice

1/26 Response for: undefined Correct! As you are doing student work analysis, consider how to find mathematical matches between the thinking statements and answers the students produce. Notice on response A that 2/52 is simplified to 1/26.

What is the perimeter of the rectangle and the area of the shaded region in the shape? -Area = 25 square units; perimeter = 30 units -Area = 50 square units; perimeter = 30 units -Area = 25 square units; perimeter = 50 units -Area = 50 square units; perimeter = 100 units

Area = 25 square units; perimeter = 30 units Correct! To find the perimeter, add up all the sides to the quadrilateral. To find the area of the shaded region, use the appropriate formula. The shaded region is a triangle and the area formula for a triangle is (b x h) / 2 or ½ (b x h).

Student A multiplied 12 ft. by 22 ft. and then multiplied ½ to determine the area of the shaded region. Student D multiplied 12 ft. by 22 ft. and then divided by 2 to determine the area of the shaded region.

Conception

1) Observations

Formative Assessment Correct! You can use observations to gather less visible information about students over a longer period of time. Since this is a type of formative assessment, you can use it to know how to address a particular student's needs.

Counting

Meaningful counting activities begin with 3-year-olds, but by the end of kindergarten (NGA Center & CCSSO, 2010), children should be able to count to 100. For children to have an understanding of counting, they must construct this idea by working through a variety of counting experiences and activities. Only the counting sequence of number words is a rote procedure. The meaning attached to counting is the key conceptual idea on which all other number concepts are developed

Module 9 MATHEMATICAL CONNECTIONS

Module 9 MATHEMATICAL CONNECTIONS

Remainders. In real contexts, remainders must be interpreted (grade four standard). Besides "left over" or "partitioned as a fraction," remainders can have three additional effects on answers: • The remainder is discarded, leaving a smaller whole-number answer. • The remainder can "force" the answer to the next highest whole number. • The answer is rounded to the nearest whole number for an approximate result.

More often than not in real-world situations, division does not result in a whole number. For example, problems with 6 as a divisor will result in a whole number only one time out of six. In the absence of a context, a remainder can be dealt with in only two ways: it can either remain a quantity left over or be partitioned into fractions (including decimal fracti

Order of operation's PMDAS (Pardon My Dear Aunt Sally)

P is parenthesis M is multiplication D is division A is addition S is subtraction

The Number Core:

Quantity, Counting, and Cardinality

Personal Assets:

Specific background information that students bring to the learning environment. Students bring interests, knowledge, everyday experiences, and family backgrounds that a teacher can draw upon to support learning.

Level 4: Rigor.

Students move from considering deductive axiomatic systems for geometry to comparing and contrasting different axiomatic systems of ge

Missconceptions

Students who are missing a concepts and are making errors in thier work. These are called conceptual errors

Use open-ended questions.

Tests in which questions have only one correct answer tend to limit what you can learn about what the student knows and can do and what they are ready to know and do!

4. The complexity of process.

This includes how quickly paced the lesson is, how many instructions you give at one time, and how many higher-level thinking questions are included as part of the task.

The degree of assistance.

This might include providing examples or partnering students

Choosing Numbers for Problems.

When selecting numbers for multiplicative story problems or activities, there is a tendency to think that large numbers pose a burden to students or that 3 * 4 is somehow easier to understand than 4 * 17. An understanding of products or quotients is not affected by the size of numbers as long as the numbers are within your students' grasp. A contextual problem involving 14 * 8 is appropriate for third-graders. When given the challenge of using larger numbers, children are likely to invent computational strategies (e.g., ten 8s and then four more 8s) or model the problem with manipulatives

Introducing Symbolism.

When students solve simple multiplication story problems before learning about multiplication symbolism, they will most likely begin by writing repeated addition equations to represent what they did. This moment is your opportunity to introduce the multiplication sign and explain what the two factors mean.

To find the mode,

determine which numbers appear the most The mode is the most frequently occurring value in the data set. The mode is the least frequently used as a measure of center because data sets may not have a mode, may have more than one mode, or the mode may not be descriptive of the data set.

A circle/Pie graph

is another way to represent data like you would in a bar graph. Example: Use it to show the percentages of students who like a certain superhero.r height.

Properties of Addition and Subtractio

Properties are generalized algebraic rules that support the understanding of how numbers, in this case, can be added or subtracted. Explicit attention to these properties, (build the terminology over time), will help children become more flexible and efficient in how they combine numbers.

A bar graph

is a simple way to represent data to your students by tracking separate pieces of related information and can be created from a line plot. Example: Students can display their favorite candies.

Two types of activities can develop familiarity with standard units:

(1) comparisons that focus on a single unit and (2) activities that develop personal referents or benchmarks for single units or easy multiples of unit

Perhaps the biggest challenge in teaching measurement is the difficulty in recognizing and sep-arating two objectives:

(1) understanding the meaning and technique of measuring a particular attribute, and (2) learning about the standard units commonly used to measure that attribute.

Student A says, "Because a yellow, blue, and yellow were already revealed, the teacher will spin one of the remaining colors."

0% chance of occuring Correct! Student A demonstrates a common misconception that probability has a memory. Here, the student provided no quantitative means of providing a reasonable calculation, and the student is certain that one of the remaining colors will result in the subsequent spin. However, since probability does not have a memory, yellow and blue could appear again.

2/5 Student A says, "Because a yellow, blue, and yellow were already revealed, the teacher will spin one of the remaining colors."

0% chance of occurring Response for: Question 4, Inline answer 1 of 4 Correct! Student A demonstrates a common misconception that probability has a memory. Here, the student provided no quantitative means of providing a reasonable calculation, and the student is certain that one of the remaining colors will result in the subsequent spin. However, since probability does not have a memory, yellow and blue could appear again.

3Here are four arguments against presenting the key word approach:

4. Key words don't work with two-step problems or more advanced problems, so using this approach on simpler problems sets students up for failure with more complex problems because they are not learning how to read for meaning.

3) Correcting the student's errors is best accomplished with a ______ activity, not a ______ like drill.

Reflective, Repetitive Correct! As has been stated, drill will not help with conceptual knowledge, which requires a degree of reflection about the relationship between certain elements. Correct! Because drill can be useful when conceptual knowledge is established, it can then help with procedural knowledge, meaning that it can help improve necessary methods and skills.

5) Connect students' interests to the content.

Relucted Students Correct! When you do this you will be allowing these reluctant learners to nurture their strengths, experience more success, and allow them more choice in their studies.

How is conceptual understanding beneficial to students in learning mathematics? A) It allows students to apply mathematical principles in different contexts. B) When approaching a problem, students are able to apply different strategies. C) It allows students to use their prior knowledge to make connections to the current content they are learning. D) Students acquire a can-do, persistent attitude. E) Students are able to reflect on and evaluate their own work.

A Correct! Because conceptual understanding is about understanding relationships and foundational ideas, this allows the students to apply the ideas and concepts in a variety of contexts and mathematical procedures.

2. A fifth-grade class is exploring the relationship between various volume formulas and are asked to fill a rectangular prism, cube, and triangular prism with water. Select two strategies that could help increase the effectiveness of this activity. A) Encourage students to collaborate and discuss their findings and strategies after the activity has been completed B) Review the properties of 3D shapes prior to the activity C) Give students step-by-step instructions for the activity D) Watch students work on the task and stop them if they are doing something wrong

A and B A) Encourage students to collaborate and discuss their findings and strategies after the activity has been completed B) Review the properties of 3D shapes prior to the activity Correct! The collaborative-learning style incorporated into the fabric of the school helps students to be resilient by aiding them with identifying their resources (peers) and testing their theories to see if they are on the right track—all while developing habits of mind that form the foundation of scholarship. In a constructivist classroom, students should be encouraged to explore multiple strategies and ways to solve a problem. Before students attempt to build a conceptual understanding, it is important to activate their prior knowledge or schemas. In mathematics, students can use schemas to organize information and relate new information to previously learned material.

Equity In Teaching

*Offers access to important mathematics with equal access to resources *Attains equal outcomes by being sensitive to individual differences *Maintains high expectations, respect, understanding, and strong support *Maximizes learning potential for all students while providing reasonable and appropriate accommodations to promote access

Some common probability and statistics misconceptions: .

*Probability has a memory, so an event that just occurred is less likely to occur next time. This is not true; in a fair independent experiment the event is equally likely for each trial. *Probability follows a pattern to predict the next outcome

D. Aces were drawn 2 times out of the last 5.

40% Correct! In response D the student converted 2/5 to 40%.

4Begin measurement activities with students making an estimate. Just as for computational estimation, students need specific strategies to estimate measures. Here are four strategies:

4The best approach to improving estimation skills is to have students do a lot of estimating. Keep the following tips in mind:

4The best approach to improving estimation skills is to have students do a lot of estimating. Keep the following tips in mind:

7. Make measurement estimation an ongoing activity. Post a daily/weekly measurement to be estimated. Students can record their estimates and discuss them for five-minutes. Invite a student or a team of student to select measurements to estimate. 8. Be precise with your language. Do not use the word measure interchangeably with the word estimate (Towers & Hunter, 2010). Randomly substituting one word for the other will cause uncertainty and possibly confusion in students

Complexity

Another strategy is to increase the sophistication of a topic by raising the level of complexity or pursuing greater rigor of content, possibly outside the regular curriculum or by connecting mathematics to other subject areas. For example, while studying a unit on place value, students can deepen their knowledge to study other numeration systems such as Roman, Mayan, Egyp-tian, Babylonian, Chinese, and Zulu. This approach provides a multicultural view of how our numeration system fits within historical number systems (Mack, 2011). In the algebra strand, when studying sequences or patterns of numbers, students can learn about Fibonacci sequences and their appearances in the natural world in shells and plant life. See the Mathematics Inte-gration Plan that can be used to help plan ways to integrate core content or create independent explorations or research projects. Using this approach, students can think about a mathematics topic through another perspective or through an historic or futuristic viewpoint.

2Contextual Problems.

Because contextual problems connect to life experiences, they are important for ELs, too, even though it may seem that the language presents a challenge to them. Some strategies to support comprehension of contextual problems include using a noun-verb word order, replac-ing terms such as "his/her" and "it" with a name, and removing unnecessary vocabulary words. A visual aid, or actual students modeling the story, would also be effective strategies for ELs and students with disabilities

Consider these definitions as you evaluate and classify each item as either a tool or non-tool.

Consider these definitions as you evaluate and classify each item as either a tool or non-tool.

This provides a way to gather in-depth information about a student's thinking processes and misunderstandings.

Diagnostic Interviews Correct! These will require that as the teacher you exercise good listening skills as you give the student a problem to work through to verbalize their thinking.

performance indicators They should contain words like "proficient" or "on target."

These are great words to include in your performance indicators because they signal to you and students where their overall standing is.

"Tape diagrams

are designed to bring forward the relational meanings of the quantities in a problem by showing the connections in context" (p. 396).

Common missconceptions

include basic ideas on number sense about place value equal sign what variables are and how to use them rational numbers Divisibility

The during phase

is still the time where students engage in the task, but they are stopping and rotating to new centers within this phase of the lesson. It is still important to ask questions and keep track of strategies students are using to later highlight.

To calculate the range

subtract the lowest value from the greatest value. Range is a measure of variability. Range of a data set is the difference between the highest and lowest data points. The interquartile range of the data is connected to the box plot described earlier. It is the difference between the lower and upper quartiles (Q3 - Q1), or the range of the middle 50 percent of the data. Let's look at an example

When students write, it helps prepare them for the class discussion that can follow the assigned activity.

This is a reason for having students write. The act of writing will help students write down their thoughts that they can refer to later on if needed.

The teacher can later use the students' writings to assess their progress and adjust their teaching as needed.

This is a reason to have students write. Writing is a valuable assessment tool that teachers can use to gauge student progress and understanding. (This will be discussed again in the Assessment unit.)

3/5 Student B says, "The probability of spinning a blue will be 1 out of the total number of sections."

This is the correct answer. Correct! Student B demonstrates the correct analysis because blue has a theoretical chance of occurring ⅙ of the time. Probability equals a part divided by the whole and there is one part in question (blue) and six total parts, thus ⅙ is the correct answer.

Histograms.

Though line and dot plots are widely used for small data sets, in many real data sets, there is a large amount of data and many different numbers. A dot plot would be too tedious to create and not illustrate the spread of data as effec-tively. In this case, a histogram is an excellent choice as data are grouped in appropriate intervals. A histogram displays numeric data in consecutive equal intervals. The number of data elements falling into that particular interval determines the height or length of each bar. Histograms differ from the other bar graphs, which are used for categorical data, and for which the order of the bars doesn't matter (Metz, 2010). Histograms effectively show the distribution of data values, especially the shape of the distribution and any outlier values

2The Relations Core: More Than, Less Than, and Equal To

Though the concept of less is logically related to the concept of more (selecting the set with more is the same as not selecting the set with less), the concept of less proves to be more difficult for children than more. A possible explanation is that young children have many opportunities to use the word more but may have limited exposure to the word less. To help children, frequently pair the idea of less with more and make a conscious effort to ask, "Which is less?" questions as well as "Which is more?" questions. In this way, the concept can be connected with the better-known idea and the term less can become familiar. Also ask, "Which has fewer?"

Cultural Assets:

Cultural backgrounds and practices that students bring to the learning environment, such as traditions, languages, world views, literature, and art that a teacher can draw upon to support learning.

1) Stay focused on the big ideas in math as derived from state and national standards.

Culturally and Ethically Diverse Students Correct! State and national standards are designed to be broad enough to be accessible to a variety of students. Focusing on standards will also keep you from the temptation to remove certain assignments or content areas that you may initially think are beyond the grasp of certain students.

6) Make the content relevant by relating it to prior knowledge and connecting it to many contexts.

Culturally and Ethically Diverse Students Correct! This is appropriate for all students, but doing this with your culturally and ethnically diverse students is especially important since they may feel isolated at times. Drawing on their unique experiences and contexts will help to pull them in.

3. A fifth-grade class has been exploring volume by using centimeter cubes to build rectangular prisms. They have developed the volume formula of V = lwh. Which activity is a recommended follow-up to this lesson? A) A practice activity—students identify geometric solids in the classroom and practice finding their surface area. B) A drill activity—students practice their multiplication facts. C) A conceptual activity—students explore the area of various 2D shapes using manipulatives. D) A drill activity—dimensions for various rectangular prisms are provided, and students compute the volume of each.

D) A drill activity—dimensions for various rectangular prisms are provided, and students compute the volume of each. Correct! The objective of the lesson was to find the volume formula for a rectangular prism. Any task associated with the objective must align. Because students have developed a conceptual understanding of the material, it is okay to use a drill activity in this situation. A drill activity should only be used after students develop a conceptual understanding. Drill activities should never be used to teach concepts.

Which is a true statement concerning all linear equations? A) The y values increase as the x values increase. B) The y values decrease as the x values increase. C) As the x values increase, the y values increase at a faster rate. D) All of the points on the graph lie on a straight line.

D) All of the points on the graph lie on a straight line. Correct! Linear functions are a subset of growing patterns and functions, which can be linear or nonlinear. But because linearity is a major focus of middle school math, and because growing patterns in elementary school tend to be linear situations, it is important to understand linear functions. Linearity can be established by looking at other representations. If you make a table for the coordinate points and the graph forms a straight line, the graph is a linear function.

Which strategy will allow the students to visualize the difference between 1/4 and 1/8? A) Peer-assisted learning B) Explicit strategy response C) Think-alouds D) CSA sequence

D) CSA sequence Correct! CSA sequence requires that the students start with concrete representations instead of numerical representations. In the example provided, the students will start with physical (concrete) representations of 1/4 and 1/8, then they will provide drawings or pictures (semi-concrete) that represent 1/4 and 1/8, and lastly they will work solely with numbers as they solve the problem.

2Choosing Numbers for Problems.

For example, a problem involving the combination of 30 and 42 has the potential to help first-and second-grader focus on sets of 10. As they decompose 42 into 40 and 2, it is not at all unreasonable to think that they will add 30 and 40 and then add 2 more. The structure of a word problem can strongly influence the type of strategy a student uses to solve multidigit problems. This thinking is especially true for students who have not been taught the standard algorithms for addition and subtraction which are not required until fourth grade

7) Peer and self-assessments

Formative Assessment Correct! Peer and self-assessments are a great way for students to reflect on their own learning and to give feedback to their peers. This type of formative assessment can, if you encourage it, help students to see what adjustments they need to make.

3) Provide opportunities to pace their own learning and move to more advanced and complex content.

Gifted Students Correct! Gifted students are typically ready to move to more advanced content. Allowing them to do so will also demand more independence and responsibility on their part.

4) Allow these students to explore topics that are outside the curriculum but still within their grasp.

Gifted Students Correct! This means that you allow and encourage these students to make connections from the math content to other content areas.

Wait silently for the student to answer

Give ample time to allow the student to think and respond. Only then should you move to rephrasing the question or probing for a better understanding of the student's thoughts. After the student gives a response (whether it is accurate or not), wait again! This second wait time is even more important because it encourages the student to elaborate on his or her initial thought and provide more information.

Include opportunities for explanations.

Give students the time and space to describe their thinking and their use of strategies. This component also can reveal strong or faulty reasoning (Fagan, Tobey & Brodesky, 2016).

2) Relational Understanding

Giving your students a problem and asking them to explain how they arrived at the solution. Correct! Anytime students explain their reasoning allows them to demonstrate and build their relational understanding

3) Instrumental Understanding

Having students compete to see who can solve the most problems. Correct! Competition and game activities are fun and can help learning, but make sure that it goes beyond demonstrating instrumental understanding.

UNIT 3 PRACTICE QUIZ

INSTRUCTIONAL STRATEGIES

Here are four arguments against presenting the key word approach:

Many problems do not have key words. A student who has been taught to rely on key words is then left with no strategy. Here's an example: Aidan has 28 goldfish. Twelve are orange and the rest are yellow. How many goldfish are yellow

Unit 4

Mathematical Communication

Unit 4 Practice Quiz

Mathematical Communication

Unit 9:

Mathematical Learning Research

2) Correct Over time, you will develop your class into a mathematical community of learners where students feel comfortable taking risks and sharing ideas, where students and the teacher respect one another's ideas even when they disagree, where ideas are defended and challenged respectfully, and where logical or mathematical reasoning is valued above all. You must teach your students about your expectations for this time and how to interact respectfully with peers. The goal for sharing ideas is to view multiple ways to solve a problem. Students should have the opportunity to share their mathematical thinking while the teacher serves as a facilitator. The teacher should avoid referring to an answer, solution, or method as correct or incorrect, and allow students to self-discover their errors through strategic competency.

Metacognition refers to conscious monitoring and regulation of your own thought process. Metacognition is connected to learning. Good problem solvers monitor their thinking regularly and adjust as needed. Metacognitive behavior can be learned, and making it part of classroom discourse is one way to make this happen. The THINK framework can be used to ensure students are developing metacognitive skills. Talk about the problems. How can it be solved? Identify a strategy to solve the problem. Notice how your strategy helped you solve the problem. Keep thinking about the problem. Does it make sense?

Student B used 13 ft. as the height of the triangle to determine the area of the shaded region. Student C multiplied 12 ft. by 22 ft. to determine the area of the shaded region.

Misconceptions:Student Response for: Question 9, Draggable box 3 of 4 Correct! Student C used the correct base and height but did not use a formula for the triangle area. The student used A = bh, which is the formula for the area of a rectangle or parallelogram.

Strategies for Teaching Operations through Contextual Problem

*Analyzing Context Problems. *Think about the Answer before Solving the Problem. *Work a Simpler Problem. *Caution: Avoid the Key Word Strategy! *Here are four arguments against presenting the key word approach:

The use of nonstandard units for beginning measurement activities is beneficial at all grade levels because they:

*Focus on the attribute being measured. *Avoid conflicting objectives in introductory lessons. *Provide a good rationale for using standard units. T *

Reducing Resistance and Building Resilience

*Give Students Choices That Capitalize on Their Unique Strengths *Nurture Traits of Resilience *Make Mathematics Irresistibl *Give Students Leadership in Their Own Learnin

Provide Clarity.

*Repeat the timeframe. *Emphasize connections. *Adapt delivery modes. *Emphasize the relevant points. * Support the organization of written work. *Provide examples and nonexamples.

Some common misconceptions in algebra:

*Some common misconceptions in algebra: *An equal sign means "the answer is." *Variables can only be used for a single unknown value.

When teaching a lesson on probability, which statement demonstrates correct student reasoning about the probability of choosing two cards with replacement when using a standard 52-card deck? Correct! With this question, you must remember there is a replacement. This means that each time an individual picks a card, it will be put back in the deck. For this reason, the denominator will always be out of 52. The answer choice that starts with, "Selecting a black queen.." is the correct answer because there are two black queens and two red kings. Choosing a black queen back to back would result in 2/52 x 2/52 = 4 /2,704. This would be the same for choosing two red kings back to back

-Selecting a black queen both times is just as likely as selecting a red king back to back. -Selecting the jack of hearts back to back would carry a 2 out of 108 chance. -Selecting a red three back to back is just as likely as choosing the nine of hearts twice. -Selecting a black card first means a red card is more likely on the second draw according to the objective of the lesson.

After working through learning activities with manipulatives, a teacher transitioned to calculating equivalent fractions using multiplication. To check for student understanding, the teacher calls two students to the board to find two equivalent fractions for 2/7. The first student provides this answer: 4/14 and 8/28. The second student answers 4/9 and 5/10. Which strategy should the teacher employ to reveal student thinking?

-Since neither solved the problem correctly, the teacher should review how to find equivalent fractions using multiplication and allow for additional practice time in cooperative learning groups. -The teacher should allow both students to retell their strategies and discuss their methods for obtaining the answers. -The teacher should thank both students, acknowledge the correct answer of the first student, and have the first student show the second how to find the correct answer. -The teacher should facilitate discourse and allow peers to discuss which student was correct and identify additional equivalent fractions

An elementary math teacher is teaching a geometry unit on quadrilaterals.Which instructional approach or student statement demonstrates a discovery-based approach to teaching this concept? Correct! Remember to think about the objective of the lesson. Here, the teacher was focused on teaching quadrilaterals or squares, rectangles, and rhombi. While some of the other answers may be true, you must always remember to think about the objective of the lesson. If you provide students with quadrilaterals and they recognize that the sum of the interior angles equals 360, this would be a discovery-based approach. A discovery-based approach allows students to discover these facts rather than a teacher explicitly stating this information.

-Students explore the idea that the distance around a circle divided by the diameter of that circle always results in 3.14, or pi. -Students are given quadrilaterals and explore their interior angles to discover that all interior angles add up to 360 degrees. -Students are given different triangles and explore their interior angles to discover that all interior angles add up to 180 degrees. -When exploring triangles, students recognize that the smallest angle is opposite the smallest leg of the triangle. -A student recognizes that rectangles and squares have similar properties, so a rectangle must be a square.

A teacher is demonstrating probability with a spinner divided into six equal sections, each in a different color. After 3 spins, the outcomes are yellow, blue, yellow. Students are asked to consider the probability of spinning a blue.Which two responses have misconceptions about probability?

-The probability of spinning a blue is ⅙ because one section of the spinner is blue and there are 6 total sections. -A blue will appear ½ of the time because the outcomes follow a pattern that goes yellow, blue, yellow, blue, etc. -The probability of blue of the next spin is 0% because blue has already appeared during this round of spins. -The experimental probability of spinning a blue in three trials is ⅓. Correct! Choice B is a misconception because probability does not occur in a pattern. Blue has the same probability of appearing each spin and is not dependent on what happened previously.Correct! Choice C is also a misconception. Probability does not have a memory, and the prior spins do not affect the chance of blue occurring again in the future.

Learning basic facts can have better results if a teacher promotes 1)_____ strategies.

1) multiple Correct! Allowing multiple strategies is important because students comes with their own unique way of thinking and diverse sets of knowledge and experiences that help them make sense of the math you are teaching.

A key to getting students to be 1)_______ is to engage them in interesting problems in which they use their 2)_______. knowledge as they search for solutions and create new ideas in the process.

1) reflective Correct! When your students are reflective they are able to make better sense of the math they are learning, which means they can continue to make connections to prior knowledge. 2) prior Correct! Making connections to prior knowledge is important for many reasons, but among them is that this helps students retain the information better as it relates to other things they already know

Related to supporting multiple approaches, it is important to allow students the time to 1) _______.. the mathematics they are exploring.

1) struggle with Correct! This is productive struggle known as disequilibrium, which means that students are trying to figure out how to make sense of the new knowledge they are exposed to. Once this new knowledge is internalized, the student can use it to make sense of future, related information.

Each box plot has these three features

1. A box that contains the "middle half" of the data, one-fourth to the left and right of the median. The ends of the box are at the lower quartile, the median of the lower half of the data, and the upper quartile, the median of the upper half of the data 2. A line inside the box at the median of the data. 3. A line (sometimes known as the whisker) extending from the end of each box to the lower extreme and upper extreme of the data. Each line, therefore, covers the upper and lower fourths of the

Box Plots. Box plots (also known as box and whisker plots) are a method for visually displaying not only the center (median) but also the range and spread of data. Importantly, a box plot highlights the interquartile range, making visible the middle 50 percent Each box plot has these three features

1. A box that contains the "middle half" of the data, one-fourth to the left and right of the median. The ends of the box are at the lower quartile, the median of the lower half of the data, and the upper quartile, the median of the upper half of the data. 2. A line inside the box at the median of the data. 3. A line (sometimes known as the whisker) extending from the end of each box to the lower extreme and upper extreme of the data. Each line, therefore, covers the upper and lower fourths of the data.

A teacher is demonstrating probability and rolls a die two times, revealing a 1 and then a 6. A group of students is then asked to determine the probability of rolling a 6 on the next roll. Match the reasoning on the left to the student response. Here are students' responses and reasoning

1. Because a 1 and 6 already appeared, the next roll will be either a 2, 3, 4, or 5. Student Reasoning 0% chance of occuring 2. The probability of rolling a 6 on the next roll would be 1 out of 6. the correct answer 3. Because a 1 and 6 have already been rolled, a 6 will appear every other time the die is rolled. 1/2 chance to occur 4. In all 220 outcomes, a 6 will appear 45 times. 9/44 chance of occuring

scribe three phases, which they found were effective in increasing elementary students' understanding of probability: Probability investigations are an excellent fit to the before, during, after lesson plan model. In the before phase, students make predictions of what they think will be likely; in the during phase, students experiment to explore how likely the event is; and in the after phase, students compile and analyze the experimental results to determine more accurately how likely the event is.

1. Concrete exploration: Students make predictions about what they think will happen and then engage in hands-on experiments. 2. Representation: Students organize and represent the data gathered in phase 1, selecting their own ways to summarize the data, which might include lists, tally charts, tables, dot plots, bar and circle graphs, or pictorial representations. 3. Construction: Students analyze their representations, and construct a model describing the probability. This is presented symbolically and diagrammatically.

Here are ways to explore area of circles that build con-ceptual understanding for the formula:

1. Cover a circle with tiles. Place 1-inch square tiles on the circle or cut out squares that can be glued on. The advantage of cutouts is that students can cut squares that are only partially inside the circle and place the extra pieces somewhere else. Then students need to get a measure of the radius from the sides of the tiles.

Implications for Instruction The collection of geometric experiences you provide are the single most important factor in moving students up this developmental ladder to higher levels of geometric thought! Many activities can be modified to span two levels of thinking, helping students move from one level to the next. Consider Clements and Sarama's (2014) four features of effective early geometry instruction:

1. Show a variety of shapes and have students compare both examples and nonexamples with a focus on critical characteristics. 2. Facilitate student discussions about the properties of shapes, having them develop essential language along the way. 3. Encourage the examination classes of shapes that goes beyond the traditional, allowing students to explore relationships and recognize different categories, orientations, and sizes. 4. Provide students with a range of geometric experiences at every level, having them use physical materials, drawings, and technology.

A simpler-problem strategy has the following steps Work a Simpler Problem.

1. Substitute small whole numbers for all relevant numbers in the problem. 2. Model the problem (with counters, drawings, number lines, bar diagrams, or arrays) using the new numbers. 3. Write an equation that solves the simpler version of the problem. 4. Write the corresponding equation, substituting back the original numbers. 5. Calculate! 6. Write the answer in a complete sentence, and decide whether it makes sense.

Students struggle to understand and interpret histo-grams (Cooper, & Shore, 2008; Kaplan, Gabrosek, Curtiss, & Malone, 2014; Meletiou-Mavrotheris & Lee, 2010). These challenges include:

1. Understanding the distinction between bar graphs and histograms 2. Finding the center of the distribution (students focus on the heights of the bars (y-axis) instead of the distribution (x-axis). 3. Interpreting a flatter histogram to mean there is less vari-ability in the data.

Student C says, "A blue will appear next because it will appear every other time.

1/2 Chance to Occur Correct! Student C demonstrates a common misconception that probability has a memory; however, this student provides additional information about his thought process. This student believes that probability occurs in patterns and that the yellow, blue, yellow pattern will continue. In this case, every other time will result in ½ or 50% chanc

4/5 Student C says, "A blue will appear next because it will appear every other time."

1/2 chance to occur Correct! Student C demonstrates a common misconception that probability has a memory; however, this student provides additional information about his thought process. This student believes that probability occurs in patterns and that the yellow, blue, yellow pattern will continue. In this case, every other time will result in ½ or 50% chance.

5/5 Student D says, "In 100 spins, the blue will appear 20 times.

1/5 chance of occuring Response for: Question 4, Inline answer 4 of 4 Correct! Student D simply indicates that the probability will be 20/100, which simplifies to ⅕.

2Here are four arguments against presenting the key word approach:

2. Key words are often misleading. Many times, the key word or phrase in a problem suggests an operation that is incorrect. The following problem shared by Drake and Barlow (2007) demonstrates this possibility: There are three boxes of chicken nuggets on the table. Each box contains six chicken nuggets. How many chicken nuggets are there in all? (p. 272 Drake and Barlow found that a student generated the answer of 9, using the words how many in all as a suggestion to add 3 + 6. Instead of making sense of the situation, the student used the key word approach as a shortcut in making a decision about which operation to select.

Strategies to Avoid There are a number of ineffective approaches for gifted students, including the following 1. Assigning more ofthe same work. This approach is the least appropriate way to respond and the most likely to result in students' hiding their ability. 2. Giving free time to early finishers. Although students may find this opportunity rewarding, it does not maximize their intellectual growth to "go beyond" and can lead to students hurrying to finish a task. 3. Assigning gifted students to help struggling learners. Routinely assigning gifted students to teach students who are not meeting expectations does not stimulate their intellectual growth and can place them in socially uncomfortable situations.

4. Providing pull-out opportunities. Unfortunately, generalized gifted programs are often unre-lated to the regular mathematics curriculum (Assouline & Lupkowski-Shoplik, 2011). Dis-connected, add-on experiences are not enough to build more complex and sophisticated understandings of mathematics. 5. Offering independent enrichment on the computer. Although there are excellent enrichment opportunities to be found on the Internet and terrific apps, the practice of having gifted students use a computer program that focuses on skills does not engage them in a way that will enhance conceptual understanding and support their ability to justify their thinking.

Which two instructional strategies can be used to effectively hold small-group and classroom discussions? A) Encourage students to think-pair-share and compare solutions and strategies. B) Trade and grade papers so students can see how others solved the problem. C) Have students share multiple methods about how to solve problems. D) Discuss the correct answer and the procedure that only one student used.

A and C Correct! Think-pair-share has students first think about how they would approach a problem, and then pair up with another student to share their ideas. This gives all students a chance to talk and to see what others think. This is also a great instructional strategy to use with ELs and other students with learning disabilities because it gives them a chance to share their thoughts in a non-threatening situation. One of the great benefits of a problem-based approach is that it allows students to come up with multiple ways to solve a particular problem while also expecting all students to meet the same learning objectives as they solve the problem. You should let your students know that a problem has multiple entry points and encourage them to find a variety of ways to solve it. You can then hold a classroom discussion to allow students to share their approaches. Both of these methods also encourage metacognition, which means that the students are thinking about their own thinking. As students develop this ability they can better detect their own thinking and problem-solving errors.

A teacher wants to model -3 + 6 to a class. Which two models shown correctly show this problem? A) Solution A B) Solution B C) Solution C D) Solution D

A and D Correct! Solution B is a visual representation of -3 + 6. The student will start at -3 and move six spots on the number line to the right, landing on +3. The two color counters in solution D are a common manipulative for representing integers. One color on the counter (red in this case) represents the negative integer, and the other color (blue in this case) represents the positive integer. By matching the negative and positive integers, a quick visual solution is provided. In this case, there are 3 blue (positive) integers that were not canceled out by a red (negative) integer. The solution is +3.

ANSWER TO ABOVE QUESTION: 1A Remember that sociocultural theory is about learning happening in social settings. And even though one scenario has the teacher assist the student, the teacher does not use a constructivist approach when she tells the student exactly what to do in mathematical terms. This does not build on the student's prior knowledge nor help her make connections and build meaning.

A and D Correct! These scenarios illustrate both theories in practice. Remember that a key part of constructivism is that students are able to build on prior knowledge and experience, whether it comes from experiences outside or inside the classroom. This is a great example because the teacher asks a question that allows students to make a connection where they can then proceed further. The teacher helping the student or the students working together and teaching each other illustrates a key aspect of sociocultural theory, which puts emphasis on learning in social, not merely solitary, settings.

2) Select all instructional strategies that can be used to help and encourage mathematical communications. A) Provide opportunities that will encourage the metacognitive process and allow students to explain their thinking B) Tell students that a math problem will be easy to solve before presenting it C) Make certain to call on students who are prepared and ready to volunteer in class D) Allow students to create posters that use student-created vocabulary before introducing precise language E) Use class time effectively by encouraging students to evaluate and compare their answers with other students

A and E A) Provide opportunities that will encourage the metacognitive process and allow students to explain their thinking E) Use class time effectively by encouraging students to evaluate and compare their answers with other students

Teaching Multiplication and Division Multiplication and division are often taught separately, with multiplication preceding division. It is important, however, to combine multiplication and division soon after multiplication has been introduced in order to help children see how these operations have an inverse relationship. In most curricula, these topics are first presented in second grade, become a major focus in third grade, with continued development in the fourth and fifth grades

A major conceptual hurdle in working with multiplicative structures is understanding that while a group contains a given number of objects these groups can also be considered as single entities (Blote, Lieffering, & Ouewhand, 2006; Clark & Kamii, 1996). Children can solve the problem, "How many apples in 4 baskets of 8 apples each?" without thinking multiplicatively simply by counting out 4 sets of 8 counters and then counting all. To think multiplicatively about this problem as four sets ofeight requires children to conceptualize each group of eight as a single entity to be counted four times. Experiences with making and counting equal groups, especially in contextual situations, are extremely use

2Counting On and Counting Back

A strategy is to allow students to bob their head as they must keep track of several things: the counting sequence, the number they are starting at and how many they need to count beyond that number (Betts, 2015). Because this "double-counting" process (Voutsina, 2016) is not easily kept in working memory, teachers can model the actions and use a think-aloud to orally share their thinking as they pull these multiple components together. Eventually, first-graders should be able to start from any number less than 120 and count on from there (NGA Center & CCSSO, 2010). Children will later realize that counting on is adding and counting back is subtracting. Frequent use of Activities 7.7, 7.8, and 7.9 is recommended

A sixth-grade teacher is teaching how to solve the following equations: 10 = 3x - 5 (x + 2) = 30 Which equation should the teacher use to assess the students' understanding? A) 2(x - 4) = 20 B) 10 - 1.5x = 20 C) -2x + 2 = 10 D) x + 5 = 10 - x

A) 2(x - 4) = 20 Correct! The objective of the lesson is to solve a two-step equation. The assessment must align with the objective and this equation matches the equations students have been practicing. The equation x + 5 = 10 - x extends this lesson by providing variables on either side of the equation and does not align with the objective. The equation -2x + 2 = 10 would require students to divide by a negative coefficient, which has not been introduced in the given equations. And the equation 10 - 1.5x = 20 involves operations with decimals and that, too, has not been introduced

An eighth-grade class has been exploring volume by using centimeter cubes to build rectangular prisms. They have developed the volume formula of V = lwh. Which activity is appropriate as a follow-up to this lesson? A) A drill activity: Dimensions for various rectangular prisms are provided, and students compute the volume of each. B) A conceptual activity: Students explore area of various 2D shapes using manipulatives. C) A practice activity: Students identify geometric solids in the classroom and practice finding their surface area. D) A drill activity: Students practice their multiplication facts.

A) A drill activity: Dimensions for various rectangular prisms are provided, and students compute the volume of each. Correct! The objective of the lesson was to find the volume formula for a rectangular prism. Any task associated with this objective must align. In this situation, because students have developed a conceptual understanding of the material, it is okay to use a drill activity. A drill activity should only be used after students develop a conceptual understanding. Drill activities should never be used to teach concepts.

8. Which statement is not proposed by the constructivist theory? A) A safe environment is necessary to gain mathematical power. B) Children understand and acquire knowledge differently. C) Children must be active learners. D) Effective learning requires reflective thinking. The constructivist learning theory argues that people produce knowledge and form meaning based on their experiences. A wonderful understanding of a concept does not always mean the student will assimilate the new information. In a constructivist classroom, students need a variety of learning opportunities to explore and build relational understandings.

A) A safe environment is necessary to gain mathematical power. Correct! When students learn in an instrumental manner, math can seem like endless lists of isolated skills, concepts, rules, and symbols that must be refreshed regularly, and they often seem overwhelming to keep straight. Constructivists talk about teaching big ideas. Big ideas are really large networks of interrelated concepts. Constructivism is a philosophy that enhances students' logical and conceptual growth. The underlying concept within the constructivism learning theory is the role which experiences—or connections with the adjoining atmosphere—play in students' education.

A second-grade teacher has 23 students with varying abilities. Which instructional strategies can be used to help provide learning opportunities for all students? A) Allow students to progress through the material at their own pace and use flexible grouping options. B) Provide additional work for students to complete when they finish an assignment. C) Pair a low-achieving student with a high-achieving student. D) Lower the learning objective for the low-achieving students so they can be successful on the task.

A) Allow students to progress through the material at their own pace and use flexible grouping options. Correct! When planning for instruction, you must consider all students in the classroom. While pairing a low-achieving student with a high achieving student is an example of peer-assisted learning, this question is about providing access to learning opportunities for all students. Teachers should instead use flexible grouping options and allow students to progress through the material at their own pace.

After having your students analyze the high temperatures from July over several years, you ask them to create a histogram to represent their data. How can you incorporate observational assessment into this lesson? A) Create a card for each student to write specific learning objectives and take notes about student understanding while walking around the classroom. You can then use this information to modify the lesson B) Have students complete a quiz at the end of the lesson where they answer simple questions about the characteristics of histograms. C) Ask each student to submit a completed histogram at the end of class and evaluate it with a rubric. D) Conduct a class discussion where discourse is promoted and questions are asked about histograms and what students learned, and record highlights of the discussion.

A) Create a card for each student to write specific learning objectives and take notes about student understanding while walking around the classroom. You can then use this information to modify the lesson Correct! All teachers can learn useful information about their students every day. During an observational assessment a teacher uses anecdotal notes or a checklist to make comments about student learning. There is a difference between holding a classroom discussion and performing an observational assessment. During an observational assessment, the teacher must be taking notes or using a checklist to keep track of student understanding.

Which activity could be used to make a mathematical connection within the mathematics curriculum when teaching a unit on multiplying fractions by whole numbers? A) Demonstrate the relationship between multiplying fractions by whole numbers and multiplying decimals by whole numbers. B) Use manipulatives to demonstrate fraction multiplication by whole numbers. C) Have students research and journal four applications of multiplying fractions by whole numbers. D) Create several application problems to provide context involving the multiplication of fractions by whole numbers.

A) Demonstrate the relationship between multiplying fractions by whole numbers and multiplying decimals by whole numbers Correct! This approach requires students to make a connection between two separate mathematical concepts, in this case, between fractions and decimals. You will almost certainly have students who are not sure how fractions and decimals relate. Activities like this will help them see the connection.

A class is working through a unit of study about plant growth. They have planted tomato plants and have placed plants under artificial light, sunlight, and in the dark. They are observing and recording growth over two weeks. How can the math and science curriculum be integrated during this unit? A) Gather data about plant growth and then convert it into a graph. B) After gathering and analyzing data about plant growth, consider how planting and growing a garden can help families save money. C) Prior to planting, work in groups to develop a hypothesis about plant growth in different types of light. D) Record observations about plant growth in science journals and predict future growth.

A) Gather data about plant growth and then convert it into a graph. Correct! This approach offers a math to science connection by requiring students to gather data, a very scientific activity, and to then put it into a graph, which will require them to apply the mathematical concepts they are learning.

Select from a variety of shapes that are similar to each other. A)Level 0: Visualization B)Level 1: Analysis C)Level 2: Informal Deduction

A) Level 0: Visualization Correct! At Level 0 students should be able to see which shapes are similar to each other by looking at them. For example, if you show your students many different types of triangles, they should still be able to recognize that they are all triangles even though they are not identical.

You have noticed that one of your students is counting objects with an accurate sequence of number words, but is not attaching one number to each object. Therefore, the student's final count is inconsistent and inaccurate. Which instruction would you plan to help this student develop a better grasp of one-to-one correspondence? A) Provide the student with a set of blocks or counters organized in a pattern or row to point at as they say each number word. B) Have the student practice recognizing number-words in written form on flash cards. C) Spread out a large quantity of math manipulatives in a pile for students to practice grouping and counting. D) After the student practices counting the number of objects in a set, have the student practice identifying the total number of objects in the set.

A) Provide the student with a set of blocks or counters organized in a pattern or row to point at as they say each number word. Correct! One-to-one correspondence is one of the first counting skills students develop after learning to count orally. This skill is best developed by helping students practice pointing to objects while saying the number words verbally. Objects should be arranged in sets, patterns, or rows for students, so they can easily keep track of which items they have already counted.

Which stakeholder is directly in charge of choosing mathematical standards? A) State departments of education B) Parent-teacher organizations C) Federal department of education D) Teacher organizations

A) State departments of education Correct! The state department of education in each state determines which standards will be taught in their schools. While the federal government may suggest standards, and even financially incentivize adopting those standards, the decision of which standard is implemented lies with each individual state.

7. What is an environment that is not desirable for mathematical instruction? A) Students are working through practice problems to learn the mathematical strategy. B) Reasoning is celebrated as students defend their methods and justify their solutions. C) Students are testing ideas, making conjectures, developing reasons, and offering explanations. D) The focus is on students actively figuring things out.

A) Students are working through practice problems to learn the mathematical strategy. Correct! Practice problems should only be used after conceptual understanding. Practice problems do not help students build problem-solving strategies. Problem-solving strategies should be encouraged and taught by using rich problems and building a classroom of respect. Teachers should use class time to discuss students' strategies and allow students time to collaborate while working through rich tasks.

During the month of October, a teacher is planning a thematic unit on fall, which includes studying harvests and pumpkins and will involve math. How can the teacher create a thematic-based lesson? A) Students can estimate how many seeds are in a pumpkin and then make conjectures based on mass. B) Students can find the mass of several pumpkins and arrange them in order. C) The teacher can collaborate with the art teacher and have students decorate a pumpkin in class. D) Students can compare and contrast how fall is celebrated around the world.

A) Students can estimate how many seeds are in a pumpkin and then make conjectures based on mass. B) Students can find the mass of several pumpkins and arrange them in order. Correct! The key to this question is making sure that you see both a fall/pumpkin aspect and a math aspect. In the second choice, students are estimating the number of seeds (math) in a pumpkin (fall) and making conjectures based on mass (math). In the last choice, students explore mass (math) of pumpkins (fall) and arrange them in order

Which two activities show a connection to a context outside the math classroom? A) Students explore how certain architectural designs influence the stability of a bridge or skyscraper. B) Students practice multiplication facts and write about the process in their math journal. C) Students use manipulatives to solve fraction problems. D) Students compare oil and gas prices from around the world and create diagrams.

A) Students explore how certain architectural designs influence the stability of a bridge or skyscraper D) Students compare oil and gas prices from around the world and create diagrams. Correct! Having students compare the oil prices from around the world and preparing graphs can help create a connection to the outside world. Relating curriculum to outside events helps contextualize the mathematical concepts. When you as the teacher attempt to make a mathematical connection outside of the classroom, you are trying to relate the math concepts to real life. Studying architectural designs can be integrated into a geometry unit. You can describe how mathematics is needed to analyze and calculate structural problems for engineering a solution to ensure that a structure will remain standing and stable. The sizes and shapes of design elements are possible to describe because of mathematical principles such as the Pythagorean theorem

A teacher wants students to write out how to use positive and negative chips to model operations with integers. How does this lesson address the Communication Standard from the Principles for School Mathematics? A) Students will use precise math language to express their reasoning. B) Students will model an abstract concept using manipulatives. C) It allows students to connect previously learned math to newly learned math content. D) Students use the manipulative to show the teacher how to add and subtract integers.

A) Students will use precise math language to express their reasoning. Correct! When students work through various math problems, it is important that you encourage them to use mathematically precise language. For example, students tend to use top number and bottom number when working with fractions. Math teachers should use good questioning techniques and respond to students by modeling how to use mathematically precise language. For example, the teacher could say, "How did you get your common denominator?"

After completing probability experiements, students displayed their results in a circle graph. How can a teacher incorporate an observational assessment into this lesson? A) Take short narrative-style notes on a card to record student understanding while circulating through the class and use that data to modify the lesson. B) Have students complete a quiz at the end of the lesson where they answer simple questions about probability. C) Conduct a class discussion to promote discourse and ask students questions about probability and what they learned. Record highlights of this discussion. D) Ask each student to explain the results of their probability experiments in a formal presentation.

A) Take short narrative-style notes on a card to record student understanding while circulating through the class and use that data to modify the lesson. Correct! All teachers can learn useful bits of information about their students every day. During an observational assessment, a teacher uses anecdotal notes or a checklist to make comments about student learning. There is a difference between holding a classroom discussion and performing an observational assessment. During an observational assessment, the teacher must be taking notes or recording observations using a checklist or other record syste

Question 3 A teacher would like to collaborate with an art teacher to integrate a math lesson. Which concept can the teacher use to incorporate a pattern on a quilt into the math lesson? A) Tessellations of polygons B) Addition of fractions C) Volume measurements D) Probability

A) Tessellations of polygons

A teacher is planning a lesson on adding and subtracting positive and negative integers. The teacher will use red and black chips as a manipulative and allow students to develop patterns for understanding the addition and subtraction problems. The following NCTM standard is associated with the lesson: "Represent the idea of a variable as an unknown quantity using a letter or a symbol." How do you rate the activity and alignment to the standard? A) The activity is effective at showing students how to add and subtract integers but the standard is not aligned. B) The activity is not effective at showing students how to add and subtract integers and the standard is not aligned. C) The activity is not effective at showing students how to add and subtract integers but the standard is aligned. D) The activity is effective at showing students how to add and subtract integers and the standard is aligned.

A) The activity is effective at showing students how to add and subtract integers but the standard is not aligned. Correct! While this lesson does a very good job of teaching students to represent integers both numerically and with a manipulative, it does not address the use of variables. Since the standard relates to variables, not integers, the lesson is not aligned.

4) A teacher has a group of low-level learners and high-ability learners. The objective of the lesson is for students to measure the length of an object using a non-standard unit of measurement. The teacher plans to deliver explicit instruction to the low-level learners on how to measure with a paper clip (an example of a non-standard unit of measurement) and plans to allow high-ability learners the opportunity to create their own non-standard measuring unit out of art supplies and measure objects using their own unit. How effective is this activity at meeting the needs of all learners? A) The lesson effectively addresses all learners and offers complexity and depth for higher ability students. B) The lesson does not address the needs of all learners because providing students with step-by-step instruction on how to measure an object with a non-standard unit will not allow the low-level learners to meet the learning objective. C) The lesson addresses the needs of all learners because all students will have the chance to complete the same assignment. D) The lesson does not address the needs of all learners because the teacher is expecting the low-level learners to measure the object with a non--standard unit.

A) The lesson effectively addresses all learners and offers complexity and depth for higher ability students. Correct! The lesson objective is to have students measure the length of an object using a non-standard unit. All students must meet this objective, and a teacher can never reduce the expectation for lower-level learners. Teachers can differentiate by considering the four elements: content, process, product, and environment. In this situation, the teacher is differentiating the content by providing the higher-level learners an opportunity to create their own non-standard unit and then use it to measure. The process by which students are reaching the objective also differs. The lower-level learners are receiving more explicit instruction. The teacher will use probing questions and model precise language as students measure with non-standard units. The same assignment is not being completed by all students because the lower-level students are measuring with a paper clip while the higher-level learners are creating a more complex product. However, all students are meeting the same objective. This type of lesson delivery provides the higher-level learners with more depth and complexity.

A student solves and submits the above fraction problem. How can this student demonstrate strategic competency? A) The student uses a manipulative and realizes that a common denominator was not used. B) The student collaborates with a peer; the peer informs him that a common denominator was not used and the error is corrected. C) The student works with other students to show them how to solve fractions. D) The student explains to another student that 5 - 3 = 2, so the answer is 1/2.

A) The student uses a manipulative and realizes that a common denominator was not used. Correct! Truly understanding math is more than just content knowledge. Strategic competence is the ability to self-regulate the learning process and make adjustments to mathematical thinking. This does not occur when a teacher or peer tells a student they have done something incorrect, but rather through the metacognitive process where students can detect their own errors.

Question: Imagine that in your classroom you worked through a series of concrete learning activities to help students understand how to find equivalent fractions using manipulatives. You then transitioned students from using manipulatives to finding equivalent fractions using representative problem-solving. After this you want to check for understanding of students' ability to calculate equivalent fractions using multiplication. You call two students to the board to solve the following problem: Find two equivalent fractions for 2/7. The first student provides the answer 4/14 and 8/28. The second student answers 4/9 and 5/10. After seeing the students' work, you now want to use an instructional strategy that reveals student thinking. Which strategy should you use?

A) The teacher should facilitate discourse and allow peers to discuss which student was correct and identify additional equivalent fractions. B) The teacher should allow both students to retell their strategies and discuss their method of obtaining the answer. C The teacher should thank both students, acknowledge the correct answer of the first student, and have the first student show the second how to find the correct answer. D) Since neither solved the problem correctly, the teacher should review how to find equivalent fractions using multiplication and allow for additional practice time in cooperative learning groups.

Which strategy will help students gain a step-by-step, metacognitive understanding of how to form a whole using 1/4 and 1/8? A) Think-alouds B) Peer-assisted learning C) Explicit strategy instruction D) CSA sequence

A) Think-alouds Correct! Think-alouds are very similar to explicit strategy instruction in that they both involve the teacher explaining and modeling the process. The difference, however, is that the teacher is trying to show the metacognitive process in approaching and solving a problem. In other words, the teacher will work on a different but similar problem with students. The students will be involved in the think-aloud process by helping to answer the teacher's questions about why she did what she did. The students will then imitate the thinking and problem-solving process that the teacher is modeling in one problem and apply it to another.

10. Students in a sixth-grade class are using models to explore how to find percents. Which question should help foster critical thinking? A) What patterns have you noticed when modeling percents? B) Will a percent always be out of 100? C) Will a percent always be between 1% and 100%? D) Can you model 50%?

A) What patterns have you noticed when modeling percents? Correct! Critical thinking can be evoked by asking open-ended questions. When asking students to explain the observed patterns, the teacher is asking an open-ended question. The other choices are closed-ended questions and can be answered with a simple yes or no. In a constructivist classroom, teachers should ask open-ended questions that cannot be answered with a simple yes or no

A teacher is explaining the process of finding the slope of a line. Which tool or manipulative could help engage the students in this process? A) Base-t en blocks B) Cuisenaire rods C) Calculator D) Isometric drawing applet

A) calculator Correct! A calculator is considered a technology tool that can be used in the math class. While a calculation should never replace basic computational skills, some math skills and objectives are beyond computation and students can use a calculator to solve these problems. A calculator can be used to solve algebraic and other math concepts and students can always check their work using a calculator.

When teaching about growing patterns, students should be encouraged to do all the following except __________. A) make a pattern when given a table or chart of numeric data B) try to determine how each frame in the pattern differs from the preceding frame C) build patterns and discuss how they can be extended in a logical manner D) build or draw the first five or six frames to discover the numeric relationship of the frame number and the object

A) make a pattern when given a table or chart of numeric data Correct! Beginning in the primary grades and extending through the middle years, students can explore patterns that involve a progression from step to step. In technical terms, these are called sequences. Analyzing growing patterns should include the developmental progression of reasoning by looking at the visuals, reasoning about the numerical relationship, and then extending to a larger case. Students' experiences with growing patterns should start with fairly straightforward patterns and then move on to patterns that are more complicated.

The equal sign is one of the most important symbols in elementary arithmetic and in all mathematics. What is a common misconception about the equal sign? A)Students understand that the equal sign means "the answer is." B)An equal sign is used to write an equation. C)An equal sign shows the relationship between two different quantities. D)Students understand the equal sign to represent equivalency.

A)Students understand that the equal sign means "the answer is." Correct! Students will have the common misconception that the equal sign means they have to solve something rather than it showing equivalency.

After watching the video "Teaching Mathematics for Understanding" and reading the pages from the textbook, what did you encounter that helped you comprehend aspects of understanding mathematics? (Select three of the following options.) A) The ability of students to explain why they solved the problem in a certain way B) Emphasis on relational understanding C) The ability of students to make connections D) Emphasis on instrumental understanding E) The ability of students to solve the math problem

A, B, and C Correct! Relational understanding is very important since it means the student not only knows how to solve a mathematical problem but also that they know why they solved it in a certain way. In other words, they can explain what they did and why. It is also important that the student makes connections between what they already know and what they need to learn. This not only increases their understanding, it helps the knowledge stick better as well. Instrumental understanding and the student's ability to solve a given math problem is the same. While this is important, it does not characterize a full understanding.

Multiplication and Division Problem Structures

As with addition and subtraction, there are problem structures that will help you in formulating and discussing multiplication and division tasks. These structures will also benefit your students in generalizing as a way to identify familiar situations.

Mean.

Ask an adult what the mean is, and they are likely to tell you something like, "The mean is when you add up all the numbers in the set and divide the sum by the number of numbers in the set." This is not what the mean is, this is how you calculate the mean. This is a reminder of the procedurally driven curriculum in our history and the need to shift to a more conceptually focused approach. Another limited conception about the mean is that it is considered the way to find a measure of center regardless of the context (McGatha, Cobb, & McClain, 1998). In fact, in the CCSS-M, sixth-grade students are expected to deter-mine when the mean is appropriate and when another measure of center (e.g., the median) is more appropriate: "Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values" (NGA Center & CCSSO, 2010, p. 39). The next section focuses on developing the concept of the mean.

1) Relational Understanding

Asking the question, "Was there something in this problem that reminded you of another problem you've done?" Correct! This question is a great way to help students make connections either to other things they have learned in class or things they have experienced outside of class.

Your fifth-grade class was asked to explore the relationship of various volume formulas and to fill a rectangular prism, cube, and triangular prism with water. Which two strategies could help increase the effectiveness of the activity? A) Give students step-by-step instructions for the activity. B) Encourage students to collaborate and discuss findings and strategies once the activity is complete. C) As you watch students working on the task, stop them if they are doing something wrong. D) Before the activity, review the properties of 3D shapes.

B and D Correct! Collaborative learning is a great constructivist teaching strategy that helps students to be resilient. It identifies their resources (peers) and tests their theories to see if they are on the right track, all while developing habits of mind that form the foundation of scholarship. In a constructivist classroom, students should be encouraged to explore multiple strategies and ways to solve a problem. Before students attempt to build a conceptual understanding, it is important to activate their prior knowledge. In teaching mathematics, you should help students relate new information to previously learned material.

When you need to plan a lesson for all learners, which two of the following strategies would you use? A) Use different learning objectives for lower- and higher-level learners. B) Use graphic organizers to help students organize work. C) Choose a step-by-step method for all students. D) Do drill exercises to help students learn new concepts. E) Include opportunities for collaboration and classroom discussion.

B and E Correct! Graphic organizers can help teachers tier lessons and provide students with structure on how to organize their thoughts. For example, lower-level learners can be provided with a more structured graphic organizer while higher-level learners can be provided with a less structured graphic organizer. Regardless, all learners can benefit from using a graphic organizer to help organize their mathematical thinking. You should never create different learning objectives for different students, only different ways for them to meet those objectives. Discussion and collaboration is heavily emphasized in constructivist teaching since it gives students a chance to share ideas and build knowledge. In addition, these teaching strategies also allow you to accommodate students since they can approach a problem in their own way. However, you will want to make sure that you are still maintaining the same learning objectives and standards for all students.

1) A sixth-grade teacher uses a think-aloud to model how to solve the following three equations. a.) 2x + 4 = 12 b.) 10 = 3x - 5 c.) 3(x + 2) = 30 Which equation can be used on an assessment to measure learning outcomes? A) x + 5 = 10 - x B) 2(x - 4) = 20 C) -2x + 2 = 10 D) 10 - 1.5x = 20

B) 2(x - 4) = 20 Correct! The objective of the lesson is to solve two-step equations. The assessment must align with the objective and the equation provided in choice B matches the equations students have been practicing. Choice A extend this lesson by providing variables in either side of the equation and does not align with the objective. Choice C would require students to divide by a negative coefficient and that has not been introduced in the equations the teacher presented. Choice D involves operations with decimals and that, too, has not been introduced.

Which process demonstrates a conceptual understanding of solving 32 + 48 = _____? A) A student uses a peer-assisted approach when solving the problem by using a manipulative. B) A student first adds 8 + 2 to get 10 and then adds 30 + 40 + 10 to get a sum of 80. C) Students use a traditional algorithm and then check their work with a calculator. D) A student collaborates with a classmate to solve the problem with a calculator.

B) A student first adds 8 + 2 to get 10 and then adds 30 + 40 + 10 to get a sum of 80. Correct! A conceptual understanding occurs when a student is able to explain a math process. Often students are taught traditional algorithms, but lack the understanding of their use. While traditional algorithms can be taught, a conceptual understanding can further be developed when students use number properties (e.g., commutative and associative) to solve math problems or use the technique called subitizing. Subitizing occurs when students group numbers together for easy computations. This often occurs when students find groups of ten as illustrated in the correct answer.

The objective of a lesson is to have students develop their own addition problems. Students have been asked to write a "join" problem. Which of the following correctly represents this problem type? A) Lillian has a dozen cookies. She gives 8 away to her friends. How many does she have left? B) Caesar has $34 and earns $15 more dollars mowing a lawn. How much money does he have? C) Sara has 12 apples. Four of her apples are red and the rest are green. How many green apples does she have? D) Stockton and Logan are catching insects in the garden. Stockton caught 4 more insects than Logan. Logan has 7 insects. How many insects does Stockton have?

B) Caesar has $34 and earns $15 more dollars mowing a lawn. How much money does he have? Correct! There are four different addition and subtraction problem types: join, separate, part-part whole, and compare. In this case, students are asked to identify a join problem. Here Caesar starts with a quantity, and more is joined. Sara is doing a part-part whole problem. Stockton and Logan are doing a compare problem. Lillian is doing a separate problem.

6. To create an environment for doing mathematics, what is the teacher's role? A) Test ideas and conjectures and tell students the results B) Create a spirit of inquiry, trust, and expectations C) Demonstrate mathematical procedures D) Focus on posing problems

B) Create a spirit of inquiry, trust, and expectations Correct! In a constructivist math class, the teacher should act as a facilitator and ask guiding questions. This process takes time, and the teacher must use class time effectively to build a classroom culture of respect and trust. Once this is established, students will feel comfortable talking about errors and learning from their mistakes.

Students in your third-grade class are asked how many small unit squares will fit in a rectangle that is 54 units long and 36 units wide. They are provided with base-10 blocks to identify a solution. While students complete this assignment, you plan to walk around the room to observe, question, and prompt the students to ensure understanding. How should you use a rubric to help your students complete this assignment? A) Use a 4, 3, 2, 1 rubric scoring system and equate this to A, B, C, D letter grades. B) Explain how the rubric will be used for evaluation before students begin the problem. C) Have students complete this activity in their learning journal. D) Use observational assessment and take notes on the students.

B) Explain how the rubric will be used for evaluation before students begin the problem. Correct! It is always a good practice to share the rubric with students before they begin an assignment. This allows them to know the specific criteria and objectives that they need to focus on when doing the assignment. There is no need to keep students in the dark about what you expect from them. Remember that rubric scores do not need to be the same as letter grades. Having students complete this in their learning journal may be okay, but this still does not tell us how you should use a rubric. Observational assessment is a form of assessment but not a rubric.

5) When working with a student with a mild learning disability, which instructional approach should be used to accommodate for the student but should not be used for the general population? A) Think-aloud B) Explicit instruction C) Think-pair-share D) Collaboration

B) Explicit instruction Correct! Students with learning disabilities have strengths that the teacher should use to help develop a conceptual understanding. The learning objective should never be minimized for students with learning disabilities and strengths should be noted; however, explicit instruction can be used to help meet the needs of students that are struggling.

Traditional problem-solving lessons include a teacher explaining the math and students practicing the math, after which they apply those skills to solving related math problems. Which answer explains why this rarely works? A) It begins where the students are. B) It assumes detailed explanations produce understanding. C) Problem-solving is too wrapped up in the lesson. D) Students do not have enough time to practice in advance.

B) It assumes detailed explanations produce understanding. Correct! Such an approach can help with instrumental but not relational understanding. Constructivism argues that people produce knowledge and form meaning based on their experiences, which help them make connections. A student may understand a concept and yet still be unable to assimilate the new information if it cannot be connected to prior knowledge and experiences.

When working with children who have learning disabilities, which of the following is important to remember? A) Learning disabled students are not as mentally capable. B) Learning disabilities should be compensated for by helping students use their strengths. C) Learning disabled students are easily remediated with more practice. D) Instructional modifications should not be made since all students need to learn the material.

B) Learning disabilities should be compensated for by helping students use their strengths. Correct! Students with learning disabilities have strengths that the teacher can encourage in order to help develop a conceptual understanding. Remember that the learning objective should never be minimized for students with learning disabilities and the students' strengths should instead be noted.

Sort shapes by recognizing common properties. A)Level 0: Visualization B)Level 1: Analysis C)Level 2: Informal Deduction

B) Level 1: Analysis Correct! At Level 1 students are able to classify shapes by identifying their properties. When you do this activity, have your students use the words "at least." For example, all triangles have at least three sides; all squares have at least four even sides with four right angles, etc.

A student wants to construct a graph to show each varying temperature of ice water after 5, 10, 15, 20, 25, and 30 minutes. What type of graph would best show this data? A) Stem-and-leaf plot B) Line graph C) Histogram D) Bar graph

B) Line graph Correct! A line graph is used to represent two related pieces of continuous data, and a line is drawn to connect the points. For example, a line graph might be used to show how the length of a flagpole shadow changed from one hour to the next during the day. The horizontal scale would be time, and the vertical scale would be the length of the shadow. Data can be gathered at specific points in time (e.g., every 15 minutes), and these points can be plotted. A straight line can be drawn to connect these points because time is continuous and data points do exist between the plotted points.

For the next set of questions, use the following scenario: Your students are given an assignment to see how many 1/4 and 1/8 fractions it takes to make up a whole. However, the teacher notices that many students are struggling to understand why 1/4 is bigger than 1/8. Question Which strategy makes use of sociocultural learning to help assist students who are struggling up to a more competent level of understanding by providing them with the knowledge they need when they need it? A) CSA sequence B) Peer-assisted learning C) Explicit strategy instruction D) Think-alouds

B) Peer-assisted learning Correct! Peer-assisted learning follows a sort of apprenticeship model that has less experienced or less knowledgeable students receive assistance from their more knowledgeable peers. An important element of this process is that knowledge is provided on an as-needed basis. This means that less knowledgeable students will work on their own as best as they can before receiving assistance. As the teacher, however, you will want to be careful who you pair students with, especially as most young students may not know exactly how to teach a concept to a peer. This strategy does, however, highlight sociocultural learning in action since students are learning in a socially interactive way that helps students increase their level of competency.

Which answer is not a reason to use a calculator in the constructivist math class? A) Provides exploratory learning opportunities B) Performs basic computations such as 7 × 3 when computation skills are the objective of the lesson C) Save time when computation skills are not the objective of the lesson D) Enhances problem-solving in the classroom

B) Performs basic computations such as 7 × 3 when computation skills are the objective of the lesson

Find the perimeter of the above rectangle and the area of the blue region. A) Perimeter = 100 units; area = 50 square units B) Perimeter = 30 units; area = 25 square units C) Perimeter = 50 units; area = 25 square units D) Perimeter = 30 units; area = 50 square units

B) Perimeter = 30 units; area = 25 square units Correct! To find the perimeter, add up all the sides to the quadrilateral. To find the area of the shaded region, use the appropriate formula. The shaded region is a triangle and the area formula for a triangle is (b x h) / 2 or ½ (b x h).

1) Which stakeholder is directly in charge of choosing mathematical standards? A) Federal department of education B) State department of education C) Parent-teacher organizations D) Teacher organizations

B) State department of education Correct! The state department of education in each state determines which standards will be taught in their schools. While the federal government may suggest standards, and even financially incentivize adopting those standards, the decision of which standard is implemented lies with each individual stat

Question 1A Even though constructivism and sociocultural theory are two different learning theories, they can both be used simultaneously in teaching mathematics. Imagine that you are teaching your students about fractions. You instruct them to explain how to divide five brownies between three friends. Which two scenarios illustrate these two theories working together? A) One student is working by himself to figure out how to solve this problem. He is not sure how to proceed and the teacher assists him by asking, "Have you ever divided candy between your friends before?" The student has. He completes the problem and can then show other students how he did it.

B) Students are working by themselves to solve this problem and must write out their solution using fractions. C) One student is working by herself and remembers a time when she actually did divide three brownies between five friends. This helps her know how to proceed. D) Students are working in groups of three and are figuring out how to solve this problem using some of the concepts about fractions that they have already learned. They must then write their answer using fractions and share it with the class. E) One student is working by herself but cannot figure out how to proceed. She then asks the teacher for help and the teacher says, "Let's first begin by writing 5 over 3 and see what happens."

Question 1 A first-grade teacher is planning a lesson on adding fractions. How can the teacher allow the students to make a math to real-world connection when teaching this lesson? A) Students can use a fraction model when solving the equations. B) Students can use a recipe and determine the total amount of ingredients. C) Students can complete a homework task that will require them to add fraction equations. D) Students can complete a quiz on adding fractions.

B) Students can use a recipe and determine the total amount of ingredients.

4) A teacher wants students to write out how to use positive and negative chips to model operations with integers. How does this lesson address the representation process standard? A) Students will use precise math language to express their reasoning. B) Students will model an abstract concept using manipulatives. C) Students will use the manipulative to show the teacher how to add and subtract integers. D) It allows students to connect previously learned math to newly learned math content.

B) Students will model an abstract concept using manipulatives. Correct! The textbook readings will provide additional information about the Principles and Standards for School Mathematics. When students work through various math problems, it is important that you encourage them to use multiple representations to model math concepts. Translating between multiple representations of a math concept builds conceptual understanding.

A teacher is presenting a lesson on a histogram. Which two statements demonstrate correct analysis of the histogram? A) Six students scored a 60 on the math test. B) The average number of students within each interval is 6. C) The median of the test scores will be 70. D) Two more students scored within the 80-89 score range than in the 60-69 score range.

B) The average number of students within each interval is 6. D) Two more students scored within the 80-89 score range than in the 60-69 score range. Correct! A histogram shows continuous data on the x-axis and you are unable to actually determine an actual score. For example, it is not possible to determine how many students actually scored a 60 on this test because 6 students scored between 60 and 69. The average number of students within each range is calculated by adding the total number of students within each range and dividing by 5, which is the total number of bars. Eight students scored within the 80-89 score range while six students scored within the 60-69 range. The difference between eight and six is two.

In your reading you also encountered the following teaching strategies: explicit strategy instruction; concrete, semi-concrete, abstract (CSA) sequence; peer-assisted learning; and think-alouds. These four strategies are not the only teaching strategies to help you provide equity in teaching, but they are good tools that can be used in many situations when students are stuck. The following questions highlight these strategies. Question Which two strategies could be part of a lesson that uses think-alouds? A) The teacher uses a peer tutoring opportunity. B) The teacher demonstrates to students how to solve a problem. C) The teacher discusses methods and strategies to help solve problems and provides reasoning for each step. D) The teacher has students use tools and manipulatives during learning activities.

B) The teacher demonstrates to students how to solve a problem. C) The teacher discusses methods and strategies to help solve problems and provides reasoning for each step. Correct! Think-aloud is an instructional strategy that has the teacher model a certain task for students while explicitly telling the students what you are doing and why. As you do this, students will imitate your process by working on a similar problem. This strategy is one that allows novice learners to learn from those who are more expert

All of the following are true about probability except __________. A) probability of an event is a measure of the likelihood of an event occurring B) probability changes based on the number of trials C)experimental probability gets closer to theoretical probability with a larger data set D) probability can be determined in two ways: theoretically and experimentally

B) probability changes based on the number of tri Correct! Theoretical probability is the likeliness of an event happening based on all possible outcomes. The theoretical probability will not change based on the number of trials. For example, the theoretical probability of rolling a 4 on a six-sided die is 1/6. The probability will not change as more trials are conducted. The theoretical probability of rolling a 4 will remain 1/6.

According to the Teaching Principle defined by NCTM, effective mathematics teaching requires __________. A) educated mathematics teachers who possess certification in their subject B) understanding what students know and need to learn C) instructional and financial support D) knowing the important ideas to teach

B) understanding what students know and need to learn Correct! According to Principles and Standards for School Mathematics (2000), what students learn about mathematics depends almost entirely on the experiences that teachers provide every day in the classroom. To provide high-quality mathematics education, teachers must (1) understand deeply the mathematics content they are teaching; (2) understand how students learn mathematics, including a keen awareness of the individual mathematical development of their own students and common misconceptions; and (3) select meaningful instructional tasks and generalizable strategies that will enhance learning. Each of these three things will be covered in this course.

Bar Graphs

Bar graphs include object graphs (real objects used in the graph), picture graphs, and regular bar graphs and are typically used to graph categorical data After object and picture graphs, bar graphs are among the first ways used to group and present data with students in preK-2.

Continuous Data Graphs

Bar graphs or picture graphs are useful for illustrating categories of data that have no numeric ordering—for example, favorite colors or TV shows. On the other hand, when data are grouped along a continuous scale, they should be ordered along a number line. Examples of such infor-mation include temperatures that occur over time, height or weight over age, and percentages of test takers' scoring in different intervals along the scale of possible score

1)Correct! An effective strategy for discussion, specifically during the problem-solving phase, is think-pair-share. Students first spend time developing their own thoughts and ideas on how to approach the task. Then they pair with a classmate and discuss each other's strategies. This provides an opportunity to test out ideas and to practice articulating them. This is also a great approach for LEP (limited English proficiency) students and students with learning disabilities because it provides a nonthreatening chance to speak about their thoughts.

Because your students will likely have a big range in where they are mathematically, it is important to use problems that have multiple entry points. This means that the task has varying degrees of challenge within it or it can be approached in a variety of ways. One advantage of a problem-based approach is that it can help accommodate the diversity of learners in every classroom because students are encouraged to use a variety of strategies that are supported by their prior experience. After the problem-solving stage, it is best to hold a classroom discussion and allow students to share how they solved the problem. A rich problem can take 15-20 minutes to discuss.

performance indicators Performance indicators can be set by looking at students' common misconceptions.

Being aware of students' common misconceptions can help you create your performance indicators because you will know which weak spots you should help students overcome.

Box Plots.

Box plots (also known as box and whisker plots) are a method for visually displaying not only the center (median) but also the range and spread of data. Importantly, a box plot highlights the interquartile range, making visible the middle 50 percent (the box). In Figure 20.15, the ages in months for 27 sixth-grade students are given, along with stem-and-leaf plots for the full class and the boys and girls separately. The stem-and-leaf is a good way to prepare for creating a box plot. To find the two quartiles, find the medians of the upper and lower halves of the data. Mark the two extremes, the quartiles, and the median. Then create the box plot on a number line

1) The meaning of some math terms are different than everyday use, thus causing confusion for these students.

Build in Math Vocabrualry lesson. Correct! Several words that are commonly used in math (e.g., irrational, factor, matrix, variable) can typically have other meanings outside a mathematical context or they are not often used in other contexts. It is important for all students, not just EL students, to be familiar with these words.

Equal-Group Problems. Equal group problems involve three quantities: the number of groups (sets or parts of equal size), the size of each group (set or part) and the total of all the groups (whole or product). The parts and wholes terminology is useful in making the connection to addi-tion. When the number of groups and size of groups are known, the problem is a multiplication situation with an unknown product (How many in all?). When either the group size is unknown (How many in each group?) or the number of groups is unknown (How many groups?), then the problem is a division situation.

But note that these division situations are not alike. Problems in which the size of the group is unknown are called fair-sharing or partition division problems. The whole is shared or distributed among a known number of groups to determine the size of each. If the number of groups is unknown but the size of the equal group is known, the problems are called measurement division or sometimes repeated-subtraction problems. The whole is "measured off" in groups the given size. Use the illustration in Figure 8.10(a) as a reference

Creativity

By presenting open-ended problems and investigations students can use divergent thinking to examine mathematical ideas—often in collaboration with others. These collaborative experi-ences could include students from a variety of grades and classes volunteering for special math-ematics projects, with a classroom teacher, principal, or resource teacher taking the lead. Their creativity can be stimulated through the exploration of mathematical "tricks" using binary num-bers to guess classmates' birthdays (Karp & Ronau, 2009) or design large-scale investigations of the amount of food thrown away at lunchtime (Ronau & Karp, 2012). A group might find mathematics in art (Bush, Karp, Lenz, & Nadler, 2017; Bush, Karp, Nadler, & Gibbons, 2016). Another aspect of creativity provides different options for students in culminating performanceof their understanding, such as demonstrating their knowledge through inventions, experiments, simulations, dramatizations, visual displays, and oral presentations. Noted researcher on mathematically gifted students, Benbow states that acceleration com-bined with depth through enrichment is best practice (Read, 2014), Then learning is not only sped up but the learning is deeper and at more complex levels.

You are teaching the concept of long division. Which two activities would encourage and promote student communication about their math thinking? A) Introduce a mathematical board game for students to play during station time. B) Allow students to practice long division using drill and practice methods. C) Have students create their own word problem that would require long division. D) Have students draw a picture when solving long division problems.

C and D Correct! Encouraging students to create a word problem about a situation can provide the teacher with information about students' mathematical thinking. For example, if you provide a student with 210 ÷ 10, the student should be able to identify the dividend and divisor in the division problem and be able to apply that in the word problem correctly. Drawing a picture is also an effective strategy that students can use to solve a problem. If students are provided with 210 ÷ 10, you would expect to see a model that looks like this: This picture illustrates the thinking of students and shows that 210 can be divided into 10 groups. Each group will receive 21. This is different than using the traditional algorithm and helps develop a conceptual understanding.

Choose two learning strategies that describe effective math learning. A) Students must be presented with step-by-step procedures. B) Students should be taught new math concepts in isolation. C) Students learn new ideas through collaboration with the teacher and peers. D) Students learn best by presenting formulas and practice problems. E) Students learn by rearranging previously learned concepts and making connections.

C and E Correct! Each of these strategies highlights important aspects of both constructivism and sociocultural learning theory. Teachers who use sociocultural strategies encourage students to frequently assess how the activity is helping them to gain understanding. By questioning themselves and their strategies, students in the sociocultural classroom ideally become expert learners.

Which student error represents a misconception about place value? A) A student writes 9 x 5 = 14. B) A student writes 10 - 5 = 50. C) A student writes six-hundredths as 0.6. D) A student writes six-hundredths as 0.06.

C) A student writes six-hundredths as 0.6. Correct! The first two answers represent a computation error while the third answer is written as 0.6 and should be written as 0.06.

6) Select all strategies that should be part of the lesson that uses think-aloud. A) A teacher provides a peer-tutoring opportunity. B) Students use tools and manipulatives during learning activities. C) A teacher demonstrates to students how to solve a problem. D) A teacher discusses methods and strategies to help solve problems and provides reasoning for each step.

C) A teacher demonstrates to students how to solve a problem. D) A teacher discusses methods and strategies to help solve problems and provides reasoning for each step. Correct! A think-aloud is an instructional strategy that involves the teacher demonstrating the steps to accomplish a task while verbalizing the thinking process and reasoning that accompany the steps. The students follow this instruction by imitating this process of talking through a solution on a different, but parallel, task. This also derives from the model in which expert learners share strategies with novice learners.

Question 2 A second-grade math teacher wants to work with a science teacher on integrating the content. How can the teacher apply a math connection? A) By exploring how fractions are used in making a recipe B) By allowing students to use manipulatives when solving problems C) By expressing the distribution of water on Earth in a circle graph D) By allowing students to create a timeline of a particular culture

C) By expressing the distribution of water on Earth in a circle graph

A math teacher would like to collaborate with an art teacher by demonstrating a math connection. How can a math teacher incorporate a stained glass window into a math lesson? A) By teaching concepts of addition of fractions B) By teaching concepts of rational numbers C) By teaching concepts of symmetry D) By teaching concepts of probability

C) By teaching concepts of symmetry Correct! Integrating math content with other areas can help students gain a better conceptual development of certain topics. A stained glass window often contains many different shapes that can be used to illustrate symmetry.

A fifth-grade teacher is planning a discovery-based lesson that will compare the concepts of perimeter and circumference. Which tool or manipulative can the teacher use to help enhance learning outcomes? A) Online graphing applet B) Virtual isometric drawing tool C) Calculator to explore patterns or relationships D) Virtual three-dimensional shapes

C) Calculator to explore patterns or relationships

A teacher is planning a discovery-based lesson on circumference. Which technology tool can the teacher use to help facilitate this lesson? A) Virtual three-dimensional shapes B) Online graphing applet C) Calculators to explore patterns and relationships D) Virtual isometric drawing tool

C) Calculators to explore patterns and relationships Correct! Calculators can be used to help students discover patterns and when teaching discovery-based lessons. Three-dimensional shapes, like the other answer options suggest, do not work here because the lesson is concerned with the concepts of circumference and not volume or surface area.

How can you demonstrate a mathematical connection when teaching a unit on making circle graphs? A) Encourage classroom discourse and use think-alouds when teaching circle graphs. B) Allow students to use compasses and protractors to create a circle graph. C) Demonstrate how ratios, parts to whole, fractions, and percent can be used to create a circle graph. D) Allow students to create and conduct an experiment to gather data.

C) Demonstrate how ratios, parts to whole, fractions, and percent can be used to create a circle graph. Correct! A math-to-math connection is formed when a teacher relates a previously learned math concept to a newly learned math concept. In this example, the teacher is relating the previously learned concept of ratios and fractions to a newly learned concept of circle graphs.

Which activity could be used to make a mathematical connection within the literacy curriculum when teaching a unit on multiplying fractions by whole numbers? A) Demonstrate the relationship between multiplying fractions by whole numbers and multiplying decimals by whole numbers. B) Create several application problems to provide context involving the multiplication of fractions by whole numbers. C) Have students research and write about four applications of multiplying fractions by whole numbers. D) Use manipulatives to demonstrate fraction multiplication by whole numbers.

C) Have students research and write about four applications of multiplying fractions by whole numbers. Correct! In this situation the teacher is attempting to form a math to reading connection by having the students research and write about a mathematical skill.

Why is problem-based learning an effective teaching method that meets the needs of the different learners in your classroom? A) It provides students an opportunity to explain their work. B) It offers the most challenge for students, thus helping to push them further. C) It allows students to approach the content in their own way. D) It is a way for students to work collaboratively in groups.

C) It allows students to approach the content in their own way. Correct! Problem-based learning is a great way for students to approach the content with their already existing skills, experience, and knowledge, thus allowing them to make use of these things and to make their own connections with the content. As you think about lesson planning that meets the needs of various types of students, be sure to make use of problem-based learning.

5. Traditional problem-solving lessons involve a teacher explaining the math, and then students practice that math. This is followed by applying the mathematics to solve problems. Why does a lesson set up as described rarely work? A) Problem-solving is too wrapped up in the lesson. B) It begins where the students' knowledge and skills are at. C) It assumes wonderful explanations produce understanding. D) Students do not have enough time to practice in advance.

C) It assumes wonderful explanations produce understanding. Correct! Constructivist learning theory is a philosophy that enhances students' logical and conceptual growth. The underlying concept within the constructivism learning theory is the role that experiences—or connections with the adjoining atmosphere—play in student education. The constructivist learning theory argues that people produce knowledge and form meaning based on their experiences. A wonderful understanding of a concept does not always mean the student will assimilate the new information. In a constructivist classroom, students need a variety of learning opportunities to explore and build a relational understanding.

9. Which statement is not included in relational and conceptual understanding? A) It improves problem-solving abilities. B) It enhances memory. C) It eliminates poor attitudes and beliefs. D) It requires less to remember.

C) It eliminates poor attitudes and beliefs. Correct! Thinking and learning processes occur in two main ways known as relational and instrumental understanding. Instrumental understanding is knowing how to get from Point A to Point B. It gives a step-by-step process that can be followed. Students with instrumental understandings know how to do a mathematical process; however, they may not understand why they are doing it. Relational understanding is knowing how to do something and why it is done. It is knowing how to take different routes to find answers and connecting not only Points A and B but also the rest of the problem or context.

Which of the following are not benefits of relational understanding? A) It enhances memory. B) It improves problem-solving abilities. C) It helps students gain proficiency in a single method. D) It requires less to remember.

C) It helps students gain proficiency in a single method Correct! Instrumental thinking is knowing how to get from Point A to Point B. It gives a step-by-step process that can be followed. Students with instrumental understanding know how to do a mathematical process but do not, however, understand why they are doing it. Relational thinking is knowing how to do something and why it is done. It is knowing how to take different routes to find answers and connecting not only A and B but also the rest of the problem or context.

Have students create minimal defining lists for certain shapes. A)Level 0: Visualization B)Level 1: Analysis C)Level 2: Informal Deduction

C) Level 2: Informal Deduction Correct! The goal in Level 2 is that students can use logical reasoning to identify a shape. With an MDL activity, the students should be able to list the "defining" properties of a shape as well as the "minimal" properties needed for it to be classified as a certain shape. For example, a square needs not only at least four equal sides but four right angles as well. This activity is different from Level 1 where they identify properties, because in level 2 students are deducing what properties make a shape. For example, "If a shape has four equal sides and four right angles, then it must be a square, which means that it is also a type of rectangle."

Study the following discussion scenario and think about which two "Talk Moves" the teacher uses. In your classroom you have your students explain how they solved the equation 63 - 27 = ____. As your students are explaining their responses, you ask two students to come up with different answers. One student, Meg, has the right answer and the other student, Mark, has the wrong answer. Without saying that Mark is wrong and Meg is right, instead you ask both of them to explain their answers. You then ask Mark what he thinks of Meg's answer and vice versa. After this short discussion, Mark realizes that he got the wrong answer and he is even able to see where his mistake was. A) Revoicing B) Rephrasing C) Reasoning D) Waiting E) Elaborating

C) Reasoning and E) Elaborating Correct! Reasoning and elaboration were used by this teacher. Reasoning is used when the teacher asks the students what they think of each others' explanations. This invites the students to pause and reflect a little further on what is being said rather than just saying that Mark is wrong. Elaboration is used when the teacher has the students explain and elaborate on their answers. Because the teacher did this, Mark was able to see why he got the wrong answer.

Which two statements accurately describe challenges facing students from diverse groups? A) Students with learning disabilities also struggle with organization. B) Lower-level students benefit from different learning goals. C) Special education students may require extended time to complete a task. D) Students from diverse backgrounds solve problems differently.

C) Special education students may require extended time to complete a task. D) Students from diverse backgrounds solve problems differently. Correct! Because students will enter your classroom with various skills and diverse backgrounds, as the teacher you should embrace, rather than push against or ignore, the different approaches students use when solving problems. Because students with disabilities typically struggle with perceptual or cognitive processing, they can benefit from a variety of accommodations which include extended time and the ability to complete fewer problems on a task. Students with mild disabilities may also benefit from explicit instruction that is a structured, teacher-led instructional approach targeting a specific strategy. Also note that providing accommodations is not the same as changing learning objectives because the accommodation helps students to meet the learning objective.

2) A teacher is planning a lesson on finding the area of rectangles. The teacher allows students to use a geoboard to create rectangles and assist in finding the area of those rectangles. The following standard is associated with the lesson: Measure and estimate liquid volumes and masses of objects. How should the teacher rate the activity and alignment to the standard? How should a teacher rate the activity and alignment to the standard? A) The activity is not effective at showing students how to find area of rectangles, and the standard is not aligned. B) The activity is not effective at showing students how to find area of rectangles, but the standard is aligned. C) The activity is effective at showing students how to find area of rectangles; however, the standard is not aligned. D) The activity is effective at showing students how to find area of rectangles, and the standard is aligned.

C) The activity is effective at showing students how to find area of rectangles; however, the standard is not aligned Correct! While this lesson does a very good job of teaching students to find the area of a rectangle, it does not address liquid or mass. Since the standard relates to volume and mass, not area, the lesson is not aligned.

3) A third-grade teacher is working with a mixed-ability classroom. What is one strategy that is beneficial for all learners? A) Make sure all students complete the same task to ensure equality. B) Use a procedural method that is easy to follow, then teach the concept of the math procedure. C) Use effective wait time and multiple methods when approaching math problems. D) Allow a gifted student to peer tutor a lower-achieving student.

C) Use effective wait time and multiple methods when approaching math problems. Correct! The key to this question is that you are looking for a strategy that is effective for all students. Using wait time after asking questions and using multiple methods to solve math problems is a constructivist approach to teaching math. Teachers do not want all students to do the same task. They want students to meet the same objective, but students can do different tasks to meet that objective. Additionally, teachers want to teach the conceptual process of math first rather than any rote procedure. Lastly, teachers should never rely on a gifted student to peer tutor.

Imagine that you are observing a teacher who has a group of low-level learners and high-ability learners in her classroom. The objective of her current lesson is for students to create a histogram. She plans to deliver explicit instruction to the low-level learners on how to create a histogram, and plans to allow the high-ability learners the opportunity to create a histogram after developing a statistical question and gathering data. How effective is this teacher's activity at meeting the needs of all learners? A) The lesson does not address the needs of all learners because providing students with step-by-step instructions on how to create a histogram will not allow the special education students to meet the learning objective. B) The lesson does not address the needs of all learners because the teacher is expecting the special education students to create a histogram. C) The lesson effectively addresses all learners and offers complexity and depth for higher ability students. D) The lesson addresses the needs of all learners because all students will have the chance to create the same product.

C) The lesson effectively addresses all learners and offers complexity and depth for higher ability students. Correct! The lesson objective is to have students create a histogram. All students must meet this objective and the teacher can never reduce the expectation for lower-level learners. Teachers can differentiate by considering the four elements: content, process, product, and environment. In this situation the teacher is differentiating the content by providing the higher-level learners an opportunity to create their own statistical question, develop a survey protocol, and then create a histogram. The process by which students are reaching the objective also differs. The lower-levels learners are receiving more explicit instruction which differs from step-by-step instruction. The teacher will use probing questions and model precise language as students create a histogram. The same product is not being created by all students because the lower-level students are creating a histogram while the higher-level learners are creating a more complex product. This type of lesson delivery provides the higher-level learners with more depth and complexity.

A teacher is planning a lesson on mean and poses the following data set to students: 6, 8, 7, 3. A student responds and says the mean is 6. The teacher then changes the data set to: 6, 6, 8, 7, 3. How will the mean change? A) The mean will increase. B) The mean will decrease. C) The mean will not change. D) The mean will increase by 1 value.

C) The mean will not change. Correct! Practice this problem using a calculator and a whiteboard and check the work. Try calculating the answers with and without adding the extra 6. What happens? Now try it again with another set of numbers. Adding the mean to a set of numbers will not affect the mean of the entire set.

1. Which type of information should not be provided by math teachers? A) Mathematical conventions B) Clarification of students' methods C) The preferred method D) Alternative methods

C) The preferred method Correct! In a constructivist classroom, students should be encouraged to explore multiple strategies and methods to solve a problem. The teacher should avoid using step-by-step instructions and reverting to a single process or method. Although the teacher may have a preferred method, students should be encouraged to explore alternative problem-solving approaches.

When you teach math lessons to your students, which of the following types of information should you not provide to your students? A) Alternative methods B) Mathematical conventions C) The preferred method D) Clarification of students' methods

C) The preferred method Correct! You should not encourage one preferred method. In using a constructivist teaching approach, students should be encouraged to explore multiple strategies and ways to solve a problem. The teacher should avoid using step-by-step instructions and reverting to a single process or method. Although the teacher may have a preferred method, students should be encouraged to explore alternative problem-solving approaches.

Which of the following represents a misconception elementary students have about probability? A) When calculating probability, the total number of possible outcomes should be the denominator. B) When performing a probability experiment, with replacement, the theoretical probability will not change. C) The probability of rolling a 6 goes down every time a 6 is rolled. D) Probability can be expressed as a fraction, decimal, or percent.

C) The probability of rolling a 6 goes down every time a 6 is rolled. Correct! In the above dice example, it would be common for young students to think that because a 1 or a 6 had already been rolled, then it is less likely that those numbers will come up again.

A student correctly reads the number 65 as "sixty-five" and has the ability to form a one-to-one correspondence to demonstrate the value of the number. But when the student is asked to write the number form, the student writes 6 + 5. If any, what type of misconception does this represent? A) The student has an idea that single digit numbers are independent of place value. B) The student has an idea that multidigit numbers are dependent of place value. C) The student has an idea that multidigit numbers are independent of place value. D) There is no misconception to note.

C) The student has an idea that multidigit numbers are independent of place value. Correct! Students need a deep understanding of the place-value pattern ("10 of these is 1 of those") to support more efficient ways of working with two-digit numbers and beyond. Place-value is difficult to teach and learn as it is often masked by successful performance on superficial tasks such as counting by ones on a 1-100 number chart. In this example, the student believes that the number 65 is made from 6 ones and 5 ones.

In your second-grade class, your students are learning about the compare subtraction model. You use the common experience of eating in the cafeteria as a context outside the mathematics curriculum. Which scenario exemplifies this subtraction model? A) Three students eating lunch are joined by two more students. B) Five students finish lunch and go out to recess. C) Three more students are at lunch than are at recess. D) Six students are eating lunch together.

C) Three more students are at lunch than are at recess. Correct! Students often struggle with comparison examples of subtraction. Rather than subtracting and finding the difference, students add the subtrahend and minuend thinking the sum is the correct answer. The concept of comparative subtraction (comparing two quantities and determining the difference) is a little more difficult to grasp than take-away subtraction and should be practiced separately.

What misconceptions do students have about variables? A) Teachers can introduce algebra concepts early by expressing unknown quantities with a variable rather than a box or question mark. B) Variables can be used for a single unknown value. C) Variables can only be used for a single unknown value. D)Variables can be used for quantities that vary.

C) Variables can only be used for a single unknown value. Correct! Variables are used within expressions or equations to help us understand patterns. In addition to being used for single unknown values, variables may also be used for quantities that vary. Early learners can benefit from using variables in simple problems.

What is the primary criteria for using drill? A) When the desired outcome is for student understanding B) To build conceptual understanding of a procedure C) When the desired outcome is to establish greater facility with a skill or strategy D) To encourage flexible strategies in doing math problems

C) When the desired outcome is to establish greater facility with a skill or strategy Correct! Drill activities are appropriate in a math class after students have a conceptual understanding of the material. While drill activities should not be used to introduce a concept, they can, however, be used to reinforce the conceptual understanding. For example, when teaching a lesson on common denominators, a teacher can use Cuisenaire rods to help develop a conceptual understanding. The teacher can then use a drill activity to assess the students' skills and understanding. Drill can also be used to assess basic multiplication facts. The conceptual understanding can be established, for example, by using area tiles to display a 5 x 2 array, and then once students develop the conceptual understanding, drill can reinforce what they have learned. Students will be able to advance to more difficult concepts once they have mastered some basic skills (e.g., multiplication tables).

A teacher should make good use of calculators by using them for all of the following except __________. A) to save time when computation skills are not the objective of the lesson B) to enhance problem-solving in the classroom C) to perform basic computations such as 7 x 3 when computation skills are the objective of the lesson D) to drill basic facts such as the multiples of 7

C) to perform basic computations such as 7 x 3 when computation skills are the objective of the lesson Correct! If your objective is to help students develop computation skills, then having them use a calculator will not help. However, you would want students to use a calculator if they are exploring patterns, conducting investigations, testing conjectures, solving problems, and visualizing solutions.

If you are teaching a unit on multiplication, which activity would help your students make a connection within the mathematics curriculum? A)Have students journal about ways multiplication is used in the real world. B)Have students use centimeter cubes to construct arrays that represent the solutions to multiplication problems. C)Have students demonstrate how multiplication and division are inverse operations. D)Have students discuss how division can be used when calculating the velocity of an object in motion.

C)Have students demonstrate how multiplication and division are inverse operations. Correct! This will help students see the connection between multiplication and division. When making math connections within the mathematics curriculum, think about how you can connect content from one chapter or unit to another.

A teacher is presenting a lesson on a histogram. Which statement demonstrates correct analysis of the histogram? A)Five students scored a 61 on the math test. B)The intervals of 50-59 and 70-79 contain the same level of frequency. C)Ten students scored above a 79 on the test. D)The median of the test scores will be 70. E) The average test score is 65.

C)Ten students scored above a 79 on the test.

Which three instructional strategies can a teacher use to help meet the needs of English learners? A) Require all students to use the English language. B) Present step-by-step instructions using formulas. C) Encourage EL students to collaborate with bilingual peers. D) Allow EL students to have access to translation devices such as electronic dictionaries. E) Allow the use of a calculator when computing number problems. F) Use a mathematics word wall with picture cues.

C, D, and F Correct! It is important for EL students to feel comfortable in the math classroom. While teachers encourage immersion, they also must remember that they need to find the student's true math ability and attempt to remove any potential barriers. While it may be uncommon to have bilingual students, this accommodation can be extremely helpful for EL students in making a successful transition. Furthermore, using dictionaries or word walls can help EL students learn the English language and will allow them to successfully demonstrate their true math ability.

Thinking about Zero

Children need to discover the number zero particularly as understanding its value is a required standard of kindergartners. Surprisingly, it is not a concept that is easily grasped without intentionally building understanding. Three-and four-year-old children can begin to use the word zero and the numeral 0 to symbolize that there are no objects in the set. With the dot plates discussed previously (see Figure 7.2), use the zero plate to discuss what it means when there is no dot on the plate. We find that because early counting often involves touching an object, zero is sometimes not included in the count. Zero is one of the most important digits in the base-ten system, and purposeful conversations about it and its position on the number line are essential. Activities 7.1, 7.2, and 7.12 are useful in exploring the number zero

REASONING FOR 1 Correct! Graphic organizers can help classroom teachers tier lessons and provide students with structure on how to organize their thoughts. For example, lower-level learners can be provided with a more structured graphic organizer while higher-level learners can be provided with a less structured graphic organizer. Regardless, all learners can benefit from using a graphic organizer to help organize their mathematical thinking.

Classroom teachers should always encourage students to share ideas and collaborate. Class discussion refers to the interactions that occur throughout the lesson. The goal is to keep the cognitive demand high while students are learning and formalizing math concepts. The value of discussion cannot be oversimplified. Discourse is an opportunity to use and model precise language and for students to share and collaborate about methods and strategies used to solve a problem.

Teaching Addition and Subtraction

Combining the use of situations and models (counters, drawings, number lines, bar diagrams) is important in helping students construct a deep understanding of addition and subtraction. Building the understanding of these operations now will support these operations with larger numbers, fractions, and decimals later on. Note that addition and subtraction are taught at the same time to reinforce their inverse relationship

Asking students to write out their solutions and then to teach it to someone else.

Communication Correct! Communication is demonstrated in a variety of ways, but the goal here is that students are able to communicate their ideas clearly. You can also assess communication by having students communicate their ideas in a variety of ways by using words, pictures, and numbers.

1Compare Problems.

Compare problems involve the comparison of two quantities. The third amount in these problems does not actually exist but is the difference between the two quantities (see Figure 8.1[d]). Like part-part-whole problems, comparison problems do not typically involve a physical action. The corresponding Compare Story Activity Page can help students model the situation. The unknown quantity in compare problems can be one of three quantities: the smaller amount, the larger amount, or the difference.ds Glossary; NGA Center & CCSSO, 2010, p. 88).

Comparison Models.

Comparison situations involve two distinct sets or quantities and the difference between them. Several ways of modeling the difference relationship are shown in Figure 8.8. The same model can be used whether the difference or one of the two quantities is unknown

In a geometry lesson, you ask students to find ways that the material relates to things they see outside the classroom.

Connections Correct! Assessing students ability to make connections should be an ongoing activity. It is important that students are able to make connections between the material they are learning in class from lesson to lesson, unit to unit, etc. Students should also be able to make connections to contexts outside the classroom. This can make learning math more fun and meaningful.

Cooperative learning is a term that describes an instructional arrangement for teaching academic and collaborative skills to groups of students. Cooperative learning is deemed highly desirable because of its tendency to reduce peer competition and isolation, and to promote academic achievement and positive interrelationships. According to the National Council of Teachers of Mathematics, learning environments should be created that promote active learning and teaching, classroom discourse, and individual, small-group, and whole-group learning.

Constructivism as a learning theory says that people construct their own understanding and knowledge of the world through experiencing things and reflecting on those experiences. When people encounter something new, they have to reconcile it with their previous ideas and experiences, sometimes changing what they believe or discarding the new information as irrelevant. In any case, people are active creators of their own knowledge. To do this they must ask questions, explore, and assess what they know.

4 QUESTION RESPONSE/REASONING Correct! Constructivist teachers encourage students to constantly assess how the activity is helping them gain understanding. By questioning themselves and their strategies, students in the constructivist classroom ideally become expert learners. Cooperative learning is a term that is used to describe an instructional arrangement for teaching academic and collaborative skills to groups of students. Cooperative learning is deemed highly desirable because of its tendency to reduce peer competition and isolation and to promote academic achievement and positive interrelationships. According to the National Council of Teachers of Mathematics, learning environments should be created that promote active learning and teaching, classroom discourse, and individual, small-group, and whole-group learning. Cooperative learning is one example of an instructional arrangement that can be used to foster active student learning, which is an important dimension of learning mathematics and highly endorsed by math educators and researchers. Students can be given tasks to discuss, problem solve, and accomplish.

Constructivism is a theory based on observation and scientific study about how people learn. It says that people construct their own understanding and knowledge of the world through experiencing things and reflecting on those experiences. When people encounter something new, they have to reconcile it with their previous ideas and experiences, maybe changing what they believe, or maybe discarding the new information as irrelevant. In any case, people are active creators of their own knowledge. To do this, they must ask questions, explore, and assess what they know. In the classroom, the constructivist view of learning can point toward a number of different teaching practices. In the most general sense, it usually means encouraging students to use active techniques (experiments, real-world problem-solving) to create more knowledge, and then to reflect on and talk about what they are doing and how their understanding is changing. Teachers make sure they understand the students' pre-existing conceptions and guide the activity to address and then build on them.

A fifth-grade teacher is teaching a lesson on dividing fractions. The class is solving ½ ÷ ¾ = ____. Sam says the answer is ⅜ and Ryan says the answer is 4/6. The teacher allows the class to share answers and possible solutions and Ryan suggests that Sam's answer is incorrect. How should a teacher respond to promote critical thinking? A) Ryan can you explain your thoughts regarding Sam's answer? B) Sam your answer is incorrect. Is there another way to solve this problem? C) Ryan your answer is correct, can you write the answer on the board? D) Sam your answer is incorrect, have Ryan show you how to solve the problem correctly.

Correct! A teacher's response should encourage critical thinking and open discussion. Here, the teacher avoids indicating whether the student is correct or incorrect and encourages the student to elaborate and justify his thinking. Through this process, students should strategically compensate for any errors they make by internalizing the situation. By allowing students to compare and analyze possible solutions, other students have the opportunity to correct errors and learn from peer assistance.

B) The teacher should allow both students to retell their strategies and discuss their method of obtaining the answer.

Correct! Remember that in this situation you are not concerned about telling students how to solve equivalent fractions, rather, you want to create discourse to help students reveal their thinking and work through it on their own. In a constructivist classroom, teachers should avoid explicitly telling students that answers are correct or incorrect. Students should have the opportunity to elaborate and explain their thought processes. Because this question targets the student thought process, the student should be the one to discuss which strategies were used.

All rubrics will have performance indicators. Which of the following is not an important element of performance indicators? A) They should contain words like "proficient" or "on target." B) Performance indicators should be shared with students beforehand. C) Performance indicators can be set by looking at students' common misconceptions. D) They should focus on the number of correct responses from the student. E) They should help focus students on the objectives.

D) Correct! They should focus on the number of correct responses from the student. This is not something that performance indicators in rubrics should focus on. When you focus on the number of correct or incorrect responses, students will likely focus only on how many they can get right and wrong in order to pass, instead of on improving their overall performance by targeting their knowledge and performance gaps. As an example, being a student at WGU, the pre-assessment for your courses is designed to help you know what your knowledge gaps are so that you can fill them in.

A seventh-grade teacher is planning a lesson that will require students to find the slope of the line and explore functions. Which tool or manipulative can the teacher use to help enhance learning outcomes? A) Base-ten blocks B) Cuisenaire rods C) Isometric drawing applet D) Graphing calculator

D) Graphing calculator

A survey was taken on how many years an individual spent in school beyond high school. What type of graph should be used to represent this data? A) Stem-and-leaf plot B) Circle graph C) Bar graph D) Histogram

D) Histogram Correct! A histogram is a type of bar graph where the categories of data must be displayed in chronological order. In a typical bar graph, categories can be moved around without changing the meaning of the graph. For example, if you surveyed favorite ice cream colors, it would not matter in which order you displayed the data. But with a histogram, the order of the categories cannot be switched around. Histograms display data with categories such as days of the week, months of the year, or numerical categories.

Determine the mean, median, mode, and range for the following data set: 18, 18, 46, 2. A) Mean: 21; median: 18; mode: 44; range: 18 B) Mean: 21; median: 16; mode: 18; range: 16 C) Mean: 21; median: 16; mode: 18; range: 16 D) Mean: 21; median: 18; mode: 18; range: 44 .

D) Mean: 21; median: 18; mode: 18; range: 44 . Correct! To determine the mean, add up all the values and divide by the number of addends. To find the median, arrange the numbers in numerical order and locate the number in the middle. To calculate the range, subtract the lowest value from the greatest value. To find the mode, determine which numbers appear the most

Students in a third-grade class are asked, "What fractional part of a hexagon pattern block piece is each triangle pattern block piece?" The teacher provides students with pattern blocks to help solve the problem. How should the teacher use a rubric to assess the progress of the students? A) Use an alternate assessment type since this task is not performance based B) Use a 4, 3, 2, 1 scoring system and equate this to a percentage out of 4 to assign grades C) Have students complete this activity in their learning journals and then complete a self-assessment using a rubric D) Provide students with a copy of the rubric before beginning the problem and explain how the rubric will be used for evaluation

D) Provide students with a copy of the rubric before beginning the problem and explain how the rubric will be used for evaluation Correct! A rubric is a framework that can be designed or adapted by the teacher for a particular group of students or a particular math task. A rubric usually consists of a scale of three to six points that is used as rating of total performance on a single task rather than a count of how many items in a series of exercises are correct or incorrect. A rubric should always be explained to students because it may contain certain criteria that students must do. Scoring is comparing work to criteria or a rubric that describes what the teacher expects the work to be. Grading is the result of accumulating scores and other information about a student's work.

When a student employs multiple representations, which of the following have they demonstrated? A) Instrumental understanding B) Ability to recall facts with automaticity C) Recursive thinking D) Relational understanding

D) Relational understanding Correct! Multiple representations refer to students' ability to show methods and solutions to a problem in a variety of ways. For example, a solution to a division problem might be represented using a picture, modeling with manipulatives, or using the standard algorithm. These are three different representations. When students are able to use multiple representations, it means they are constructing knowledge and have developed a relational understanding. With relational understanding, students know how a concept they have learned is connected to other things within and outside of math.

Two students (Ally and John) are working on how to solve the area of a triangle. Ally says the formula is (b x h)/2 while John says the formula for the area is b x h. How can the teacher respond in a way that will promote understanding and communication? A) The teacher should tell Ally she is correct. B) The teacher should ask a student to explain why Ally's formula is correct or incorrect. C) The teacher should ask John to show Ally why her answer is wrong. D) The teacher should ask Ally to explain and elaborate on her thinking.

D) The teacher should ask Ally to explain and elaborate on her thinking Correct! As a teacher you should serve as a facilitator, especially in a constructivist classroom, so that students will be more willing to share their ideas during discussion without the risk of being judged. When you say, "That is correct," there is no reason for students to think about and evaluate the response. Consequently, you will not have the chance to hear and learn from students about their thought process. Teachers should also use praise cautiously. Praise offered to correct solutions or excitement over interesting ideas suggests that the student did something unusual or unexpected.

Avoid revealing whether a student's answer is right or wrong.

Often facial expressions, tone of voice, or body language can give students an idea as to whether their answer is correct or incorrect. Instead, use a response such as "Can you tell me more?"

Imagine that you assign your students to write out their answers to problems using words, numbers, and pictures. A parent complains that the extra writing assignments belong in a writing class, not a math class. Which response is not a reason that you should give this parent as a justification for having students write in your math class? (Remember that your ideas here should be based on content from the videos and textbook.) A) Writing helps students reflect on, explain, and defend their thought process and answers. B) The teacher can later use the students' writings to assess their progress and adjust their teaching as needed. C) When students write, it helps prepare them for the class discussion that can follow the assigned activity. D) The writing activities will help improve students' grammar and sentence structure.

D) The writing activities will help improve students' grammar and sentence structure. Correct! It may be true that the writing will help students to improve their overall writing skills, but this is not mentioned in the textbook nor is it a primary reason to include writing assignments in a math class. In addition, you may have your kindergarten class draw pictures instead of write words, and drawing pictures will clearly not improve their grammar.

3) According to the teaching principles defined by NCTM, what does effective mathematics teaching require? A) Having educated mathematics teachers who possess certification in their subject B) Having instructional and financial support C) Knowing the important main ideas to teach D) Understanding what students know and need to learn

D) Understanding what students know and need to learn Correct! According to Principles and Standards for School Mathematics (2000), what students learn about mathematics depends almost entirely on the experiences that teachers provide every day in the classroom. To provide high-quality mathematics education, teachers must (1) understand deeply the mathematics content they are teaching; (2) understand how students learn mathematics, including a keen awareness of the individual mathematical development of their own students and common misconceptions; and (3) select meaningful instructional tasks and generalizable strategies that will enhance learning. Be sure to review all the principles and standards outlined in Principles and Standards for School Mathematics. These are summarized in Chapter 1.

Students in a sixth-grade class are working through a unit on fractions, decimals, and percentages. They are using manipulative models to explore how to find percent when given a decimal. The teacher wants them to recognize the relationship between decimals and percents. Which question could help foster critical thinking? A) Will a percent always be out of 100? B) Will a percent always be between 1% and 100%? C) Can you model 50%? D) What patterns have you noticed when modeling percent?

D) What patterns have you noticed when modeling percent? Correct! Since your goal here is to ask a question that fosters critical thinking, this can be done by asking open-ended questions. When asking students to explain the observed patterns, the teacher is asking an open-ended question. The other choices are closed-ended questions and can be answered with a simple yes or no. In a constructivist classroom, you want to ask open-ended questions that cannot be answered with a simple yes or no.

There are many dynamic geometry software programs that you can use in your classroom. Take a few minutes to explore the following websites: GeogebraDesmos Geometry Complete the following sentence. These geometry programs __________. A) are simply drawing packages B) do not preserve shapes and lines if dragged or pulled C) are designed for high school classrooms only D) allow students to create and manipulate shapes

D) allow students to create and manipulate shapes Correct! The Geometer's Sketchpad and GeoGebra are both open-source dynamic geometry programs that allow students to create, manipulate, and measure geometry shapes. As students do this, they can see how different geometric shapes relate to each other.

UNit 5 Practice test

DIFFERENTIATED MATHEMATICS INSTRUCTION

3) Correct: Encouraging students to create a word problem about a situation can provide the teacher with information about students' mathematical thinking. For example, if you provide students with 100 ÷ 10, students should correctly identify the dividend and divisor in the division problem and be able to apply that in the word problem correctly.

Drawing a picture is one effective strategy students can use to solve a problem. If students are provided 100 ÷ 10, you would expect to see a model that looks like this: 10 circles in 2 rows of five. Each circle has 10 red dots. The sum of the circles is 10. This model illustrates the thinking of students and shows that 100 can be divided into 10 groups. Each group will receive 10. This is different than using the traditional algorithm and helps develop a conceptual understanding.

How is a conceptual understanding beneficial to students when learning mathematics? A) It allows them to use their prior knowledge to make connections to the current content they are learning. B) When students approach a problem, they are able to apply different strategies. C) The student acquires a "can-do," persistent attitude. D) Students are able to reflect on and evaluate their work. E) It allows the student to apply mathematical principles in different contexts.

E) It allows the student to apply mathematical principles in different contexts Correct! Because conceptual understanding is about understanding relationships and foundational ideas, this allows the student to apply the ideas and concepts in a variety of contexts. A: This is relational understanding B: This is strategic competence. C: This is productive disposition. D: This is adaptive reasoning.

Give opportunities for students to share their thinking without interruption.

Encourage stu-dents to use their own words and ways of writing things down. Correcting language or spelling can sidetrack the flow of students' explanation

Depth

Enrichment activities go in a depth beyond the topic of study to content that is not specifically a part of your grade-level curriculum but is an extension of the original mathematics. For example, while studying place value both using very large numbers or decimals, students can stretch their knowledge to study other bases such as base five, base eight, or base twelve. This extension provides a view of how our base-ten numeration system fits within the broader system of number theory. Other times the format of enrichment can involve studying the same topic as the rest of the class while differing on the means and outcomes of the work. Examples include group investigations, solving real problems in the community, writing data-based letters to outside audiences, or identifying applications of the mathematics learn

Five cards are pulled from a fair deck of 52 cards and the results occur in the following ways: ace of diamonds, ace of hearts, 9 of hearts, 10 of spades, 2 of hearts. Students are then asked what the theoretical probability is on an ace being drawn next. Select the student responses with their reasoning.

Five cards are pulled from a fair deck of 52 cards and the results occur in the following ways: ace of diamonds, ace of hearts, 9 of hearts, 10 of spades, 2 of hearts. Students are then asked what the theoretical probability is on an ace being drawn next. Select the student responses with their reasoning.

2Compare Problems.

For each of these situations, two examples are provided: one problem in which the difference is stated in terms of "how many more?" and another in terms of "how much less?" (or how many fewer). Note that initially the language of "more," "less," and "fewer" may confuse students and may present a challenge in interpreting the relationships between the quantities. You can find more examples of compare problems as well as the other problem types in the Common Core State Standards (Table 1 in the CCSS Mathematics Standard

Unit 2 Summary In this unit, you explored state and national standards. While the United States does have national standards, each state has ultimate control over the standards they choose. Take note of the five math process/practice skills (Principles and Standards for School Mathematics, PSSMs) to understand key ideas about each process skill. For example, representation means to model and use multiple representations, communication means that students are given opportunities to use precise math language and notation, reasoning and proof means that students are evaluating conjectures and analyzing evidence, connections means that students are developing relationships between mathematics being learned and the real-world/other content areas, and problem-solving means that students are taking time to self-assess or monitor and reflect on their mathematical work.

In a lesson on fractions that requires students to represent their math thinking with a model, a teacher could directly link the lesson to the representation process standard. Lastly, during this unit you explored math standards and rated the level of effectiveness and alignment to a standard. A lesson only needs to address a small portion of the standard to be effective.

Unit 3 Summary

In this unit you explored the constructivist and sociocultural learning theories. Students learn best in a social learning environment where they can share ideas with classmates and discuss different ways to approach math. Students also learn by rearranging previously learned concepts and by making connections to other areas of math. They can show a conceptual understanding when they subitize numbers not by doing traditional algorithms or rote processes. Teachers should look for rounding techniques

Unit 8 Summary

In this unit, you explored how math is applied to real-life situations and made connections to other math topics. Students should realize that math is not taught in isolation, so it is important to relate math concepts to real life. For example, one can relate a stained-glass window to concepts like area and perimeter or even symmetry and shapes. Furthermore, it is important to form relational learning by connecting math concepts to other math concepts. A math-to-math connection is formed when teachers show students how a math concept is related to another math concept. This helps students understand that math is not isolated and what they previously learned is important to what they are learning now.

Response to the above question as to why the other options are incorrect.

Incorrect. Revoicing, rephrasing, and waiting are great "Talk Moves," but they were not used by the teacher in this scenario. Revoicing involves the student or teacher restating the other person's statement as a question. This can help make sure ideas are understood correctly. Rephrasing is when students restate other students' ideas in their own words. Waiting can be used to allow students to pause and think about the problem at hand.

Line Plots and Dot Plots

Line plots (also called dot plots) are counts of things along a numeric scale on a number line. Both terms are used in the CCSS-M standards, with line plots introduced in grade 2 using whole-number units and progressing to displaying data in fractions of a unit in grade 5. In middle school, the term dot plot replaces line plot, but the only difference is that line plots use Xs to represent each data point and dot plots use dots

Unit 7 Summary During this unit you explored math tools and manipulatives for the elementary math classroom. When selecting appropriate tools, look for opportunities for students to actually explore and discover math ideas. Furthermore, be comfortable with tangrams, Cuisenaire rods or centimeter rods, or even pizza. These can be used when teaching fractions. Calculators are encouraged in a constructivist classroom and are considered technology. Usually calculators are used when exploring and discovering math topics (patterns). However, calculators should never replace a computational objective or computational skills.

Moreover, two colored counters or a number line could help with integers or negative numbers, or absolute value. Lastly, geoboards can be used to assist the kinesthetic learner in the concept of area and perimeter, and currency/plastic money or base-ten blocks can be used to teach place value. Pattern blocks can be used to teach patterns and basic shapes; however, pattern blocks are not good manipulatives for similar shapes. When teaching similar shapes, students should be able to make polygons larger or smaller and pattern blocks into predetermined shapes. Instead, a teacher should use grid paper, a virtual geometric app, or geoboards.

Unit 10 Summaryy Unit Summary In this unit, you explored graphical displays and measures of center. It is important to know the basics about bar graphs, stem-and-leaf plots, line graphs, circle graphs, scatter plots, histograms, and how to calculate the mean, median, and mode. A stem-and-leaf plot is helpful when attempting to see each individual piece of data.

On the other hand, a histogram has ranges, which means it is impossible to actually determine a specific data point on a histogram. Histograms can be used to analyze the variability of the data. The flatter a histogram is, the less variability in the data set. A line graph is best to show change over time and a bar graph is best to show categorical data. A circle graph should be used to compare parts to wholes, ratio, or fractional comparison.

Translation Tasks.

One important assessment option is what we refer to as a transla-tion task. Using four possible representations for concepts, students are asked, for example, to demonstrate understanding using words, models, numbers and word problems. As students flexibly move between these representations, there is a better chance that a concept will be integrated into a rich web of ideas.

Picture Graphs.

Picture graphs (also called pictographs) move up a level of abstraction by using a drawing or picture of some sort that represents what is being graphed. The picture can represent one piece of data or it can represent a designated quantity. Students need to learn to interpret the scale for the pictograph

Problem-Based Tasks.

Problem-based tasks are tasks that are connected to actual problem-solving activities used in instruction. High quality tasks permit every student to demonstrate their abilities (Rigelman & Petrick, 2014; Smith & Stein, 2011). They also include real-world or authentic contexts that interest students or relate to recent classroom events. Of course, be mindful that English language learners may need support with contexts, as challenges with language should not overshadow the attention to their mathematical ability. Problem-based tasks have several critical components that make them suitable components of the formative assessment process

Asking your students to show the results of their data by using a bar graph, a pie chart, and some physical objects of their choosing.

Representations Correct! Representation is a skill that students can demonstrate by representing problems and/or solutions using a variety of models. They should also be able to explain how these different representations are connected and how they help them think about the problem and/or solution.

This provides a record of how students perceive their strengths and weaknesses.

Self-Assessments Correct! It is important that you encourage your students to be open and honest when doing self-assessments. These are also a way to help students take greater responsibility of their learning.

Change Problems—Separate and Take From.

Separate problems are commonly thought of take away or take from problems in which part of a quantity is physically being removed from the start amount (see Figure 8.1[b]). Notice that in separate problems the start amount is the whole or the largest amount, whereas in the join problems, the result is the largest amount (whole). Have students use the Separate Story Activity Page as a graphic organizer

performance indicators Performance indicators should be shared with students beforehand.

Sharing performance indicators beforehand is extremely important so students know what they are trying to achieve.

This should be used as a way for students to demonstrate procedural knowledge and how to apply it outside the classroom.

Tests Correct! In giving tests, however, you will want to go beyond just knowing right and wrong answers. Refer back to page 70 to the four suggestions about how to make tests more valuable.

Translation Tasks continued

So, what is a good way of structuring a translation task? With use of template based on a format for assessing concept mastery from Frayer, Fredrick, and Klausmeier (1969) (see Figure 5.5) and a Translation Task Activity Page, you can give students a computational equation and ask them to: • Tell a story or write a word problem that matches the equation. • Illustrate the equation with materials, models, or drawings. • Explain their thinking about the process of arriving at an answer or describe the mean-ing of the operation

2Introducing Symbolism

Some care should be taken with the equal sign, as it is a relational symbol meaning "is the same as" and is not an operations symbol. However, many children think of it as a symbol that signals that the "answer is coming up next" which is an obstacle to understanding equations (Byrd, McNeil, Chesney, & Matthews, 2015). In fact, children often interpret the equal sign in much the same way as pressing the = on a calculator: "give me the answer.

The following considerations can help maximize the value of your tests

Students can use models to work on test questions when those same models have been used during instruction to develop concepts. Simple drawings can be used to repre-sent counters, base-ten pieces, fraction pieces, and the like (see Figure 5.9). Provide examples in class of how to draw the models before you ask students to draw on a test.

Level 3: Deduction. (Note: Levels 3 and 4 are beyond the scope of this book)

Students move from considering the relationships between properties of geometric objects to developing deductive axiomatic systems for geometr

How structured the task is

Students with special needs, for example, benefit from highly structured tasks, but gifted students often benefit from a more open-ended tasks.

4) End of unit tests

Summative Assessment Correct! An end of unit test measures what your students have learned during an entire unit. You hopefully have used sound instruction and formative assessments to prepare them for these tests, which are also likely to make up a significant part of their grade. It should be noted, however, that you can use an end of unit test to adjust instruction for the next unit and to prepare your students for state assessments or end of term tests.

2) State assessments

Summative Assessment Correct! High-stakes tests that come from the state or federal government are meant to see whether students are meeting certain objectives and standards. The best way you can prepare your students for these tests is to provide instruction and formative assessments that focus on building conceptual knowledge and the big ideas that are in these standards.

Unit 5 Summary 2

Teachers must be careful with broad generalization and avoid assuming that special education students will also struggle with organization. They must make certain that the classroom is set up for a high level of collaboration where students feel comfortable making mistakes and learning from each other. Moreover, the learning objective should never be changed for lower-level students. If the objective of the lesson is to measure angles with a protractor, all students must have the opportunity to measure angles with a protractor. That being said, teachers can incorporate accommodations. Practical accommodations can include reducing the amount of problems, extending time, or the friendly number approach. Also, when planning for ELs, it is helpful to incorporate word walls, translate documents to their native language, use simple language, and encourage bilingual students to collaborate with EL students.

The Associative Property for Addition.

The associative property for addition states that when adding three or more numbers, it does not matter whether the first pair is added first or if you start with any other pair of addends (expected of first graders in CCSS-M). This property allows for a great deal of flexibility so students can change the order in which they group numbers to work with combinations they know. For example, knowing this property can help students identify "combinations of ten" from the numbers they are adding by mentally grouping numbers differently from just reading the expression from left to right

1The Relations Core: More Than, Less Than, and Equal To

The concepts of "more," "less," and "same" are basic relationships contributing to children's overall understanding of number. Almost any kindergartener can choose the set that is more if presented with two sets that are obviously different in number. In fact, Baroody (1987) states, "A child unable to use 'more' in this intuitive manner is at considerable educational risk" (p. 29). Classroom activities should help children build on and refine this basic notion.

Student B says, "The probability of spinning a blue will be 1 out of the total number of sections."

The correct Answer Correct! Student B demonstrates the correct analysis because blue has a theoretical chance of occurring ⅙ of the time. Probability equals a part divided by the whole and there is one part in question (blue) and six total parts, thus ⅙ is the correct answer.

Mode. .

The mode is the most frequently occurring value in the data set. The mode is the least frequently used as a measure of center because data sets may not have a mode, may have more than one mode, or the mode may not be descriptive of the data set. Median

Understanding is hard to define, but it can be explained as a measure of the quality and quantity of connections that an idea has with existing ideas. The extent that a student under-stands why an algorithm works or understands relationships is their depth of understanding

The extent that a student under-stands why an algorithm works or understands relationships is their depth of understanding

Imagine that you are a teaching supervisor over a new teacher at your elementary school. This new teacher is familiar with constructivism but he chooses not to use it in his classroom because, he tells you, "a constructivist approach takes too long for students to learn the material. Students have a lot to learn, so it is better if math is taught through direct instruction where I can simply explain the procedures rather than having students waste time trying to figure out what things mean."

The following sentences are statements you could tell this teacher about constructivism and how people learn.

Problem Difficulty

The structure of some problem types is more difficult than others. For example, problems in which a physical action is taking place, as in join and separate prob-lems with result-unknown or the both-parts unknown problems are easier because children can directly model the physical action or act out the situation. That is why these types start being presented in kindergarten. However, the join or separate problems in which the start is unknown (e.g., Sandra had some pennies) are often the most difficult, probably because students attempting to model the problems directly do not know how many counters to put down to begin. Instead, they often use trial and error (Carpenter, Fennema, Franke, Levi, & Empson, 2014) to determine the unknown

1Choosing Numbers for Problems.

The structure of the problem will change the challenge of the task, but you can also vary the problem's complexity by the numbers you choose to use. In general, the numbers in the problems should align with the students' number development. But, if at first a stu-dent struggles with a problem, use smaller numbers to see if it is the size of the numbers causing the obstacle. You can also intensify the challenge by increasing the size of the numbers. Rather than wait until students develop techniques for computing numbers, use word problems as an opportunity for them to learn about number and computation simultane-ously.

Calculators used for exploration and discovery

Tool Correct! When used for exploration and discovery, calculators are considered a tool. For example, students could use a calculator as a tool to explore the relationship between the circumference and diameter of a circle. However, when teaching a lesson about a computational skill (i.e., adding two-digit numbers), a calculator is not an appropriate tool because it is used for straightforward computational purposes rather than explorative purposes.

True or False In a probability experiment, an event is a subset of the sample space. True False

True Correct! A subset of an event is considered to have happened. In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. Visit module 13 for more information about probability.

UNIT 9UNIT SUMMARY In this section you learned about teaching strategies and common misconceptions for basic algebra, number sense, probability, and geometry. Here are some common misconceptions some teachers encounter in the elementary math classroom.

UNIT 9UNIT SUMMARY In this section you learned about teaching strategies and common misconceptions for basic algebra, number sense, probability, and geometry. Here are some common misconceptions some teachers encounter in the elementary math classroom.

Attribute: Area

Units: index cards, squares of paper, tiles How many cards will cover the surface of the buck

Attribute: Weight

Units: objects that stretch the spring in the scale How many units will pull the spring as far as the bucket will?

4) EL students are unable to fully explain ideas.

Use revoicing and press for details Correct! It may sound cruel to press students for more explanation when they already have trouble with language, but keep in mind that revoicing consists of restating what students are trying to explain, which can help affirm their ideas. When you press them for more explanation, you are also allowing them to explain more and develop their language and math abilities even further. A positive approach is clearly necessary here.

Lessons Built on Context or Story Problems What might a good lesson built around word problems look like? The answer comes more naturally if you think about students not just solving the problems but also using words, pictures, and numbers to explain how they solved the problems and justify why they are correct. In a single lesson, focus on a few problems with in-depth discussions rather than multiple problems with little elaboration. Students should use whatever physical materials or drawings they feel will help them.

Whatever they record should explain what they did well enough to allow someone else to understand their thinking. A particularly effective approach is having students correct each others' written solutions. By using anonymous work, students can analyze the reasoning used, assess the selection of the operation, find the mistakes in computation and identify errors in copying numbers from the problem.

Unit 5 Summary 1

When planning for all learners, think about the entire class and incorporate broad strategies into instruction. In a constructivist math class, it is best to avoid presenting a step-by-step approach that is easy to follow to the entire class. Instead, students should be encouraged to explore multiple strategies and methods of solving a problem. Overall general strategies for all learners would consist of think-alouds, formative assessments, graphic organizers, real-world tasks for students to complete, or simply providing wait time for students to think after asking a question. It is important to only use step-by-step (explicit) instruction when working with lower-level or special education students. Step-by-step instructions can be used as an accommodation for lower-achieving students, but teachers should never start with this type of instruction for all students.

FORMATIVE ASSESSMENT Notes. Observing how students solve story problems will provide information about their understanding of number, strategies they may be using to answer basic facts, and methods they are using for multidigit computation. Look beyond the answers they get on a worksheet. For example, a child who uses counters and counts each addend and then recounts the entire set for a join-result-unknown problem (this approach is called count all ) needs to develop more sophisticated strategies.

With more practice, they will count on from the first set. This strategy will be modified to count on from the larger set; that is, for 4 + 7, the child will begin with 7 and count on, even though 4 is the start amount in the problem. Eventually, students use facts retrieved from memory, and their use of counters fades completely or counters are used only when necessary. Observations of children solving prob-lems can inform what numbers to use in problems and what questions to ask that will focus students' attention on more efficient strategies.

To determine the mean Ask an adult what the mean is, and they are likely to tell you something like, "The mean is when you add up all the numbers in the set and divide the sum by the number of numbers in the set." This is not what the mean is, this is how you calculate the mean. This is a reminder of the procedurally driven curriculum in our history and the need to shift to a more conceptually focused approach.

add up all the values and divide by the number of addends. Another limited conception about the mean is that it is considered the way to find a measure of center regardless of the context (McGatha, Cobb, & McClain, 1998). In fact, in the CCSS-M, sixth-grade students are expected to deter-mine when the mean is appropriate and when another measure of center (e.g., the median) is more appropriate: "Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values" (NGA Center & CCSSO, 2010, p. 39). The next section focuses on developing the concept of the mean.

Manipulatives

are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics (e.g., connecting cubes) or for other purposes (e.g., buttons)" (Van de Walle et al., 2019, p. 20). They are more concrete and provide insights into new and abstract relationships. Consider each of the concepts and the correspond-ing model in Figure must impose on the tool in order to see the concept.

To find the median

arrange the numbers in numerical order and locate the number in the middle. Mean Absolute Deviation. Whereas the range relates to the median, the mean absolute deviation (MAD) relates to the mean. It is the mean of the how far away each data point is from the mean, telling us the "average" distance from the mean for the data set. MAD, therefore, describes how spread out the data are (Kader & Jacobbe, 2013). In other words, a large MAD means that there is a lot of deviation (difference) between data points and the mean, so the data are spread out. In the CCSS-M, MAD is introduced in sixth grade, with the intent being that it is explored in an informal manner to develop a deeper understanding of variability. Let's use the previous data set to explore MAD.

A number line A number line is also a shift from counting a number of individual objects in a collection to counting continuous length units. There are, however, ways to support young learners as you introduce and model number lines (see Figure 8.6). A number line measures distances from zero the same way a ruler does. If you don't emphasize the unit (length), children may focus on the tick marks or numerals instead of the spaces (a misunderstanding that becomes apparent whentheir answers are consistently off by one

is an essential model, but it can initially present conceptual difficulties for children below second grade and students with disabilities, This initial confusion is partially due to their difficulty in seeing the unit, which is a challenge when it appears in a continuous line.

The median

is the middle value in an ordered set of data. Half of all values lie at or above the median and half at or below. The median is easier to understand and to compute and is not affected, as the mean is, by one or two extremely large or extremely small values outside the range of the rest of the data. The most common misconception using the median emerges when students neglect to order the numbers in the data set from least to greatest. The median and the mean first appear as standards in the sixth grade in the CCSS-M (NGA Center & CCSSO, 2010).

Conceptual knowledge

refers to connected knowledge: "mental connections among mathematical facts, procedures, and ideas

Procedural knowledge

refers to how to complete an algo-rithm or procedure.

Practice

refers to varied tasks or experiences focused on a particular concept or procedure. Practice, as defined here, has numerous benefits, as this chapter has described. Students have an increased opportunity to develop conceptual ideas, alternative and flexible strategies for solving, and making connections between concepts and procedures. Importantly, using worthwhile tasks as practice sends a clear message that mathematics is about figuring things o

In Level 2, Informal Deduction.

students can now logically order shapes according to their properties and come up with definitions for shapes. In this example, students can identify which shapes are parallelograms and can even label a square as a type of parallelogram. Students move from considering the properties of shapes to explor-ing the relationships between properties ofgeometric objects.

In Level 1 Analize

students can recognize and name properties and classify them. In this example, students can find the squares and rectangles. Students move from considering classes ofshapes rather than individual shapes by focusing on the properties of shape

In Level 0 Visualizaion

students visually recognize similar shapes and should be able to group them together. In this example, students can search out the triangles. Students move from considering shapes and what they "look like" to developing classes or groupings of shapes that seem to be "alike."

Moving from Level 1 to Level 2. Level 2 thinking is expected to begin in grade 5, when students classify shapes based on their properties in categories and subcategories (NGA Center & CCSSO, 2010). Instructional considerations supporting students' movement from level 1 to level 2 are:

• Challenge students to explore and test examples. Ask questions that involve reasoning, such as "If the sides of a four-sided shape are all congruent, will you always have a square?" and "Can you find a counterexample?" • Encourage the making and testing of hypotheses or conjectures. "Do you think that will work all the time?" "Is that true for all triangles or just equilateral ones?" • Examine properties of shapes to determine necessary and sufficient conditions for a shape to be a particular shape. "What properties must diagonals have to guarantee that a quadrilateral with these diagonals will be a square?" • Use the language of informal deduction. Say: all, some, none, if . . . then, what if, and so on. • Encourage students to attempt informal proofs. As an alternative, require them to make sense of informal proofs that you or other students have suggested

When considering what to differentiate, first consider the learning profile of each student. Second, consider what can be differentiated across three critical elements: Third, consider how the physical learning environment might be adapted. This might include the seating arrangement, specific grouping strategies, and access to materials.

• Content (what you want each student to know) • Process (how you will engage them in thinking about that content) • Product (what they will show, write, or tell to demonstrate what they have learned

Here are several reasons for including estimation in measurement activities:

• Develops familiarity with standard units. If you estimate the height of the door in meters before measuring, you must think about the size of a meter. • Promotes multiplicative reasoning through using a benchmark. The width of the building is about one-fourth of the length of a football field—perhaps 25 yards.

Problem-Based Tasks. Continued

• Focus on an important mathematics concept or skill aligned to valued learning targets. • Stimulate the connection of students' previous knowledge to new content. • Allow multiple solution methods or approaches with a variety of tools. • Offer opportunities for students to correct themselves along the way. • Provide occasions for students to confront common misconceptions. • Encourage students to use reasoning and explain their thinking. • Create opportunities to observe students' use of mathematical processes and practices. • Generate data for instructional decision making as you listen to your students' thinking.

Moving from Level 0 to Level 1. Instructional considerations that support students moving from level 0 to level 1 are as follows:

• Focus on the properties of shapes rather than on simple identification. As new geometric con-cepts are learned, students should be challenged to use these attributes to classify shapes. • Challenge students to test ideas about shapes using a variety of examples from a particular cate-gory. Say, "Let's see if that is true for other rectangles," or "Can you draw a triangle that does not have a right angle?" Question students to see whether observations made about a particular shape apply to other shapes of the same kind. • Provide ample opportunities to draw, build, make, put together (compose), and take apart (decompose) shapes in both two and three dimensions. Build these activities around under-standing and using specific attributes or properties. • Apply ideas to entire classes of figures (such as all rectangles or all prisms) rather than to individ-ual models. For example, find w

These websites provide a lot of interesting data • The Official Olympic Records contains all Olympic records, providing information about the games and events, such as the medalists in every Olympic event since Athens 1896. • data.gov is the home of the U.S. governments open data and provides access to all data sets available from the U.S. government, all on one site. • The USDA Economic Research Service Food Consumption site offers wonderful data sets on the availability and consumption of hundreds of foods. Annual per capita esti-mates often go back to 1909.

• Google Public Data Explorer makes large data sets available to explore, visualize, and interpret. • Better World Flux provides data related to the progress of countries and the world over the years, highlighting interesting trends and patterns. • NCTM Illuminations State Data Map is a source that displays state data on population, land area, political representation, gasoline use, and so on. • The Central Intelligence Agency (CIA) World Fact Book provides demographic infor-mation for every nation in the world: population, age distributions, death and birth rates, and information on the economy, government, transportation, and geography. • U.S. C

Here are several reasons for including estimation in measurement activities:

• Helps students focus on the attribute being measured and the measuring process. Think how you would estimate the area of the cover of this book using playing cards as the unit. To do so, you have to think about what area is and how the units might be placed on the book cover. • Provides an intrinsic motivation for measurement activities. It is interesting to see how close you can come in your estimate to the actual measure

•Science is about inquiry and is full of measurements and data, and therefore provides excellent opportunities for interdisciplinary learning experiences. For example, consider this short list of idea

• How many plastic bottles or aluminum cans are placed in the school's recycling bins over a given week? • How many times do different types of balls bounce when each is dropped from the same height? • How many days does it take for different types of bean, squash, and pea seeds to germi-nate when kept in moist paper towels? • Which brand of bubble gum will give you the largest bubble? • Do some liquids expand more than others when frozen? • Does a plant grow faster when watered with water, soda, or milk?

2There are four important principles of iterating units of length, whether they are nonstandard or standard (Dietiker, Gonulates, Figueras, & Smith, 2010). Units must be:

• Measuring ropes: Cut rope into 1-meter lengths. These ropes can measure the perimeter and the circumference of objects such as the teacher's desk, a tree trunk, or a pumpkin. • Drinking straws: Straws can easily be cut into smaller units and can be linked together with a long string. The string of straws can be a bridge to a ruler or measuring tape. • Short units: Connecting cubes, toothpicks, and paper clips are useful nonstandard units for measuring shorter lengths. Cuisenaire rods are also useful, as they are easy to place end to end and are also metric (centimeters)

Reasons to Use Experiments

• Provide a connection to counting strategies (lists, tree diagrams) to increase confidence that the probability is accurate • Provide an experiential background for examining the theoretical model (when you begin to sense that the probability of two heads is 1 analysis in Figure 21.4 seems more reasonable) 4 instead of 1 3 through experiments, the • Help students see how the ratio of a particular outcome to the total number of trials begins to converge to a fixed number (for an infinite number of trials, the relative fre-quency and theoretical probability would be the same) • Help students learn more than students who do not engage in doing experiments

Here is a list of the money ideas and skills typically required in the primary grades:

• Recognizing coins and identifying their value • Counting and comparing sets of coins • Creating equivalent coin collections (same amounts, different coins) • Selecting coins for a given amount • Making change • Solving word problems involving money (starting in second grade

The following are additional examples of probabilities of independent events. Any one of these could be explored as part of a full lesson.

• Rolling an even sum with two dice • Spinning blue twice on a spinner • Having a tack or a cup land up when each is tossed once • Getting at least two heads from tossing four coins • Rolling two dice and getting a difference that is no more than 3

◆ Measurement involves comparing an attribute of an item or situation with a unit that has the same attribute. Lengths are compared to units of length, areas to units of area, time to units of time,◆ Estimation of measures and the development of benchmarks or referents for frequently used units of measure help students increase their familiarity with units, preventing errors and aiding in the meaningful use of measurement. ◆ Measurement instruments (e.g., rulers, protractors) group multiple units so that you do not have to iterate a single unit multiple times.

◆ Area and volume formulas provide a method of measuring these attributes by using only measures of length. ◆ Area, perimeter, and volume are related. For example, as the shapes of regions or three-dimensional objects change while maintaining the same areas or volumes, there is an effect on the perimeters or surface areas.

BIG IDEAS ◆ Statistics is a different field of study from mathematics; although mathematics is used in statistics, statistics is concerned with analysis of data and the resulting practical implications. With statistics, the context always matters. ◆ Doing statistics involves a four-step process: formulating questions, collecting data, analyzing data, and interpreting results.

◆ Different types of graphs and other data representations provide different information about the data and, hence, the population from which the data were taken. The choice of graphical representation can affect how well the data are understood. ◆ Measures that describe data with numbers are called statistics. ◆ The shape of data provides a "big picture" of the data rather than a collection of numbers. Graphs and statistics can provide a sense of the shape of the data, including how spread out or how clustered they are.

◆ What makes shapes alike and different can be determined by geometric properties (i.e., defining characteristics). Shapes can be classified into a hierarchy of categories according to the properties they share. ◆ Transformations help students think about the ways properties change or do not change when a shape is moved in a plane or in space. These changes can be described in terms of translations, reflections, rotations, dilations, the study of symmetries, and the concept of similarity.

◆ Shapes can be described in terms of their location in a plane or in space. Young children use language such as above, below, and next to while later students use coordinate systems used to describe these locations more precisely. The coordinate view also offers ways to understand certain properties of shapes and can be used to measure distance, an important application of the Pythagorean theorem. ◆ Visualization includes the recognition of shapes in the environment, developing relationships between two-and three-dimensional objects, the ability to draw and recognize objects from different viewpoints, and to mentally change the orientation and size of shapes.

◆ Algebra is a useful tool for generalizing arithmetic and representing patterns in our world. Explaining the regularities and consistencies across many problems gives students the chance to generalize.

◆ The methods we use to compute and the structures in our number system can and should be explored. For example, the generalization that a + b = b + a tells us that 83 + 27 = 27 + 83 without the need to compute the sums on each side of the equal sign.

BIG IDEAS ◆ Probability is based on two foundational ideas: variability (discussed in Chapter 20) and expectation (prediction of what will happen). ◆ Probability that an event will occur is on a continuum from impossible (0) to certain (1). A probability of 1 2 indicates an even chance of the event occurring. ◆ For some events, the exact probability can be determined by analyzing the sample space. A probability determined in this manner is called a theoretical probability.

◆ The relative frequency of outcomes (e.g., from experiments or simulations) can be used as an estimate of the probability of an event. The larger the number of trials, the better the estimate will be. ◆ Simulation is a technique used for answering real-world questions or making decisions in complex situations in which an element of chance is involved; a model is designed that has the same probabilities as the real situation to explore how likely an event is

STATISTICAL QUESTIONS FOR INTERPRETING DATA ● What do the numbers (symbols) tell us about our class (or other population)? ● If we gathered the same kind of data from another class (population), how would that data look? What if we asked a larger group, how would the data look? ● How do the numbers in this graph (population) compare to this graph (population)? ● Where are the data "clustering"? How much of the data are in the cluster? How much are not in the cluster? About what percent is or is not in the cluster?

● What kinds of variability might need to be considered in interpreting these data? ● Would the results be different if . . . [change of sample/population or setting]? (Example: Would gathered data on word length in a third-grade book be different from a fifth-grade book? Would a science book give different results from a reading book?) ● How strong is the association between two variables (scatter plot)? How do you know? What does that mean if you know x? If you know y? ● What does the graph not tell us? What might we infer? ● What new questions arise from these data? ● What is the maker of the graph trying to tell us?