Mathematics Methods and Instruction for Students with Mild/Moderate Exceptionalities - D237
Traditional problem-solving lessons involve a teacher explaining the math, and then students practicing that math. This is followed by applying the mathematics to solve problems. Why does a lesson set up as described rarely work?
It assumes wonderful explanations produce understanding.
Which statement is not included in relational and conceptual understanding?
It eliminates poor attitudes and beliefs.
A teacher has designed a statistics lesson that requires students to visually present their analysis of the data.How can students use technology or tools to enhance their presentations?
Use a spreadsheet to create a graph of the data
A third-grade teacher is working with a mixed-ability classroom. What is one strategy that is beneficial for all learners?
Use effective wait time and multiple methods when approaching math problems.
First-year algebra students are learning how to solve two-step equations. The teacher notices that the students are not using precise mathematical language.Which instructional strategy should the teacher employ to encourage students to use precise mathematical language when completing this task?
Use procedural and conceptual questioning using mathematical vocabulary
A sixth-grade class is given the following problem: The temperature in Duluth, Minnesota, at 4:00 p.m. was 10°F. The temperature dropped steadily so that by 10:00 p.m. it had dropped 12°F. What is the temperature at 10:00 p.m.? Which representation models this problem?
Vertical chart 4-10; 10-(-2).Ten vertical lines are shown. The top of each vertical line contains a yellow circle placed in the line.
When teaching a unit on geometry and symmetry, which tool or manipulative can be used to enhance learning outcomes?
Virtual dynamic software
A teacher wants students to communicate mathematically outside of the classroom. The teacher gives a homework assignment that requires the students to participate in mathematical discourse.Which tool could the teacher assign the students to use from home to meet this objective?
Virtual flashcards
Students in a sixth-grade class are using models to explore how to find percents. Which question should help foster critical thinking?
What patterns have you noticed when modeling percents?
A student is tasked with selecting two cards with replacement from a fair poker deck of 52 cards.Which option demonstrates correct student reasoning about possible results of this experiment?
"Getting two kings is just as likely as getting two 10s."
A student constructed a graph to show how many of a town's residents have completed five different levels of education (elementary school, high school, two years of college, four years of college, graduate school).Which student explanation shows correct mathematical thinking?
"I used a bar graph because it compares the number of people in each level of education."
Which two demonstrate correct student geometric reasoning?
"It seems that, in triangles, longer sides are opposite larger angles."" "I have measured many different circles' circumferences and diameters, and the ratio of these always seems to be the same number. I predict that if I measure more, the ratio will have the same value."
Two students worked together to compute 3/4 - 5/8. Student A thinks the difference is 1/0 and student B thinks it is 1/8.Which teacher response is appropriate to evaluate conceptual understanding?
"Student A, will you please explain the process you used?"
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?
2(x - 4) = 20
A lesson for sixth-grade students focuses on solving the following equations:3x + 7 = 18 14 = 2x - 8 5(x + 1) = 20Which equation should the teacher use to assess these students' understanding of the lesson?
3(x - 2) = 15
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 responses have misconceptions about probability?
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 with the next spin is 0% because blue has already appeared during this round of spins.
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 a recommended follow-up to this lesson?
A drill activity—dimensions for various rectangular prisms are provided, and students compute the volume of each.
Students are learning about the concept of the perimeter of a rectangle.Which resource would help to effectively teach this concept?
A ruler to measure side lengths of at least five rectangles
Which statement is not proposed by the constructivist theory?
A safe environment is necessary to gain mathematical power.
What is a common misconception about place value?
A student writes six hundred three as 63.
Select all strategies that should be part of a lesson that uses think-aloud.
A teacher demonstrates to students how to solve a problem. A teacher discusses methods and strategies to help solve problems and provides reasoning for each step.
What is an example of a peer-assisted teaching strategy?
A teacher explains how to solve a complex math problem.
What is an appropriate use of manipulatives from web-based technology in mathematical instruction?
A teacher shows a website and demonstrates how to effectively interact with virtual manipulatives.
A student is given the following problem: 17 + 42 + 13When the student is asked how she might solve this problem using mental math, she replies, "Since 42 added to 13 is the same as 13 added to 42, I would add 17 and 13 to get 30 then add 30 to 42 to get 72."What algebraic thinking does this student response demonstrate?
Addition is commutative and associative
Which strategy teaches for understanding of mathematics?
Allow the use of a variety of manipulatives to model a problem
In a sixth-grade math class, students are reviewing operations involving fractions. The teacher wants to use an instructional strategy that will encourage students to build on another student's prior description about how to add two fractions.How should the teacher do this?
Ask other students to add to and provide examples of the first student's description
The mean of a data set is 12. How can the mean increase?
By adding a 14 to the data set
A tenth-grade math teacher wants to work with a science teacher on integrating the content. How can the teacher apply a math connection?
By expressing the distribution of water on Earth in a circle graph
Which instructional strategy support a classroom environment that encourages mathematical communication?
Include assessment opportunities for students to explain their thinking
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?
Calculator to explore patterns and relationships
To create an environment for doing mathematics, what is the teacher's role?
Create a spirit of inquiry, trust, and expectations
Which tool can be used to teach a lesson on fractions?
Cuisenaire rods
Select all the planning strategies for all learners.
Include opportunities for collaboration and classroom discussion Use graphic organizers to help students organize work
Which activity could be used to make a mathematical connection within a mathematics curriculum when teaching a unit on multiplying decimals?
Demonstrate how multiplying fractions is related to multiplying decimals
Which activity could be used to make a mathematical connection within a mathematics curriculum when teaching a unit on basic probability?
Demonstrate how probabilities can be displayed in a pie graph
Students in an eighth-grade class work on the following problem:A student has taken five quizzes. The scores were 72%, 86%, 78%, 81%, and 68%. What would the student need to score on the sixth quiz to achieve an 80% average?How should the teacher use a rubric to assess the progress of students' work on this problem?
Describe the expectations of the rubric to the students before they begin
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.
Encourage students to collaborate and discuss their findings and strategies after the activity has been completed Review the properties of 3D shapes prior to the activity
Students in a third-grade class are asked, "How many small unit squares will fit in a rectangle that is 54 units long and 36 units wide?" The teacher provides students with base-ten blocks to help solve the problem. How should the teacher use a rubric to assess the progress of the students?
Explain how the rubric will be used for evaluation before students begin the problem
True or False Theoretical probability changes based on the number of trials.
False
A ninth-grade teacher is planning a unit of study on equivalent fractions. Select all tools or manipulatives that can enhance learning outcomes.
Fraction circles Cuisenaire rods
The students in a class studied polygons, including classifying polygons as concave, convex, and regular.Which performance task will effectively assess understanding of what was taught?
Given a variety of polygons, arrange the polygons by concave, convex, or regular and then describe the characteristics of each.
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?
Graphing calculator
Which resource can help a student visualize triangles with corresponding sides?
Grid paper
Which instructional strategy is useful for facilitating effective class and small-group discussions about mathematics?
Have students compare solution strategies
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.
Have students draw a picture when solving long division problems Have students create their own word problem that would require long division
Select all instructional strategies that can be used to effectively hold small group and classroom discussions.
Have students share multiple methods about how to solve the problem Encourage students to think-pair-share and compare solutions and strategies
A seventh-grade class includes students who lack motivation and students with mild learning disabilities. Which two instructional supports would meet the needs of this group of students as they study math?
Help transitioning from a problem to a particular representation Opportunities to discuss problems and ideas with other students
Which question demonstrates an effective technique to elicit mathematical discussion and thinking?
How did you figure out the solution?
A teacher has assigned this contextual (story) problem to her class.The Cooper family is having a holiday party for their 3 children. Susan, 10 years old, Noah, 7 years old, and Conner, 5 years old, will each be allowed to have as many guests as their age. If Mrs. Cooper wants to make sure there are 2 cupcakes for each child at the party, how many cupcakes must she bake? If each box of cupcake mix makes 24 cupcakes, how many boxes must she buy?The teacher makes this statement: "Use any representation you wish to understand the problem."How does this statement make mathematics accessible to all learners?
It encourages the students to approach the problem from individual perspectives.
A group of students is given the equation 6/8 = 3/4. They are asked to explain the relationship.Which student response demonstrates relational understanding of the concept?
It's like 2 quarters equals 5 dimes.
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?
Line Graph
What is a teacher-created instructional material that promotes student reflection of thinking related to math concepts?
Math journals
Students are working in groups on the following problem:Teresa has twice as many marbles as Patrice. Together they have 36 marbles. How many marbles does each girl have?Mena states, "Teresa has 12 marbles and Patrice has 24 marbles."How should the teacher respond to promote an understanding of Mena's thinking?
Mena, can you explain how you got your answer?
Which process has a student demonstrated when they use multiple representations?
Moving from instrumental to relational learning
Which two statements describe common mathematical learning theories?
New ideas are accommodated in the brain by reworking old ideas. New ideas are made accessible through peer and teacher support.
Which answer is not a reason to use a calculator in the constructivist math class?
Performs basic computations such as 7 × 3 when computation skills are the objective of the lesson
What are two strategies for planning mathematics lessons for all learners?
Provide graphic organizers and tables for students to organize their work. Include think-pair-share opportunities for students to discuss math concepts
Select all instructional strategies that can be used to help and encourage mathematical communications.
Provide opportunities that will encourage the metacognitive process and allow students to explain their thinking Use class time effectively by encouraging students to evaluate and compare their answers with other students
When a student employs multiple representations, what have they demonstrated?
Relational understanding
Given the following table: x12345y1491625 Which representation models this information?
Scatter plot shown with 5 dots. The y-axis spans from 0 to 30 in increments of 5. The x-axis spans from 0 to 6 in increments of 1. The first dot is at 1 on the x-axis and just above 0 on the y-axis. The second dot is at 2 on the x-axis and just below 5 on the y-axis. The third dot is at 3 on the x-axis and just below 10 on the y-axis. The fourth dot is at 4 on the x-axis and just above 15 on the y-axis. The fifth dot is at 5 on the x-axis and 25 on the y-axis.
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?
Selecting a black queen both times is just as likely as selecting a red king back-to-back.
Which activity will help students make a mathematical connection to contexts outside the math curriculum?
Sort and classify flowers by the symmetry of the petals
Select all statements that describe challenges that students from diverse groups may face.
Special education students may require extended time to complete a task. Students from diverse backgrounds may solve problems differently.
Which stakeholder is directly in charge of choosing mathematical standards?
State department of education
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?
Stem-and-leaf plot
During the basketball season, a player scored 12, 14, 11, 17, 21, 18, 17, and 12 points in the first eight games.Which representation of the data shows the player's points scored in each game?
Stem-and-leaf plot
Student A and Student B are learning how to add multi-digit numbers. Their incorrect attempts to add 542 and 374 are shown here. (Image Description) Student A and Student B both add 542 and 374. Both students stacked the numbers of the problem vertically to find the sum. Writing is neat and place values are correctly aligned. Student A's response is 8116. Student B's response is 905, and this student carried a 1 to the hundreds place. Which statement explains the confusion that led to the incorrect sums for these students?
Student A struggles to understand place value. Student B struggles to add single digit numbers accurately.
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?
Students are given quadrilaterals and explore their interior angles to discover that all interior angles add up to 360 degrees.
What is an environment that is not desirable for mathematical instruction?
Students are working through practice problems to learn the mathematical strategy.
A first-grade teacher is planning a lesson on adding whole numbers. How can the teacher allow the students to make a math to real-world connection when teaching this lesson?
Students can use a recipe and determine the total amount of ingredients.
Which two statements describe learning difficulties that students from diverse groups may encounter?
Students from other countries often solve problems or illustrate concepts differently. Students who are English Language Learners (ELL) require more time to solve problems.
Select two learning theories that describe effective math learning.
Students learn new ideas through collaboration with their teacher and peers. Students learn by rearranging previously learned concepts and making connections.
What is an important characteristic of the process of teaching and learning mathematics?
Students of all ages should frequently be asked to provide rationale for their answers.
A seventh-grade teacher is presenting a unit on various holidays.Which activity could be used by the teacher help students make mathematical connections as they work through the unit?
Students take a recipe from their family's traditional celebration of a specific holiday and adjust the ingredients to serve a larger group.
A lesson is being designed to address the communication standard from Principles and Standards for School Mathematics (PSSM). The lesson requires that students write out directions for using fraction bars to model several addition problems.How does this lesson meet the communication standard?
Students use the language of mathematics to express ideas precisely.
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 process standard?
Students will use precise math language to express their reasoning.
A teacher is teaching a lesson in an inclusive classroom on measuring angles with a protractor. She has a range of students with varying levels of ability. All students are given a protractor and a worksheet with pictures of many different angles. Which approach supports students with special needs while still serving other students in the class as they learn to measure angles with a protractor?
Students with special needs receive extra support to learn to measure angles with a protractor
After analyzing high temperatures from July over several years, students created histograms to display the data. How can a teacher incorporate an observational assessment into this lesson?
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.
An eighth-grade class has several students with moderate learning disabilities.Which instructional strategy would help the teacher provide equitable learning opportunities for this diverse class?
Teach multiple representations of ideas for one problem at a time
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?
Tessellations of polygons
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?
The activity is effective at showing students how to find the area of rectangles, but the standard is not aligned.
A teacher writes a fifth-grade lesson plan in which students will use a pan balance to demonstrate equalities in equations. The students will use counters and the pan balance to make the equations balance for ten open-sentence equations (15 + 34 = n + 21).The following standard was included within the plan:NCTM (National Council of Teachers of Mathematics): Principles & Standards for School Mathematics:Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.How would an administrator rate the learning activity for its overall quality and alignment with the standard?
The activity was effective in demonstrating the concept of equality in equations, but the standard did not align with the lesson.
A teacher has a group of learners of mixed abilities. The objective of the lesson is for students to create a histogram. The teacher plans to deliver explicit instruction to learners who are struggling with the concepts on how to create a histogram and plans to allow learners who are mastering the concepts the opportunity to create a histogram after developing a statistical question and gathering data. How effective is this activity at meeting the needs of all learners?
The lesson addresses the needs of all learners because all students will have the chance to create the same product.
A teacher has a class of students with special needs and students with high ability. The teacher grouped the students by ability level for a lesson on making a plot line to display a data set of measurements in fractions. In the lesson, the teacher is prepared to provide step-by-step instructions for the students with special needs about how to construct a plot line. The teacher has also planned to engage the high-ability students in constructing an activity to gather data and then create a plot line to display the data. How effectively does this lesson plan address the needs of all students?
The lesson effectively addresses the needs of all students. The complexity of the tasks for the different groups of students is based on ability.
Which type of information should not be provided by math teachers?
The preferred method
According to the teaching principles defined by NCTM, what does effective mathematics teaching require?
Understanding what students know and need to learn
Students are creating bar graphs from a survey about pet care.How can the teacher incorporate observational assessment into this lesson?
Use a list of specific content objectives to take notes about student understanding while circulating through the class and then use the data to modify the lesson
During an in-class assignment covering the day's lesson on similar triangles, a teacher noticed several students making the following error.(Image Description) Two similar acute triangles, one small and one large. The small triangle has the length of all three sides labeled: the left measures 4 inches, the right measures 5 inches, and the base measures 7 inches. The large triangle has two sides labeled: the left measures 12 inches and the right measures 15 inches. The base is left unlabeled. The student work is listed below the triangles in five vertically stacked lines of text. The first line shows 4 over 7 equals x over 12. The second line shows 4 times 12 equals 7x. The third line shows 48 equals 7x. The fourth line shows 48 over 7 equals x. The fifth line shows x equals 6.86 inches.Which part of the lesson should be retaught to help students understand the material?
The properties of similar figures
A teacher provides students with a jar of marbles. There are 24 white marbles, 10 blue marbles, and 6 yellow marbles in the jar. The teacher asks students to predict the chances of a blindfolded student reaching into the jar and pulling out a yellow marble. A student answers that the chances are one out of six, which could be demonstrated as the ratio 1/6. Which conclusion should the teacher reach after analyzing the student's response?
The student correctly answered the problem that the probability of pulling a yellow marble is one out of six but incorrectly provided the ratio.
A student is asked to compute 2 + 3 x 5 and responds that the answer is 25.What can the teacher conclude about the student's algebraic thinking?
The student does not follow the correct order of operations.
A student is asked to add two fractions and writes the following: (Image Description) 1 over 2 plus 1 over 3 equals 2 over 5.The student is unsure whether this solution is correct.Which action shows strategic competence?
The student uses fraction strips to model adding 1/2 and 1/3 and notes that it is equivalent to 5/6 not 2/5.
Which scenario demonstrates a context outside of mathematics curriculum where probability could be taught?
The students look at the batting average of a baseball player and determine the likelihood of the player getting a hit.
A teacher asks her students to solve this problem:What is the mean of the numbers 12, 18, 16, 24 and 10?Which two instructional strategies could the teacher use to encourage students to create their own representation of their mathematical thinking?
The teacher asks each student to create a picture to demonstrate how to determine the solution. The teacher asks the students to work in cooperative groups to create a word problem using the scores and explain to the class how they found the solution.
Which two strategies could be part of a lesson plan that uses a think-aloud strategy rather than peer-assisted learning?
The teacher discusses alternative methods of solving the problem. The teacher talks through the steps of a solution, identifying the reasoning at each step.
The objective of a lesson is for students to solve word problems involving the multiplication of multidigit numbers. Although one student is able to solve multidigit multiplication problems, the student struggles with solving word problems.How can this student's needs be accommodated?
The teacher lets the student use a chart with the math operations vocabulary matched to the math symbols while practicing word problems.
Which two statements describe teaching for all students?
The teacher looks for ways to make tasks more relevant to students with varied backgrounds. The teacher scaffolds tasks to provide access to higher-level thinking for students.
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?
The teacher should allow both students to retell their strategies and discuss their methods for obtaining the answers.
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?
Think-aloud
A second-grade class is learning about the part-plus-part-equals-whole addition model. Their teacher decides to use the common experience of eating in the cafeteria as a context outside the mathematics curriculum to teach this concept.Which scenario exemplifies this addition model?
Three students eating lunch are joined by two more students.
How does a teacher use a rubric to analyze student performance?
To assign a score to a mathematical task based on a preset scoring framework
True or False In a linear relationship, all the points on a graph lie on a straight line.
True
True or False In a probability experiment, an event is a subset of the sample space.
True