Teaching Strategies for Math
A consumer takes out a loan for $500 that charges 10% annual interest. What is the total cost of the loan if the consumer pays it back one year from the date of origination?
A $550 The total cost of the loan is the original amount ($500) plus the cost in interest ($50, 10% of the loan)
Mr. Sexton has been trying a variety of teaching methods to engage his class, but it seems to make things more out of control. How can he increase engagement while maintaining an orderly classroom?
A Establish a daily procedure for class and vary the activities used for instruction. Students will be more on task when provided a routine to follow . B
Graphic / Pictorial Representation
A graph or picture that serves as a visual model of a mathematical equation. Number line
Numeric Representation
A model using numbers to display a mathematical concept.
Symbolic Representation
A model using symbols or variables to display a mathematical concept. Formula
Coordinate Plane
A plane, often divided into a grid, with a horizontal x-axis and a vertical y-axis that intersect at the origin
Which of the following does not have 8 vertices?
A rectangular pyramid. IT only has 5 vertices.
Word Wall
An on-going bulletin board with common terms used frequently in the classroom. Vocabulary words are added as they are introduced
A teacher is teaching her students how to determine the mean of a data set. She finds that several students multiply the sum by the number of numbers instead of dividing by the number of numbers. What should she do to help students master this concept?
B Give each student a different number of cubes and ask them to determine the average number of cubes by sharing so that everyone has an equal number of cubes. This helps students scaffold from a concrete understanding of the material to a more abstract way of determining the value.
A seventh-grade teacher has been teaching her students about areas of regular polygons. She has already introduced them to the topic at the concrete level by having students interact with magnet tiles. In which way should the teacher continue teaching the topic?
C Have students draw diagrams that demonstrate the way to determine area of polygons. After learning about a topic at the concrete level students should move onto the representational level where they draw figures and diagrams to model the topic.
The Payday Lending industry has faced increased criticism and scrutiny for which of the following practices?
C charging high interest rates Payday lenders charge high interest rates which locks many low-income individuals into cycles of debt and repayment, a practice for which these companies have faced much criticism.
Mrs. Adamson's student asks her how much space a cube takes up. Mrs. Adamson said to answer this question, the student would need to calculate the volume of the cube. Which of the following measurable attributes is the formula for a cube based upon?
C length Before the volume of a cube can be calculated, the length, width, and height must be measured. Length is the best answer.
Which of the following describes the most helpful first step(s) in determining whether a given number is prime or composite?
Find the square root of the nearest integer that is the perfect square closest to the given number.
Injective Functions / One-to-One (1-1) Functions
Functions in which each output is mapped to and produced by only one input. Each injective function has an inverse.
Constructivism
Learning new behaviors by adjusting our current view of the world Research projects
Cognitivism
Learning new behaviors by connecting current knowledge with new knowledge Teaching fractions by talking about pizza slices
Auditory Learning
Learning primarily by hearing things Lectures
Visual Learning
Learning primarily by seeing things Written examples
Kinesthetic Learning / Tactile Learning
Learning primarily by touching things or doing an activity create and act out plays or skits
Behaviorism
Learning theory rooted in the notion that all behaviors are learned through interaction with the environment
Auditory Methods
Lessons using materials for students to listen to. Ex: speeches, music, or direct instruction
Ms. Davis teaches a fifth-grade math class primarily composed of English language learners (ELL). Which of the following can support her ELL students? Select all answers that apply.
Make a word wall. A word wall is helpful for English language learners to visualize a word. Use gestures, pictures, and models to explain terms. Use of gestures, pictures, and models are helpful to English language learners.
Mr. Swan wants to ensure that his students truly understand the material he is teaching. When students get questions incorrect on a test, he presents them the opportunity to correct their answers for half credit. He asks students questions such as "what if I changed this number?" and "why did you do this?" What process is Mr. Swan trying to get his students to engage in?
Metacognition is reflecting on one's thought process to deepen understanding. This is what Mr. Swan is attempting to do.
Manipulatives
Objects used by students to illustrate and explore mathematical concepts, such as to represent numbers in an equation Blocks, Coins
Mrs. Nadir's students are great at determining the surface area of cubes. They struggle with determining the surface area of rectangular prisms. What should she do to help her students be successful?
Reinforce how to determine the area of rectangles and then procedurally add the areas of the faces of a block.
A seventh-grade teacher is teaching her students to determine the area of circles. She gives them the formula area = πr2 and several word problems to complete. She notices many students are getting quadruple the area that they should. What went wrong?
She introduced the content at the abstract level; she should have introduced it at the concrete level. Students were confusing the radius with the diameter. If they had be introduced to these topics at the concrete level they would not be confused. Next students should master the content at the representational level before finally progressing to the abstract level.
In which of the following activities would students demonstrate the use of mathematical terminology as a precise means of expressing mathematical ideas?
Students identify various parts of mixed fractions. By having students call out the numerator, denominator, and integer portions of a mixed fraction, they are demonstrating the use of mathematical terminology as a precise means of expressing mathematical ideas.
English Language Learners (ELLs)
Students who are learning the English language, or for whom English is not a first language
Mr. Kopko's class is learning about long division. He asks students to use the calculator to check their results. On the following division problem several students are confused by their results. 22/7 = 3R1 = 3.142857 How should Mr. Kopko instruct students to deal with remainders?
Tell them to multiply the decimal part of the solution by the denominator to see if it matches the remainder. Multiplying the decimal part of the solution by the denominator should match the remainder and allow the students to check their solutions.
Criterion-Referenced Tests
Tests in which a standard has been set for the test taker to achieve in order to pass the test. A multiple choice or short answer test on the content of a unit of study in which a 70% is needed to pass.
Norm-Referenced Tests
Tests that compare an individual's performance/achievement to a group called the "norm group." An IQ test
Learning Style
The manner in which a student learns best Visual Learning
Which of the following would not be an appropriate way to represent 3/4?
The point on this number line represents 7/4 not 3/4.
Texas Essential Knowledge and Skills (TEKS)
The state foundation curriculum developed by the State Board of Education, that requires all students to demonstrate the knowledge and skills necessary to read, write, compute, problem solve, think critically, apply technology, and communicate across all subject areas
Triangle Inequality Theorem
The sum of the lengths of any two sides of the triangle must greater than the third side.
SAS Postulate (Side-Angle-Side)
Two sides and the angle of one triangle are congruent to THE other two sides and the angle of another triangle, then the triangles are congruent.
SSS Postulate (Side-Side-Side)
Two triangles are congruent if all three sets of corresponding sides are congruent.
Abstract Thinking
Using numbers or letter variables in an equation 13x = y
Concrete Representations
Using physical pieces to represent mathematical problems Manipulatives
Which of the following questions can be used to demonstrate the interdisciplinary connections between math and other subjects?
Where have you seen this before? This question emphasizes how the content connects to other material.
Pie Chart
a graph in which a circle is divided into sectors that each represent a proportion of the whole. Pie charts are helpful when displaying the relative distribution of categories.
Number Line
a straight line where each number is equal distance from the next one
reflection in y-axis (Rule)
causes a sign change in x
Compartmentalized Teaching
concepts taught one at a time in isolation of other concepts (no longer recommended)
Which representation is best for demonstrating probabilities?
graphical images There are many types of graphs that can display probabilities: circle graphs, histograms, etc.
Heterogeneous Group
group comprised of individuals working on various levels A small group of students with varying academic abilities working together on a science project is a heterogeneous group.
Flexible Grouping
grouping students based on their learning needs or interests After reviewing the student test results, a teacher can use flexible grouping to organize groups based on students' areas of weakness.
Integrated Teaching
multiple concepts are used in problem-solving at once (current best practices)
Reflection in the y=x line
switch x and y
Metacognition
the ability to think about one's own thought process
Interest is best defined as:
the cost associated with borrowing from the bank which issues a credit card. Using a credit card is essentially taking out many small loans, and interest is the cost of borrowing the money from the bank which issued your credit card.
In a 30-60-90 triangle
the shorter leg is half the length of the hypotenuse. The angle between the shorter side and the hypotenuse is always 60°.
Concrete Operational Stage
the third stage of Piaget's Theory of Cognitive development, occurring from 7 years old to adolescence, in which children begin to think logically and use inductive reasoning
Ms. Mansfield gives her students the following exit ticket at the end of the class period. The teacher is using the exit ticket to assess student mastery of which of the following learning goals?
to compare and contrast the attributes of a shape and its dilation when graphed on a coordinate grid In the ticket, ΔA′B′C′ΔA′B′C′ is a dilation, with scale factor of 3, of ΔABCΔABC. Student answers will enable the teacher to determine whether students
The formula to find the sum of the measures of the interior angles of a regular polygon
(n−2)180
A teacher draws a 10x8 grid on the board without any "X"s in the grid. The teacher then writes "X"s in one and a half rows of the grid. The teacher asks the students to create an equation to represent blocks left on the grid, if the "X"s represent the blocks that have been removed. Which of the following is the best equation in response to the teacher's question?
80 - 15 There are 80 blocks on the grid. If 15 are removed, then the equation 80 - 15 represents this.
Mr. Jones is hosting a career day for his sixth-grade class. Jeremy tells him that his father cannot come present because his job has nothing to do with math because he did not attend college. Mr. Jones asks Jeremy what his dad does for a living. Jeremy says his dad is a carpenter. What should Mr. Jones tell Jeremy?
A Carpentry involves a lot of math including accurate measurements, determination of angles, and geometry. B Carpentry involves algebra to determine how much raw material to buy. This is a true statement, but the other statements are also true. C Mathematics is used by people throughout their lives, whether or not they attend college.
Mr. Erikson has his friend Ted, who is an architect, come present to the class about how he uses math in his job. What is Ted likely to talk about?
A How geometric figures are a part of most buildings. Geometry is often used by architects and this is the right choice.
Mrs. Lin wants her students to be familiar with various terms relating to polygons. An English teacher suggests making her students define the word and then write a sentence with them in context. Is this a good practice for a math teacher?
A No, math vocabulary terms relating to polygons are better described visually. This is true. Having a more visual activity or project would help students understand the terms better.
Mr. Feeny has been teaching fifth-grade math for thirty years. He will only accept answers from his students that follow his algorithmic procedures. If a student determines a correct solution by any method other than the way they were taught in class they will not receive credit. How could Mr. Feeny improve his teaching practice?
A Teach students multiple varied ways to achieve the right answer and accept any correct answer as long as there is mathematically reasonable supporting work. This is the best choice. Students should be encouraged to think through a variety of ways to solve problems.
Effective differentiated instruction occurs when, depending on each student's abilities:
A students receive different levels of scaffolding on assignments. Students should be expected to master all appropriate mathematics content, but some students will need more scaffolding, or teacher help, than others as they learn the material.
A teacher plans a three-part learning exercise for her students: Students use a balance to learn about mathematical equivalence. When the same number of tokens are added or removed from each side, the balance remains level. Students practice similar problems with pictures of tokens and a balance. Students see algebraic representations of the form: x+6−6=10x+6−6=10. The teacher most likely planned this activity to accomplish which of the following goals?
A to provide instruction along a continuum from concrete to abstract This learning activity moves from a concrete exercise in step one to a visual representational activity in step two, and finishes with abstract examples using algebraic expressions.
Ms. Valerie is teaching a unit on percentage and sets up a mock store in her classroom where all of the items are marked on sale. She marks each item with a discount of 15% off, 20% off, or 30% off and establishes a tax rate of 7%. Ms. Valerie then has each student choose three items and calculate the final price, including the discount and taxes. The teacher most likely planned this activity to demonstrate her understanding of how to do which of the following?
apply mathematics to real life and a variety of professions Ms. Valerie's lesson shows students how to apply percentages to real life in the sales profession.
In a unit on personal finance, a sixth-grade teacher wants students to be able to identify the difference between fixed and variable costs. Which of the following examples would best highlight this difference?
categorizing the expenses of a local restaurant into expenses that depend on the number of customers and expenses that do no not depend on the number of customers This will help students highlight the differences between variable and fixed costs because the student is actively having to categorize an expense into one of the two categories.
reflection in x-axis (Rule)
causes a sign change in y
A teacher engages her class in a discussion of the coordinate plane. The students are asked to identify the quadrants, the coordinate axes, and the mathematical notation for various points in the plane. Students are asked to develop a way to quickly identify the quadrant in which various points lie. Which of the following objectives is the teacher most likely trying to address with this lesson?
developing precise mathematical language when expressing mathematical ideas
Non-proportional Manipulatives
objects that are not proportional to each other with respect to shape and size. Often all of the items are the same size. counting tokens
Proportional Manipulatives
objects that are proportional to each other with respect to shape and size tangrams
Joshua is learning about volumes of three-dimensional figures. First, his teacher explains what volume is. Then, she writes the formula for area of a cube on the board v = s3. Next, she has the students recite "the volume of a cube is the side length cubed". Finally, she has students take six-sided dice of various sizes and measure them to determine their volume. Which best describes the teaching method is the teacher attempting to use?
task variety The teacher is using task variety because she is presenting the same material in multiple ways.
Mr. Zammit is teaching his class about shapes. One of his students incorrectly labels all rectangles as squares and all rectangular prisms as cubes. Which of the following should Mr. Zammit do in this situation? Select all that apply
A Require his student to use correct mathematical vocabulary. Correct vocabulary is a very essential part of mathematics. B Make an analogy to help the student understand his mistake, for example calling every rectangle a square is like calling every fruit an apple. Connecting to a real world example can help the student understand his mistakes and improve his future use of math terms.
Mrs. Matthews is teaching her sixth-grade class about areas of regular geometric figures. How should she best introduce this topic to her students?
C Give students pattern blocks to manipulate. Tell them the area of the smallest figure is one and ask them to determine the area of the larger figures. This method would allow students to engage with manipulatives they had used in earlier years and connect them to newer knowledge. This would be a good engaging experience that allows them to connect prior knowledge to new learning goals.
A sixth-grade teacher is teaching her students about circles. She wants them to learn about circumference. She gives the students the formula c = dπ and a worksheet to complete. Many students fail to get the problems correct. What went wrong?
C She introduced the content at the abstract level; she should have introduced it at the concrete level. Students should be introduced to material at the concrete level. Next students should master the content at the representational level before finally progressing to the abstract level.
A sixth-grade teacher discovers that each student in his class receives an allowance from their parents. Which of the following examples would best demonstrate to the students the power of saving their allowance instead of spending all of their allowance?
C Show students the expected return of 5% allowance savings over a 10-year period. Demonstrating how much money students could make by saving their allowance would best demonstrate the power of saving money.
Which activity can emphasize the interdisciplinary connections that math has to other subjects?
D A career fair where professionals talk about their use of math in their job. This activity will show how math connects to many other subjects.
Symbolic Stage / Representational Stage
Drawing pictures or symbols to represent numbers in an equation Squares
Ms. Smith is beginning a unit on area and perimeter. She begins the unit with an activity that requires students to use a list of formulas provided to find areas of several different quadrilaterals and triangles. The entire class is having difficulties with this assignment. What should Ms. Smith do to remedy the situation?
Have the students find areas of quadrilaterals on a geoboard or grid. The best idea for helping her students is to drop down cognitive a level to remediate. Since working with formulas is an abstract process, using the pictorial level (grid) or even the concrete level (geoboards) is a good option.
A fifth-grade teacher is beginning a unit on equivalent fractions with her students. If this is an introductory lesson, which of the following activities would be the most effective in helping the students understand the concept of equivalent fractions?
use pattern blocks to model different fractions equivalent to ½ Since this is an introductory activity, concrete, proportional manipulative materials like this should be used for concept development. It is important not to rush past this step and to use a variety of different materials to develop and reinforce understanding of this concept.
Anytown School District provides a 50 multiple-choice question mathematics assessment to all students. The students complete the assessment, the tests are scored, and the scores are compared throughout the school district. Which of the following mathematics component is most likely the goal of this type of assessment?
accuracy The assessment most likely is designed to measure the students' accuracy of answering questions since it is multiple choice.
The formula for permutations is: nPr = n!(n−r)!(n−r)!n! while the formula for combinations is: nCr= n!r!(n−r)!r!(n−r)!n!. As a warm-up, Ms. Caulkins asks her students to compare these two formulas and identify the differences and similarities before she has taught what the formulas are used for. What is her goal in this exercise?
activating prior knowledge of reducing fractions Activating prior knowledge allows students to build on prior concepts with new material.
Ms. Bindle is teaching her students about congruent triangles. She wants students to be able to write basic geometric proofs. Which of the following is the best way to assess their mastery at the end of the unit?
an open ended assessment where they write three proofs on their own
Which of the following would be the least appropriate use for handheld calculators in the classroom?
answering computation questions on a test of operational skills All suggestions are valid, except allowing students to use calculators on tests of their operational skills such as addition, subtraction, multiplication, and division.
Two color counters are often used to model addition and subtraction of integers. The red counters represent negative integers; the yellow represent positive integers.
C -2 The problem pictured above is -7 + 5 = -2. Using the chips, a red and a yellow chip are paired together to form a "zero-pair"-a model of -1 + 1 = 0. Pairs are matched and removed from the group leaving two red chips unmatched. This results in a representation of an answer of -2.
Mr. Sherlock is teaching his students to convert units such as miles per hour to other units such as meters per second. Which is the best first activity for his students to master this concept?
C Give students index cards with stickers on each that represent the numerators and denominators of the fractions in the conversion. Show students how to line up the index cards so identical stickers appear on the top of one card and the bottom of the other to make sure unwanted units cancel. This activity teaches students how to cancel units while converting units. It is also at the concrete level because it involves tangible manipulatives.
Mrs. Jones is teaching a lesson on slope-intercept form. She requires each student to find the slope and y-intercept of a set of graphs, then put them into a formula that describes the graph. The students work one problem at a time and Mrs. Jones circulates to check their work. If a student has the correct answer, Mrs. Jones gives them a checkmark and they move on to the next question. If the student has the wrong answer, Mrs. Jones directs them to the incorrect portion of their work and they revise their answer. Mrs. Jones continues to circulate the room until all students have finished the assignment. Which of the following learning theories best matches the activity Mrs. Jones uses with her students?
D Behaviorism learning theory The students are learning by receiving feedback with positive reinforcement.
Mr. Ginger is teaching his students about solving single variable equations. He decides to host a team competition where students are placed in groups of four and earn points for the team if they solve the equation the fastest. Only the fastest team can receive any points. Team A receives 160 points, Team B receives 40 points, Team C and Team D both receive 0 points. How can he improve this activity?
D Give all teams points if they receive the correct answer with a bonus for solving it fastest. This gives all students a fair chance to demonstrate mastery.
A parent is complaining about the math homework. They feel that their ELL child is at a disadvantage because they cannot afford internet at home and the homework is best completed using online software. The teacher is providing students time in class to complete the work that requires online resources, however, this student has not been using it stating that he will do it at home. What is the best strategy the teacher should use to prepare for meeting with the parent?
D Open communication with the parents that does not involve educational jargon. Open communication that encourages parental involvement is best.
Which of the following is not an advantage of using a credit card?
D There are no fees for late payments. Credit card companies do charge late fees if bills are not paid on time.
Students in Mr. Tan's class are using multicolored fraction tiles. Which of the following would NOT be an appropriate activity for Mr. Tan to ask his students to perform with the tiles?
D using the tiles to add positive and negative fractions There is typically not a way to differentiate between positive and negative fractions using fraction tiles.
Mr. Francis has been teaching his students about how dimensional change affects volume and area. He asks students what would happen to the volume of a cube if he doubles the side length. Several students say it will quadruple. How should he address this misunderstanding?
Give a problem with real numbers and have students show their calculations.
Ms. Miller is a student teacher in a fourth-grade classroom. She has heard that group work is important so she wants to plan for group activities. On her first day student teaching, she briefly says "today we'll be doing group work about fractions" before she sends the students to stations. How could she best improve her teaching?
Give a whole group lesson on fractions before breaking into groups. Ms. Miller did not give enough whole group instruction before letting the students work in small groups. At a minimum she should give a whole group lesson relating to the content of the day before sending students to work at stations.
Mr. Owens wants encourage his students to write about math. As a homework assignment, He asks his students to write about why it is beneficial to take the diagonal across the park using mathematical reasoning. Some of the responses are only one sentence long. How can he improve this activity in the future? Select all answers that apply.
Give students a sample response to a prompt about a prior topic that exemplifies properties of good writing. Model writing responses to word questions in class Give students a rubric for how the assignment will be graded.
Mrs. Blue wants her students to be able to write two column geometric proofs. Which is the most appropriate way to determine their mastery?
Give students an open ended exam where they write multiple two column proofs. Asking students to write a proof is the best way to determine if they can write a proof.
Mr. Kolbein wants his students to practice sorting numbers by order of magnitude. He thinks a game would be most engaging, and writes various numbers on index cards, one per card. He then designs a game in which pairs of students compete against other pairs of students to see which pair can sort number cards the fastest. Which of the following additional activities would increase the engagement and learning value of the game?
Give the students certain criteria to follow, but have each pair of students choose their own numbers and make one set of number cards (and keys) for groups to use during the game.
Students in Mrs. Rogers algebra class are learning basic statistics. They first learn to determine mean, median, and mode by traditional methods. Then students are taught how to enter data into Excel to determine the same information. Several students do not have Excel available at home. How should Mrs. Rogers accommodate these students?
Give time in class to complete assignments that require Excel.
Ms. Neuhaus is teaching her class about nets. As an introductory activity she has all students build three dimensional figures from nets of a cube, a tetrahedron, a cone, a cylinder, and a rectangular prism. She finds that this takes three instructional periods. How could she improve her approach?
Group students in groups of five and have each member make one net and then compare.
Mr. Potter wants his students to learn about different types of angles such as corresponding angles, alternate interior angles, vertical angles, and corresponding angles. Which activity below is likely to ensure most students are able to understand how to identify them in a diagram?
Have students create a diagram where they trace the various angle types in different color markers and define them. This will help students identify angle pairs on a diagram.
A teacher is teaching students to graph linear equations. He wants students to use the graphing calculator to check their answers. He gives students graphing calculators and begins teaching how to graph linear equations. He finds that many students are unable to graph equations without the calculator. How can he improve his instruction?
In future years, teach students how to graph using pencil and paper before giving out graphing calculators. Calculators are meant to be a tool and it is essential that students can graph by paper and pencil methods.
Use the multiplication problem below to answer the question that follows. 13⋅12=(10⋅12)+(3⋅12)=120+36=15613⋅12=(10⋅12)+(3⋅12)=120+36=156 A student in a mathematics class wants to know whether the method above for multiplying integers is correct. Of the following, which is the best teacher response to the student?
It is a correct method that works every time. The method works every time, and is an example of using the distributive property to make larger numbers easier to multiply.
When planning a unit on quadratic systems, a teacher includes problems in which a student has to solve two linear equations with two unknowns, a topic covered earlier in the year. This intentional revisiting of solving systems of linear equations shows that the teacher understands which of the following concepts?
It is important to plan mathematical instruction as a series of interconnected concepts and procedures. By spiraling old content into new, the teacher connects previous learning to new applications.
Tactile Methods
Lessons using materials for students to touch and handle. Encouraging students to take notes, use study sheets, build dioramas or models
Visual Methods
Lessons using materials for students to view. maps, images, political cartoons, multimedia presentations and graphs
A mathematics teacher is beginning a unit on multiplication of fractions. Which of the following is the least appropriate way to model this process?
Model the process with base-10 blocks. Base-10 blocks are not appropriate for multiplication of common fractions, although they are excellent manipulatives for modeling concepts such as addition/subtraction of positive integers and decimals and multiplication of integers and decimals.
Maria has recently moved from Mexico City to the U.S. She is a secondary student who speaks little English, but who came from her school in Mexico City with excellent grades. Which of the following would be the most appropriate accommodation for Maria's math teacher to use with Maria?
Pair Maria with another student who speaks Spanish, to clarify instructions in Spanish as needed. If possible, pairing Maria with a student who speaks Spanish would be an excellent strategy.
Which of the following cannot form a regular tessellation?
Pentagon
Mrs. Herschend decided not to give a test about ratios and instead had her students do a project to display their knowledge. She has decided that she will do this for every unit going forward. What is the main disadvantage to this approach?
Students need to practice test taking skills periodically. For standarized testing
Graphing Calculator
a device that is capable of solving advanced mathematical equations, plotting graphs and performing other tasks with variables Graphing calculators can be used for a variety of purposes such as graphing equations or solving systems of equations
If a plane that is neither parallel nor perpendicular to the base B passes through the cube, which of the following could be the shape of the intersection of the plane and the cube?
a parallelogram with diagonal greater than the diagonal of B
Think-Aloud
a teaching strategy in which a teacher states his/her thoughts aloud to demonstrate how the students should go about solving a problem or understanding a text Math teachers model thinking by reading a problem aloud and verbalizing figuring out what it is asking what needs to be done. Language arts teachers ask themselves questions about the text as they read aloud.
Compass
a tool used to draw circles or arcs
Protractor
a tool used to measure angles
Bar Graph
a visual representation of data which compares values in different categories the number of people who prefer each genre of movie
Line Graph
a visual representation of data which shows change over time or in response to a manipulated variable
Ms. Mueller uses pizza to introduce a topic on multiplying fractions. Images of a pizza are taped to pie-shaped sections of cardboard to represent various fractions. Students can "cut" the fractional pizza slices into halves, fourths, and eighths to show the effects of fraction multiplication in answer to questions such as "What is one eighth of one half?" When finished, the students draw their own "pizza pieces" to represent similar problems. Which of the following best describes the instructional strategies demonstrated by the teacher with this activity?
development of mathematical instruction that transitions between concrete, symbolic, and abstract representations Ms. Mueller is using a concrete example of "pizza parts" that she can follow with symbolic and abstract representations.
The teacher provides a word problem for her students: Sandra is making sandwiches for her family's camping trip. She has 72 slices of turkey, 48 slices of cheese, and 96 pieces of lettuce. What is the greatest number of sandwiches she can make if each sandwich has the same filling of turkey, cheese, and lettuce? Which of the following mathematical concepts is she most likely teaching in this lesson?
greatest common factor
Homogeneous Group
group comprised of individuals working on the same level A small group of students reading a book together on the same reading level is a homogeneous group.
Mr. Fischer, a bilingual teacher, teaches a mathematics class composed of native English speakers and English language learners (ELLs). He has introduced a new topic with new vocabulary words in which he presented the vocabulary words with several examples. Which of the following strategies should Mr. Fischer use next to check each student's understanding of the vocabulary words?
having students write a definition for each term in their own words in their native language It is best to have the students construct their own definition in their native language so Mr. Fischer can assess their knowledge of the vocabulary words.
A math teacher plans her instructional delivery method on resolving the difficulty students have distinguishing between mode and median. She plans to have students first work alone calculating the mode and median of sets of performance results from the school track team. Next, her students will work in groups of 2 or 3 to discuss and interpret their results, and record a summary of the significance of the results on whiteboards. Finally, the groups will present their summaries to the class, along with a teacher-led discussion of the findings. By planning such an activity, the teacher demonstrates that she understands:
how to apply a variety of instructional delivery methods that can help students develop their mathematical thinking. The teacher's plans show she understands how to use individual, small-group, and large-group instruction methods to help students develop their mathematical thinking.
Teachers in the Leeman school district would like to increase their students' mathematics proficiency by emphasizing algebraic thinking, problem-solving, and communication within middle school math classes. Which of the following would best accomplish this goal?
implementing a standards-based program on higher-order thinking Standards-based programs focus on deep understanding and content mastery. They emphasize the importance of the student's thought process and problem-solving skills.
The mathematics teacher and art teacher work together to create an interdisciplinary lesson using tessellations, which are basic geometric shapes set to a repeating pattern. The students cover a large piece of poster board with the patterns they create. Which of the following mathematical concepts is most closely reflected in this activity?
infinity
A teacher wants to use concrete manipulatives to help her students learn about areas of regular polygons. Which of the following is the most suitable manipulative?
magnetic tiles Magnetic tiles can show how various polygons are composed of smaller regular polygons.
A teacher wants to introduce a lesson on probability and simulations to her students. Which manipulative would not be good to use?
ruler A ruler is not a manipulative used for probability simulations.
Which of the following is NOT considered a benefit of cash?
security
As an exploratory activity within a unit on writing the equations of patterns, a teacher plans to have students use colored tiles to build the terms of the sequence shown. Which of the following is the best instructional delivery method for this topic?
structured small group instruction In a small group, every student can handle the manipulatives yet still collaborate with peers to make sense of the task and discuss each other's questions.
At the end of a lesson on factoring, Ms. Wilson gave her class an exit ticket. After she reviewed the responses on the exit ticket, Ms. Wilson realized that many of her students were still struggling with the concept of factoring. Which of the following strategies would be best for Ms. Wilson to use in her next lesson on factoring to help the students solidify their conceptual understanding of factoring?
using manipulatives to show factoring as the reverse, or un-doing, of distribution This activity uses concrete manipulatives to demonstrate the concept of factoring. Students can use prior knowledge of distribution to make connections to factoring.
Verbal Representation
word problems and verbal descriptions of how to solve a problem or what the solution means "We know that Sam gets $10 each week for allowance, so let's make that a constant. And we know that Sam wants to save $150 to buy a new bicycle, so that's a constant, too. But, what we don't know is how long does Sam need to save - let's make that X. So the equation is 10x = 150."
