Math Examples

Question 1

A 6th grade teacher is teaching students how to solve the following equations:
a.) 2x + 4 = 12
b.) 10 = 3x – 5
c.) (x +2) = 30
The teacher wants to form an assessment to assess the students’ understanding. Which equation should the teacher use?

A). x + 5 = 10 – x
B). 2(x – 4) = 20
C). -2x + 2 = 10
D). 10 – 1.5x = 20

Answer 1

21.) B

The objective of the lesson is to solve two-step equation. The assessment must align with the objective and the equation provided in choice B matches the equations student 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 student to divide by a negative coefficient and that has not been introduced in the above equations. Choice D involves operations with decimals and that, too, has not been introduced.

Question 2

Students are creating histograms after analyzing the high temperatures from July over several years. How can the teacher incorporate an observational assessment into this lesson?

A). Have students complete a quiz at the end of the lesson where they answer simple questions about the characteristics of histograms.
B). Create a card for each student on which to take short narrative style notes using a list of specific content objectives to take notes about student understanding while circulating through the class. Then use that data to modify the lesson.
C). Conduct a class discussion where discourse is promoted and questions are asked about histograms and what children learned. Record highlights of this discussion.
D). Ask each student to submit a completed histogram at the end of class, and evaluate it with a rubric.

Answer 2

B.) 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 records using a checklist or other record system. .

Question 3

23.) 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, and they are provided with Base Ten Blocks to identify a solution. How should a teacher use a rubric to assess the progress of the students?

A). Use a 4, 3, 2, 1 scoring system and equate this to A, B, C, D letter grades to assign grades
B). Use an alternate assessment type since this task is not performance based
C). Explain how the rubric will be used for evaluation before students begin the problem
D). Have students complete this activity in their learning journals

Answer 3

C.) 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 – comparing work to criteria or 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.

Question 4

24.) How can a teacher use a mathematical connection to teach a unit about Halloween? Choose 2.

A). Students can compare and contrast how Halloween is celebrated around the world
B). Students can estimate how many seeds are in a pumpkin and then make conjectures based on mass
C). A teacher can collaborate with the art teacher and have student decorate a pumpkin in class
D). Students can find the mass of several pumpkins arrange them in order

Answer 4

B and D.) The term integration literally means “to combine into a whole.” Thus, when integrating curricula, the emphasis in on a comprehensive understanding of a “whole,” rather than many unrelated “parts.” With Constructivism, teachers and students are working together to build an education based upon what students’ experiences are and what they know, so that learning becomes meaningful. A math to math connection is formed when a teacher relates a previously learned math concept to a newly learned math concept.

Question 5

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). Use manipulatives to demonstrate fraction multiplication by whole numbers
B). Create several application problems to provide context involving the multiplication of fractions by whole numbers
C). Demonstrate the relationship between multiplying fractions by whole numbers and multiplying decimals by whole numbers
D). Have students research and journal four applications of multiplying fractions by whole numbers

Answer 5

D.) In this question the teacher is attempting to form a math to reading connection. This means the teacher must relate a math concept to a reading skill. In this question, the teacher is having student research and write about a mathematical skill.

Question 6

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 complete a homework task on adding fractions
B). Students complete a quiz on adding fractions
C). The students use a fraction model to help solve problems
D). The students use a recipe with ½ cup of flour, ¼ cup of milk, and 1/3 cup of butter to determine the total amount of ingredients

Answer 6

D.) Relating curriculum to outside events helps contextualize the mathematical concepts. When teacher attempt to make a mathematical connection outside of the classroom they are trying to relate the math concepts to real life. In this example, student are making a connection to a recipe and using math to determine a new portion size,

Question 7

Which 2 activities show a connection to context outside of the classroom?

A). Students use manipulatives to solve fraction problems
B). Students compare oil and gas prices from around the world and create diagrams
C). Student practice multiplication facts and write about the process in their math journals
D). Students explore how certain architectural designs influence the stability of a bridge or skyscraper

Answer 7

B.) Relating curriculum to outside events helps contextualize the mathematical concepts. When teacher attempt to make a mathematical connection outside of the classroom they are trying to relate the math concepts to real life. Having student compare the oil prices from around the world and preparing graphs can help create a connection to the outside world.

D.) Allowing student to explore certain architectural design can be integrated into a geometry unit. Mathematics is needed to analyze and calculate structural problems in order to engineer a solution that will assure that a structure will remain standing and stable. The sizes and shapes of the elements of a design are possible to describe because of mathematical principles such as the Pythagorean Theorem.

Question 8

A second grade class is learning about the compare subtraction model. Their teacher decides to use the common experience of eating in the cafeteria as a context outside the mathematics curriculum. Which scenario exemplifies this subtraction model?

A). Six students are eating lunch together
B). Three more students are at lunch than at recess
C). Three students eating lunch are joined by two more students
D). Five students finish lunch and go out to recess

Answer 8

B.) 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.

Question 9

How can a teacher demonstrate a mathematical connection when teaching a unit on making circle graphs?

A). Allow students to create and conduct an experiment to gather data
B). Allow students to use compasses and protractors to create circle graphs
C). Demonstrate how ratios, parts to whole, fractions, and percents can be used to create a circle graph
D). Encourage classroom discourse and use think-alouds when teaching about circle graphs

Answer 9

C.) 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.

Question 10

A math teacher wants to work with a social studies teacher on integrating the content. How can a teacher apply a math connection?

A). By demonstrate the complexity of creating various graphs
B). By allowing students to use manipulatives when solving problems
C). By allowing students create a timeline of the Native American cultures
D). By exploring how ratios, fractions, and percents can be expressed in a circle graph

Answer 10

C). Integrating math content with other core areas can help students gain a better conceptual develop of certain topics. Here, the teacher is trying to show students that math can be used when building and constructing timelines.

Question 11

A teacher would like to collaborate with an art teacher to integrate a math connection. How can a math teacher incorporate a stained glass window into a math lesson?

A). By teaching concepts of symmetry
B). By teaching concepts of probability
C). By teaching concepts of addition of fractions
D). By teaching concepts of rational numbers

Answer 11

A). Integrating math content with other areas can help students gain a better conceptual develop of certain topics. A stained glass window often contains many different shapes which can be used to illustrate symmetry.

Question 12

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

Answer 12

B and C

Grid paper can be used to help student draw shapes and see relationships between the transformation. C.) Dynamic geometry programs allow students to create shapes on the computer screen and then manipulate and measure them by dragging vertices.

Question 13

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 ask Ally to explain and elaborate on her thinking.
B). The teacher should ask a student to explain why Ally’s formula is correct or incorrect.
C). The teacher should tell Ally she is correct.
D). The teacher should ask John to show Ally why her answer is wrong.

Answer 13

A

Teachers should serve as facilitator because student 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

Question 14

Which two tools/manipulatives can be used to teach a lesson on fractions?
A). Cuisenaire rods
B). Graph paper
C). A protractor
D). Dice
E). Fraction Circles

Answer 14

A and E
Cuisenaire rods are mathematics learning aids for students that provide a hands-on elementary school way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors.

E.) Fraction circles are a set of nine circles of various colours. Each circle is broken into equal fractional parts and uses the same-sized whole.

Question 15

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). Conduct a class discussion where discourse is promoted and questions are asked about histograms and what children learned, and record highlights of the discussion.
B). 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.
C). Have students complete a quiz at the end of the lesson where they answer simple questions about the characteristics of histograms.
D). Ask each student to submit a completed histogram at the end of class and evaluate it with a rubric.

Answer 15

B

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.

Question 16

Choose two of the following 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.

Answer 16

C and E

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

Question 17

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). Before the activity, review the properties of 3-D shapes.
B). As you watch students working on the task, stop them if they are doing something wrong.
C).Give students step-by-step instructions for the activity.
D). Encourage students to collaborate and discuss findings and strategies once the activity is complete.

Answer 17

A and D

Before students attempt to build a conceptual understanding, it is important to activate the prior knowledge or schema of students’. In mathematics, students can use schemas to organize information and relate new information to previously learned material. 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

Question 18

A teacher is demonstrating probability and would like students to compare the probability of rolling a dice and flipping a coin. Which two statements below represent a conceptual understanding of probability?

A). Flipping a tails and then a head on a fair coin is just as likely as rolling a one and then a six on a fair six-sided dice.
B). Flipping a tails and then a heads is just as likely as rolling an even number and then rolling an odd number on a fair six-sided dice.
C). Rolling a two, four, or five on a fair six-sided dice is less likely than flipping a head and then a tail on a two sided coin.
D). Flipping a tail on a two sided coin is less likely than than rolling a one, two, three, or four on a fair six sided dice.

Answer 18

Student 2: Flipping a tails and then a heads is just as likely as rolling an even number and then rolling an odd number on a fair six-sided dice.
Writing this narrative out would look like this: 1/2 x 1/2 = 3/6 x 3/6
When you see an AND, multiply the events together. This student is correct because those two outcomes are just as likely.

Student 4: Flipping a tail on a two sided coin is less likely than than rolling a one, two, three, or four on a fair six sided dice.
Writing this narrative out would look like this: 1/2 < 1/6 + 1/6 + 1/6 + 1/6. This student is correct because 1/2 is less likely than 4/6.

Question 19

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 and the standard is aligned.
B). The activity is effective at showing students how to add and subtract integers but 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 not effective at showing students how to add and subtract integers and the standard is not aligned.

Answer 19

B

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

Question 20

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). Bar graph
B). Line graph
C). Stem and leaf
D). Histogram

Answer 20

B

A line graph is used to show change over time. We have units of time in minutes

Question 21

A survey was taken on how many years people spent in school beyond high school. What type of graph should be used to represent this data?

A). Bar graph
B). Stem-Leaf plot
C). Histogram
D). Circle graph

Answer 21

C

A histogram uses ranges of continuous data on the x-axis. Once the data is collected, we can create ranges of data that will appear on the x-axis.