Middle Grades Mathematics 5–9
For Testing until 12/31/2023

Test Format and Sample Questions

The Middle Grades Mathematics 5– to 9 subject area test consists of approximately 50 multiple-choice questions. You will have two and one-half hours to complete the test.

Each of the questions will contain four response options. You will choose the best response out of the available options, and indicate A, B, C, or D.

A scientific calculator, such as the TI-30 XIIS, TI-30 XS, or similar model, will be provided at the test site through September 30, 2023. Beginning October 1, 2023, the scientific calculator will be provided on-screen during the test. Examinees may not bring their own calculator.

The table below presents types of questions on the exam and directs you to examples of these formats among the sample items that follow.

Table of Question Formats

Type of Question	Sample Question
Word Problem Apply mathematical principles to solve a real-world problem and choose the best response option.	Question 2
Graphics Examine a question involving a number line, a geometric figure, graphs of lines or curves, a table, or a chart, and select the best response option.	Question 4
Scenario Examine a situation, problem, or case study. Then answer a question, make a diagnosis, or recommend a course of action by selecting the best response option.	Question 10
Direct Question Choose the response option that best answers the question.	Question 15
Command Select the best response option.	Question 20

Sample Questions

The following questions represent both the form and content of questions on the examination. These questions will acquaint you with the general format of the examination; however, these sample questions do not cover all of the competencies and skills that are tested and will only approximate the degree of examination difficulty.

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Competency 1—Knowledge of problem-solving and reasoning skills

DIRECTIONS: Read the question and select the best response.

Use the given number game to answer the following question.

Pick a number.
Add 5 to the number.
Next, multiply 4.
Subtract 12.
Divide by 4.

If x represents the starting number, and y represents the final number, which equation is equivalent to the number game?

Choose an answer

1. y = 2  y equals 2 
2. y = 2x  y equals 2 x 
3. y = x − 2  y equals x minus 2 
4. y = x + 2  y equals x plus 2 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 1—Knowledge of problem-solving and reasoning skills

DIRECTIONS: Read the question and select the best response.

John knows that he can mow a rectangular lawn that measures 20 yards by 40 yards in 2 hours. He estimates that it should take him about 4 hours to mow a rectangular lawn that measures 40 yards by 80 yards. Is John's estimate reasonable?

Choose an answer

1. Yes, because doubling the size of the field will double the amount of mowing time.
2. No, because John will be able to mow a longer distance without changing direction, resulting in a shorter mowing time.
3. Yes, because when the perimeter of the field is doubled, the total distance that the mower will need to travel is doubled.
4. No, because doubling the dimensions of the field will increase the area of the field by 4, resulting in a mowing time of about 8 hours.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 1—Knowledge of problem-solving and reasoning skills

DIRECTIONS: Read the question and select the best response.

In general, division can be defined by which equation, assuming b ≠ is not equal to 0?

Choose an answer

1. a ÷ b = c if and only if c • b = a  a divided by b equals c if and only if c times b equals a 
2. a ÷ b = c if and only if c • a = b  a divided by b equals c if and only if c times a equals b 
3. a ÷ b = c if and only if c = a − b  a divided by b equals c if and only if c equals a minus b 
4. a ÷ b = c if and only if b = c ÷ a  a divided by b equals c if and only if b equals c divided by a 

AnswerEnter to expand or collapse answer. Answer expanded

Correct Response: A.

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Competency 2—Knowledge of mathematical manipulatives and models and instructional technology

DIRECTIONS: Read the question and select the best response.

A number cube is rolled 50 times with the results shown in the table.

Results of Number Cube Rolling Experiment

Outcome	1	2	3	4	5	6
Frequency	10	9	11	12	8	0

What is the experimental probability of rolling a 4?

Choose an answer

1. ⁴/₁₂ 4 over 12 
2. ⁴/₅ 4 over 5 
3. ¹²/₅₀ 12 over 50 
4. ¹⁰/₅₀ 10 over 50 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 2—Knowledge of mathematical manipulatives and models and instructional technology

DIRECTIONS: Read the question and select the best response.

A teacher is going to develop the concept of squaring a binomial expression. Which of the following models would be the most appropriate to use?

Choose an answer

1.  a drawing of a rectangle, followed by a plus sign, followed by a rectangle. Both rectangles have dimensions of height equal to "a" and width equal to "b." 
2.  a drawing of a rectangle, followed by a multiplication sign, followed by a rectangle. Both rectangles have dimensions of height equal to "a" and width equal to "b." 
3.  a drawing of one square partitioned into 4 inner quadrilaterals, by means of a single interior vertical line and a single interior horizontal line. In the upper left is a big square, and in the lower left is a small square. The outer square is labeled with a height of "a plus b" and a width of "a plus b." 
4.  a drawing of a rectangle, followed by a multiplication sign, followed by a second rectangle, followed by a plus sign, followed by a square. Both rectangles have dimensions of height equal to "a" and width equal to "b." The square has sides equal to "b." 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 3—Knowledge of assessment in mathematics

DIRECTIONS: Read the question and select the best response.

Which of the following is  start uppercase LEAST end uppercase  likely to be used as a formative assessment for teaching students about decimals?

Choose an answer

1. students taking a semester exam which includes concepts on decimals
2. students listing numbers between 3.4 and 3.5 as a warm-up activity at the board
3. students participating in individual student-teacher interviews on decimal concepts
4. students writing an explanation of why there are many values between two numbers

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 3—Knowledge of assessment in mathematics

DIRECTIONS: Read the question and select the best response.

In solving the linear equation 5x −minus 9 = 3x + 3, a student wrote 5x + 3x −minus 9 = 3x −minus 3x + 3 as the next step. Which of the following statements identifies the error the student made?

Choose an answer

1. The student added and subtracted the same amount on both sides of the equation.
2. The student multiplied the same amount on both sides of the equation.
3. The student subtracted the same amount on both sides of the equation.
4. The student added the same amount on both sides of the equation.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 3—Knowledge of assessment in mathematics

DIRECTIONS: Read the question and select the best response.

Identify the primary goal of formative assessments.

Choose an answer

1. ranking students
2. evaluating teachers
3. making instructional decisions
4. measuring student achievement

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 4—Knowledge of connections among mathematical concepts

DIRECTIONS: Read the question and select the best response.

Which of the following concepts would students need to understand before starting a lesson on simplifying fractions?

Choose an answer

1. least common multiple
2. order of operations
3. greatest common factor
4. solving proportions

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 4—Knowledge of connections among mathematical concepts

DIRECTIONS: Read the question and select the best response.

A student wants to purchase a wallpaper border for a bedroom. The border is sold in rolls 8 yards long and 10 inches wide. The room measures 12 feet by 13 feet. By performing the following operations, the student determines that 6 rolls of border are needed.

Equation: 2 open parenthesis 13 plus 12 close parenthesis over 8 equals 6.25.

Which statement best explains the student's mathematical error?

Choose an answer

1. The student rounded incorrectly and thought only 6 rolls were needed.
2. The student found the perimeter instead of the area of the room.
3. The student did not change all of the dimensions to the same units.
4. The student used an incorrect formula for perimeter.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 5—Knowledge of number sense, operations, and proportionality

DIRECTIONS: Read the question and select the best response.

Identify the proportion that would correctly solve the following problem.

Chris and Kelly will divide a profit of $400 400 dollars  in the ratio 5: to 3. If Chris receives the smaller amount, how much will Chris receive?

Choose an answer

1. ⁵/₈  =  ⁴⁰⁰/_x  5 over 8 equals 400 over x 
2. ³/₈  =  ^x/₄₀₀  3 over 8 equals x over 400 
3. ³/₅  =  ^x/₄₀₀  3 over 5 equals x over 400 
4. ³/₅  =  ⁴⁰⁰/_x  3 over 5 equals 400 over x 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 5—Knowledge of number sense, operations, and proportionality

DIRECTIONS: Read the question and select the best response.

The height TR of a tree may be measured by using similar triangles. A mirror is placed at point M so that the top of the tree is sighted in the mirror by a person standing at point P. The person's eye is at point E. Given the measurements shown in the diagram what is the length of TR?

A graphic is constructed as follows. There is a horizontal line segment labeled with left endpoint R and right endpoint P. A point on this line segment somewhere between the endpoints is labeled M. Line segment R-M has a measurement of 4.0 meters; line segment M-P has a measurement of 0.6 meters. A vertical line segment rises from point R ending at point T, so that line segment T-R is perpendicular to line segment R-P. A vertical line segment rises from point P ending at point E, so that line segment E-P is perpendicular to line segment R-P. Line segment EP has a measurement of 1.8 meters. A line segment connects T to M, forming right triangle R-T-M to the left, and a line segment connects M to E, forming right triangle M-E-P to the right.

Choose an answer

1. 5.2 m
2. 7.2 m
3. 9.0 m
4. 12.0 m

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 5—Knowledge of number sense, operations, and proportionality

DIRECTIONS: Read the question and select the best response.

A school has a total enrollment of 1720 students. One day 129 students were absent. What percent were absent?

Choose an answer

1. 0.075% percent 
2. 7.5% percent 
3. 12.9% percent 
4. 13.3% percent 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 6—Knowledge of foundations of algebra

DIRECTIONS: Read the question and select the best response.

Which of the following rules represents a function?

Choose an answer

1. {(x, y): y² = x²} left curly bracket left paren x comma y right paren colon y squared equals x squared right curly bracket
2. {(x, y): y = x²} left curly bracket left paren x comma y right paren colon y equals x squared right curly bracket
3. {(x, y): |y| = x²} left curly bracket left paren x comma y right paren colon the absolute value of y equals x squared right curly bracket
4. {(x, y): y² = x} left curly bracket left paren x comma y right paren colon y squared equals x right curly bracket

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 6—Knowledge of foundations of algebra

DIRECTIONS: Read the question and select the best response.

What is the expression 2^–3 × 2²2 to the power of negative 3 times two squared equivalent to?

Choose an answer

1. 4
2. 2
3. ¹/₂1 half
4. ¹/₄1 fourth

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 6—Knowledge of foundations of algebra

DIRECTIONS: Read the question and select the best response.

Which graph represents the following inequality?

5z +plus 10 >greater than 0

Choose an answer

1. A number line with increments marked at the integers from negative twenty to 0. An open circle is located at the increment two spaces to the left of 0, and a dark line beginning at this open circle travels to the right, ending in a boldface arrow pointing right, just to the right of 0.
2. A number line with increments marked at the integers from negative twenty to 0. An open circle is located at the increment two spaces to the left of 0, and a dark line beginning at this open circle travels to the left, ending in a boldface arrow pointing left at the increment one space to the right of negative twenty.
3. A number line with increments marked at the integers from negative twenty to 0. An open circle is located at negative fifteen, and a dark line beginning at this open circle travels to the right, ending in a boldface arrow pointing right at the increment one space to the left of 0.
4. A number line with increments marked at the integers from negative twenty to 0. An open circle is located at negative fifteen, and a dark line beginning at this open circle travels to the left, ending in a boldface arrow pointing left at the increment negative twenty.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 6—Knowledge of foundations of algebra

DIRECTIONS: Read the question and select the best response.

Which of the following lists of numbers is arranged in ascending order?

Choose an answer

1. √ 12 the square root of 12, 5, 6, √ 31 the square root of 31
2. 5, √ 12 the square root of 12, 6, √ 31 the square root of 31
3. √ 12 the square root of 12, 5, √ 31 the square root of 31, 6
4. 5, √ 12 the square root of 12, √ 31 the square root of 31, 6

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 7—Knowledge of algebraic thinking

DIRECTIONS: Read the question and select the best response.

The graph of the equation f(x) = 2^xf left paren x right paren is equal to 2 to the power of x is reflected over the x-axis and translated three units to the right to become g(x)g left parenx right paren, as depicted below.

The graph illustrates the scenario described in the question. It shows the graph of F of X equals two to the X power. It shows the graph of G of X as being reflected over the X axis and translated three units to the right.

What is the equation for the graph of g(x)g left paren x c right paren?

Choose an answer

1. g(x) = –2^{(x − 3)} g left paren x right paren equals negative 2 to the power of left paren x minus 3 right paren
2. g(x) = –2^x − 3 g left paren x right paren equals negative 2 to the power of x minus 3
3. g(x) = 2^{–(x − 3)} g left paren x right paren equals 2 to the power of negative left paren x minus 3 right paren
4. g(x) = 2^–x − 3 g left paren x right paren equals 2 to the power of negative x, minus 3

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 7—Knowledge of algebraic thinking

DIRECTIONS: Read the question and select the best response.

Two friends decide to join a gym. One friend joins a gym that charges a one time member fee of $5050 dollars and $2020 dollars for each month. The other friend joins a gym that charges a one time member fee of $8080 dollars and $1818 dollars for each month. After how many months will both friends have paid the same total amount?

Choose an answer

1. 15
2. 20
3. 25
4. 30

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 7—Knowledge of algebraic thinking

DIRECTIONS: Read the question and select the best response.

Solve.

9 = √ y − 4  + 7  9 equals the square root of y minus 4 plus 7 

Choose an answer

1. –negative8
2. 0
3. 4
4. 8

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 7—Knowledge of algebraic thinking

DIRECTIONS: Read the question and select the best response.

Simplify the given expression.

 open parentheses negative 3 x to the power of 2 n all over x to the power of 3 n times y close parentheses cubed 

Choose an answer

1. ^–27_xⁿy  negative 27 over x to the power of n times y 
2. ^–27_x³ⁿy³  negative 27 over x to the power of 3 n times y cubed 
3. ^–9_xⁿy  negative 9 over x to the power of n times y 
4. ^–9_x³ⁿy³  negative 9 over x to the power of 3 n times y cubed 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 8—Knowledge of data analysis, statistics, and probability

DIRECTIONS: Read the question and select the best response.

To the nearest tenth, find the mean weight of a group of pears with individual weights of 4.7, 4.0, 6.2, 6.5, 6.1, 6.5, 4.7, and 6.0 ounces.

Choose an answer

1. 4.7  ozounces
2. 5.6  ozounces
3. 6.1  ozounces
4. 6.2  ozounces

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 8—Knowledge of data analysis, statistics, and probability

DIRECTIONS: Read the question and select the best response.

Choose the most appropriate way to represent the given data to determine if the speed is constant.

Time (minutes)	Distance (meters)
0	0
1	13
2	26
3	39
4	52
5	65
6	78

Choose an answer

1. box-and-whisker plot
2. circle plot
3. line plot
4. double line plot

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 8—Knowledge of data analysis, statistics, and probability

DIRECTIONS: Read the question and select the best response.

In a random poll of shoppers at a mall, 32 out of 40 shoppers surveyed live in the same town where the mall is located. On average, 3250 shoppers visit the mall each day. Estimate the number of out-of-town shoppers that visit the mall each day.

Choose an answer

1. 100
2. 256
3. 320
4. 650

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 9—Knowledge of two-dimensional geometry

DIRECTIONS: Read the question and select the best response.

Use the diagram to answer the following question.

There is a triangle labelled A B C. Between A and B there is point D. Distance A D is the same as distance D B. Between A C there is point E. Distance A E is the same as distance E C. Between B C there is point F. Distance B F is the same as distance F C. D E F is the triangle formed by these points within A B C.

If AB = 3x – 2 and EF = x + 1, find AB.

Choose an answer

1. 1
2. 4
3. 7
4. 10

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 9—Knowledge of two-dimensional geometry

DIRECTIONS: Read the question and select the best response.

Two sides of a triangle measure 4 feet and 9 feet. What must be true of the length of the third side?

Choose an answer

1. It is greater than 4 ft but less than 9 ft.
2. It is greater than 5 ft but less than 9 ft.
3. It is greater than 4 ft but less than 13 ft.
4. It is greater than 5 ft but less than 13 ft.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 9—Knowledge of two-dimensional geometry

DIRECTIONS: Read the question and select the best response.

What is the distance from point S to point T?

A graphic of a Cartesian plane superimposed on graph paper is shown. The Y axis has labels at negative ten, negative 5, 5, and 10. The X axis, has labels at negative 10, negative 5, 5, and 10. Point T on the graph is located at 4 comma 5. Point S on the graph is located at negative 5 comma negative 2.

Choose an answer

1. 4
2. 10
3. √ 130 the square root of 130
4. √ 136 the square root of 136

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 9—Knowledge of two-dimensional geometry

DIRECTIONS: Read the question and select the best response.

The radius of the largest circle is 8 inches. Find the area of the shaded region.

There are two circles, both centered on point O. One circle is larger than the other. Both circles are bisected by a line that goes upwards from left to right. The length of the segment of the diameter line between the exterior circle and the interior circle is marked as being the same as the length of the segment from the interior circle to point O. There is also a ray extending from point O downwards left to right to the exterior circle. The angle between the ray and the diameter line that is pointing downwards is marked as sixty degrees. On the other side of the ray extending to the diameter line, the area between the interior and exterior circle is shaded.

Choose an answer

1. ^4π₃ in²  4 pi over 3 inches squared 
2. ^8π₃ in²  8 pi over 3 inches squared 
3. 4π in²  4 pi inches squared 
4. 16π in²  16 pi inches squared 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 10—Knowledge of measurement and spatial sense

DIRECTIONS: Read the question and select the best response.

The scale on a blueprint drawing is ¹/₄ 1 quarter inch =equals 1 foot. If on the blueprint a living room is 3 ³/₄ and 3 fourths inches wide, what is the actual width of the room?

Choose an answer

1. 3 ³/₄ and 3 and 3 fourths  ft
   
2. 4 ft
3. 12 ³/₄ and 3 fourths  ft
4. 15 ft

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 10—Knowledge of measurement and spatial sense

DIRECTIONS: Read the question and select the best response.

A gardener uses a piece of wire fencing 6 meters long to enclose a circular region for a garden. What is the radius of the garden?

Choose an answer

1. ³_π 3 over pi meters
2. ⁶_π 6 over pi meters
3. 9π 9 pi meters
4. 36π 36 pi meters

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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