Mathematics 6–12
For Testing until 12/31/2023

Test Format and Sample Questions

The Mathematics 6– to 12 subject area test consists of approximately 50 multiple-choice questions. You will have two and one-half hours to complete the test.

Each question will contain four response options, and you will indicate your answer by selecting 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
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 2
Direct Question Choose the response option that best answers the question.	Question 3
Sentence Completion Select the response option that best completes the sentence.	Question 4
Command Select the best response option.	Question 5
Word Problem Apply mathematical principles to solve a real-world problem and choose the best response option.	Question 16
Selection Choose the response option that contains all the elements needed to correctly answer the question.	Question 25

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 algebra

DIRECTIONS: Read the question and select the best response.

Which of the following is a solution of the given equation?

3(n + 2) − 4(n − 1) = n + 133 times n plus 2 minus 4 times n minus1 equals n + 13

Choose an answer

1. – ¹¹/₂ negative 11 over 2
2. – 3 negative 3
3. – ³/₂ negative 3 over 2
4. 2

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 1—Knowledge of algebra

DIRECTIONS: Read the question and select the best response.

Which of the following is the slope of the given graph?

A graphic of a Cartesian plane superimposed on graph paper is shown. The Y axis has labels at negative ten, negative five, five, and ten. The X axis has labels at negative ten, negative five, five, and ten. A line sloping downward from left to right is indicated, and it goes through point negative two comma eight and through point two comma negative two.

Choose an answer

1. – ⁵/₂ negative 5 over 2
2. – ³/₂ negative 3 over 2
3. – ²/₅ negative 2 over 5
4. ³/₂ 3 over 2

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 1—Knowledge of algebra

DIRECTIONS: Read the question and select the best response.

What is the solution set of the given equation?

0 = 2x + 7 + 5 zero equals the square root of 2 x plus seven plus 5

Choose an answer

1. {9}open bracket 9 close bracket
2. {–9}open bracket negative 0 close bracket
3. {–16}open bracket negative 16 close bracket
4. ∅ empty 

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 2—Knowledge of advanced algebra

DIRECTIONS: Read the question and select the best response.

The numbers represented in the given set

√64 , ²⁵/₉ , √81 , √1 the square root of 64, the square root of 25 over 9, the square root of 81, the square root of 1

can all be classified as

Choose an answer

1. whole numbers.
2. integers.
3. rational numbers.
4. irrational numbers.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 2—Knowledge of advanced algebra

DIRECTIONS: Read the question and select the best response.

Expand the given expression.

(q − 2r)³ open parentheses q minus 2 r close parentheses cubed

Choose an answer

1. q³ − 3q²r + 3qr² − 8r³q cubed minus 3 q squared r plus 3 q r squared minus 8 r cubed
2. q³ − 6q²r − 6qr² − 8r³ q cubed minus 6 q squared r minus 5 q r squared minus 8 r cubed
3. q³ − 2q²r + 4qr² − 8r³q cubed minus 2 q cubed r plus 4 q r squared minus 8 r cubed
4. q³ − 6q²r + 12qr² − 8r³q cubed minus 6 q squared r plus 12 q r squared minus 8 r cubed

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 2—Knowledge of advanced algebra

DIRECTIONS: Read the question and select the best response.

Find the 16^thsixteenth term in the geometric sequence 1024, –512, 256, …

Choose an answer

1. – ¹/₆₄ negative 1 over 64
2. – ¹/₃₂ negative 1 over 32
3. ¹/₆₄ 1 over 64
4. ¹/₃₂ 1 over 32

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 2—Knowledge of advanced algebra

DIRECTIONS: Read the question and select the best response.

What is the product of the given matrices?

There are two matrices side by side. The left matrix is two by three, with entries as follows: In order in the first row is three, negative two, and one. In order in the second row is zero, four, and negative five. The right matrix is three by two, with entries as follows: In order in the first row is one and zero, In order in the second row is negative two and three. In order in the third row is three and four.

Choose an answer

1. This is a two by two matrix, with entries as follows. In order in the first row is two and negative two. In order in the second row is negative twenty-three and negative eight.
2. This is a two by two matrix, with entries as follows. In order in the first row is ten and negative two. In order in the second row is negative twenty-three and negative eight.
3. This is a two by two matrix, with entries as follows. In order in the first row is ten and one. In order in the second row is negative seven and thirty two.
4. This is a two by two matrix, with entries as follows. In order in the first row is two and negative seven. In order in the second row is twenty-three and thirty-two.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 3—Knowledge of functions

DIRECTIONS: Read the question and select the best response.

Which of the following specifies an even function?

Choose an answer

1. {(x,y): y = –x³}open bracket open parentheses x comma y close parentheses colon y equals negative x cubed close bracket
2. {(x,y): y = (x − 1)²}open bracket open parentheses x comma y close parentheses colon y equals open parentheses x minus 1 close parentheses squared close bracket
3. {(x,y): y = x² − 1 }open bracket open parentheses x comma y close parentheses colon y equals the square root of x squared minus 1 close bracket
4. {(x,y): y = ¹_x}open bracket open parentheses x comma y close parentheses colon y equals 1 over x close bracket

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 3—Knowledge of functions

DIRECTIONS: Read the question and select the best response.

Which equation describes the translation of the graph of the given parabola so that its vertex is at (3, –4)?

A graphic of a Cartesian plane with vertical axis Y and horizontal axis X is shown. The curve on the plane is an upward opening parabola with vertex at the origin. The graph is labeled Y equals X squared.

Choose an answer

1. y + 4 = (x − 3)²y plus 4 equals open parentheses x minus 3 close parentheses squared
2. y − 4 = (x + 3)²y minus 4 equals open parentheses x plus 3 close parentheses squared
3. y + 3 = (x − 4)²y plus 3 equals open parentheses x minus 4 close parentheses squared
4. y − 3 = (x + 4)²y minus 3 equals open parentheses x plus 4 close parentheses squared

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 4—Knowledge of geometry

DIRECTIONS: Read the question and select the best response.

Two similar right circular cylinders have a scale factor of 2:3. The larger cylinder has a surface area of 252π pi square feet. Find the surface area of the smaller right circular cylinder.

Choose an answer

1. 112 ft²feet squared
2. 168 ft²feet squared
3. 112πpi ft²feet squared
4. 567πpi ft²feet squared

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 4—Knowledge of geometry

DIRECTIONS: Read the question and select the best response.

A right triangle has legs of length 5 and x +plus 2 and a hypotenuse of length x +plus 3. What is the perimeter of the triangle?

Choose an answer

1. 12
2. 25
3. 30
4. 34

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 4—Knowledge of geometry

DIRECTIONS: Read the question and select the best response.

Find the measure of angle y in the given figure.

A graphic of an irregular convex hexagon with the measures of some interior and some exterior angles is shown. Beginning in the upper left area and travelling clockwise around vertex by vertex and side by side, the angles are labeled as follows: First there is an exterior angle of forty degrees against the first side. At the next end of this first side is an interior angle of ninety-five degrees. At the third vertex is an exterior angle of eighty degrees against the third side. At the next end of the third side is an interior angle of one hundred sixty degrees. At the fourth vertex there is an exterior angle as measured against the third side (and not against the fourth side) of seventy degrees. At the fifth vertex is an exterior angle as measured against the fourth side (and not against the fifth side) of measure Y.

Choose an answer

1. 40°degrees
2. 65°degrees
3. 70°degrees
4. 80°degrees

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 4—Knowledge of geometry

DIRECTIONS: Read the question and select the best response.

Which of the following is an example of the construction of two parallel lines?

Choose an answer

1. A. A drawing of two parallel horizontal lines, cut by a transversal is shown. The point at which the transversal intersects with the top line has two angles, opposite each other, labeled X below the line and Y above the line.
2. B. A drawing of two parallel horizontal lines, cut by a transversal is shown. The point at which the transversal intersects with the top line has the obtuse angle above the line labeled as X, and the point at which the transversal intersects the bottom line has the acute angle above the line labeled as Y.
3. C. A drawing of two parallel horizontal lines, cut by a transversal is shown. The point at which the transversal intersects with the top line has the acute angle above the line labeled as X, and the point at which the transversal intersects the bottom line has the acute angle above the line labeled as Y.
4. D. A drawing of two parallel horizontal lines, cut by a transversal is shown. The point at which the transversal intersects with the top line has the acute angle above the line labeled as X, and the point at which the transversal intersects the bottom line has the obtuse angle below the line labeled as Y.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 5—Knowledge of coordinate geometry

DIRECTIONS: Read the question and select the best response.

Identify the equation of a circle with center (4, –2)open parentheses 4 comma negative 2 close parentheses and a radius of 6 units.

Choose an answer

1. (x + 4)² + (y − 2)² = 6open parentheses x plus 4 close parentheses squared plus open parentheses y minus 2 close parentheses squared equals 6
2. (x + 4)² + (y − 2)² = 36open parentheses x plus 4 close parentheses squared plus open parentheses y minus 2 close parentheses squared equals 36
3. (x − 4)² + (y + 2)² = 36open parentheses x minus 4 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 36
4. (x − 4)² + (y + 2)² = 6open parentheses x minus 4 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 6

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 5—Knowledge of coordinate geometry

DIRECTIONS: Read the question and select the best response.

M is the midpoint of start vinculum AB end vinculum. If the coordinates of A are (–4, 6)open parentheses negative 4 comma 6 close parentheses and the coordinates of M are (2, –2)open parentheses 2 command negative 2 close parentheses, what are the coordinates of B?

Choose an answer

1. (–2, 4)open parentheses negative 2 comma 4 close parentheses
2. (–1, 2)open parentheses negative 1 comma 2 close parentheses
3. (6, –8)open parentheses 6 comma negative 8 close parentheses
4. (8, –10)open parentheses 8 comma negative 10 close parentheses

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 6—Knowledge of trigonometry

DIRECTIONS: Read the question and select the best response.

Suppose that an object attached to a coiled spring is pulled down 6 inches from its equilibrium position and released. It takes 4 seconds for it to complete one oscillation. Which of the following equations models the position of the object at time t?

Choose an answer

1. s open parentheses t close parentheses equals negative 6 cosine open parentheses pi over 2 t close parentheses
2. s open parentheses t close parentheses equals 6 cosine open parentheses pi over 2 t close parentheses
3. s open parentheses t close parentheses equals negative 6 sine open parentheses pi over 2 t close parentheses
4. s open parentheses t close parentheses equals 6 sine open parentheses pi over 2 t close parentheses

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 6—Knowledge of trigonometry

DIRECTIONS: Read the question and select the best response.

Find the equation represented by the given graph.

A graphic of a Cartesian plane with vertical axis Y and horizontal axis X is shown. The Y axis has labels at negative five and positive five, with notches at all the integers in-between. The X axis has labels at pi over two, pi, three pi over two, and two pi. A smooth curve begins at four on the Y axis, travels down through pi over two on the X axis, continues traveling down to a minimum point the same distance below the axis as the beginning point was above the axis, then travels back up through three pi over two on the X axis, up to a high point above two pi (the same height as the initial point).

Choose an answer

1. y equals sine open parentheses 4 x plus pi over 2 close parentheses
2. y equals 4 sine x minus pi over 2
3. y equals 4 sine open parentheses x plus pi over 2 close parentheses
4. y equals sine 4 x minus pi over 2

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 7—Knowledge of statistics and probability

DIRECTIONS: Read the question and select the best response.

What is the median of the given list of data points?

25, –negative 2, –negative 3, 10, –negative 10, 25, –negative 4, –negative 4, 26, 11, –negative 4, –negative 20, 12, 8

Choose an answer

1. –negative 4
2. 3
3. 5
4. 6

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 7—Knowledge of statistics and probability

DIRECTIONS: Read the question and select the best response.

Which of the following could be inferred from the given box plot?

Figure: There is a number line, going from left to right, starting with the number 800 and going through the number 2400, with increments marked at 1000, 1200, 1400, 1600, 1800, 2000, and 2200. Below this number line is the title “Male (A1:A8) versus Female (B1:B8) College Admission Test Scores.” Above the number line there are two box plots. The topmost is labeled A1:A8, and the leftmost whisker of the box plot begins between 1000 and 1200 on the number line, and the leftmost boundary of the box is at 1400. There is a center line through the box at 1600, and the rightmost boundary of the box is between 1800 and 2000. The rightmost whisker of the box plot ends between 2000 and 2200. The bottom box is labeled B1:B8, and the leftmost whisker of the box plot begins between 1200 and 1400 on the number line, and the leftmost boundary of the box is between 1400 and 1600. There is a center line through the box at 1600, and the rightmost boundary of the box is between 1600 and 1800. The rightmost whisker of the box plot ends between 2200 and 2400.

Male (A1:A8) vs. Female (B1:B8) College Admission Test Scores

Choose an answer

1. The males have outscored the females for this set of scores.
2. The males have more variance in their scores.
3. The lowest score for this set of scores belongs to a female tester.
4. The females have more variance in their scores.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 7—Knowledge of statistics and probability

DIRECTIONS: Read the question and select the best response.

At a furniture store, the selection of couches consists of three different shapes (A, B, and C), two sizes (small and large), and five colors (red, orange, yellow, green, blue). When ordering a couch, a customer must choose one shape, one size, and one color. If each of the possible choices is equally likely, what is the probability that a customer will choose a small green couch in shape B?

Choose an answer

1. ¹/₁₈₀1 over 180
2. ¹/₃₀1 over 30
3. ¹/₁₀1 over 10
4. ¹/₅1 over 5

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 8—Knowledge of calculus

DIRECTIONS: Read the question and select the best response.

Evaluate the integral from 3 to negative 1 of open parenthesis x cubed plus 1 close parenthesis d x.

Choose an answer

1. 20
2. 22 ¹/₂ 22 and a half
3. 23 ¹/₄ 23 and a fourth
4. 24

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: D.

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Competency 8—Knowledge of calculus

DIRECTIONS: Read the question and select the best response.

The graph of the derivative of a function is given. Identify the graph of the function.

Figure: A graphic of a Cartesian plane is shown. The Y axis has labels at negative one point five, negative one, negative point five, zero, point five, one, and one point five. The X axis has labels at negative three pi over two, negative pi, negative pi over two, zero, and pi over two. The derivative of a function is represented by a wave that is represented as follows from left to right: beginning at negative two pi, negative one below the X axis before intersecting the X axis at negative three pi over two, above the X axis going up until negative pi, one, intersecting the X axis again at negative pi over two, below the X axis and intersecting the Y axis at negative one, going up and intersecting the X axis again at pi over two and then above the X axis to pi, one.

Choose an answer

1. A: A graphic of a Cartesian plane is shown. The Y axis has labels at negative one point five, negative one, negative point five, zero, point five, one, and one point five. The X axis has labels at negative three pi over two, negative pi, negative pi over two, zero, and pi over two. A wave is presented that begins on the X axis at negative two pi and goes below the X axis before intersecting the X axis at negative pi, above the X axis until intersecting the X axis again at the origin, and then below the X axis before returning to the X axis at pi. The high point of the graph is negative pi over two, one, and the low points are negative three pi over two, negative one, and pi over two, negative one.
2. B: A graphic of a Cartesian plane is shown. The Y axis has labels at negative one point five, negative one, negative point five, zero, point five, one, and one point five. The X axis has labels at negative three pi over two, negative pi, negative pi over two, zero, and pi over two. A wave is presented that begins at the X axis at negative two pi and then goes above the X axis before intersecting the X axis at negative pi , below the X axis until intersecting the X axis again at the origin, and then above the X axis before returning to the X axis at pi. The high points of the graph are negative three pi over two, one, and pi over two, one, and the low point is negative pi over two, one.
3. C: A graphic of a Cartesian plane is shown. The Y axis has labels at negative one point five, negative one, negative point five, zero, point five, one, and one point five. The X axis has labels at negative three pi over two, negative pi, negative pi over two, zero, and pi over two. A wave is presented that begins above the X axis at negative two pi, one before intersecting the X axis at negative three pi over two, below the X axis until intersecting the X axis again at negative pi over two, above the X axis and intersecting the Y axis at one before intersecting the X axis again at pi over two and then below the X axis to pi, negative one. The high points of the graph are negative two pi, one, and zero, one, and the low points of the graph are negative pi, negative one, and pi, negative one.
4. D: A graphic of a Cartesian plane is shown. The Y axis has labels at negative one point five, negative one, negative point five, zero, point five, one, and one point five. The X axis has labels at negative three pi over two, negative pi, negative pi over two, zero, and pi over two. A wave is presented that begins below the X axis at negative two pi, negative one before intersecting the X axis at negative three pi over two, above the X axis until intersecting the X axis again at negative pi over two, below the X axis and intersecting the Y axis at negative one before intersecting the X axis again at pi over two and then above the X axis to pi, one. The high points of the graph are negative pi, one and pi, one, and the low points of the graph are negative two pi, negative one, and zero, negative one.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 8—Knowledge of calculus

DIRECTIONS: Read the question and select the best response.

Find the derivative of the function y = ln x.

Choose an answer

1. e^–xe to the power of negative x
2. x^–1x to the power of negative one
3. x
4. e^xe to the power of x

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 9—Knowledge of mathematical reasoning

DIRECTIONS: Read the question and select the best response.

Suppose a and b are odd prime numbers and c and d are even composite numbers. Which of the following statements would always be true?

Choose an answer

1. a +plus c is a prime number
2. c +plus d is a prime number
3. a +plus b is a composite number
4. b +plus d is a composite number

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 9—Knowledge of mathematical reasoning

DIRECTIONS: Read the question and select the best response.

Which of the following is the inverse of this statement?

If you are a pianist, then you are a musician.

Choose an answer

1. If you are a musician, then you are a pianist.
2. If you are not a pianist, then you are not a musician.
3. If you are not a musician, then you are not a pianist.
4. If you are a pianist, then you are not a musician.

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 10—Knowledge of instruction and assessment

DIRECTIONS: Read the question and select the best response.

While checking student work during class, a teacher noticed the following examples on a student paper.

This is a sample of student work. The directions for the activity read Simplify Completely. Underneath there are two columns. The left column contains the problems and the right column contains the student answers. Problem one is the square root of two hundred. The given answer is two times the square root of ten. Problem two is the square root of seventy-five. The given answer is three times the square root of five. Problem three is the square root of fifty-two. The given answer is thirteen times the square root of two.

Which of the following is the most appropriate way to remediate the student's misconception?

Choose an answer

1. interviewing the student
2. telling the student to work the problems again
3. assigning a remediation worksheet for homework
4. providing the correct answers

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: A.

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Competency 10—Knowledge of instruction and assessment

DIRECTIONS: Read the question and select the best response.

Students are given a checklist to use for problem solving. This is a form of which type of assessment?

Choose an answer

1. performance assessment
2. formative assessment
3. objective assessment
4. summative assessment

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: B.

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Competency 10—Knowledge of instruction and assessment

DIRECTIONS: Read the question and select the best response.

Which of the following materials would be most appropriate for simulating random selection without replacement?

Choose an answer

1. balance scales
2. fraction bars
3. playing cards
4. number cubes

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 10—Knowledge of instruction and assessment

DIRECTIONS: Read the question and select the best response.

Which problem solving strategy would be most effective for solving the given problem?

The water level in a reservoir increased by 18%18 percent in January. In February, the level decreased by 2.5 feet. In March, the water level increased by 12%12 percent to a depth of 48.3 feet. What was the water level at the beginning of January?

Choose an answer

1. looking for a pattern
2. making an estimate
3. working backwards
4. solving a simpler problem

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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Competency 10—Knowledge of instruction and assessment

DIRECTIONS: Read the question and select the best response.

When assessing students’ knowledge of factoring, which of the following is an accommodation a teacher might use for an ESE student?

Choose an answer

1. having students work in a group on a worksheet
2. telling the student how to factor
3. giving the student extended time to complete the task
4. providing a take home test for the student

AnswerEnter to expand or collapse answer.Answer expanded

Correct Response: C.

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