Grade 8

Expressions, Equations, and Inequalities

In the 8th grade, instructional time focuses on developing skills in the following three areas: (1) formulating and reasoning about expressions, equations, and systems of equations; (2) the introduction of functions and understanding the basics of how to use functions to describe quantitative relationships; and (3) analyzing two- and three-dimensional space and figures to find congruence, similarity, as well as an understanding of how to apply the Pythagorean Theorem.

Below, you will find examples of the types of question and answer sets students who are operating at a Grade 8 math level in the target language should be able to understand, along with their corresponding NYS Next Generation Learning Standard.

Expressions, Equations, and Inequalities

Example 1: NGLS Standard NY-8.EE.1

Which expression is equivalent to open parentheses 15 to the third closed parentheses times open parentheses 15 to the negative seventh closed parentheses? A. 15 to the negative 21. B. negative 15 to the fourth. C. one over fifteen to the fourth. D. one over fifteen to the negative fourth

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This question is asking the student to work with radicals and integer exponents. To show mastery of this skill, the student must know and apply the properties of integer exponents to solve the equation.

Correct Answer: C

Reproduced from: NYS Grade 8 Test Item 14, 2022

Originally aligned to: CCSS Standard 8.EE.A.1

Example 2: NGLS Standard NY-8.EE.6

The graph of a line is shown on the coordinate plane below. A coordinate plane is shown with an x-axis ranging from negative 6 to 6 and a y-axis ranging from negative 6 to 6. A straight line is drawn on this plane, passing through the y-axis at approximately (0, 1) and the x-axis at approximately (-2, 0). The line also passes through points such as (2, 2) and (4, 3). Which equation represents the graph of the line? A. y = one half times x plus one. B. y = one half times x minus two. C. y = 2x plus 1. D. y = 2x minus 2.

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This question is asking the student to understand the connections between proportional relationships, lines, and linear equations. To show mastery of this skill, the student must derive the equation y = mx + b for the given line, which intercepts the vertical axis at b, in order to determine which equation is the correct choice.

Correct Answer: A

Reproduced from: NYS Grade 8 Test Item 33, 2023

Example 3: NGLS Standard NY-8.EE.8b

A line is graphed on the coordinate plane below. A standard coordinate plane is shown with x and y axes ranging from negative 5 to 5. A straight line is graphed, passing through the points (0, -4) and (4, -2). Line y = negative x + 2 will be graphed on the same coordinate plane to create a system of equations. What is the solution to that system of equations? A. (−2,4). B. (0,−4) C. (2, −4) D. (4,−2)

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This question is asking the student to analyze and solve linear equations. To show mastery of this skill, the student must solve systems of two linear equations in order to determine the appropriate answer choice.

Correct Answer: D

Reproduced from: NYS Grade 8 Test Item 11, 2021

Originally aligned to: CCSS Standard 8.EE.C.8b

Functions

Example 1: NGLS Standard NY-8.F.1

A set of ordered pairs is shown below. {(−3,3), (1,1), (4,2), (−1,−1), (−2,−1), (3,1), (−2,3)} Which ordered pair should be removed to make the set a function? A. (−3,3). B. (4,2). C. (−2,−1). D. (3,1)

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This question is asking the student to demonstrate understanding of the graph of a function as a set of ordered pairs consisting of an input and the corresponding output. To show mastery of this skill, the student must evaluate the given pairs to determine which one does not belong.

Correct Answer: C

Reproduced from: NYS Grade 8 Test Item 38, 2024

Example 2: NGLS Standard NY-8.F.3

Which equation represents a linear function? A. y = one half times x minus three. B. y = x squared plus five. C. y = x squared plus two x. D. y = one fifth times x cubed

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This question is asking the student to recognize examples of functions that are linear and non-linear. To show mastery of this skill, the student must determine which equation is an equivalent of y = mx + b.

Correct Answer: A

Reproduced from: NYS Grade 8 Test Item 20, 2024

Example 3: NGLS Standard NY-8.F.5

A function has the following properties: It is increasing and linear when the value of 𝑥 is between −5 and −3. It remains constant when the value of 𝑥 is between −3 and 0. It is decreasing and linear when the value of 𝑥 is between 0 and 3. It is increasing and nonlinear when the value of 𝑥 is between 3 and 5. Which graph best represents this function?Graph A: Shows a function that increases linearly from x=-5 to x=-3, remains constant from x=-3 to x=0, decreases linearly from x=0 to x=3, and then increases linearly from x=3 to x=5. Graph B: Shows a function that increases linearly from x=-5 to x=-3, remains constant from x=-3 to x=0, decreases linearly from x=0 to x=3, and then increases nonlinearly (curving upwards) from x=3 to x=5. Graph C: Shows a function that increases linearly from x=-5 to x=-3, then decreases linearly from x=-3 to x=0, remains constant from x=0 to x=3, and then increases nonlinearly from x=3 to x=5. Graph D: Shows a function that increases nonlinearly from x=-5 to x=-3, remains constant from x=-3 to x=0, decreases linearly from x=0 to x=3, and then increases linearly from x=3 to x=5.

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Correct Answer: A

Reproduced from: NYS Grade 8 Test Item 2, 2017

Originally aligned to: CCSS Standard 8.F.B.5

Geometry

Example 1: NGLS Standard NY-8.G.6

Which list of numbers could represent the side lengths of a right triangle? A. 5, 10, 13. B. 5, 12, 17. C. 10, 24, 26. D. 10, 24, 68

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This question is asking the student to understand and apply the Pythagorean Theorem. To show mastery of this skill, the student must use the Pythagorean theorem to determine which set of numbers could represent the side lengths of a right triangle.

Correct Answer: C

Reproduced from: NYS Grade 8 Test Item 30, 2024

Example 2: NGLS Standard NY-8.G.9

A manufacturer makes cylindrical cans in two sizes. The dimensions of each can are shown in the diagram below. Diagram 1 - LARGE CAN: A cylinder with a height of 10 centimeters and a diameter indicated as 3.5 centimeters. Diagram 2 - SMALL CAN: A cylinder with a height of 6 cm and a diameter indicated as 7 cm. What is the difference between the volumes, in cubic centimeters, of the large can and the small can in terms of pi? A. 4 pi. B. 49 pi. C. 73.5 pi. D. 155.5 pi

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This question is asking the student to solve mathematical problems involving volume of cylinders. To show mastery of this skill, the student must determine the volumes of each cylinder and then compare the results to determine the difference between the large and small cans.

Correct Answer: B

Reproduced from: NYS Grade 8 Test Item 35, 2024

Statistics and Probability

Example 1: NGLS Standard NY-8.SP.1

Which scatter plot best represents a linear association between 𝑦 and 𝑥 ? Graph A: Shows a positive, non-linear association. The points start near the origin and curve upwards, indicating that as x increases, y also increases at an accelerating rate. Graph B: Shows a scattered distribution of points with no clear linear or non-linear association. The points are spread out and do not follow a discernible trend. Graph C: Shows a non-linear association. The points initially increase with x, reach a peak, and then decrease as x continues to increase, forming a curve that resembles an inverted U-shape. Graph D: Shows a negative, linear association. The points generally follow a downward sloping line, indicating that as x increases, y decreases.

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This question is asking the student to investigate patterns of association in bivariate data. To show mastery of this skill, the student must interpret the given scatter plots to determine which one best represents a linear association, regardless of outliers.

Correct Answer: D

Reproduced from: NYS Grade 8 Test Item 2, 2023

Example 2: NGLS Standard NY-8.SP.2

Which statement best describes the data in a scatter plot where the 𝑦-values are decreasing as the and 𝑥-values are increasing? A. The data can best be modeled by a vertical line. B. The data can best be modeled by a horizontal line. C. The data can best be modeled by a line with a positive slope. D. The data can best be modeled by a line with a negative slope.

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This question is asking the student to investigate patterns of association in bivariate data. To show mastery of this skill, the student must determine that if the y-values of the given scatter plot are decreasing while the x-values are increasing, this indicates that the data can best be modeled by a line with a negative slope.

Correct Answer: D

Reproduced from: NYS Grade 8 Test Item 18, 2021

Originally aligned to: CCSS Standard 8.SP.A.2

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