Grade 9

Algebra: Seeing Structure in Expressions

Algebra: Arithmetic with Polynomials and Rational Expressions

Algebra: Reasoning with Equations and Inequalities

The NGLS organizes standards by course at the high school level. In the 9th grade, the MLS covers a variety of topics taken primarily from Algebra I. A small percentage of the test will assess Geometry knowledge as well. In Algebra I, students build on prior experiences with linear equations, graphical and algebraic methods, functions and data.

Below, you will find examples of the types of question and answer sets students who are operating at a Grade 9 math level in the target language should be able to understand, along with their corresponding NYS Next Generation Learning Standard.

Algebra I

Number and Quantity

Example 1: NGLS Standard AI-N.RN

Which expression results in an irrational number? A. the square root of 3 times the square root of 3. B. negative two thirds plus one fourth. C. five times the square root of eighty one. D. one third plus the square root of three.

View Rationale and Additional Information

This question is asking the student to use properties and operations to understand the different forms of rational and irrational numbers. To show mastery of this skill, the student must be able to perform all four arithmetic operations using rational numbers and square roots and be able to categorize the sum or product of rational or irrational numbers.

Correct Answer: D

Reproduced from: NYS Algebra I Test Item 7, August 2024

Algebra: Seeing Structure in Expressions

Example 1: NGLS Standard AI-A.SSE.A

When factored, the expression 𝑥 cubed minus 36 𝑥 is equivalent to Options A. open parentheses x plus 6 closed parentheses times open parentheses x minus six closed parentheses. B. open parentheses x plus eighteen closed parentheses times open parentheses x minus 18 closed parentheses. C. X times open parentheses X plus 6 closed parentheses times open parentheses X minus 6 closed parentheses. D. X times open parentheses X plus eighteen closed parentheses times open parentheses X minus 18 closed parentheses

View Rationale and Additional Information

This question is asking the student to interpret the structure of the given expression. To show mastery of this skill, the student must identify ways to rewrite the expression through factoring.

Correct Answer: C

Reproduced from: NYS Algebra I Test Item 1, January 2025

Algebra: Arithmetic with Polynomials and Rational Expressions

Example 1: NGLS Standard AI-A.APR.A

Which expression is equivalent to 3 times open parentheses 𝑥 squared minus 2 𝑥 plus 3 closed parentheses minus open parentheses 4 x squared plus 3 𝑥 minus 1 closed parentheses? A. negative X squared plus X plus two. B. negative X squared minus 8 X plus 7. C. negative X squared minus 3 X plus 8. D. negative X squared minus 9 X plus 10

View Rationale and Additional Information

This question is asking the student to perform arithmetic operations on polynomials. To show mastery of this skill, the student must use the order of operations to determine an equivalent expression.

Correct Answer: D

Reproduced from: NYS Algebra I Test Item 3, August 2024

Example 2: NGLS Standard AI-A.APR.B

The zeros of the function F of X equals X times open parentheses X minus 5 closed parentheses times open parentheses 3 X plus 6 closed parentheses are A. Zero, negative five, and 2. B. zero, 5, and negative 2. C. negative 5 and 2, only. D. 5 and negative 2, only

View Rationale and Additional Information

This question is asking the student to understand the relationship between zeros and factors of polynomials. To show mastery of this skill, the student must identify the zeros of the given polynomial functions, provided the suitable factorizations.

Correct Answer: B

Reproduced from: NYS Algebra I Test Item 9, June 2024

Algebra: Creating Equations

Example 1: NGLS Standard AI-A.CED.A

At Adelynn’s first birthday party, each guest brought 1 dollar in coins for her piggy bank. Guests brought nickels, dimes, and quarters for a total of 28 dollars. There were twice as many dimes as nickels and twelve more quarters than nickels. Which equation could be used to determine the number of nickels, X, that her guests brought to her party? A. point zero five x plus point ten X plus point two five X equals 28. B. point zero five X plus point ten times open parentheses 2 X closed parentheses plus point two five open parentheses X plus twelve closed parentheses equals 28. C. point zero five times open parentheses two X closed parentheses plus point ten X plus point two five times open parentheses X plus twelve closed parentheses equals 28. D. point zero five times open parentheses X plus twelve closed parentheses plus point ten times open parentheses two X closed parentheses plus point two five X equals 28.

View Rationale and Additional Information

This question is asking the student to create an equation in one variable to represent a real-world context. To show mastery of this skill, the student must determine which equation could be used to determine how many nickels, x, were brought to Adelynn's party.

Correct Answer: B

Reproduced from: NYS Algebra I Test Item 4, August 2024

Example 2: NGLS Standard AI-A.CED.A

During summer vacation, Ben decides to sell hot dogs and pretzels on a food cart in Manhattan. It costs Ben fifty cents for each hot dog and forty cents for each pretzel. He has only one hundred dollars to spend each day on hot dogs and pretzels. He wants to sell at least two hundred items each day. If H is the number of hot dogs and P is the number of pretzels, which inequality would be part of a system of inequalities used to determine the total number of hot dogs and pretzels Ben can sell? A. H plus P is less than or equal to two hundred. B. H plus P is greater than or equal to two hundred. C. zero point five zero H plus zero point four zero P is greater than or equal to two hundred. D. zero point five zero H plus zero point four zero P is less than or equal to two hundred.

View Rationale and Additional Information

This question is asking the student to create inequalities to represent a real-world context. To show mastery of this skill, the student must determine which equation represents how many hot dogs and pretzels Ben would need to sell in order to achieve his goal of selling greater than or equal to 200 items per day.

Correct Answer: B

Reproduced from: NYS Algebra I Test Item 24, Jan 2023

Algebra: Reasoning with Equations and Inequalities

Example 1: NGLS Standard AI-A.REI.B

Which equation has the same solution as X squared plus six X minus 18 equals zero? A. open parentheses X plus 3 closed parentheses squared equals twenty four. B. open parentheses X plus three closed parentheses squared equals twenty seven. C. open parentheses X plus six closed parentheses squared equals twenty four. D. open parentheses X plus six closed parentheses squared equals 27.

View Rationale and Additional Information

This question is asking the student to solve quadratic equations in one variable. To show mastery of this skill, the student must use factoring to determine which equation has the same solution as the given equation.

Correct Answer: B

Reproduced from: NYS Algebra I Test Item 8, August 2024

Example 2: NGLS Standard AI-A.REI.D

The graph shows a coordinate plane with x and y axes. Line 1 is solid: This line passes through approximately (0,-1) and (1,1). The region to the right is shaded. Line 2 is dashed: This line passes through approximately (0,4) and (1,1). The region to the right is shaded. There is a crosshatch pattern where it overlaps with the other shaded area. Which point is a solution to this system? A. (1,1) B. (2,−2) C. (1,8) D. (4,2)

View Rationale and Additional Information

This question is asking the student to solve a system of inequalities graphically. To show mastery of this skill, the student must understand that the overlapping areas of the graph represent all points that could be a solution to the system.

Correct Answer: D

Reproduced from: NYS Algebra I Test Item 7, January 2025

Functions

Example 1: NGLS Standard AI-F.IF.B

A bookstore owner recorded the number of books sold and the profit made selling the books. A table is shown with two columns and five rows. In the first column, labeled "Books Sold" the quantities are 100, 250, 300, and 350. The second column, labeled "profit" has the quantities fifty dollars, two hundred seventy five dollars, three hundred fifty dollars and four hundred twenty five dollars. What is the average rate of change, in dollars per book, between 100 and 350 books sold? A. zero point five zero. B. zero point six seven. C. one point five zero. D. two point zero zero

View Rationale and Additional Information

This question is asking the student to interpret functions that arise in applications in terms of the context. To show mastery of this skill, the student must calculate the average rate of change given the table of information provided.

Correct Answer: C

Reproduced from: NYS Algebra I Test Item 10, August 2024

Example 2: NGLS Standard AI-F.BF.A

Jack started a new fitness program. The first day he did 10 push-ups. The program required him to increase the number of push-ups each day by doing 9 less than twice the number from the previous day. Which recursive formula correctly models Jack’s new program, where N is the number of days and A sub N is the number of push-ups on the 𝑛th day? A. A sub one equals ten and A sub N equals two A sub N minus 1 end sub minus nine. B. A sub one equals ten and A sub N equals 9 minus 2 A sub N minus 1. C. A sub one equals 10 and A sub N equals 2 times open parentheses N minus one closed parentheses minus nine. D. A sub one equals ten and A sub N equals 9 minus 2 times open parentheses N minus one closed parentheses

View Rationale and Additional Information

This question is asking the student to build a function that models a relationship between two quantities. To show mastery of this skill, the student must identify the correct set of formulas used to model Jack's new program.

Correct Answer: A

Reproduced from: NYS Algebra I Test Item 19, August 2023

Example 3: NGLS Standard AI-F.LE.B

A landscaping company charges a set fee for a spring cleanup, plus an hourly labor rate. The total cost is modeled by the function C of X equals fifty five X plus eighty. In this function, what does 55 represent? A. the set fee for the cleanup. B. the hourly labor rate for a cleanup. C. the profit earned by the company for one cleanup. D. the number of hours of labor required for one cleanup

View Rationale and Additional Information

This question is asking the student to interpret expressions for functions in terms of the situation they model. To show mastery of this skill, the student must understand which answer possibility represents the hourly labor rate for a cleanup.

Correct Answer: B

Reproduced from: NYS Algebra I Test Item 5, January 2025

Geometry

Example 1: NGLS Standard G-SRT.B

Given the information marked on the diagrams below, which pair of triangles can not always be proven congruent? A. Diagram shows a large triangle A B D with a line segment B C forming two right-angled triangles, A B C and D B C. A right-angle symbol is at C. Triangles listed: triangle A B C and triangle D B C. B. Diagram shows two intersecting triangles, E F G and H I G, forming an X shape. Angle F and Angle I are marked congruent. Side E F and Side H I are marked congruent. Triangles listed: triangle E F G and triangle H I G. C. Diagram shows a quadrilateral J K L M with a diagonal J L. Sides J K and J M are marked congruent with single tick marks. Sides K L and M L are marked congruent with double tick marks. Triangles listed: triangle K L J and triangle M J L. D. Diagram shows two triangles, N O P and R S P, sharing a common vertex P. Lines O N and S R are perpendicular to the base N R. Angle N P O and Angle S P R are marked congruent with arc symbols. Sides O P and S P are marked congruent with double tick marks. Triangles listed: triangle N O P and triangle R S P.

View Rationale and Additional Information

This question is asking the student to use congruence and similarity criteria for triangles to prove relationships in geometric figures. To show mastery of this skill, the student must determine which answer can not always be proven congruent.

Correct Answer: D

Reproduced from: NYS Geometry Test Item 16, June 2022

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