What to expect when taking grade 6 high-stakes math exams

By Michael Avidon, math editor

A good part of the work we do here is to write and edit items (questions) for high-stakes statewide math exams. Those items cannot be released to the public, so the items here are not from actual exams, but are written in a similar style and align with Common Core standards. Solutions, and additional useful information, are found in the last section.

Types of Items

In the past, all items on standardized tests were multiple choice (typically, a question with four answer choices), and many of the items in these documents will be in that format.

With the advent of online exams, other formats are also used. Some of these are known as technology enhanced items (TEIs). Such items cannot be properly presented in a document, but some are presented in a mock format. These are described below.

Multiple select/multiple response: appears to have the same format as multiple choice (with 4 or more options), but has more than one correct answer. The test-taker is asked to select all of these. Sometimes these items state how many options to select and sometimes they do not.

Multiple choice drop-down/Inline choice: will have any number of drop-down menus in the middle of sentences, equations, labeling a diagram, etc. The test-taker must choose one option from each menu to correctly complete the sentences, equations, etc.

In this document, the existence of a drop-down menu is indicated with braces:

Start of sentence {option 1/option 2/option 3} end of sentence.

MORE IN THIS SERIES: Grade 7 assessment test | Grade 8 assessment | Printable version of grade 6 test

Drag and drop: have blank spaces or boxes in sentences, equations, etc. in which are placed correct options (e.g., numbers, words) that are found elsewhere (the bottom or side). The test-taker points to options, clicks on them, and drags them across the screen, dropping them into the appropriate spaces. In this document, underlined blank spaces will generally indicate drop zones and a labeled list of options will be presented at the bottom.

Fill in the blank: The test-taker usually has to provide a whole number or decimal.

Constructed response: requires a complete, hand-written solution.

Hot spot: a student must point and click to indicate a correct answer (e.g., select a point on a number line).

Sample Test Items

Standard: 6.EE.A.2b

Item Type: Hot spot/selectable text

1. Click on the coefficient in the expression: 3x² + 4 + y(y+y)

Standard: 6.EE.A.2c

Item Type: Multiple choice

2. Which of the following expressions has the greatest value when p = 2?

• A. 3³ – p⁴
• B. p³ + 7
• C. 2p + 10¹
• D. 3² + p²

Standard: 6.EE.A.3

Item Type: Multiple select

3. Select all of the expressions that are equivalent to 24x + 32y.

• A. 20x + 4x + 16y × 2y
• B. 8(3x + 4y)
• C. 8(x + x + x + y + y + y + y)
• D. (24 + 32)(x + y)
• E. 4(6x + 8y)

Standard: 6.EE.B.5

Item Type: Multiple select

4. Which values for p make ^p ⁄ ₂ < 10 true? Select all that apply.

• A. 10
• B. 22
• C. 6
• D. 18
• E. 30
• F. 20

Standard: 6.EE.B.6

Item Type: Multiple choice

5. Derrick has 15 pretzels and gives his friend p pretzels. Which expression represents the number of pretzels Derrick has left?

• A. 15 – p
• B. 15 + p
• C. p – 15
• D. 15/p

Standard: 6.EE.B.7

Item Type: Multiple choice

6. What is the solution to 3.4n = 5.44?

• A. n = 0.625
• B. n = 1.6
• C. n = 2.04
• D. n = 8.84

Standard: 6.EE.C9

Item Type: Multiple choice

7. Felipe walks at a pace of 85 steps per minute. Which equation shows the number of steps, s, he takes in m minutes?

• A. m = s + 85
• B. s = m + 85
• C. m = 85s
• D. s = 85m

Standard: 6.G.A.1

Item Type: Two-part multiple choice

8. Part A.

The front side of a barn, shown below, needs to be repainted.

Which expression could be used to find the area, in square feet, of the front side of the barn?

1. (11 x 14) + (^{4 x 14} ⁄ ₂)
2. (11 x 14) + (^{4 x 11} ⁄ ₂)
3. (14 x 11) + (4 x 11)
4. (14 x 11) + (4 x 14)

Part B.

What is the area of the front side of the barn?

1. 176 square feet
2. 182 square feet
3. 198 square feet
4. 210 square feet

Standard: 6.G.A.2

Item Type: Multiple choice

9. A swimming pool is in the shape of a right rectangular prism. The pool is 6 feet deep but the water is filled to only 3.4 feet deep. The length of the pool is 28 feet and the width is 15 feet. What is the volume of the unfilled part of the pool?

• A. 1,092 cubic feet
• B. 1,428 cubic feet
• C. 1,512 cubic feet
• D. 2,520 cubic feet

Standard: 6.G.A.3

Item Type: Multiple choice

10. Three vertices of the parallelogram ABCD are shown on the coordinate plane below.

What are the coordinates of point C?

1. (-8, 6)
2. (6, -8)
3. (6, 8)
4. (8, 6)

Standard: 6.G.A.4

Item Type: Fill in the blank

11. Casey made a gift box in the form of a cube from the net shown below. The net is made up of five squares and two identical right triangles.

The total area of the net is 24 square inches.

It follows that x = _____ inches.

Standard: 6.NS.A.1

Item Type: Multiple choice

12. What is the length of a rectangular table with a width of ¾ meter and area of ⅝ square meters?

1. ⅛ meter
2. ¹⁵ ⁄ ₃₂ meter
3. ⅚ meter
4. ⁶ ⁄ ₅ meter

Standard: 6.NS.B.3

Item Type: Multiple choice

13. A store manager buys 45 toy cars. The cost for all the cars is $742.50, and each car costs the same. What is the cost for one car?

• A.$18.72
• B. $17.28
• C. $16.94
• D. $16.50

Standard: 6.NS.B.4

Item Type: Multiple choice

14. Which pair of numbers has a greatest common factor of 2 and a least common multiple of 30?

• A. 5 and 6
• B. 6 and 10
• C. 2 and 15
• D. 10 and 18

Standard: 6.NS.C.6b

Item Type: Multiple select

15. Cassidy graphed the point (–3, 4) on a coordinate plane.

Which statements are true? Select all that apply.

1. Cassidy’s point is in Quadrant I.
2. Cassidy’s point is in Quadrant II.
3. The point (4, –3) is a reflection of Cassidy’s point across the y-axis.
4. The point (–3, –4) is a reflection of Cassidy’s point across the x-axis.
5. A reflection of Cassidy’s point across the y-axis results in a point in Quadrant I.
6. The point (–4, 3) is a reflection of Cassidy’s point across the x-axis, then the y-axis.

Standard: 6.NS.C.8

Item Type: Multiple choice

16. Hakeem puts an X at the point (–2, 5) on a coordinate plane. Elena puts an X at the point (3, 5). How many units apart are Hakeem and Elena’s points?

• A. 0
• B. 1
• C. 5
• D. 10

Standard: 6.RP.A.2

Item Type: Multiple choice

17. A box of 6 bagels costs $4.50. What is the unit cost?

• A. $0.75 per bagel
• B. $0.85 per bagel
• C. $0.90 per bagel
• D. $1.33 per bagel

Standard: 6.RP.A.3b

Item Type: Multiple choice

18. Marco typed 108 words in 3 minutes. If he continues typing at this rate, what is the number of words he will type in 8 minutes?

• A. 180
• B. 264
• C. 288
• D. 352

Standard: 6.SP.B.4

Item Type: Multiple choice

19. The histogram below displays the ages of people at a party.

Which frequency chart displays the same data as the histogram?

A.

Ages	0-10	11-20	21-30	31-40	41-50	51-60
Frequency	2	4	14	16	8	4

B.

Ages	0-10	11-20	21-30	31-40	41-50	51-60
Frequency	2	4	13	16	7	3

C.

Ages	0-10	11-20	21-30	31-40	41-50	51-60
Frequency	2	4	14	16	8	3

D.

Ages	0-10	11-20	21-30	31-40	41-50	51-60
Frequency	2	4	12	16	6	3

Standard: 6.SP.B.5a

Item Type: Multiple select

20. Which of the following represents a data set that includes exactly 20 observations? Select all that apply.

A.

B.

C.

D. The number of students that can do 20 push-ups in less than two minutes

E. A pictogram that shows the favorite choices of twenty students for 5 movie types

Solutions and Explanations

1. Click on the coefficient in the expression: 3x² + 4 + y(y+y)

Key: 3

Solution:

The parts of the expression separated by + or – signs are called terms. The first term is 3x², and this particular term is also called a monomial. A monomial is a product of a constant and variables. The constant factor in a monomial is called a coefficient. So 3 is a coefficient.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

The “4” is a constant term. It is not multiplying any variables, so it would not be called a coefficient. The factor of y in the third term is not a constant, so it is not a coefficient.

2. Which of the following expressions has the greatest value when p = 2?

Key: B

Solution:

Substituting into each of the four options yields the following values:

• A. 3³ – 2⁴ = 27 – 16 = 11
• B. 2³ + 7 = 8 + 7 = 15
• C. 2 x 2 + 10¹ = 4 + 10 = 14
• D. 3² + 2² = 9 + 4 = 13

The greatest value, 15, is produced by option B.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

Remember that the exponent (the little number up and to the right) tells you how many times to multiply the base (the number written on the line). For example, 2⁴ = 2 x 2 x 2 x 2. Also, compute exponents first, then multiply, then add or subtract (Go back to each computation above and check that they were done this way.) This is called “order of operation.”

3. Select all of the expressions that are equivalent to 24x + 32y.

Key: B, C, E

Solution:

Simplify each option by adding like terms, using multiplication, and using the Distributive Property:

• A. 20x + 4x + 16y × 2y
• B. 8(3x + 4y)
• C. 8(x + x + x + y + y + y + y) This is the same as B.
• D. (24 + 32)(x + y)
• E. 4(6x + 8y)

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

The Distributive Property says a(b +c) = a x b + a x c. This was used in a “forward” manner several times, and in “reverse” in option A.

Any operation inside parentheses must be done first. This was done in options C and D, adding “like terms” or constants first. You cannot combine the unlike terms in B and E.

Adding like terms is a shorthand way of using the Distributive Property. In option A, you could say that 20x and 4x are like terms (same variables multiplied by constants). Adding like terms means adding the constants and copying the variable(s), so you obtain 24x.

4. Which values for p make ^p ⁄ ₂ < 10 true? Select all that apply.

Key: A, C, D

Solution:

You can substitute each value into the inequality. For p = 10, you get ¹⁰ ⁄ ₂ < 10 or 5 < 10, which is true. For p = 22, you get ²² ⁄ ₂ < 10 or 11 < 10. This is false. Likewise, you should find that 6 and 18 yield true inequalities, while 30 and 20 yield false inequalities.

Alternatively, you could rewrite the inequality by multiplying both sides by 2, as you would for an equation, to obtain p < 20. This tells you directly that all the options less than 20 make the inequality true.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

An inequality or an equation is a statement written using mathematical notation and often contains a variable. The inequality in this item says “An unknown number divided by 2 is less than 10.” Any value that can replace the variable/unknown number to create a true statement is called a solution to the equation or inequality. As we showed, replacing p by 10 creates a true statement; replacing p by 22 creates a false statement.

5. Derrick has 15 pretzels and gives his friend p pretzels. Which expression represents the number of pretzels Derrick has left?

Key: A

Solution:

If you start with 15 of something and give away 5, for example, then you have 15 – 5 = 10 left. If you start with 15 and give away p, then you have 15 – p left.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

First, any mathematical expression, no matter how complicated, is the representation of a single number. This is distinctly different from an equation or inequality, which is a statement about the relation between two expressions/numbers.

It may be helpful to temporarily replace unknown numbers with specific numbers to determine what expression represents the unknown amount, as was done in the solution.

6. What is the solution to 3.4n = 5.44?

Key: B

Solution:

Divide each side by 3.4: ^3.4n ⁄ _3.4 = ^5.44 ⁄ _3.4 → n = 1.6

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

You can rewrite an equation by performing the same arithmetic operation on each side. Each equation is a statement about the relation between numbers; when you perform these operations and get a new equation, this is a new statement, but contains the same information written in a different format. For example, an equation might say your brother is 5 years older than you, or that you are 5 years younger than your brother; it is the same information written in a different way.

7. Felipe walks at a pace of 85 steps per minute. Which equation shows the number of steps, s, he takes in m minutes?

Key: D

Solution:

After 2 minutes, he walks 85 x 2 steps; after 3 minutes, he walks 85 x 3 steps; after m minutes, he walks 85 x m steps. So s = 85m.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

A proportional relationship, as in this item, is always represented by an equation in the form y = kx, where k is a constant. You will learn more about these in grade 7.

8. Part A.

The front side of a barn, shown below, needs to be repainted.

Which expression could be used to find the area, in square feet, of the front side of the barn?

Part B.

What is the area of the front side of the barn?

Key: A, B

Solution:

Part A. Split the pentagon with a horizontal line segment into an 11 by 14 foot rectangle and a triangle with 14 foot base. The height of the triangle is 15 – 11 = 4 feet. The area of the pentagon is the area of the rectangle (11 x 14) plus the area of the triangle (½ x 14 x 4). This is equivalent to option A.

Part B. The area of the rectangle is 154 square feet. The area of the triangle is 28 square feet. The total area is 182 square feet.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

A general theme that is found throughout mathematics is to take a new problem and reduce it to a combination of old/simpler problems (if you can).

In this case, finding the area of a pentagon is “reduced” to finding the area of a rectangle plus the area of a triangle.

9. A swimming pool is in the shape of a right rectangular prism. The pool is 6 feet deep but the water is filled to only 3.4 feet deep. The length of the pool is 28 feet and the width is 15 feet. What is the volume of the unfilled part of the pool?

Key: A

Solution:

The unfilled part is also a right rectangular prism, with length 28 feet, width 15 feet, and height 6 – 3.4 = 2.6 feet. The volume of the unfilled part is 28 x 15 x 2.6 = 1092 cubic feet.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

A prism is any three-dimensional object whose cross sections are identical polygons. A rectangular prism has cross sections that are rectangles (a triangular prism has cross sections that are triangles). A right rectangular prism has cross sections that are at right angles to the other faces (as opposed to an oblique prism, which is tilted). The volume of any prism is the area of its base (a cross section) times its height.

10. Three vertices of the parallelogram ABCD are shown on the coordinate plane.

What are the coordinates of point C?

Key: D

Solution:

AB is a horizontal segment and is parallel to CD, so that is also horizontal. Therefore, the y-coordinate of C is 6. Now the length of AB is 8, so that is the length of CD. This implies that the x-coordinate of C is 8 or –8. But the name of the parallelogram (ABCD) tells you the order that the vertices are in. So C must be in Quadrant I, and its coordinates must be (8, 6).

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

By definition, opposite sides of a parallelogram are parallel. It is a fact that opposite sides are equal in length. (It is also true that opposite angles are equal and consecutive angles are supplementary, but you may not need to know that in grade 6.)

If you start at A and go 3 units right and 5 units up, you end at D. Likewise, if you start at B and go 3 units right and 5 units up, you end at C.

11. Casey made a gift box in the form of a cube from the net shown below. The net is made up of five squares and two identical right triangles.

The total area of the net is 24 square inches.

Key: It follows that x = 2 inches.

Solution:

The net is equivalent to 6 identical squares, each with area 4 square inches and side length 2 inches. The triangles must have legs equal to the sides of the squares.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

The net of one object can be shown in multiple ways. Often, the net of a cube is shown as 6 squares. When the net shown folds up, the two triangles align to form a square that is parallel to the dashed square. Even if you drew 6 squares, you could arrange them numerous ways and still have a net of a cube. Try it with paper and scissors.

12. 12. What is the length of a rectangular table with a width of ¾ meter and area of ⅝ square meters?

Key: C

Solution:

The area divided by the width is the length: ⅝ ÷ ¾ = ⅝ x ⁴ ⁄ ₃ = ²⁰ ⁄ ₂₄ = ⅚ meter.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

Dividing by a fraction is the same as multiplying by its reciprocal.

13. A store manager buys 45 toy cars. The cost for all the cars is $742.50, and each car costs the same. What is the cost for one car?

Key: D

Solution:

The cost of one car is $742.50 ÷ 45 = $16.50.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

You can use long division with decimals just as you can with whole numbers. Make sure to carefully align decimal points.

14. Which pair of numbers has a greatest common factor of 2 and a least common multiple of 30?

Key: B

Solution:

Determine the gcf and lcm for each pair:

1. 5 and 6 have an lcm of 30, but have a gcf of 1.
2. 6 and 10 have an lcm of 2•3•5 = 30 and a gcf of 2.
3. 2 and 15 have an lcm of 30, but have a gcf of 1.
4. 10 and 18 have a gcf of 2, but have an lcm of 2•5•3² = 90.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

You can factor any number into powers of primes. For example, 10 = 2 ¹ x 5¹ and 18 = 2¹ x 3². Compare the powers (“no” power of a prime is the same as having the power 0) . Take the lesser power of each prime as a factor for the gcf: 2¹ x 3⁰ x 5⁰ = 2 x 1 x 2 = 2. Take the greater power of each prime as a factor for the lcm: 2¹ x 3² x 5¹ = 2 x 9 x 5 = 90.

15. Cassidy graphed the point (–3, 4) on a coordinate plane.

Which statements are true? Select all that apply.

Key: B, D, E

Solution:

Quadrant I is the top right corner of the coordinate plane and has positive coordinates.

Quadrant II is the top left corner and has negative x and positive y-coordinates, so the point is in Quadrant II.

The reflection of the point across the y-axis is in Quadrant I and has coordinates (3, 4).

The reflection of the point across the x-axis is in Quadrant III and has coordinates (-3, -4).

Reflecting the point across both axes results in a point in Quadrant IV with coordinates (3, -4).

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

In the center of the coordinate plane is a point called the origin and labeled (0, 0). All other points are labeled “relative” to this one. In order to distinguish points to the left from points to the right of the origin, mathematicians made a decision to call the direction to the right positive and the direction to the left negative. This is shown by the first coordinate of a point, so the point (–3, 4) is 3 units to the left of center. In order to distinguish points above from points below the origin, mathematicians made a decision to call the direction above positive and the direction below negative. This is shown by the second coordinate of a point, so the point (–3, 4) is 4 units above center.

16. Hakeem puts an X at the point (–2, 5) on a coordinate plane. Elena puts an X at the point (3, 5). How many units apart are Hakeem and Elena’s points?

Key: C

Solution:

A horizontal line segment that is 2 + 3 = 5 units long connects the points.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

Distance is always a positive number. The x-coordinates of –2 and 3 show that the two points are 2 units to the left and 3 units to the right of center respectively. The identical y-coordinates of 5 show that the points are both 5 units above center, and therefore lie on a horizontal line.

17. A box of 6 bagels costs $4.50. What is the unit cost?

Key: A

Solution:

The unit cost is $4.50 ÷ 6 = $0.75.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

Unit cost equals total cost divided by the number of units, or the cost for buying one unit (here, one bagel). The store may not sell one bagel for 75¢, but this computation allows you to compare the cost for one bagel when buying in bulk to the cost when buying one.

18. Marco typed 108 words in 3 minutes. If he continues typing at this rate, what is the number of words he will type in 8 minutes?

Key: C

Solution:

Marco’s unit rate of typing is 108 ÷ 3 = 36 words per minutes. In 8 minutes, he would type 36 x 8 = 288 words.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

In the previous item, you found a unit cost, which could also be called a “unit rate” in a very general sense: the amount of one quantity divided by the amount of another quantity. In item #17, the quantities were total cost and number of bagels; in this item, the quantities are number of words and number of minutes. The word rate is commonly used when the divisor is a quantity of time.

19. The histogram displays the ages of people at a party. Which frequency chart displays the same data as the histogram?

Key: B

Solution:

The height of each bar is the number of people (frequency) in that age group. The heights are 2, 4, 13, 16, 7, and 3. Only one chart has these numbers.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

The word frequency means how often something happens.

20. Which of the following represents a data set that includes exactly 20 observations?

Key: A, B, E

Solution:

The numbers of observations, if they can be determined, are given:

1. the total number of dots = 5 + 2 + 3 + 7 + 3 = 20
2. the total of the heights of the bars = 1 + 3 + 9 + 5 + 2 = 20
3. cannot be determined from a box and whisker chart
4. The observations would be the number of students, which is not given.
5. The observations are the choices of the 20 students.

Commentary (formulas, ideas, mistakes, misunderstandings, advice):

Different methods of representing data have different advantages and disadvantages. Option A has a dot plot, which shows every piece of data (this is similar to a line plot and to a bar graph). Option B has a histogram. This is similar to a dot plot in that the size of each “category” is shown by height. However, because a histogram has intervals, this display partially summarizes the data (e.g., you know that there are 9 pieces of data in the 51 to 75 category, but you don’t know exactly which 9 numbers they are). Option C shows a box and whisker plot, which summarizes data to a greater extent than a histogram, but clearly displays the summary data (minimum, 1^st quartile, median, 3^rd quartile, and maximum) and does so better than the preceding displays.