# 3 Mathematics New York State Testing Program

```New York State
Testing Program
Mathematics
Common Core Sample Questions
3
The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and nonprofit
organizations or institutions to reproduce these materials for their own use, but not for sale, provided copyright notices are retained as they appear in
this publication. This permission does not apply to mass distribution of these materials, electronically or otherwise.
1
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: CR
1
Part A: Fill in the blanks below with whole numbers greater than 1 that will make
the number sentences true.
1.
2.
3.
4.
5.
63 ÷ ___ = 7
63 = 21 × ___
21 = ___ × 7
7 × (___ × ___ ) = 21 × 7
(21 × 3) ÷ ___ = 7
Part B: If the product of two whole numbers greater than 1 is 63, what could the
two whole numbers be? _______, ________
Key: Part A
1.
2.
3.
4.
5.
63 ÷ 9 = 7
63 = 21 × 3
21 = 3 × 7
7 × (3 × 7) = 21 × 7 or 7 × (7 × 3) = 21 × 7
(21 × 3) ÷ 9 = 7
Key: Part B
7,9 (or 9,7) or 3,21 (or 21,3).
Aligned CCLS: 3.OA.4, 3.OA.5, and 3.OA.6
Commentary: This question aligns with CCLS 3.OA.4, 3.OA.5, and 3.OA.6 and assesses
the student’s ability to determine the unknown whole numbers in multiplication and
division equations. The question also assesses the student’s ability to apply properties of
operations as strategies to multiply or divide and to understand division as an unknown
factor problem.
Rationale:
Part A: Errors in A1, A2, and A3 are most likely due to errors in computing single-digit
number facts or difficulty in non-forward execution. Errors in A4 and A5 may be
attributed to errors in applying properties of operations as strategies to multiply and
divide. Errors in A1 and A5 may also be due to not recognizing division as an unknownfactor problem.
Part B: Errors likely arise from limited application of associative and/or distributive
properties of multiplication to generate multiple combinations of whole numbers for a
given product.
2
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
2
Two groups of students from Douglas Elementary School were walking to the
library when it began to rain. The 7 students in Mr. Stem’s group shared the
3 large umbrellas they had with Ms. Thorn’s group of 11 students. If the same
number of students were under each umbrella, how many students were under
each umbrella?
A
B
C
D
16
10
18
21
Key: A
Aligned CCLS: 3.OA.8, 3.OA.2
Commentary: This question aligns to CCLS 3.OA.8 and assesses the student’s ability to
solve a two-step word problem using addition and division of whole numbers. It also
aligns to 3.OA.2 because it assesses the ability to partition a number into equal groups.
Rationale: The total number of students in the two groups is 18 (7 + 11), so 18 must be
divided into 3 equal groups; therefore, 6 students are in each group. Selecting Options B
and D could indicate relating of the numbers in the problem with incorrect operations
(adding 7 and 3 in B and multiplying 7 and 3 in D) and therefore a lack of understanding
of the problem. Selecting Option C indicates that a student had knowledge of how to
begin the problem, by adding the two groups together, but then forgot to divide the
students into 3 equal groups.
3
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
3
Tommy made 6 rows of blocks, with each row containing 8 blocks. How many
blocks did Tommy have altogether?
You may use the space below to draw a picture of the problem.
A
B
C
D
14
36
48
64
Key: C
Aligned CCLS: 3.OA.1
Commentary: This question aligns to CCLS 3.OA.1, as it assesses the student’s ability to
interpret the product created when each of the 6 rows of blocks contains 8 blocks.
Rationale: Selecting Option C as the correct answer shows that a student is able to
visualize the abstract concept of 6 rows with 8 in each row and is able to count the total
number of blocks, either by adding them together (individually or in sets) or by finding the
product of 6 × 8. Selecting Option A likely indicates that the wrong operation was chosen
to model the situation (adding 6 + 8) as well as an incorrect visualization of what the
problem was asking. Selecting Option B or D could indicate that students performed an
error in addition or multiplication or used a single dimension for their calculations (6 rows
of 6 each or 8 rows of 8 each).
4
Common Core Sample Questions
Domain: Number and Operations—Fractions
Item: CR
Give a fraction that represents each point on the string compared to the whole.
4
A
B
C
D
E
Point A _______
Point B _______
Point C _______
Identify another fraction that is equivalent to the fraction represented by
point A. ________
Identify another fraction that is equivalent to the fraction represented by
point C. ________
Key
A
1
, or fraction equivalent
4
B
1
, or fraction equivalent
2
C
3
, or fraction equivalent
4
D any fraction equivalent to
1
, but not the answer given in A
4
E any fraction equivalent to
3
, but not the answer given in C
4
Aligned CCLS: 3.NF.2a, 3.NF.2b, 3.NF.3b
Commentary: This question aligns to CCLS 3.NF.2a and assesses the student’s ability to
represent a fraction in the form
1
on a number line. It also aligns to 3.NF.2b and
b
assesses the student’s ability to represent a fraction
a
on the number line and also
b
aligns to 3.NF.3b, assessing the ability to generate simple equivalent fractions.
Rationale: Incorrect responses for A, B, or C may be due to incorrect division of the
whole into four equal parts and constructing an incorrect denominator. An incorrect
5
Common Core Sample Questions
answer for D or E while having a correct answer for A, B, and/or C likely indicates a
difficulty in constructing equivalent fractions or limited knowledge of equivalent
fractions. An incorrect answer for D or E could also be a result of incorrect answers for A
and/or C.
6
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
5
A third grade class decided to sell boxes of cookies to help raise money for a
school trip. Each box has two bags of cookies inside, and each bag holds 14
each student need to sell?
A
B
C
D
28
56
112
224
Key: C
Aligned CCLS: 3.OA.3
Commentary: This question is aligned to CCLS 3.OA.3 and assesses a student’s ability to
solve a multiplication word problem.
Rationale: Selecting Option C as the correct answer indicates that the student has
accurately understood that each of the 4 boxes contains two bags and that each of
those bags holds 14 cookies as well as the correct operation, multiplication, to
determine the total number of cookies (4 x 2 x 14 = 112). Option A does not include the
information from the stem and only gives the total amount of cookies in a single box (2 x
14 = 28). Option B is the result of multiplying the number of boxes by the number of
cookies (4 x 14 = 56), but excludes that each box contains two bags of 14 cookies.
Option D is the result of multiplying all the numbers in the problem using “two” twice
(2 x 2 x 4 x 14 = 224), rather than a single time.
7
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
6
There were 54 apples set aside as a snack for 3 classes of students. The teachers
divided up the apples and placed equal amounts on 9 separate trays. If each of
the 3 classes received the same number of trays, how many apples did each class
get?
A
B
C
D
2
6
18
27
Key: C
Aligned CCLS: 3.0A.2, 3.OA.3
Commentary: This question is aligned to CCLS 3.0A.2 and assesses a student’s ability to
interpret whole-number quotients of whole numbers. It is also aligned to 3.OA.3 and
assesses the ability to divide whole numbers less than 100 when solving word problems
in situations involving equal groups.
Rationale: The correct response, Option C, is arrived at by determining the number of apples
per tray (
9
54
= 6 per tray) and then determining the number of trays per class ( = 3 trays per
3
9
class). Therefore, each class received 18 apples (3 trays × 6 apples per tray = 18 apples).
Option C could also be arrived at by dividing the total number of apples by the number of
classes (54 apples ÷ 3 classes = 18 apples). Option A could be arrived at by determining the
number of apples per tray (
54
= 6 per tray) and incorrectly dividing the number of apples per
9
tray by the number of classes. Option B could represent the number of apples per tray (
54
=
9
6 per tray) rather than the number of apples for the entire class. Option D could represent the
multiplication of three classes by the number of trays (3 x 9 = 27).
8
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
7
There are 3 large picture frames. Each picture frame contains exactly 2 pictures.
What fraction represents just one picture out of all the pictures in the frames?
A
1
3
B
2
3
C
2
5
D
1
6
Key: D
Aligned CCLS: 3.0A.1, 3.NF.1
Commentary: This question is aligned to CCLS 3.0A.1 and assesses the student’s ability
to interpret the setting in order to find the total number of pictures in the frames. It also
aligns to 3.NF.1 because it assesses the student’s ability to understand the fraction
1
as
6
the quantity formed by 1 part when the whole is partitioned into 6 equal parts.
Rationale: Option D is correct since there are 2 pictures per frame and 3 frames, with a
total of six pictures (2 x 3). One picture would represent
1
of all the pictures in the
6
frames. Option A is incorrect because the denominator does not accurately represent
the whole. Option B indicates the incorrect creation of a fraction using the given
numbers in the problem. Option C is incorrect because the denominator is the result of
adding the numbers rather than multiplying them to determine the total number of
pictures in all three frames.
9
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
8
20 ÷ n = 5 and n × 5 = 20. What is n?
A
B
C
D
4
5
8
15
Key: A
Aligned CCLS: 3.0A.6
Commentary: This question is aligned to CCLS 3.0A.6 and assesses the student’s ability
to understand division as an unknown-factor problem.
Rationale: Since 20 ÷ 4 = 5, and 4 × 5 = 20, it follows that n = 4. If answer choice B was
selected, it is possible that the student confused the use of 5 within the problem or
made an error in whole number division or multiplication. Answer choice C may be
selected if students add the value of n twice or double the value of n given the presence
of two n’s in the stem. Answer choice D indicates a misunderstanding of the operations
in the problem or a false assumption of addition and subtraction (20 – 15 = 5 and 15 + 5
= 20). If answer choice C or D was selected, the student probably did not know the
correct number sentence to make the statement true, or could not divide correctly.
10
Common Core Sample Questions
Domain: Number and Operations—Fractions
Item: MC
9
Three students are sharing a box of 8 crayons. Jari has 2 of the crayons on his
desk, Nora has 3 of the crayons on her desk, and Tommy has 1 of the crayons on
his desk. If the rest of the crayons are still in the box, what fractional part of the
crayons is still in the box?
A
C
D
1
8
2
8
3
8
6
8
Key: B
Aligned CCLS: 3.NF.1
Commentary: The question is aligned to CCLS 3.NF.1 and assesses the student’s ability
to understand a fraction
1
a
as the quantity formed by a parts of .
b
b
Rationale: Option B is the correct answer. If a total of 2 + 3 + 1 crayons (6) are on the
students’ desks, then 8 − 6 = 2 crayons are still in the box. This can be represented
2
fractionally as of the box. Option A could indicate a student selected the unit fraction
8
for a single crayon rather than answering the specific question of the stem. Option C
likely indicates a student created a fraction from the number of students with the
number of crayons (3 students per 8 crayons) or could indicate an error in calculating
the number of crayons remaining in the box. Option D may indicate that students
represented the fractional part of crayons out of the box, rather than in the box.
11
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: CR
10
Part A: What number sentence can be represented by the picture below?
Use the blanks below to create your number sentence.
______×______ = ______
Part B: Put the circles below into eight equally sized groups and write an
equation to represent the picture.
oooooooooooooooooooooooo
Key
Part A 4 × 6 = 24 or 6 × 4 = 24
Part B o o o
ooo ooo ooo ooo ooo ooo ooo
3 × 8 = 24 or 8 × 3 = 24, or
24
=8
3
Aligned CCLS: 3.OA.1, 3.OA.2
Commentary: This question is aligned to CCLS 3.OA.1 and assesses the student’s ability
to interpret products of whole numbers. Part B aligns to CCLS 3.OA.2 and assesses the
student’s ability to partition a set into equal groups and represent the partition as an
equation.
Rationale:
Part A: Recognizing that 6 stars are in each box and that there are 4 boxes, the
student writes the multiplication sentence 6 × 4 = 24 or 4 × 6 = 24.
Part B: After grouping the circles into 8 groups of 3 each, one of the family of facts
3 × 8 = 24, 8 × 3 = 24, and
24
= 8 is identified.
3
12
Common Core Sample Questions
Mathematics
Common Core Sample Questions
These materials are copyrighted. Reproducing any part of these materials by anyone other than New York State school personnel is illegal.
13
Common Core Sample Questions
New York State
Testing Program
Mathematics
Common Core Sample Questions
4
The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and nonprofit
organizations or institutions to reproduce these materials for their own use, but not for sale, provided copyright notices are retained as they appear in
this publication. This permission does not apply to mass distribution of these materials, electronically or otherwise.
1
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: CR
1
Candy wants to buy herself a new bicycle that costs \$240. Candy has already saved
\$32, but she needs to make a plan so she can save the rest of the money she needs.
She decides to save the same amount of money, x dollars, each month for the next
four months.
Part A: Write an equation that helps Candy determine the amount of money
she must save each month.
Equation __________________________________
Part B: Solve the equation to find the amount of money she must save each month
to meet her goal of buying a bicycle.
Key:
(240 32)
Part A:
Part B:
4
(240 32)
4
208
4
= x or 32 + 4x = 240 or equivalent equation
=x
=x
52 = x
or other valid process.
AND
Aligned CCLS: 4.OA.3
Commentary: This question aligns to CCLS 4.OA.3 and assesses a student’s ability to solve
a multi-step word problem posed with whole numbers. It also assesses the ability to
represent a problem using an equation with a letter standing for the unknown quantity.
2
Common Core Sample Questions
Rationale: In Part A the equation includes the subtraction of \$32 from \$240 to identify
how much is needed to be saved in four months and the division of the remaining
amount, \$208, by four to represent the amount to be saved each month. Likely errors
may include dividing \$240 by four without subtracting the already saved amount of \$32
( 240 60 ) or using \$32 dollars as the amount of money saved during the first month
4
and dividing the remaining amount by three ( (240 32) 69.33 ). In Part B errors may
3
occur during the computation of the equation in Part A or may be the result of accurate
computations based on an inaccurate equation from Part A.
3
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: CR
2
Students from three classes at Hudson Valley Elementary School are planning a boat
trip. On the trip, there will be 20 students from each class, along with 11 teachers
and 13 parents.
Part A: Write an equation that can be used to determine the number of boats, b,
they will need on their trip if 10 people ride in each boat.
Equation: b =______________________________________
Part B: How many boats will be needed for the trip if 10 people ride in each boat?
Part C: It will cost \$35 to rent each boat used for the trip. How much will it cost to
rent all the boats needed for the trip?
4
Common Core Sample Questions
Key:
Part A: b =
[20(3) 11 13]
10
Part B: Work:
b=
84
10
b=8R4
The number of boats needed is 8 + 1 = 9 boats
Part C: Total Cost = 35 × 9 = 315
Aligned CCLS: 4.OA.3
Commentary: This question aligns to CCLS 4.OA.3 and assesses a student’s ability to solve a
multi-step word problem posed with whole numbers. It also tests the student’s ability to
represent the problem using an equation, with a letter standing for the unknown quantity.
It tests a student’s ability to interpret the remainder of the division problem and use this
interpretation properly to determine the number of boats as well as the total cost.
Rationale: The equation in Part A includes a calculation for the number of students who
went on the trip (20 × 3 = 60) plus the 11 teachers and 13 parents, bringing the total to 84
individuals on the trip. The number of boats, b, needed is the sum of all individuals
divided by the number of people able to sit in a single boat. In Part B, students perform
the calculation—84 is divided by 10, to get 8 R 4. The remainder of 4 indicates that an
additional boat is needed, so the number of boats needed is 8 + 1 = 9 boats. In Part C, the
total cost is the number of boats required multiplied by the cost per boat,
\$35 × 9 = \$315.
5
Common Core Sample Questions
Domain: Number and Operations—Fractions
Item: CR
3
Elena, Matthew, and Kevin painted a wall. Elena painted
Matthew painted
5
of the wall and
9
3
of the wall. Kevin painted the rest of the wall.
9
Part A: Use the box below to represent the wall. Show the fraction of the wall that
Kevin painted.
Part B: What fraction of the wall did Kevin paint? ____________________
Key:
Part A:
Kevin
1
9
AND
Part B:
1
9
6
Common Core Sample Questions
Aligned CCLS: 4.NF.3d
Commentary: This question is aligned with CCLS 4.NF.3d and assesses a student’s ability to
solve word problems involving addition and subtraction of fractions referring to the same
whole and having like denominators by using visual fraction models and equations to
represent the problem.
Rationale: The wall should be partitioned into nine equal pieces, and the portion Kevin
painted should be indicated. Determining the fraction Kevin painted may be solely
completed via the visual model. However, the use of equations may also be used to
determine the fraction Kevin painted. Matthew and Elena completed
entire job represents 1 whole,
5
3
8
+ = . Since the
9
9
9
8
8 1
is subtracted from 1 (1 − = ).
9
9 9
7
Common Core Sample Questions
Domain: Measurement and Data
Item: CR
4
The area of Ken’s rectangular garden is 480 square feet. The garden is 24 feet wide.
What is the length of fencing Ken will need to buy in order to fence in the garden
completely on all four sides?
Key:
Length of the garden:
480
24
= 20 feet
Perimeter: 2 × (20 + 24) = 88 feet
Aligned CCLS: 4.MD.3
Commentary: This question aligns to CCLS 4.MD.3 because it assesses a student’s ability to
apply the area and perimeter formulas in a real-world situation.
Rationale: Using the formula area = length × width, the length of the garden can be found
by solving the equation (480 = length × 24), dividing the area by the width of the garden:
480
24
= 20. Calculating the length of fencing needed to surround the garden on all four sides
requires the use of both length and width: 2 × (20 + 24) = 88 feet.
8
Common Core Sample Questions
Domain: Measurement and Data
Item: CR
5
Lisa recorded the approximate amount of time, in hours, it took her to do her
homework each day for 15 days.
1
4
,
1
2
3
, 0, 1,
4
,
1
4
,
1
2
, 0, 1,
1
3 1 1 3
, , , , 0,
4 4 4 4
4
In the space below, create a line plot to represent Lisa’s data. Be sure to label the
x-axis and title the line plot.
_______________________________________
Key:
Lisa’s Approximate Homework Times (in hours)
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
0
1
1
1
3
4
2
4
Aligned CCLS: 4.MD.4
Commentary: This question aligns to CCLS 4.MD.4. It assesses a student’s ability to
display a data set of measurements in fractions of a unit and make a line plot to display
that data.
Rationale: The set of data contains five unique values, including the fractions
3
4
1
4
,
1
2
, and
. All of these values are plotted on the number line, and each occurrence of that
number receives an x to represent each unique occurrence. The axis should be properly
labeled, and the line plot titled.
9
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
6
Which of the number patterns below follows the rule subtract 7 to get to the
next number?
A
B
C
D
79, 72, 56, 51, 47, 44
66, 60, 53, 45, 36, 26
51, 44, 37, 30, 23, 16
43, 36, 29, 24, 19, 12
Key: C
Aligned CCLS: 4.OA.5
Commentary: This question is aligned to CCLS 4.OA.5 and assesses a student’s ability to
generate a number pattern, based upon a given rule.
Rationale: The correct answer is Option C, because each successive term is created by the
rule “subtract 7.” The pattern in Option C is “subtract 7,” or 51 – 44 = 7,
44 – 37 = 7, and so on. An answer of Option A or D would most likely indicate that the
student did not test to see if the pattern explained how every term in the sequence was
generated. Selecting Option B would most likely indicate a mistake in subtraction or
application of the rule.
10
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
7
The first five terms in a shape pattern are shown below. The rule of the pattern is
the number of circles increases by three. Which of the following would be true of
the 6th term?
A
B
C
D
The number of circles in the 6th term would be a multiple of four.
The number of circles in the 6th term would be a prime number.
The number of circles in the 6th term would be an even number.
The number of circles in the 6th term would be divisible by five.
Key: C
Aligned CCLS: 4.OA.5
Commentary: This question is aligned with CCLS 4.OA.5 and assesses a student’s ability to
identify apparent features of the pattern that were not explicit in the rule itself.
Rationale: The correct answer is Option C, because the addition of three more circles to the
fifteen in the 5th term will result in an even number, 18. Option A could indicate a
misunderstanding of the concept of multiple. Option B could indicate a misunderstanding of
the concept of prime number, or may indicate an inaccurate association between the value
of three in the rule with three being a prime number. Option D may indicate an application
of the claim in Option D on the fifth term rather than on the unrepresented sixth term.
11
Common Core Sample Questions
Domain: Operations and Algebraic Thinking/Number and Operations—Fractions
Item: MC
8
A high school basketball team scored a total of 108 points in their final game. Joanne
scored exactly
1
of all the points the team scored, and Renee scored 23 points. How
3
many points were scored by the rest of the team?
A
B
C
D
36
49
59
85
Key: B
Aligned CCLS: 4.OA.3, 4.NF.4c
Commentary: This question is aligned with CCLS 4.NF.4c and 4.OA.3. It assesses a
student’s ability to multiply a fraction by a whole number and to solve a multi-step word
problem using addition and subtraction of whole numbers.
Rationale: The correct choice, Option B, is found by computing
1
x 108 = 36, which is the
3
number of points scored by Joanne. Since Renee scored 23 points, the total points scored by
the two girls is 36 + 23 = 59 points. Then 108 – 59 = 49, the number of points scored by the
rest of the team. If students miss the last step, they may select Option C. If they miss the last
two steps, they may select Option A. If they simply subtract 108 – 23, they will get Option D.
12
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
9
and 3 times as many peanut butter cookies. What is the total number of cookies
that Jim baked?
A
B
C
D
11
24
32
40
Key: C
Aligned CCLS: 4.OA.2
Commentary: This question is aligned to CCLS 4.OA.2 because it assesses a student’s ability
to multiply in order to solve a word problem that also includes a multiplicative comparison
(3 times as many).
Rationale: Option C is the correct answer. Jim baked 3 × 8 = 24 peanut butter cookies, and
since he also baked 8 sugar cookies, the total number of cookies baked is
24 + 8 = 32. Selecting Option B as the correct answer indicates that the 8 sugar cookies were
Option A indicates a simple addition (8 + 3 = 11) that does not incorporate the claim that
there are 3 times as many peanut butter cookies as sugar cookies. Option D may indicate
miscalculations throughout the process.
13
Common Core Sample Questions
Mathematics
Common Core Sample Questions
These materials are copyrighted. Reproducing any part of these materials by anyone other than New York State school personnel is illegal.
14
Common Core Sample Questions
New York State
Testing Program
Mathematics
Common Core Sample Questions
5
The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and nonprofit
organizations or institutions to reproduce these materials for their own use, but not for sale, provided copyright notices are retained as they appear in
this publication. This permission does not apply to mass distribution of these materials, electronically or otherwise.
Domain: Operations and Algebraic Thinking
1
Common Core Sample Questions
Domain: Operations and Algebraic Thinking
Item: MC
1
Which mathematical expression is equivalent to the number sentence below?
Subtract 15 from 45, and then divide by 3.
A
B
C
D
15-45 ÷ 3
(45-15) ÷ 3
(15-45) ÷ 3
3 ÷ (45-15)
Key: B
Aligned CCLS: 5.OA.2
Commentary: This question aligns to CCLS 5.OA.2 because the student must translate
from the statement to numerical expression, without evaluation.
Rationale: Option B is correct because it accurately shows the subtraction of 15 from 45
and the division of that difference by 3. Option A is incorrect because it does not
incorporate an understanding of order of operations. Option C is incorrect because it
subtracts 45 from 15 rather than 15 from 45 as directed by the given number sentence.
It is likely due to a direct adoption of the order of the numbers in the number sentence
and/or a misunderstanding of “15 from 45” (45-15). Option D correctly represents the
order of subtraction, but rather than dividing the difference by 3 as in Option B it
incorrectly divides 3 by the difference.
2
Common Core Sample Questions
Domain: Number and Operations in Base Ten
Item: MC
2
How many times greater is the value of the digit 5 in 583,607 than the value of the
digit 5 in 362,501?
A
B
C
D
10 times
100 times
1,000 times
10,000 times
Key: C
Aligned CCLS: 5.NBT.1
Commentary: This question aligns with CCLS 5.NBT.1 because it requires that the student
understand that the digits have different place values in the numbers provided in the
stem and that the digits differ in the two numbers by a factor of 10, as place value
suggests.
Rationale: Option C is correct. The value of 5 in 583,607 is 500,000; whereas the value of
5 in 362,501 is 500. Hence, the former is
500, 000
500
1,000 times the latter. The alternate
distractors represent miscalculations or lack of understanding of place value.
3
Common Core Sample Questions
Domain: Number and Operations in Base Ten
Item: MC
3
Nail Type By Letter Length (inches)
F
1.241
G
1.236
H
1.274
J
0.944
K
0.942
Based on the table above, which of the following comparisons of nail length is true?
A F>H
B J<K
C J>H
D G<F
Key: D
Aligned CCLS: 5.NBT.3
Commentary: This question aligns to CCLS 5.NBT.3 because it requires students to compare
two decimals to thousandths based on meanings of the digits in each place, using >, =, and
<.
Rationale: Option D is correct because nail G is shorter than nail F. Nail G is of shorter
length because while they have identical values in the ones and tenths place, nail G has a
smaller value in the hundredths place. Option A could be the result of assuming that the top
row in the table represents the largest value. Option B could represent an incorrect
comparison of the value of two decimals starting with the thousandths place rather than
the tenths place. Option C may indicate an incorrect comparison between numbers of
different place values, the tenths place with the ones place.
4
Common Core Sample Questions
Domain: Number and Operations in Base Ten
Item: MC
4
1(10,000) + 2(1,000) + 4(100) + 3(10) + 2(1) + 5(1/10) + 3(1/100).
Which number below is one-tenth of the expanded form above?
A
B
C
D
12422.53
1243.253
12432.53
12432.43
Key: B
Aligned CCLS: 5.NBT.3a and 5.NBT.2
Commentary: This question aligns with CCLS 5.NBT.3a and 5.NBT.2 because it requires
students to write decimals in equivalent forms and apply understanding of place value.
Rationale: Option B is correct.
1(10,000) + 2(1,000) + 4(100) + 3(10) + 2(1) + 5(1/10) + 3(1/100) in standard form is written
as 12432.53. One-tenth of this value moves the decimal point one position to the left to
yield 1243.253. Option A indicates an incorrect change in the tens place (decreasing the
digit in the tens place by one) rather than finding one-tenth of the number. Option C is the
correct form of the original number in standard form, but does not include the movement
of the decimal to represent one-tenth of the given number. Option D involves no movement
of the decimal and incorrectly changes the digit in the tenths place rather than taking onetenth of the number.
5
Common Core Sample Questions
Domain: Geometry
Item: MC
5
The graph below represents the location of four scientists collecting samples of
different species of plants. Dr. Schmidt is located at (3, 2), Dr. Hodge is located at
(4, 3), Dr. Alvarez is located at (6, 4), and Dr. Logan is located at (2, 4). If they want
to collect one more sample from plants located at (3, 6), which scientist is the
closest?
A
B
C
D
Dr. Hodge
Dr. Schmidt
Dr. Alvarez
Dr. Logan
Key: D
Aligned CCLS: 5.G.2
Commentary: This question aligns to CCLS 5.G.2 and assesses the student’s ability to graph
points in the first quadrant and to interpret the coordinate values of the points in the
context of the situation.
Rationale: Option D is the correct answer because Dr. Logan is the closest to the
position (3,6). Selecting Option C could indicate an incorrect plotting of the point (3,6) as
(6,3), which would place Dr. Alvarez closer. Selecting Option B could indicate a false
assumption about the proximity of the two coordinates due to their sharing an x-value.
6
Common Core Sample Questions
Selecting Option A could indicate a false assumption about Dr. Hodge’s central location,
that he is the closest to each of the other scientists.
7
Common Core Sample Questions
Domain: Measurement and Data
Item: MC
6
What is the volume, in cubic inches, of the school locker below?
A
B
C
D
2880 cubic inches
2580 cubic inches
390 cubic inches
360 cubic inches
Key: A
Aligned CCLS: 5.MD.5b
Commentary: This question aligns to CCLS 5.MD.5b and assesses the student’s ability to
apply the formula to find the volume of a rectangular prism in a real-world context.
Rationale: Option A is correct. It shows correct application of the volume formula
(12 x 8 x 30 = 2880). Option B is incorrect and would result from incorrect multiplication of
each of the dimensions of the locker. Option C is incorrect as a result of using the incorrect
operations. Option D is incorrect and would result from the student using the area formula
for a rectangle.
8
Common Core Sample Questions
Domain: Number and Operations—Fractions
Item: MC
7
Which equation below gives the correct value of the following sum?
3 14
+
8 12
A
B
C
D
3
8
+
9
24
3
7
6
+
+
=
28
24
14
12 12
10
14
=
=
37
24
17
12
3 14 17
+ =
8 12 20
Key: B
Aligned CCLS: 5.NF.1
Commentary: This question aligns to CCLS 5.NF.1 and assesses the student’s ability to add
fractions with unlike denominators.
Rationale: Option B correctly represents the creation of equivalent fractions as well as the
addition of fractions with now similar denominators. Selecting Option A suggests that
together. Selecting Option C indicates an awareness of the need for like denominators,
but does not create equivalent fractions to achieve like denominators. Selecting Option D
suggests that addition of fractions is accomplished by adding numerators together and
denominators together.
9
Common Core Sample Questions
Domain: Measurement and Data
Item: CR
8
Carson needs to purchase 5.6 meters of tape for a project. If each roll of tape
contains 80 cm and costs \$5, what is the total cost of the tape that Carson must
Show all work.
Key:
560 centimeters ÷ 80 cm = 7 rolls
7 rolls x \$5 per roll = \$35
Aligned CCLS: 5.MD.1
Commentary: This item is aligned to CCLS 5.MD.1 and assesses the student’s ability to
convert among different-sized standard measurement units (meters to centimeters), and
use the conversion in solving multi-step, real-world problems.
Rationale: The original 5.6 meters may be multiplied by a factor of 100 (to represent the
100 centimeters that compose each meter) in order to achieve similar units throughout the
following calculations (cm may be converted into meters as well). To determine the number
of rolls needed, 560 cm is divided by the 80 cm per roll to yield 7 rolls. The total cost is the
number of rolls needed, 7, times the price per roll, \$5. The total cost of 5.6 meters is \$35.
10
Common Core Sample Questions
Domain: Number and Operations—Fractions
Item: CR
9
Half of a school auditorium is needed to seat 3 equal-sized fifth grade classes.
Part A: Make a visual fraction model to represent the whole auditorium when
each class is seated in separate sections.
Part B: Write an expression to determine what fractional part of the auditorium
one fifth grade class will need.
Part C: What fraction of the auditorium will one of the fifth grade classes need?
Key:
Part A:
Auditorium
Auditorium
or equivalent model.
Part B:
Part C:
1
2
÷ 3 or
1
2
x
1
3
1
6
Aligned CCLS: 5.NF.7c
Commentary: This question is aligned to CCLS 5.NF.7c and assesses a student’s ability to
solve a real-world problem involving the interpretation of division of a unit fraction by a
non-zero whole number and computation of the quotient.
Rationale:
Part A: The correct answer correctly divides the auditorium in halves, and then divides one
half into thirds (or other equivalent models).
Part B: A correct response illustrates the ability to represent a real-world problem using a
mathematical expression, recognizing the dividend as
11
1
2
(auditorium) and the divisor as
Common Core Sample Questions
3 (classes), and to perform the proper computation of
should be computed to arrive at
1
2
÷ 3 or
1
2
÷
1
3
. The expression
1
.
3
Part C: The student uses the visual model or expression to determine the fractional part of
1
the auditorium needed by one fifth grade class, .
6
2
x
1
12
Common Core Sample Questions
Mathematics
Common Core Sample Questions
These materials are copyrighted. Reproducing any part of these materials by anyone other than New York State school personnel is illegal.
New York State
Testing Program
Mathematics
Common Core Sample Questions
6
The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and nonprofit
organizations or institutions to reproduce these materials for their own use, but not for sale, provided copyright notices are retained as they appear in
this publication. This permission does not apply to mass distribution of these materials, electronically or otherwise.
1
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: MC
1
A grocery store sign indicates that bananas are 6 for \$1.50, and a sign by the
oranges indicates that they are 5 for \$3.00. Find the total cost of buying 2
bananas and 2 oranges.
A
B
C
D
\$0.85
\$1.70
\$2.25
\$4.50
Key: B
Aligned CCLS: 6.RP.3b, 6.RP.2
Commentary: This question aligns to CCLS 6.RP.3b and 6.RP.2 because students must
find the unit price of each banana and each orange to determine the total cost of two of
each item.
Rationale: Option B is correct; two bananas cost \$0.50 and two oranges cost \$1.20.
Option A is the sum of the unit price of a banana and the unit price of an orange. Option
C is half the sum of the given sale prices. Option D is the sum of the given sale prices.
2
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: MC
2
Jeremy has two 7-foot-long boards. He needs to cut pieces that are 15 inches
long from the boards. What is the greatest number of 15-inch pieces he can cut
from the two boards?
A
B
C
D
15
10
11
12
Key: B
Aligned CCLS: 6.RP.3d
Commentary: This question aligns to CCLS 6.RP.3d because it assesses a student’s ability
to use ratios for converting measurement units and to use reasoning skills and
proportional thinking to make sense of the problem.
Rationale: Option B is correct. Converting from feet to inches, the length of one of the
boards is 7 × 12 = 84 inches. Thus, the largest number of 15-inch-long pieces that Jeremy
can cut from one board is 5, because dividing 84 by 15 yields a quotient of 5 and a
remainder of 9. It follows that the greatest number of pieces that Jeremy can cut
from the two boards is 5 + 5 = 10. Option A is the number of sections from one board.
Options C and D represent miscalculations and/or not understanding the context.
3
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: CR
3
The new floor in the school cafeteria is going to be constructed of square tiles
that are either gray or white and in the pattern that appears below:
Part A: What is the ratio of gray tiles to white tiles?
Part B: What is the ratio of white tiles to the total number of tiles in the
pattern?
Part C: If the total cost of the white tiles is \$12, what is the unit cost per white
tile?
Key:
Part A: 10 to 8, 5:4, or other equivalent ratio
Part B: 8 to 18, 4:9, or other equivalent ratio
Part C: \$1.50 per white tile
Aligned CCLS: Part A and Part B: 6.RP.1; Part C: 6.RP.2
Commentary: This question aligns to CCLS 6.RP.1 and 6.RP.2 as it assesses a student’s
ability to apply the concept of ratio in a real-world situation. It requires that the student
understand the concept and make sense of the situation.
Rationale:
Part A: The correct answer is a ratio of 10 gray tiles to 8 white tiles, or simplified, the
ratio will be 5 gray tiles to 4 white tiles.
Part B: The correct answer is a ratio of 8 white tiles to 18 total tiles, or simplified, the
ratio will be 4 white tiles to 9 tiles, in total.
Part C: Counting the tiles by color in the pattern above, it is found that there are
8 white tiles. If 8 white tiles cost \$12, then the cost per white tile is \$1.50.
4
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: CR
4
A clothing store offers a 30% discount on all items in the store.
Part A: If the original price of a sweater is \$40, how much will it cost after the
discount?
Part B: A shopper bought three of the same shirt and paid \$63 after the 30%
discount. What was the original price of one of the shirts?
Part C: Every store employee gets an additional 10% off the already discounted
price. If an employee buys an item with an original price of \$40, how
much will the employee pay?
Key:
Part A: \$28
Part B: \$30
Part C: \$25.20
Aligned CCLS: 6.RP.3c
Commentary: This question aligns to CCLS 6.RP.3c because it assesses a student’s ability to
work with percents, namely, finding a percent of a quantity in a contextual situation.
5
Common Core Sample Questions
Rationale:
Part A: The correct answer is \$28. Since 30% of 40 is 30 (40) 12 , the cost of the
100
sweater after the 30% discount is \$40 – \$12 = \$28.
Part B: The 30% discount means the shopper pays 70% of the price, or 0.7, so
63 = \$90. Since the original price of three shirts is \$90, the original price of one shirt
.7
would be \$30. The correct answer is \$30.
Part C: The correct answer is \$25.20. As shown in Part A’s rationale, applying a 30%
discount on an item that originally cost \$40 brings the price of the item down to \$28.
Applying the additional 10% employee discount on the already reduced price will
further reduce the price by 10 (28) = \$2.80, and so the final price of the item will be
100
\$28 – \$2.80 = \$25.20.
6
Common Core Sample Questions
Domain: Expressions and Equations
Item: MC
5
Represent the following expression algebraically:
A number, x, decreased by the sum of 2x and 5
A
B
C
D
(2x + 5) – x
x – (2x + 5)
x – 2x + 5
(x + 2x) – 5
Key: B
Aligned CCLS: 6.EE.2a, 6.EE.2b
Commentary: This question aligns to CCLS 6.EE.2a and 6.EE.2b because it requires the
translation from words to a multi-step algebraic expression. It also requires the
conceptualization of part of the expression as a single entity using parentheses.
Rationale: Option B is correct and is consistent with the relationship between the
minuend (x) and subtrahend (2x + 5). The expression in Option A confuses the minuend
and subtrahend, identifying the minuend incorrectly as (2x + 5). The expression in
Option C is incorrect and does not take into account the expression the sum of 2x and 5
as a single entity (2x + 5), joined through subtraction. The expression ignores the
subtraction of each term in the subtrahend, not just the term 2x. The expression in
Option D incorrectly identifies the sum of x and 2x as an expression.
7
Common Core Sample Questions
Domain: Expressions and Equations
Item: MC
6
The expression 63 × 42 is equivalent to which of the following numerical
expressions?
A
B
C
D
18 × 8
(6 × 4)5
246
216 × 16
Key: D
Aligned CCLS: 6.EE.1
Commentary: This question aligns to CCLS 6.EE.1 because it assesses a student’s ability
to translate mathematical statements that include exponents in equivalent form.
Rationale: Option D is correct. The mathematical expression in Option D correctly
interprets the exponential form of each factor: 63 = 216 and 42 = 16. Option A uses
exponents as the multiplier. Option B confuses the order of operations. Option C
misuses both the base and exponent.
8
Common Core Sample Questions
Domain: Expressions and Equations
Item: CR
7
What is the value of 2x3 + 4x2 – 3x2 – 6x when x = 3?
Show all work.
Key: 45
Aligned CCLS: 6.EE.2c
Commentary: This question aligns to CCLS 6.EE.2c because it assesses a student’s ability
to evaluate an algebraic expression when the variable is defined.
Rationale: Substituting x = 3 into the expression yields 2(33) + 4(32) – 3(32) – 6(3), which
simplifies to 45.
9
Common Core Sample Questions
Domain: Expressions and Equations
Item: CR
8
The figure below is a square with dimensions given.
2x – 1 in.
Part A: What is the perimeter of the square in terms of x?
Perimeter = ________
Part B: If the length of each side of the square is doubled, what would be the
perimeter of this new square, in terms of x?
Perimeter = ________
Part C: If x = 5, what would be the ratio of the area of the original square to the
area of the new square?
Key:
Part A: 8x – 4 or 4(2x – 1)
Part B: 16x – 8 or 4(4x – 2)
Part C: 81:324, 1:4, or any equivalent ratio
Aligned CCLS: 6.EE.2a, 6.EE.2c, 6.EE.3, 6.EE.7, 6.RP.1
Commentary: This question aligns to CCLS 6.EE.2a, 6.EE.2c, 6.EE.3, 6.EE.7, and 6.RP.1
because it assesses a student’s understanding of the simplification of algebraic expressions
as well as the concept of a ratio and the use of ratio language to describe the relationship
between two quantities. While the concept of perimeter and area is assessed at the thirdgrade level, using the concept within an algebraic form creates an on-grade-level question.
10
Common Core Sample Questions
Rationale:
Part A: Since the length of each side of the square is 2x – 1, the perimeter of the square is
the sum of the lengths of the sides of the square, or 4 times the length of each side. So
the perimeter of the square would be 4(2x – 1) = 8x – 4.
Part B: If the length of each side of the square is doubled, the length of each side
of the new square would be 2(2x – 1), or 4x – 2 inches. The perimeter would be
4 times the length of each side, so the perimeter of the new square would be
4(4x – 2) = 16x – 8.
Part C: If x = 5, the length of each side of the original square would be 9 inches. The area
of the square is equal to 9 × 9, or 81 square inches. The length of each side of the new
square is 18 inches, so the area of the new square is 324 square inches. The ratio of the
area of the original square to the area of the new square is 81:324 or 81 .
324
This could also be represented in simplified form as 1:4, 1 to 4, or 1 .
4
11
Common Core Sample Questions
Domain: Geometry
Item: MC
9
Triangle PQR and triangle QRS have vertices P(–9,7), Q(4,7), R(4,–3), and
S(10,–3).
What is the area, in square units, of quadrilateral PQSR which is formed by the
two triangles?
A 30
B 65
C 95
D 190
Key: C
Aligned CCLS: 6.G.1, 6.G.3
Commentary: This question aligns to CCLS 6.G.1 and 6.G.3 because it requires students
to determine the length of a side joining points with the same first coordinate or the
same second coordinate, and to use these side lengths to find the areas of the two
triangles.
Rationale: Option C is correct. Option A is the area of triangle QRS. Option B is the area
of triangle PQR. Option D is the incorrect area of the trapezoid (created by both
triangles) mistakenly found by (6 + 13) × 10.
12
Common Core Sample Questions
Domain: Geometry
Item: MC
10
Find the volume, in cubic feet, of the right rectangular prism pictured below.
8 5
16
B 19
C 48 3
16
D 66 1
2
A
Key: D
Aligned CCLS: 6.G.2
Commentary: This question aligns to CCLS 6.G.2 because it asks students to find the
volume of a right rectangular prism with fractional edge lengths.
1
Rationale: Option D correctly identifies the volume of the prism ( 2 3 × 8 × 3 ). Option
8
2
A is the area of the front or rear face. Option B is the area of the top or bottom face.
Option C is what students might find if they were to work with the whole numbers and
fractions separately.
13
Common Core Sample Questions
Domain: Geometry
Item: CR
11
Triangle ADE is inside rectangle ABCD. Point E is halfway between points B and C
on the rectangle. Side AB is 8 cm and side AD is 7 cm.
Part A: What is the area of triangle ADE? Show your work.
Part B: What is the ratio of the area of triangle ABE to the area of triangle ADE?
Part C: What is the ratio of the area of triangle CDE to the area of rectangle
ABCD?
Key:
Part A: 28 sq cm
Part B: 14 to 28, 1:2, or other equivalent answer
Part C: 14 to 56, 1:4, or other equivalent answer
14
Common Core Sample Questions
Aligned CCLS: 6.G.1, 6.RP.1
Commentary: This question aligns to CCLS 6.G.1 and 6.RP.1 because it assesses a
student’s ability to decompose polygons and use the information given to determine the
area of a part of the polygon. The question also assesses a student’s ability to use ratio
language to describe a ratio relationship between two quantities.
Rationale:
Part A: Using the formula to find the area of the triangle, the base of triangle ADE is
8 cm and its height is 7 cm. The area is 1 (7 × 8) = 28 sq cm.
2
Part B: The area of triangle ABE is 14 sq cm and the area of triangle ADE is 28 sq cm.
The ratio of the area of triangle ABE to the area of triangle ADE is 14:28, 1:2, or other
equivalent ratio.
Part C: The area of triangle CDE is 14 sq cm and the area of rectangle ABCD is 56 sq
cm. The ratio of the area of triangle CDE to the area of rectangle ABCD can be
represented by 14:56, 1:4, or other equivalent ratio.
15
Common Core Sample Questions
Domain: Geometry
Item: CR
12
A closed box in the shape of a rectangular prism has a length of 13 cm, a width of
5.3 cm, and a height of 7.1 cm.
Part A: Draw a net of the box and find its surface area in square centimeters.
Part B: A smaller box has dimensions that are half the measurements of the
original. Find the ratio of the surface area of the original box to the
surface area of the smaller box.
Key:
Part A: Answers may vary but should display figures similar to the diagram below:
Surface area: 2(13 × 5.3) + 2(13 × 7.1) + 2(5.3 × 7.1) = 397.66 cm2
Part B: 4:1
Aligned CCLS: 6.G.4, 6.RP.1, 6.RP.2
Commentary: This question aligns to CCLS 6.G.4, 6.RP.1, and 6.RP.2 because it asks
students to draw and use the net of a solid polyhedron to determine its surface area,
and then to find the ratio of this surface area to the surface area of a box with
dimensions that are half the size of the original.
16
Common Core Sample Questions
Rationale:
Part A: The net can be represented in a variety of configurations, as long as there are
four long rectangles all connected with two small rectangles each connected to one of
the longer rectangles. The surface area is the sum of the areas of all the faces of the
rectangular prism.
Part B: The surface area of the smaller box is 99.415 cm2. Students may use this
measure in a ratio with the surface area of the original box and divide out the
common factors. Other valid processes that result in 4:1 are also acceptable.
17
Common Core Sample Questions
Domain: Statistics and Probability
Item: MC
13
Fuel efficiency can be measured by how far, in miles, a car can travel using a
gallon of gas. The histogram below shows the fuel efficiency levels, in miles per
gallon, of 110 cars. What is the closest percentage of cars with an efficiency level
greater than or equal to 20 miles per gallon?
A
B
C
D
25%
36%
40%
44%
Key: B
Aligned CCLS: 6.SP.4, 6.RP.3c
Commentary: This question aligns to CCLS 6.SP.4 and 6.RP.3c because it is about
analyzing data from a histogram. In this process, the ability to find a percentage of a
quantity is also tested.
Rationale: Based on the data shown in the histogram, of the 110 cars considered, there
are 25 + 15 = 40 cars that have efficiency levels greater than or equal to 20 miles per
gallon. Thus, the percentage of these cars is
40
= 0.363636… = 36.36 %. Option B, 36%, is the closest one. Options A and C count
110
the frequency instead of the percentage. Option D is a misread of the histogram.
18
Common Core Sample Questions
Mathematics
Common Core Sample Questions
These materials are copyrighted. Reproducing any part of these materials by anyone other than New York State school personnel is illegal.
New York State
Testing Program
Mathematics
Common Core Sample Questions
7
The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and nonprofit
organizations or institutions to reproduce these materials for their own use, but not for sale, provided copyright notices are retained as they appear in
this publication. This permission does not apply to mass distribution of these materials, electronically or otherwise.
1
Common Core Sample Questions
Domain: Expressions and Equations
Item: MC
1
When
5
1
1
x 1 is subtracted from 1 x
8
3
4
1
5 , the result is
6
5
5
x 3
8
6
5
1
x 6
8
2
5
5
x 3
8
6
5
1
x 6
8
2
A
B
C
D
Key: B
Aligned CCLS: 7.EE.1
Commentary: This question aligns to CCLS 7.EE.1 because it assesses a student’s ability
to apply properties to subtract linear expressions with rational coefficients.
Rationale: Option B indicates the correct conversion between mixed numbers and
improper fractions, the creation of like denominators to add fractions with unlike
denominators, and the correct performance of operations, specifically subtraction.
1
1 x
4
5
x
4
5
31
6
1
6
5
x
4
5
x
8
31
6
4
3
5
x
8
and
10
x
8
31
6
5
x
8
39
6
5
x
8
6
5
x
8
1
1
3
5
x
8
4
3
8
6
1
2
2
Common Core Sample Questions
Selecting Option A indicates the addition of 4 rather than the subtraction of 4 from
3
31 ,
6
3
which is a result of the operation of subtraction not being applied to both of the
5
1
x 1 . Selection of Option C indicates an incorrect order of
8
3
1
1
5
1
subtraction between the two expressions [( x 1 ) - (1 x 5 )] as well as an error
8
3
4
6
in subtraction. Selecting Option D indicates an incorrect order of subtraction between
5
1
1
1
the two expressions [( x 1 ) - (1 x 5 )], but an accurate execution of
8
3
4
6
subtraction.
terms in the expression
3
Common Core Sample Questions
Domain: Expressions and Equations
Item: MC
2
Which expression below is equivalent to
A
B
C
D
4
x
3
2
4 ?
3
4
(x 2)
3
1
(4 x 6)
3
2
(2 x 4)
3
2
(2 x 7)
3
Key: D
Aligned CCLS: 7.EE.1
Commentary: This question aligns to CCLS 7.EE.1 because it assesses a student’s ability
to apply properties of operations to factor linear expressions with rational coefficients.
Again, the intent of 7.EE.1 is being aware of the property used to factor a linear
expression with rational coefficients, not just on the ability to factor a linear expression.
Rationale: The mathematical expression in Option D is the only choice that represents a
4
2 4
14 2
x 4
x
correct factorization of the linear expression,
= (2 x 7) .
3
3 3
3 3
Selecting Option A is the result of pulling out what appears to be a superficial similarity
between the two terms of the expression given the presence of a 4 and 3 in both terms.
Selecting Option B may indicate the selection of the unit fraction as the common factor
and a misunderstanding of factoring 1 from the mixed number 4 2 . Option C indicates
3
3
that factoring out a common term is accomplished by subtraction rather than division
2
[ 2 subtracted from both terms yields (2 x 4) ].
3
3
4
Common Core Sample Questions
Domain: Expressions and Equations
Item: MC
3
At a discount furniture store, Chris offered a salesperson \$600 for a couch and a
chair. The offer includes the 8% sales tax. If the salesperson accepts the offer,
what would be the price of the furniture, to the nearest dollar, before tax?
A
B
C
D
\$552
\$556
\$592
\$648
Key: B
Aligned CCLS: 7.RP.3, 7.EE.3
Commentary: This question aligns to CCLS 7.RP.3 because it assesses the use of
proportional relationships to solve multi-step percent problems involving tax. This also
item aligns to 7.EE.3 because it assesses the student’s ability to solve a multi-step, reallife problem with positive rational numbers and assesses the student’s ability to
Rationale: Option B is correct. The student may use proportions or linear equations to
solve this problem. To solve this algebraically, the student would need to show that the
original offer for the couch and chair, x, has been increased by 8%. The equation below
shows a possible setup for showing the unknown original value of the couch and chair,
x, with the 8% (written as a decimal) tax on the original value of the couch, which yields
the purchase price of \$600. The student must change the percent to a decimal and then
divide \$600 by 1.08. The answer then needs to be rounded to the nearest dollar.
x + .08x = 600
1.08x = 600
x = 555.56
Selecting Option A indicates a misunderstanding of when the tax is applied in the
problem. The tax is applied to an unknown value of the couch and is included in the
offer. In option A the 8% is incorrectly subtracted from the \$600 that already includes
the tax [600-(.08 x 600) = 552]. Selecting Option C indicates the subtraction of
\$8 rather than the application of 8% from the \$600 purchase price. Selecting Option D
indicates the incorrect application of the 8% sales tax to the \$600
(1.08 x 600 = 648).
5
Common Core Sample Questions
Domain: Expressions and Equations
Item: CR
4
A framed picture 24 inches wide and 28 inches high is shown in the diagram
below.
The picture will be hung on a wall where the distance from the floor to ceiling is 8 feet.
1
The center of the picture must be 5 feet from the floor. Determine the distance from
4
the ceiling to the top of the picture frame.
Key: 1
7
feet, 1 foot 7 inches, or 19 inches and appropriate work is shown.
12
Aligned CCLS: 7.EE.3
Commentary: This question aligns to CCLS 7.EE.3 because it assesses a student’s ability
to convert measurement units between forms that are appropriate, apply properties of
operations, and determine if their answer is reasonable.
6
Common Core Sample Questions
Rationale:
7
or an equivalent answer is correct. The student may choose to convert all
1
12
measures to inches, all measures to feet, or use a combination of both inches and feet.
1
1
3
The center of the picture is 5 feet from the floor, which is 8 5
2 feet from the
4
4
4
1
1
1
2
1
ceiling. The student must then subtract the
feet from the center of the
2
3
6
3
1
7
1
1
picture to the top of the picture: 2
feet.
4
6
12
7
Common Core Sample Questions
Domain: Expressions and Equations
Item: CR
5
Mandy’s monthly earnings consist of a fixed salary of \$2800 and an 18%
commission on all her monthly sales. To cover her planned expenses, Mandy
needs to earn an income of at least \$6400 this month.
Part A: Write an inequality that, when solved, will give the amount of sales
Mandy needs to cover her planned expenses.
Part B: Graph the solution of the inequality on the number line.
Key:
Part A: Mandy must sell at least \$20,000 in goods this month in order to cover her
planned expenses.
Part B:
Aligned CCLS: 7.EE.4b
Commentary: This question aligns to CCLS 7.EE.4b because it assesses a student’s ability
to write and solve a linear inequality based on a word problem, represent the solution
graphically, and interpret it in the context of the problem.
Rationale: Let x represent the amount of sales Mandy needs to make this month. It
follows that
2800 + 0.18x ≥ 6400
0.18x ≥ 3600
x ≥ 20,000
8
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: MC
6
Appliances at Discount City Store are on sale for 70% of the original price. Eli has
a coupon for an 18% discount on the sale price. If the original price of a
microwave oven is \$500, how much will Eli pay for the oven before tax?
A
B
C
D
\$440
\$287
\$260
\$240
Key: B
Aligned CCLS: 7.RP.3
Commentary: This question aligns to CCLS 7.RP.3 because it assesses a student’s ability
to compute successive percents.
Rationale: Option B is correct. This involves the application of 70% to the original price
of \$500 followed by the application of the 18% coupon on the sale price.
.70(500) = 350
350(.82) = 287
or
.70(500) = 350
350(.18) = 63
or
500 x 0.7 x 0.82 = 287
350
63
287
Selecting Option A indicates the addition of the two percents, 70% and 18%, and their
sum applied to the \$500 (.88 x 500 = 440). Selecting Option C indicates an application of
the 18% coupon to the original price and the subtraction of that from the 70% off
discount price [(500 .7) – (500 x .18) = 260]. Selecting Option D indicates an application
of each reduction in price independently and then the addition of these two separate
reductions [(.3 x 500) + (.18 x 500) = 240].
9
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: MC
7
Last summer, a family took a trip to a beach that was about 200 miles away from
their home. The graph below shows the distance driven, in miles, and the time,
in hours, taken for the trip.
What was their average speed from hour 1 to hour 4?
A 25 miles per hour
1
B 33 miles per hour
3
C 66 2 miles per hour
3
D 100 miles per hour
Key: B
Aligned CCLS: 7.RP.2b
Commentary: This question is aligned to CCLS 7.RP.2b because it assesses a student’s
ability to identify the constant of proportionality from a graph.
Rationale: Option B is correct. The student calculates that the family travels 100 miles
100 miles
1
33 miles per hour. Selecting
during this 3-hour period. Their speed is
3 hours
3
Option A indicates the division of the distance traveled, 100 miles, from hour 1 to hour 4
by four rather than three ( 100 25 ). Selecting Option C suggests the division of the
4
distance of the whole trip by the 3-hour time interval ( 200 66 2 ). Selecting Option D
3
10
3
Common Core Sample Questions
indicates the division of the distance traveled, 100 miles, by one hour or the failure to
divide the distance by time at all.
11
Common Core Sample Questions
Domain: The Number System
Item: MC
8
The numerical expression
A
B
C
D
5
6
2
6
3
1
2
3
is equal to
4
25
12
17
12
20
12
43
12
Key: A
Aligned CCLS: 7.NS.1d and 7.NS.2c
Commentary: This question aligns with CCLS 7.NS.1d and 7.NS.2c because it assesses a
student’s ability to apply properties of operations to add, subtract, and multiply rational
numbers.
Rationale: Option A is correct. The fraction is the result of a correct use of the order of
operations and execution of those operations with rational numbers.
5
6
2
6
3
5
6
5
6
2 11
3 2
11
3
3
4
17
6
25
12
1
2
3
4
3
4
3
4
or
5
12
2
3
6
3
6
4
10
48
4
9
12
12
12
12
10
48
4
9
12
25
12
Selecting Option D indicates a sign error that results in the addition rather than the
subtraction of two values (incorrectly adds 17 3 rather than subtracts 17 3 ).
6
4
6
4
Options B and C represent miscalculations or other process errors.
12
Common Core Sample Questions
Domain: Expressions and Equations
Item: CR
9
When John bought his new computer, he purchased an online computer help
service. The help service has a yearly fee of \$25.50 and a \$10.50 charge for each
help session a person uses. If John can only spend \$170 for the computer help
this year, what is the maximum number of help sessions he can use this year?
Key: 13 sessions
Aligned CCLS: 7.EE.4b
Commentary: This question is aligned to CCLS 7.EE.4b because it assesses a student’s
ability to write and solve a linear inequality based on a word problem with a real-world
application.
Rationale: If x represents the number of online help service sessions per year, then
10.5x + 25.50 ≤ 170
10.5x ≤ 144.50
x ≤ 13.76
13 sessions
13
Common Core Sample Questions
Domain: Statistics and Probability
Item: CR
10
During an experiment, the two spinners below will be spun.
Represent the sample space for this experiment. How many of these outcomes consist
of a green and an A?
Key: 6
The sample space is
(Green, A)
(Green, B)
(Green, A)
(Green, B)
(Green, A)
(Green, C)
(Yellow, A)
(Yellow, B)
(Yellow, A)
(Yellow, B)
(Yellow, A)
(Yellow, C)
(Green, A)
(Green, B)
(Green, A)
(Green, B)
(Green, A)
(Green, C)
(Red, A)
(Red, B)
(Red, A)
(Red, B)
(Red, A)
(Red, C)
Aligned CCLS: 7.SP.8b
Commentary: This question is aligned to CCLS 7.SP.8b because it assesses a student’s
ability to represent a sample space and to count the outcomes that meet certain
criteria.
Rationale: After the sample space is represented, the student can list or otherwise
identify the six outcomes that satisfy the conditions listed.
14
Common Core Sample Questions
Domain: Geometry
Item: MC
11
A scale drawing for a construction project uses a scale of 1 inch = 4 feet. The
dimensions of the rectangular family room on the scale drawing are 7.5 inches
by 12 inches.
What will be the actual area of the floor of the family room after the
construction project is completed?
A
B
C
D
90 square feet
156 square feet
360 square feet
1440 square feet
Key: D
Aligned CCLS: 7.G.1
Commentary: This question is aligned to CCLS 7.G.1 because it assesses a student’s
ability to use scale drawings and compute the area of a figure.
Rationale: Option D is correct. If the dimensions of the family room on the scale drawing
are 7.5 inches by 12 inches, the dimensions of the actual room will be 30 feet by 48 feet.
Therefore, the actual area of the family room will be 30 x 48 = 1440 square feet.
Selecting Option A indicates the multiplication of the dimensions from the scale drawing
(7.5 x 12 = 90). Selecting Option B indicates the perimeter, not the area, of the actual
room (30 + 30 + 48 + 48 = 156). Selecting Option C is the result of multiplying the
dimensions from the scale drawing (7.5 x 12 = 90) and attempting to scale the product
by multiplying by 4 (90 x 4 = 360).
15
Common Core Sample Questions
Domain: Statistics and Probability
Item: CR
12
During an experiment, three coins were tossed once.
Part A: Give the sample space to show all possible outcomes for tossing three
coins one time, using the letter H when a coin faces “heads” up, and the
letter T when it faces “tails” up.
or 3 tails?
Part C: During a math class, each of 24 students tossed three coins once. Based
on your answer to part B, how many students would you expect to get a
result of 3 heads or 3 tails?
Key:
Part A:
H
H
H
H
H
T
T
H
H
T
H
H
T
T
T
H
T
T
H
T
T
H
T
T
Part B: 2
Part C: 6
Aligned CCLS: 7.SP.8b
Commentary: This question aligns with CCLS 7.SP.8b because it assesses a student’s
ability to create a sample space of possible outcomes and to draw conclusions
(inferences) from it (7.SP.1).
Rationale: Part A:
H
H
H
H
H
T
H
T
H
T
H
H
T
T
T
H
T
T
H
T
T
H
T
T
This is the full sample space because it shows all possible outcomes of having 3 heads,
3 tails, 2 heads and 1 tail, 2 tails and 1 head.
16
Common Core Sample Questions
Part B: 2 is correct because there is one way out of 8 that gives 3 heads and one way out
of 8 that gives 3 tails.
Part C: 6 is correct because you would expect 2/8 of the 24 trials to yield 3 heads or 3
tails.
17
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: CR
13
While on vacation, a group can rent bicycles and scooters by the week. They get
a reduced rental rate if they rent 5 bicycles for every 2 scooters rented. The
reduced rate per bicycle is \$15.50 per week and the reduced rate per scooter is
\$160 per week. The sales tax on each rental is 12%.
The group has \$1600 available to spend on bicycle and scooter rentals. What is
the greatest number of bicycles and the greatest number of scooters the group
can rent if the ratio of bicycles to scooters is 5:2?
Key: 15 bicycles and 6 scooters
Aligned CCLS: 7.RP.3
Commentary: This question is aligned with CCLS 7.RP.3 because students have to use
ratios to determine the number of bicycles and scooters.
Rationale: The student may find the cost of 5 bikes plus tax = \$86.80 and the cost of
2 scooters plus tax = \$358.40 for a total of \$445.20, 1600 = 3.59, so 3 groups of 5
445.20
bikes and 2 scooters = 15 bikes and 6 scooters.
or
The student may make a table of values
Number of Bikes
5
10
15
20
Number of Scooters
2
4
6
8
Value of Bikes
77.50
155
232.50
960
310
Value of Scooters
320
640
Value of Bikes and
Scooters
397.50
795
1192.50
1590
Cost with Tax
445.20
890.40
1335.60
1780
18
1280
Common Core Sample Questions
Domain: Ratios and Proportional Relationships
Item: CR
14
David is making his own strawberry yogurt. In David’s mixture, the number of
strawberries is proportional to the amount of milk, in cups. David uses 4 cups of
milk for every 14 strawberries in his mixture.
Which equation represents the relationship between s, the number of
strawberries, and m, the number of cups of milk he uses?
A
s
B
s
C
s
D
s
1
m
10
7
m
2
2
m
5
9
m
2
Key: B
Aligned CCLS: 7.RP.2c
Commentary: This question is aligned with CCLS 7.RP.2c because the student represents
proportional relationships by equations.
Rationale: Option B is correct. The proportional relationship 4:14 is equivalent to 2:7
and is the constant of proportionality that relates the number of strawberries needed
per cups of milk. Options A, C, and D all use the wrong proportion.
19
Common Core Sample Questions
Mathematics
Common Core Sample Questions
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20
Common Core Sample Questions
New York State
Testing Program
Mathematics
Common Core Sample Questions
The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and nonprofit
organizations or institutions to reproduce these materials for their own use, but not for sale, provided copyright notices are retained as they appear in
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Grade 8 Mathematics 1 Common Core Sample Questions Domain: Functions Item: CR 1 A trainer for a professional football team keeps track of the amount of water players consume throughout practice. The trainer observes that the amount of water consumed is a linear function of the temperature on a given day. The trainer finds that when it is 90°F the players consume about 220 gallons of water, and when it is 76°F the players consume about 178 gallons of water. Part A: Write a linear function to model the relationship between the gallons of water consumed and the temperature. Part B: Explain the meaning of the slope in the context of the problem. Key: Part A: y = 3x – 50 Part B: For every one degree increase in temperature, the number of gallons consumed increases by 3. Aligned CCLS: 8.F.4 Commentary: This question aligns to CCLS 8.F.4 because it assesses a student’s ability to construct a function that models a linear relationship from a description of a relationship between two values (x,y) and interpret the rate of change. Rationale: The correct answer indicates the ability to construct a function to model a linear relationship. Given that water consumption is a function of temperature, the values cited in the problem are understood as coordinate pairs that can be related by a linear function. Part A: 220  178 42
 3
90  76 14
y = 3x +b 220 = 3(90) + b ‐50 = b 3
Part B: The slope indicates 3 gallons per degree ( ), which shows that for every
1
temperature increase in one degree, the number of gallons of water consumed would increase by three. Grade 8 Mathematics 2 Common Core Sample Questions Domain: Expressions and Equations Item: MC 2 Which of the following expressions is not equivalent to A
B
C
D
1
?
25
153 × 5–5 15–1 × 5–1 15–3 × 5 15–2 × 54 Key: D Aligned CCLS: 8.EE.1 Commentary: This question aligns to CCLS 8.EE.1 because it assesses a student’s ability to apply properties of exponents to rewrite exponential expressions. Rationale: Selecting Option D could indicate that student recognizes the incorrect addition of exponents or confusion on the concept of equivalence (54 x 5–2 = 25). Options A, B, and C involve the correct application of the properties of integer exponents. Grade 8 Mathematics 3 Common Core Sample Questions Domain: Expressions and Equations Item: CR 3 A computer can do 1000 operations in 4.5 × 10–6 seconds. How many operations can be done by this computer in one hour? Express your answer in scientific notation. Key: 8 × 1011 Aligned CCLS: 8.EE.4 Commentary: This question aligns to CCLS 8.EE.4 because it assesses a student’s ability to perform operations with numbers expressed in scientific notation. Rationale: The computer works at the rate of the 1000 operations in 4.5 × 10–6 seconds, or 2.2 × 108 multiplications per second (1000/4.5 × 10–6). Application of the conversion of 1 hour = 3600 seconds [( 2.2 × 108) x 3600] gives the number of operations (8 × 1011) the computer can complete in one hour. Grade 8 Mathematics 4 Common Core Sample Questions Domain: Expressions and Equations Item: MC x y 4 –8
–42
–3
–17
0 –2
6 28 If a line contains the points in the table above, the equation of the line is A
B
C
D
y = –2x + 5 y = 2x – 5 y = 5x – 2 y = –5x – 2 Key: C Aligned CCLS: 8.EE.6 Commentary: This question aligns to CCLS 8.EE.6 because a student uses y = mx + b to write the equation of a line given its slope and the y‐intercept. Rationale: Option C is correct. The equation of a line can be represented in slope‐
intercept form (y = mx + b) if the slope and y‐intercept is known or can be found. The slope, m, can be found by performing the following with any two pairs of the given points: m  28  (2)  30  5 . The y‐intercept, b, is given in the table as ‐2 (0,‐2). 60
6
Accurately substituting these values into the slope‐intercept form of the equation gives y = 5x – 2. Option C can also be determined by testing each of the options to determine which equation is satisfied by the set of points in the table. Selecting Option A indicates confusion in the proper location of these two values in a slope‐intercept form. Selecting Option B also indicates confusion in the proper location of these two values in a slope‐intercept form, as well as possible sign errors for the values of both the slope and the y‐intercept. Selecting Option D indicates an incorrect calculation of slope from the given table. Grade 8 Mathematics 5 Common Core Sample Questions Domain: Expressions and Equations Item: MC 5 If a line passes through the two points above, the equation of the line is A
B
C
D
y = –2x + 5 y = 2x – 5 y = 5x – 2 y = –5x – 2 Key: C Aligned CCLS: 8.EE.6 Commentary: This question aligns to CCLS 8.EE.6 because a student uses y = mx + b to write the equation of a line given its slope and the y‐intercept. Grade 8 Mathematics 6 Common Core Sample Questions Rationale: Option C is correct. The student can determine the slope graphically or algebraically and can identify (0, ‐2) as the y‐intercept from the graph. Algebraically the slope can be determined by m  8  ( 2)  10  5 . Accurately substituting these values 20
2
into the slope‐intercept form of a linear equation gives y = 5x – 2. Grade 8 Mathematics 7 Common Core Sample Questions Domain: Geometry/Expressions and Equations Item: CR 6 In the diagram below, ∆ABC is similar to ∆ART. Part A: What is the scale factor from ∆ABC to ∆ART? Part B: If the slope of AC is –2, what is the value of x for coordinate C? Part C: Using the information from parts A and B, what is the length of RT? Key: Part A: AB 8

AR 5
Part B: 4 Part C: 2.5 Aligned CCLS: 8.G.4, 8.EE.6, and 8.EE.7b Commentary: This question aligns to CCLS 8.G.4, 8.EE.6, and 8.EE.7b because it assesses the construction and application of a similarity ratio, the creation of a linear equation, and solving a linear equation with one variable. Grade 8 Mathematics 8 Common Core Sample Questions Rationale: Part A: The ratio of side AB to side AR is determined by AB 8  0 8


AR 8  3 5
Part B: The y‐intercept is (0,8) and the given slope of –2 yields the resulting linear equation for segment AB of y = –2x + 8. Solving this equation for y = 0 yields the following value for C: 0 = –2x + 8 –8 = –2x
4 = x x = 4 Part C: The length of side BC is the difference in x‐values between point B and 8
point C, 4 – 0 = 4. The ratio of side BC to side RT is . Using these two 5
pieces of information the solution to side RT can be found by solving the 8 4
proportion  .
5 x
8 4

5 x
8x = 20 x = Grade 8 Mathematics 20 5
= = 2.5 8 2
9 Common Core Sample Questions Domain: Expressions and Equations Item: CR 7 In the coordinate plane below, ∆ABC is similar to ∆AEF. What is the value of x ?
Key: x = 4 Aligned CCLS: 8.EE.6 Commentary: This question aligns to CCLS 8.EE.6 because it assesses the student’s understanding that slope is the same along a line between any two distinct points. 11  2
3
to find the slope of AC  . Next, the 60
2
3 6
8 2 6
 , and then the student will set ratios equal 
student finds the slope, FA 
x0 x
2 x
to find x = 4.
Rationale: The student can compute Grade 8 Mathematics 10 Common Core Sample Questions Domain: Expressions and Equations Item: MC 2
1
1
(2 x 1)  2  7  x
3
3
2
8 Which step would not be a possible first step for solving this equation algebraically? A multiplying every term in the equation by six 1
B subtracting 2 from 7
3
1
C subtracting x from 2x 2
2
D multiplying –1 by
3
Key: C Aligned CCLS: 8.EE.7b Commentary: This question aligns to CCLS 8.EE.7b because it assesses the student’s ability to use the distributive property and to combine like terms when solving an equation. 2
, distribution or
3
1
some other algebraic beginning that would be necessary before subtracting x from 2
2x. Options A, B, and D all represent reasonable starting points. Rationale: Option C is correct. Given that 2x is multiplying a factor of Grade 8 Mathematics 11 Common Core Sample Questions Domain: Expressions and Equations Item: CR 9 David currently has a square garden. He wants to redesign his garden and make it into a rectangle with a length that is 3 feet shorter than twice its width. He decides that the perimeter should be 60 feet. Determine the dimensions, in feet, of his new garden. Show your work. Key: 11 feet wide and 19 feet long Aligned CCLS: 8.EE.7b Commentary: This question aligns to CCLS 8.EE.7b because it assesses the student’s ability to find the perimeter of a rectangle by expanding expressions using the distributive property and collecting terms. Rationale: Width = 11 and length = 19 produces a rectangle with a perimeter of 60. The length is 3 feet shorter than twice the width. Let w = width 2w – 3 = length 2(w + 2w – 3) = 60 2w + 4w – 6 = 60 6w = 66 w = 11 2w – 3 = 19 Other processes may also result in the correct answer. Grade 8 Mathematics 12 Common Core Sample Questions Domain: Functions Item: CR 10 The three different linear functions below are represented in three different
ways, as shown.
Which function has the greatest rate of change? Does any pair of functions have the same rate of change? Justify your answer. Key: The linear function in I has the greatest rate of change of the three given functions. 3
The linear functions in II and III each have a rate of change of . 2
Aligned CCLS: 8.F.2 Commentary: This question aligns to CCLS 8.F.2 because it assesses a student’s ability to recognize and compare properties of functions represented in different ways: table of values, graphically, and algebraically. Rationale: I – The rate of change is 2. II and III – The rate of change for each is
. 2
Common Core Sample Questions Domain: Functions Item: MC 11 Of the four linear functions represented below, which has the greatest rate of
change?
Key: D Aligned CCLS: 8.F.2 Commentary: This question aligns to CCLS 8.F.2 because it assesses a student’s ability to compare rates of changes for functions represented in different ways. Rationale: Option D is correct because the rate of change is 5 ; in Option A it is 2, in
2
Option B it is 4 , and in Option C it is 7 .
3
3
Mathematics
Common Core Sample Questions
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Grade 8 Mathematics 15 Common Core Sample Questions ```