Mathematics Common Core Sample Questions

Transcription

1 New York State Testing Program Mathematics Common Core Sample Questions Grade3 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. Grade 3 Mathematics 1 Common Core Sample Questions

2 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. Key: Part A Key: Part B = = = ( ) = (21 3) = 7 Part B: If the product of two whole numbers greater than 1 is 63, what could the two whole numbers be?, = = = (3 7) = 21 7 or 7 (7 3) = (21 3) 9 = 7 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. Grade 3 Mathematics 2 Common Core Sample Questions

3 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 16 B 10 C 18 D 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. Grade 3 Mathematics 3 Common Core Sample Questions

4 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 14 B 36 C 48 D 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). Grade 3 Mathematics 4 Common Core Sample Questions

5 Domain: Number and Operations Fractions Item: CR 4 Give a fraction that represents each point on the string compared to the whole. A Point A B Point B C Point C D Identify another fraction that is equivalent to the fraction represented by point A. E Identify another fraction that is equivalent to the fraction represented by point C. Key A B C 1, or fraction equivalent 4 1, or fraction equivalent 2 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 b on a number line. It also aligns to 3.NF.2b and assesses the student s ability to represent a fraction a b on the number line and also 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 Grade 3 Mathematics 5 Common Core Sample Questions

6 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. Grade 3 Mathematics 6 Common Core Sample Questions

7 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 cookies. If each student needed to sell 4 boxes of cookies, how many cookies did each student need to sell? A 28 B 56 C 112 D 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. Grade 3 Mathematics 7 Common Core Sample Questions

8 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 2 B 6 C 18 D 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 ( 54 9 = 6 per tray) and then determining the number of trays per class ( 9 = 3 trays per 3 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 9 = 6 per tray) and incorrectly dividing the number of apples per 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). Grade 3 Mathematics 8 Common Core Sample Questions

9 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 B C D 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 6 as 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 6 of all the pictures in the 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. Grade 3 Mathematics 9 Common Core Sample Questions

10 Domain: Operations and Algebraic Thinking Item: MC 8 20 n = 5 and n 5 = 20. What is n? A 4 B 5 C 8 D 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 = 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. Grade 3 Mathematics 10 Common Core Sample Questions

11 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 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 a b as the quantity formed by a parts of 1 b. Rationale: Option B is the correct answer. If a total of crayons (6) are on the students desks, then 8 6 = 2 crayons are still in the box. This can be represented fractionally as 8 2 of the box. Option A could indicate a student selected the unit fraction 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. Grade 3 Mathematics 11 Common Core Sample Questions

12 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. o o o o o o o o o o o o o o o o o o o o o o o o Answer: Key Part A 4 6 = 24 or 6 4 = 24 Part B o o o o o o o o o o o o o o o o o o o o o o o o 3 8 = 24 or 8 3 = 24, or 24 3 = 8 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 3 = 8 is identified. Grade 3 Mathematics 12 Common Core Sample Questions

13 Grade 3 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. Grade 3 Mathematics 13 Common Core Sample Questions

14 New York State Testing Program Mathematics Common Core Sample Questions Grade4 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. Grade 4 Mathematics 1 Common Core Sample Questions

15 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. Show your work. Answer $ Key: Part A: (240 32) 4 = x or x = 240 or equivalent equation Part B: (240 32) 4 = x = x 52 = x or other valid process. AND Answer: $52 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. Grade 4 Mathematics 2 Common Core Sample Questions

16 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 ( ) or using $32 dollars as the amount of money saved during the first month 4 and dividing the remaining amount by three ( ( ) ). In Part B errors may 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 Grade 4 Mathematics 3 Common Core Sample Questions

17 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? Show your work. Answer: boats 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? Show your work. Answer: $ Grade 4 Mathematics 4 Common Core Sample Questions

18 Key: Part A: b = [20(3) ] Part B: Work: 84 b = 10 b = 8 R 4 The number of boats needed is = 9 boats Answer: 9 boats Part C: Total Cost = 35 9 = 315 Answer: $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 = 9 boats. In Part C, the total cost is the number of boats required multiplied by the cost per boat, $35 9 = $315. Grade 4 Mathematics 5 Common Core Sample Questions

19 Domain: Number and Operations Fractions Item: CR 3 Elena, Matthew, and Kevin painted a wall. Elena painted 5 9 of the wall and Matthew painted 3 9 of the wall. Kevin painted the rest of the wall. 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 Grade 4 Mathematics 6 Common Core Sample Questions

20 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 = 8. Since the 9 entire job represents 1 whole, 8 9 is subtracted from 1 (1 8 9 = 1 9 ). Grade 4 Mathematics 7 Common Core Sample Questions

21 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? Show your work. Answer: feet Key: Length of the garden: = 20 feet Perimeter: 2 ( ) = 88 feet Answer: 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 = 20. Calculating the length of fencing needed to surround the garden on all four sides 24 requires the use of both length and width: 2 ( ) = 88 feet. Grade 4 Mathematics 8 Common Core Sample Questions

22 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, 0, 1, 3 4, 1 4, 1 2, 0, 1, 3 1, 4 4, 1 4, 3 4, 0, 1 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 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 1 4, 1 2, and 3. All of these values are plotted on the number line, and each occurrence of that 4 number receives an x to represent each unique occurrence. The axis should be properly labeled, and the line plot titled. Grade 4 Mathematics 9 Common Core Sample Questions

23 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 79, 72, 56, 51, 47, 44 B 66, 60, 53, 45, 36, 26 C 51, 44, 37, 30, 23, 16 D 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 = 7, = 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. Grade 4 Mathematics 10 Common Core Sample Questions

24 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. Grade 4 Mathematics 11 Common Core Sample Questions

25 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 3 1 of all the points the team scored, and Renee scored 23 points. How many points were scored by the rest of the team? A 36 B 49 C 59 D 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 3 1 x 108 = 36, which is the number of points scored by Joanne. Since Renee scored 23 points, the total points scored by the two girls is = 59 points. Then = 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 , they will get Option D. Grade 4 Mathematics 12 Common Core Sample Questions

26 Domain: Operations and Algebraic Thinking Item: MC 9 Jim baked sugar cookies and peanut butter cookies. He baked 8 sugar cookies and 3 times as many peanut butter cookies. What is the total number of cookies that Jim baked? A 11 B 24 C 32 D 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 = 32. Selecting Option B as the correct answer indicates that the 8 sugar cookies were not added to the number of peanut butter cookies to get the total number of cookies. 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. Grade 4 Mathematics 13 Common Core Sample Questions

27 Grade 4 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. Grade 4 Mathematics 14 Common Core Sample Questions

28 New York State Testing Program Mathematics Common Core Sample Questions Grade5 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 Grade 5 Mathematics 1 Common Core Sample Questions

29 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 (45-15) 3 C (15-45) 3 D 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. Grade 5 Mathematics 2 Common Core Sample Questions

30 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 10 times B 100 times C 1,000 times D 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 1,000 times the latter. The alternate 500 distractors represent miscalculations or lack of understanding of place value. Grade 5 Mathematics 3 Common Core Sample Questions

31 Domain: Number and Operations in Base Ten Item: MC 3 Nail Type By Letter Length (inches) F G H J K 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. Grade 5 Mathematics 4 Common Core Sample Questions

32 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 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 One-tenth of this value moves the decimal point one position to the left to yield 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. Grade 5 Mathematics 5 Common Core Sample Questions

33 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 Dr. Hodge B Dr. Schmidt C Dr. Alvarez D 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. Grade 5 Mathematics 6 Common Core Sample Questions

34 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. Grade 5 Mathematics 7 Common Core Sample Questions

35 Domain: Measurement and Data Item: MC 6 What is the volume, in cubic inches, of the school locker below? A 2880 cubic inches B 2580 cubic inches C 390 cubic inches D 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. Grade 5 Mathematics 8 Common Core Sample Questions

36 Domain: Number and Operations Fractions Item: MC 7 Which equation below gives the correct value of the following sum? A B C D = = = = 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 addition of fractions is accomplished by adding numerators together and denominators 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. Grade 5 Mathematics 9 Common Core Sample Questions

37 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 buy? Show all work. Answer: $ 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. Grade 5 Mathematics 10 Common Core Sample Questions

38 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: or 1 2 x 1 3 Part C: 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 1 2 (auditorium) and the divisor as Grade 5 Mathematics 11 Common Core Sample Questions

39 3 (classes), and to perform the proper computation of or The expression 3 should be computed to arrive at 1 2 x 1 3. Part C: The student uses the visual model or expression to determine the fractional part of the auditorium needed by one fifth grade class, 1 6. Grade 5 Mathematics 12 Common Core Sample Questions

40 Grade 5 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.

41 New York State Testing Program Mathematics Common Core Sample Questions Grade6 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. Grade 6 Mathematics 1 Common Core Sample Questions

42 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 $0.85 B $1.70 C $2.25 D $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. Grade 6 Mathematics 2 Common Core Sample Questions

43 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 15 B 10 C 11 D 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 = 10. Option A is the number of sections from one board. Options C and D represent miscalculations and/or not understanding the context. Grade 6 Mathematics 3 Common Core Sample Questions

44 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? Answer: Part B: What is the ratio of white tiles to the total number of tiles in the pattern? Answer: Part C: If the total cost of the white tiles is $12, what is the unit cost per white tile? Answer: $ 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. Grade 6 Mathematics 4 Common Core Sample Questions

45 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? Show your work. Answer: 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? Show your work. Answer: 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? Show your work. Answer: 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. Grade 6 Mathematics 5 Common Core Sample Questions

46 Rationale: Part A: The correct answer is $28. Since 30% of 40 is 30 (40) , the cost of the 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 $ 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 = $ Grade 6 Mathematics 6 Common Core Sample Questions

47 Domain: Expressions and Equations Item: MC 5 Represent the following expression algebraically: A number, x, decreased by the sum of 2x and 5 A (2x + 5) x B x (2x + 5) C x 2x + 5 D (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. Grade 6 Mathematics 7 Common Core Sample Questions

48 Domain: Expressions and Equations Item: MC 6 The expression is equivalent to which of the following numerical expressions? A 18 8 B (6 4) 5 C 24 6 D 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: 6 3 = 216 and 4 2 = 16. Option A uses exponents as the multiplier. Option B confuses the order of operations. Option C misuses both the base and exponent. Grade 6 Mathematics 8 Common Core Sample Questions

49 Domain: Expressions and Equations Item: CR 7 What is the value of 2x 3 + 4x 2 3x 2 6x when x = 3? Show all work. Answer: 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(3 3 ) + 4(3 2 ) 3(3 2 ) 6(3), which simplifies to 45. Grade 6 Mathematics 9 Common Core Sample Questions

50 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? Answer: 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. Grade 6 Mathematics 10 Common Core Sample Questions

51 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 This could also be represented in simplified form as 1:4, 1 to 4, or 4 1. Grade 6 Mathematics 11 Common Core Sample Questions

52 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. Grade 6 Mathematics 12 Common Core Sample Questions

53 Domain: Geometry Item: MC 10 Find the volume, in cubic feet, of the right rectangular prism pictured below. A B 19 C D 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. Rationale: Option D correctly identifies the volume of the prism ( ). 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. Grade 6 Mathematics 13 Common Core Sample Questions

54 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 Grade 6 Mathematics 14 Common Core Sample Questions

55 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 2 1 (7 8) = 28 sq cm. 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. Grade 6 Mathematics 15 Common Core Sample Questions

56 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. Answer: 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. Answer: Key: Part A: Answers may vary but should display figures similar to the diagram below: Part B: 4:1 Surface area: 2(13 5.3) + 2(13 7.1) + 2( ) = cm 2 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. Grade 6 Mathematics 16 Common Core Sample Questions