Grade 4 Math Algebra Questions &Answers (keys/scoring guides)

additional materials [1]

Additional Materials [2]

Additional Materials [3]

Additional Materials [4]

 

 

  • What number should be put in the box to make the number sentence above true?

 

Answer:_________________________

 

 

 

Scoring Guide

Sample Correct Responses:

29

OR

29 − 8 = 21

OR

29 in the box and response area blank.

 

Score &Description

Correct

Correct response

Incorrect

Incorrect response

  • Write the next two numbers in the number pattern.

 

             1    6   4   9   7   12   10   ____    ____

 

 

 

Write the rule that you used to find the two numbers you wrote.

 

Scoring Guide

Sample Correct Responses:

15,13

Add 5 to previous number for even places and subtract 2 from previous number to get odd places

 

OR

 

+5,−2

 

OR

 

The odd terms increase by 3 and the even terms increase by 3. ("Every other number skips 3"is acceptable.)

 

 

Score &Description

Correct

Correct response

Partial

Gives 15 and 13 but not the rule

OR

Just gives evidence that they know the rule but omits or gets incorrect the 15 and/or 13

OR

Gives 8 and 13 with rule −2,+ 5

Incorrect

Incorrect response

 

 

  • What number does  n represent in the table?
    • 2
    • 3
    • 4

 

Key

 B

  • Kiara set her beads on a table to make a repeating pattern. Some of the beads rolled off the table.

 

                                                    Here is what was left. 

 

 

 

Which of the following should Kiara use to replace the missing beads in the pattern?

 

 

 

 

 

 

 

    •  

 

 

 

 

 

Key

 C

 

 

  • A pattern of dots is shown above. How many dots would be in the 6th picture?

 

Answer:________________

 

Explain how you found your answer.

 

Scoring Guide

Sample Correct Responses:

21 dots. The 5th picture would have 15 dots,5 more than the 4th. The 6th picture would have 21 dots,6 more than the 5th. An acceptable explanation may be given by drawings,words,and/or numbers.

Pictures drawn by the students need to follow the pattern of dots shown in the first four pictures in order to be considered correctly drawn. Just having the correct number of dots (but not following the pattern) is not sufficient to earn full credit.

 

 

Score &Description

Correct

Correct response:21 dots with a correct explanation.

There are two types of correct explanations.

 

• First type:The geometric explanation. For this,the student has presented correct drawings for the 5th and 6th pictures,the numbers 15 and 21,and a correct supporting explanation.

• Second type:The numerical explanation. For this,the student has not presented any drawings for the 5th and 6th pictures,but does have the correct numbers (15 and 21) with a correct supporting explanation like "It goes +2 + 3 + 4 + 5 + 6."

 

Partial 2

Partially correct response

21 dots with no explanation,a partially correct explanation,or a partially complete explanation.

 

Examples of partially correct or partially complete explanations include the following:

 

• You add one more dot to the bottom.

• You add 1 each time.

• It increased by 1 each time.

• I counted the numbers in order.

 

Partial 1

Partially correct response

Does not get 21 but has a correct explanation.

Examples of correct explanations include the following:

 

• Because the bottom keeps changing to the next number

• It goes + 2 + 3 + 4 + 5 + 6.

 

Incorrect

Incorrect response

 

 

  • The weights on the scale above are balanced. Each cube weighs 3 pounds. The cylinder weighs  N pounds. Which number sentence best describes this situation?
    • 6 +  N = 12
    • 6 +  N = 4
    • 2 +  N = 12
    • 2 +  N = 4

 

Did you use the calculator on this question?

 

Key

 A

 

 

  • The table shows how the "In"numbers are related to the "Out"numbers. When 38 goes in,what number comes out?
    • 41
    • 51
    • 54
    • 77

 

Did you use the calculator on this question?

 

Key

 D

  • Paco had 32 trading cards. He gave  N trading cards to his friend. Which expression tells how many trading cards Paco has now?
    • 32 +  N
    • 32 -  N
    • N –32
    • 32 ÷  N 

 

Key

 B

 

 

  • In the pattern shown above,which of the following would go into the blank space?

 

 

 

 

 

 

 

Key

 A

14,26,38,______ ,______

 

  1. The numbers in the pattern above are increasing by 12. Which of these numbers is part of the pattern?
    • 52
    • 58
    • 60 
    • 62

 

Key

 D

 

 

  1. Which of the figures below should be the fourth figure in the pattern shown above?

 

 

 

 

Key

 A

  1. From the starting point on the grid below,a beetle moved in the following way. It moved 1 block up and then 2 blocks over,and then continued to repeat this pattern. Draw lines to show the path the beetle took to reach the right side of the grid.

 

Scoring Guide

Sample Correct Responses:

 

Score &Description

Correct

Correct response (Note:may continue pattern beyond endpoint. i.e.,   ).

Partial

After an incorrect start,drawing picks up and completes the correct pattern. A minimum of three consecutive correct iterations must be shown.

Incorrect

Incorrect response.

Question 13 refers to Additional Materials [2]                                                  

The following question refers to the number tiles or the paper strip. Please remove the 10 number tiles and the paper strip from your packet and put them on your desk.

 

 

  1. Jan entered four numbers less than 10 on his calculator. He forgot what his second and fourth numbers were. This is what he remembered doing.

 

 

 

 

List a pair of numbers that could have been the second and fourth numbers.

(You may use the number tiles to help you.)

 

 

________  , ________

 

 List a different pair that could have been the second and fourth numbers.

 

 

________  , ________

 

 

Scoring Guide

Sample Correct Responses:

The second and fourth numbers could be:

 

Score &Description

Correct

Correct response (must not include an incorrect response on the answer lines).

Partial

Gives only one correct combination for the 2nd and 4th numbers.

Incorrect

Incorrect response.

 

                             3,6,5,8,7,10,9,?

 

 

  1. In the number pattern above,what number comes next?

Answer:_________

 

Scoring Guide

Sample Correct Responses:

12

 

Score &Description

Correct

Correct response (correctly extends pattern with given numbers).

Incorrect

Incorrect response (i.e.,writing more than one number in answer space).

  1. N stands for the number of hours of sleep Ken gets each night. Which of the following represents the number of hours of sleep Ken gets in 1 week?
    • N + 7
    • N –7
    • N x 7
    • N ÷ 7

 

Key

 C

                              

 

 

  1. Which rule describes the pattern shown in the table?

 

Key

 C

  1. Peter wrote down a pattern of A’s and B’s that repeats in groups of 3. Here is the beginning of his pattern with some of the letters erased. Fill in the missing letters.

 

 

 

 

Scoring Guide

Sample Correct Responses:

 

Score &Description

Correct

Correct response

Incorrect

Any incorrect response

In this question the student needed to fill in missing entries in a pattern of the letters "A"and "B."The student was told that the pattern repeats in groups of 3,and enough information was given for the student to find the one correct sequence.

 

  1. The objects on the scale above make it balance exactly. According to this scale,if   balances   ,then   balances which of the following?

 

Key

 B

  1. What are all the whole numbers that make 8 -   >3 true?
    • 0,1,2,3,4,5
    • 0,1,2,3,4
    • 0,1,2
    • 5

 

Key

 B

The following question refers to the situation described below.

 

A school yard contains only bicycles and wagons like those in the figure above.

 

 

  1. On Monday there were 3 bicycles and 2 wagons in the school yard. How many wheels were in the school yard?

Answer:

__________________________________________________________________

 

 

 

Did you use the calculator on this question?

 

Scoring Guide

Sample Correct Responses:

3 bicycles and 2 wagons have 6 + 8 = 14 wheels

 

Possible acceptable responses include:14,14 wheels,14 wheels in yard

 

Score &Description

Correct

Correct response

Partial

At least one correct conclusion,such as 3 bicycles implies 6 wheels,but answer not correct

Incorrect

Incorrect response

In this question the student needed to determine the total number of wheels on 3 bicycles and 2 wagons. The student needed to know or observe from the picture that each bicycle has 2 wheels and each wagon has 4 wheels,and then determine and carry out an appropriate method for determining the total number of wheels. Students were permitted to use a calculator.

The following question refers to the situation described below.

 

A school yard contains only bicycles and wagons like those in the figure above.

 

 

  1. On Tuesday the total number of wheels in the school yard was 24. There are several ways this could happen.

 

a.

How many bicycles and how many wagons could there be for this to happen?

 

Number of bicycles ________

 

Number of wagons ________

b.

Find another way that this could happen.

 

Number of bicycles ________

 

Number of wagons ________

 

 

 

Did you use the calculator on this question?

 

Scoring Guide

Sample Correct Responses:

Any  two of the following correct responses:

 

0 bicycles,6 wagons

2 bicycles,5 wagons

4 bicycles,4 wagons

6 bicycles,3 wagons

8 bicycles,2 wagons

10 bicycles,1 wagon

12 bicycles,0 wagons

 

Score &Description

Correct

Two correct responses

Partial

One correct response,either for part a or part b

OR

same correct response in both parts

Incorrect

Incorrect responses

In this question the student was told that there were bicycles and wagons in a school yard and that there were a total of 24 wheels. The student was asked to find two different combinations of bicycles and wagons that would have a total of 24 wheels. The student needed to know or observe from the picture that each bicycle has 2 wheels and each wagon has 4 wheels,and then determine two ways there could be a total of 24 wheels. Students were permitted to use a calculator.

  1. The table below shows how the chirping of a cricket is related to the temperature outside. For example,a cricket chirps 144 times each minute when the temperature is 76°.

Number Of Chirps Per Minute

Temperature

144

76°

152

78°

160

80°

168

82°

176

84°

What would be the number of chirps per minute when the temperature outside is 90° if this pattern stays the same?

Answer:_________________________

Explain how you figured out your answer.

 

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

 

Did you use the calculator on this question?

 

Scoring Guide

Sample Correct Responses:

200

For every 2° that the temperature increases,the number of chirps increases by 8.

 

Score &Description

Extended

Answers 200 with explanation that indicates number of chirps increases by 8 for every temperature increase of 2°.

Satisfactory

Gives explanation that describes ratio,but does not carry process far enough (e.g.,gives correct answer for 86° (184) or 88° (192) or carries process too far (answers 208)).

OR

Answers 200 and shows 184 86°,192 88°,200 90° but gives no explanation.

OR

Answers 200 with explanation that is not stated well but conveys the correct ratio.

OR

Gives clear description of ratio and clearly has minor computational error (e.g.,adds incorrectly).

Partial

Answers between 176 and 208,inclusive,with explanation that says chirps increase as temperature increases.

OR

Answers between 176 and 208,inclusive,with explanation that they counted by 8 (or by 2).

OR

Uses a correct pattern or process (includes adding a number 3 times or showing 184 and 86 in chart) or demonstrates correct ratio.

OR

Has half the chart with 200 on the answer line.

OR

"I added 24"(with 200 on answer line).

Minimal

Answers between 176 and 208,inclusive,with no explanation or irrelevant or incomplete explanation.

OR

Has explanation that number of chirps increases as temperature increases but number is not in range.

OR

Has number out of range but indicates part of the process (e.g.,I counted by 8′s)

OR

Explanation—as temperature increases the chirps increase but number is out of range.

Incorrect

Incorrect response.

This question required students first to recognize a pattern and then extend the pattern for three more values. In addition,students were asked to explain how they arrived at the answer. The pattern was linear in two variables—number of chirps and temperature. Both algebraic and numerical reasoning were used to obtain the answer to this question. Students were permitted to use a calculator.

 

 

M = 2

K = 6

L = 3

 

 

 

 

  1. What is K + L –M?
    • 1
    • 5
    • 7
    • 11

 

Key

 C

A point is shown on the grid below. The coordinates of the point are (2,5).

 

 

  1. On the same grid draw the point with coordinates (4,7) and the point with coordinates (8,0).

 

Scoring Guide

Sample Correct Responses:

 

Score &Description

Correct

Correct response (both points are correctly drawn)

Partial

Point (4,7) only is correctly drawn

OR

Point (8,0) only is correctly drawn

 

Incorrect

Incorrect response

In this question the student was asked to locate 2 points on a coordinate grid. A sample point illustrated how the ordered pair is related to the axes for the first and second numbers in the pair.

  1. N stands for the number of stamps John had. He gave 12 stamps to his sister. Which expression tells how many stamps John has now?
    • N+12
    • N-12
    • 12-N
    • 12 x N

 

Key

 B

 

  1. In the pattern above,which figure would be next?

 

Key

 B

  1. If    represents the number of newspapers that Lee delivers each day,which of the following represents the total number of newspapers that Lee delivers in 5 days?
    •  
    •  

 

Key

 B

 

 

Puppy’s Age

Puppy’s Weight

1 month

10 lbs.

2 months

15 lbs.

3 months

19 lbs.

4 months

22 lbs.

5 months

?

 

 

 

  1. John records the weight of his puppy every month in a chart like the one shown above. If the pattern of the puppy’s weight gain continues,how many pounds will the puppy weigh at 5 months?
    • 30
    • 27
    • 25
    • 24

 

Key

 D

 

  1. Children’s pictures are to be hung in a line as shown in the figure above. Pictures that are hung next to each other share a tack. How many tacks are needed to hang 28 pictures in this way?
    • 27
    • 28
    • 29
    • 56

 

Key

 C

 

  1. If the pattern shown continues,could 375 be one of the products in this pattern?

 

 

Explain why or why not.

 

 

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

 

Did you use the calculator on this question?

 

Scoring Guide

 

Score &Description

Correct

Correct reason.

e.g,Because 375 is not divisible by 2 (or is not even)

     OR

Because 375 is between two of the numbers in the pattern.

 

Accept:"Because 375 is uneven."

 

 

Not acceptable:

"Because you’re counting by 2′s."

     OR

"Because you’re multiplying by 2."

 

Incorrect

Any incorrect or incomplete reason.

  1. In 1990 a school had 125 students. Each year the number of students in the school increases by 50. Fill in the table to show the number of students expected for each year.

 

 

Year

Number of Students

1990

125

1991

     

1992

     

1993

     

 

 

 

 

Did you use the calculator on this question?

 

Scoring Guide

 

Score &Description

Correct

All three years correct.

 

 

 

Year

Number of Students

1990

125

1991

175

1992

225

1993

275

 

 

Incorrect #3

Only two years are correct.

Incorrect #2

Only one year is correct.

Incorrect #1

Incorrect or incomplete response for all three years.

*The use of more than one incorrect category in this question enabled NAEP to gather data on common student errors. Any response that fell into one of the incorrect categories earned no credit.

 

  1. What is a rule used in the table to get the numbers in column  B from the numbers in column  A?
    • Divide the number in column  A by 4.
    • Multiply the number in column  A by 4.
    • Subtract 9 from the number in column  A.
    • Add 9 to the number in column  A.

 

Did you use the calculator on this question?

 

Key

 A

 

  1. Suppose 120 is a number in column  A of the table. Use the same rule to fill in the number in column  B.

 

 

 

 

Did you use the calculator on this question?

 

Scoring Guide

 

Score &Description

Correct

30

Incorrect #4

129

Incorrect #3

111

Incorrect #2

480

Incorrect #1

Any incorrect response other than those described in Incorrect #2,Incorrect #3,Incorrect #4.

*The use of more than one incorrect category in this question enabled NAEP to gather data on common student errors. Any response that fell into one of the incorrect categories earned no credit.

  1. On the grid below,the dot at (4,4) is circled. Circle two other dots where the first number is equal to the second number.

 

 

 

 

Scoring Guide

 

Score &Description

Correct

Any two (or more) of the correct dots should be circled,and no others.

Incorrect #2

One correct dot circled and no incorrect dots circled.

Incorrect #1

Any incorrect response other than as described in Incorrect #2.

*The use of more than one incorrect category in this question enabled NAEP to gather data on common student errors. Any response that fell into one of the incorrect categories earned no credit.

 

  1. On the map above,which ordered pair gives the location of the house?
    • (D,1)
    • (D,4)
    • (A,1)
    • (A,3)

 

Key

 D

 

  1. A pattern of shapes is to be repeated many times. The figure above shows one completed pattern and the beginning of the next. What shape comes next?

 

Did you use the calculator on this question?

 

Key

 A

  1.  

17 ×   = 204

 

In the number sentence above,what number belongs in the ?

 

Answer: ____________________

 

 

Did you use the calculator on this question?

 

Scoring Guide

 

Score &Description

Correct

12    or    17 × 12 = 204

Incorrect

Incorrect Answer

  1. The table below shows some number pairs. The following rule was used to find each number in column B.

Rule:Multiply the number in column A by itself and then add 3.

 

Fill in the missing number,using the same rule.

 

 

 

 

  A  

  B  

Example:

2

7 = (2 x 2) + 3

 

3

12

 

5

28

 

8

        

 

 

 

 

Did you use the calculator on this question?

 

Scoring Guide

 

Score &Description

Correct

67    or    (8 x 8) + 3

Incorrect

Incorrect Answer

 

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