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Pennsylvania Standard 2.8.8 (Algebra and Functions)

NCTM Standard:  Algebra (Understanding patterns, relations and functions)

Created by:      Tracy Durica

                        Judy Durrell

                        Lindsay Good

                        Janice Loraw

                        Jennifer Taylor

 

Penny Pincher

 

Problem

 

Chris wants to save money in his bank.  Below is a chart showing how many pennies he put in his bank each day. 

 

 

DAY

NUMBER OF PENNIES

1

2

2

5

3

10

4

17

 

 

A.      How many pennies did Chris put in his bank on the 15th day if this pattern continues?

 

B.      Chris bought a shirt for $10 on the 15th day after he put pennies in his bank.  How much money was left in his bank?

 

 

For full credit, you must do the following:

 

1.           show OR describe each step of your work, even if you did it in your head       (“mental math”) or used a calculator.

AND

            2.        write an explanation stating the mathematical reason(s) why you chose each of    your 

                       steps.

 

 

 

 

Solution

 

Look for three procedures:

·        Find a pattern

·        Find the total savings

·        Find the amount of money left in the bank

 

A.      To find the pattern, one can complete a chart like the following by adding consecutive odd numbers to the number of pennies from the day before.

 

DAY

PENNIES

 

DAY

PENNIES

1

2

 

9

82

2

5

 

10

101

3

10

 

11

122

4

17

 

12

145

5

26

 

13

170

6

37

 

14

197

7

50

 

15

226

8

65

 

 

 

 

OR

 

To find the pattern, one can generalize the pattern to the expression  where  represents the day.  Therefore, one would find that  . 

 

 

B.      To find the total savings, add the following values:

 

 

Therefore, after the 15th day, there is $12.55 in the bank.

 

OR

 

Therefore, after the 15th day, there are 1,255 cents in the bank.

 

AND

 

To find the amount remaining, complete the following subtraction problem:

 

.   Therefore, there is $2.55 left in the bank.

 

Solution (cont.)

 

 

The “Why” part of the students’ solutions must include a description of the pattern used to find the amount of pennies, finding the sum to get the total amount of money in the bank (by addition) and finding the amount of money remaining (by subtraction). 

 

Note: $ sign is a label.  Thus, 2.55 is not $2.55 and would prevent a “4” score from becoming a “5”, and a “4” score would drop to a “3”.  This error would not affect other score points.  Using the dollar sign and the cent sign in one answer would also be considered a labeling error ($2.55 ¢ )

Rubric

 

 

5

Correct answer with all correct procedures/calculations shown OR described AND an explanation that tells “why” each step was taken.

·        Explanation need not address money conversion.

·        If the student uses the expression , they must define . Omission of defining  will not affect any other score level.

 

4A

Correct answer with all correct procedures/calculations shown OR described AND insufficient or no explanation.

4B

Correct answer with 2 out of 3 correct procedures/calculations shown OR described AND some explanation.

 

3A

Correct answer with 2 out of 3 correct procedures/calculations shown OR described AND no explanation.

3B

Correct answer with 1 out of 3 correct procedures/calculations shown OR described AND some explanation.

3C

Incorrect answer with all correct procedures/calculations shown OR described AND some explanation BUT one calculation or copying error carried through.

 

2A

Correct answer with 1 out of 3 correct procedures/calculations shown OR described AND no explanation.

2B

Incorrect answer with at least 2 correct procedures/calculations shown OR described AND some or no explanation.

2C

Incorrect answer with all procedures/calculations shown OR described AND no explanation BUT with no more than 2 calculation or copying errors.

 

1A

Correct answer with procedures/calculations and explanations that are not legible or not understandable or missing or incorrect.

1B

Incorrect answer, with, in most instances, one correct and critical procedure shown OR described OR explained.  If there is a calculation with that procedure, it must be correct. 

 

0A

Incorrect answer with no correct procedures, calculations or explanations shown or described. 

0B

Statements such as “???” or “I don’t know”

 

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