# The Craft of Word Problems

## CONTENTS OF CURRICULUM UNIT 04.05.07

- Introduction
- Definitions of Math Properties
- Commutative Properties of Addition and Multiplication
- Inverse Properties of Addition and Subtraction
- Associative Properties
- Distributive Property
- Reviewing Properties
- Rules for Properties of the Real Numbers
- Lesson Plan I
- Lesson Plan II
- Lesson Plan III
- Appendix A: Using Basic Properties To Solve Problems In Math
- Appendix B Glossary of Math Terms for Basic Properties
- Reading List
- Teacher Resources
- Bibliography

### Unit Guide

## Using Basic Properties to Solve Problems in Math

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## Distributive Property

The distributive property of multiplication allows you to distribute a factor, a, to two different addends, b and c: a(b +c) = ab + ac. The distributive property is used a great deal when computing mentally. For example, how would you mentally multiply 3 times 58? One approach is to think of 58 as 50 + 8 and use the distributive property in the following manner:3 x 58 -> 3 x (50 + 8) -> (3 x 50) + (3 x 8) -> 150+ 24 -> 174.

The distributive property of multiplication can also be applied to subtraction when b -> c. In this case, since 60 -> than 2, it works like this: 3 x 58 -> 3 x (60-2) -> (3 x 60) - (3 x 2) -> 180- 6 -> 174.

The distributive property is often used in connection with coming up with an easier calculation and then make adjustments. For example to compute 3 x 58, 58 can be rounded to 60 = (3 x 60), but 2 groups of 3 must be subtracted from the total 180 to get back to the actual value. The distributive property may be also applied to division expressions. For example, 132/12, If 132 is written is rewritten as 120 + 12, both can be divided by 12:

132/12 -> (120 + 12) /12 -> (120/12) + (12/12) -> 10 + 1 -> 11

The point about the Distributive Rule is, that it lets you compute the sum of products of a given number with a collection of numbers by first adding the collection, then doing one multiplication rather than having to compute each product and then add. This is used all the time in computing sale tax or discounts.

Properties of whole numbers are used extensively when computing. The particular numbers involved in a calculation determine when it makes sense to use the commutative, associative, or distributive properties or some combination of them. The situation is more complex for division and the distributive properties. (b + c) /a = (b/a) + (c/a) for all numbers except a = 0, but it is not true that a/(b +c) = (a/b) + (a/c). 24/ (4 + 2) = 24/6 = 4, but (24/4) + (24/2) = 6 + 12 = 18. Neither subtraction nor division is commutative or associative. Any selection of numbers will provide a counter example. For example (11- 6)-3 = 2 but 11- (6-3) = 8 or (48/6) /2 = 4 but 48/ (6/2) =16. Multiplication does distribute over subtraction: a x (b-c) = (a x b) - (a x c) for all a, b, and c or (b-c) x a = (b x a) - (c x a) for all a, b, c.

The following are questions for exhibiting the rules of arithmetic; see if you can identify the properties of each problem.

1. Choose a number between 1 and 10. Add 4 and double the result. Subtract 3, then, multiply by 3. Subtract 5 times one more than the original number. Tell me the answer and I will tell you your original number. (It is ten less than the answer.) How does this work? This problem can be represented by this equation (3 (2 (x + 4) -3) - 5 (x + 1). This problem demonstrates the distributive property, as well as the other Rules, because the distributive property of multiplication allows you to distribute a factor, a, to two different addends, b and c: a(b +c) = ab + ac, such as the problem above.

The calculation described in this problem is 3(2 (4+ x) -3) - 5(x + 1), we can use the Rules to rewrite the expression as follows:

= 3 (2 (4 + x) - 3) - 5(x + 1)

= 3 (8 + 2 x - 3) -5 x - 5

= 3 (2 x + 5) - 5x -5

= 6x + 15 - 5x - 5

= (6 - 5)x + 15 -5

= x + 10

2. Choose a number. Add 2, double the result. Subtract 2, double again. Divide by 4. Subject your origin number. I will tell you the answer. (It is 1.)

This can be justified by manipulations similar to those of the previous problem. The instructions say to do the calculation (2 (2 (x + 2) - 2)) x ¼ - x = E. We can manipulate this using the Rules as follows:

E = 2 (2x + 4 - 2)( ¼) - x

= 2 (2x + 2) (1/4) - x

= (4x +4) (1/4) - x

= (x + 1) - x

= 1

3. Ames sells flashlights for $7.95 and batteries for $ 3.95. It offers 10 % discount if you buy them together. One Saturday they sold 17 flashlight/ battery combinations, and on Sunday they sold 19. What was their total value of sales of the flashlight/ battery combinations for the weekend? 17(($3.95 + $7.95) - $1.19) + 19 (($3.95 + $7.95) -$1.19) = 36 x $10.61= $385.56. The total value of sales of the flashlight/battery combination for the weekend is $385.56 and can be solved by using the distributive property in the form

(a+ b) (c + d - e) = ac + ad - ae + bc + bd - be. This calculation uses the Distributive Rule on both factors (the total price with discount, and the number of units sold).

4. Suppose you have an uncle who gives you $500 on each birthday for three years in a row, and that you put it in a savings account. Suppose that the first year, the bank pays 4 % interest, the second year it pays 2% interest, and the third year it pays 6% interest. What is the total amount in your savings account at the end of three years? (Assume you opened the account the first time you got the money and that the birthday presents were the only deposits you made.) You can solve this problem in the following manner:

**
First Year:
**

500

x .04

-----

20.00

+ 500.00

-----

$520.00

**
Second Year:
**

520 First Year

+500 second year

-----

1020.00

x .02

-----

$20.40

-----

$1020.00

+ $20.40

-----

$1040.40

**
Third Year:
**

$1040.40

+$500.00

-----

$1540.40

x .06

-----

$93.4240

+$1540.40

-----

$ 1632.82

The value after 3 years can be computed in stages. It may also be (and should be) represented as a single compound expression:

= (( 500. 1.04 + 500). 1.02 + 500) 1.06

= 500.( (1.04 + 1) . 1.02 + 1 (1 .06)

= 500.( 2.04.1.02 + 1) . 1.06

= 500. (2.0808 + 1) . 1.06

= 500. (3.265648) = 1000 .1.632824

= 1,632.824

= 1,632.82

This problem involves use of all the properties, especially the associative property of multiplication and the distributive rule. The associative property states that the way in which three or more addends or factors are grouped before being added or multiplied does not affect the sum or product.

5. Janelle likes to bake. She has 3 aluminum muffin pans, each of which holds 8 muffins, and 2 cast iron pans which also hold 8 muffins each. She also has two stainless steel muffin pans which hold 12 muffins each. If Janelle fills all her muffins pans at once, how many muffins would that be?

Let: aluminum pans: 3 x 8, cast iron: 2 x 8, stainless steel: 2 x 12- This problem can be solved by using the associative property.

= (3 x 8) + (2 x 8) + (2 x 12) = 24 + 16 + 24 = 64 or

= (5 x 8) + 2 x 12

= 40 + 24

= 64

There would be 64 muffins.

6. Prunella is shopping for party supplies. Plastic tableware costs $2.50 per package. Plastic cups are $3.00 per package, plates are also $3.00, and plastic tablecloths are $3.50 each. Prunella gets two packages of spoons, two packages of forks, one package of knives, three packs of cups, one of plates, and two tablecloths. Everything is subject to 6% tax. How much will the party cost Prunella?

((2 (2 x $2.50) + 2.50 + (3 x 3.00) + 3.00 + (2 x 3.50)

10.00 + 2.50 + 9.00 + 3.00 + 7.00 = $31.50

$31.50 x .06 = $1.89

$31.50 + $1.89 = $33.39

The cost of the party is $ 33.39. (Commutative, Associative and

Distributive Properties) In particular, you use the Distributive Property to compute tax only on the total, rather than on each item separately.