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Properties of Addition and Multiplication
Using the properties of whole-number operations can make computation easier. Among the
properties of addition are the commutative property and the associative property.
Commutative Property of Addition
Changing the order of addends does not
change the sum.
a1b5b1a
3145413
Associative Property of Addition
Changing the grouping of addends does not
change the sum.
a 1 (b 1 c) 5 (a 1 b) 1 c
13 1 (7 1 4) 5 (13 1 7) 1 4
You can use properties of addition when you shop. For example, if you want to buy books that
cost $15, $37, and $25, you can arrange the prices in a way that you could add them mentally.
Example A
Use the properties of addition to find the value of (15 1 37) 1 25.
Step 1: Use the commutative property to reorder the addends.
(15 1 37) 1 25 5 25 1 (15 1 37)
Step 2: Use the associative property to group the addends.
5 (25 1 15) 1 37
Step 3: Add inside the parentheses.
5 40 1 37
Step 4: Add.
5 77
Solution: 15 1 37 1 25 5 77
The commutative and associative properties can also be used for multiplication.
Commutative Property of Multiplication
Changing the order of factors does not
change the product.
a3b5b3a
6395936
Associative Property of Multiplication
Changing the grouping of factors does not
change the product.
a 3 (b 3 c) 5 (a 3 b) 3 c
2 3 (5 3 13) 5 (2 3 5) 3 13
Example b
Use properties of multiplication to find the value of (4 3 9) 3 5.
Step 1: Use the commutative property to reorder the factors.
(4 3 9) 3 5 5 5 3 (4 3 9)
Step 2: Use the associative property to group the factors.
5 (5 3 4) 3 9
Step 3: Multiply inside the parentheses.
5 20 3 9
Step 4: Multiply.
5 180
Solution: 4 3 9 3 5 5 180
© 2014 College Board. All rights reserved.
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SpringBoard Course 1, Unit 1
Properties of Addition and Multiplication (continued)
The distributive property of multiplication is used to break apart numbers to make them easier to
multiply. The distributive property can be used over addition or subtraction.
Distributive Property of Multiplication
over Addition
To multiply a sum by a number, you can
multiply each addend by the number and
add the products.
For any numbers a, b, c,
a(b 1 c) 5 ab 1 ac
9 3 34 5 9(30 1 4) 5 (9 3 30) 1 (9 3 4)
Distributive Property of Multiplication
over Subtraction
To multiply a difference by a third number, you
can multiply each of the two numbers by the
third and then find the difference of the products.
For any numbers a, b, c,
a(b 2 c) 5 ab 2 ac
34 3 9 5 34(10 2 1) 5 (34 3 10) 2 (34 3 1)
Example C
There are 57 boxes of pencils in stock at the supply store. Each box contains 8 pencils.
Use the distributive property of multiplication over addition to find the total number of pencils.
Step 1: Rewrite the expression breaking 57 into 50 1 7.
57 3 8 5 (50 1 7) 3 8
Step 2: Use the distributive property.
5 (50 3 8) 1 (7 3 8)
Step 3: Multiply inside each set of parentheses.
5 400 1 56
Step 4: Add the products.
5 456
Solution: There are 456 pencils.
Example D
There are 63 rows in a theater. Each row has 17 seats. Use the distributive property of multiplication over
subtraction to find the total number of seats.
Step 1: Rewrite the expression breaking 17 into 20 2 3.
63 3 17 5 63(20 2 3)
Step 2: Use the distributive property.
5 (63 3 20) 2 (63 3 3)
Step 3: Multiply inside each set of parentheses.
5 1,260 2 189
Step 4: Subtract the products.
5 1,071
Solution: There are 1,071 seats.
© 2014 College Board. All rights reserved.
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SpringBoard Course 1, Unit 1
Properties of Addition and Multiplication (continued)
PRACTICE
Identify the property.
1.56 1 79 5 79 1 56
2.7 3 14 5 (7 3 20) 2 (7 3 6)
3.(5 3 8) 3 7 5 5 3 (8 3 7)
4.17 1 (23 1 38) 5 (17 1 23) 1 38
Write the number that makes the equation true.
5.28 3 9
28 3 9 5 (20 3 9) 1 (
3 9)
6.42 3 38
42 3 38 5 (42 3 40) 2 (42 3
)
Use a property to find the value of each expression.
7.20 3 34 3 5
8.17 1 56 1 83
9.9 3 47
10.5 3 8 3 40
11.35 3 68
12.(65 1 80) 1 20
13. Calista took 36 rolls of film that each took 24 photos. How can you use the distributive
property of multiplication over addition to find the total number of photos that
Calista took?
© 2014 College Board. All rights reserved.
3
SpringBoard Course 1, Unit 1