# Lesson 1-4 Properties and Mental Math

```1-4
Learn to use number
properties to compute
mentally.
Vocabulary
Commutative
Property
Associative
Property
Distributive
Property
Properties and
Mental Math
Mental math means “doing math
extremely good at mental math.
When she was asked to multiply
7,686,369,774,870 by 2,465,099,745,779,
she took only 28 seconds to multiply
the numbers mentally and gave
18,947,668,177,995,426,462,773,730!
Many mental math strategies use
number properties that you already know to
make equivalent expressions that may be
easier to simplify.
COMMUTATIVE PROPERTY (Ordering)
Words
Numbers
18 9 9 18
numbers in any order.
The Commutative and
Associative Properties
do not apply to
subtraction or division.
EXAMPLE
ASSOCIATIVE PROPERTY (Grouping)
1
Words
Numbers
When you are only adding or
only multiplying, you can group
any of the numbers together.
( 17 2 ) 9 17 ( 2 9 )
( 12 2 ) 4 12 ( 2 4 )
Using Properties to Add and Multiply Whole Numbers
A
22
15 2 2 15
Simplify 12 4 18 46.
12 4 18 46
Look for sums that are multiples of 10.
12 18 4 46
( 12 18 ) ( 4 46 )
50
30
80
Use the Commutative Property.
Chapter 1 Whole Numbers and Patterns
Use the Associative Property to make
groups of compatible numbers.
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B
Simplify 5 12 2.
5 12 2
12 5 2
12 ( 5 2 )
12 10
The expressions in
each row of Example
1B are equivalent
expressions because
they have the same
value.
Look for products that are multiples of 10.
Use the Commutative Property.
Use the Associative Property to group
compatible numbers.
120
Use mental math to multiply.
To multiply a number by a sum, such as 6 (10 4), you can use the
order of operations, or you can use the Distributive Property.
DISTRIBUTIVE PROPERTY
Words
Numbers
To multiply a number by a sum,
multiply by each number in
6 (10 4) (6 10) (6 4)
60
24
84
When you multiply two numbers, you can “break apart” one of the
numbers into a sum and then use the Distributive Property.
EXAMPLE
2
Using the Distributive Property to Multiply
Use the Distributive Property to find each product.
A
B
4 23
4 23 4 ( 20 3 )
( 4 20 ) ( 4 3 )
80
12
92
8 74
8 74 8 (70 4)
(8 70) (8 4)
560 32
592
“Break apart” 23 into 20 3.
Use the Distributive Property.
Use mental math to multiply.
“Break apart” 74 into 70 4.
Use the Distributive Property.
Use mental math to multiply.
Think and Discuss
1. Give examples of the Commutative Property and the Associative
Property.
2. Name some situations in which you might use mental math.
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1-4 Properties and Mental Math
23
1-4
Exercises
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MR10 1-4
Exercises 1–30, 31, 39, 41, 45, 51
GUIDED PRACTICE
See Example 1
See Example 2
Simplify.
1. 13 ⴙ 9 ⴙ 7 ⴙ 11
2. 19 ⴙ 18 ⴙ 11 ⴙ 32
3. 25 ⴙ 7 ⴙ 13 ⴙ 5
4. 5 ⴛ 14 ⴛ 4
5. 4 ⴛ 16 ⴛ 5
6. 5 ⴛ 17 ⴛ 2
Use the Distributive Property to find each product.
7. 5 ⴛ 24
8. 8 ⴛ 52
9. 4 ⴛ 39
10. 6 ⴛ 14
11. 3 ⴛ 33
12. 2 ⴛ 78
13. 9 ⴛ 12
14. 2 ⴛ 87
INDEPENDENT PRACTICE
See Example 1
See Example 2
Simplify.
15. 15 ⴙ 17 ⴙ 3 ⴙ 5
16. 14 ⴙ 7 ⴙ 16 ⴙ 13
17. 6 ⴙ 21 ⴙ 14 ⴙ 9
18. 5 ⴛ 25 ⴛ 2
19. 2 ⴛ 32 ⴛ 10
20. 6 ⴛ 12 ⴛ 5
Use the Distributive Property to find each product.
21. 3 ⴛ 36
22. 4 ⴛ 42
23. 6 ⴛ 71
24. 2 ⴛ 94
25. 6 ⴛ 23
26. 5 ⴛ 25
27. 6 ⴛ 62
28. 7 ⴛ 21
29. 8 ⴛ 41
30. 2 ⴛ 94
PRACTICE AND PROBLEM SOLVING
Extra Practice
See page EP2.
Use mental math to find each sum or product.
31. 8 ⴙ 13 ⴙ 7 ⴙ 12
32. 2 ⴛ 25 ⴛ 4
33. 4 ⴙ 22 ⴙ 16 ⴙ 18
34. 5 ⴛ 8 ⴛ 12
35. 5 ⴙ 98 ⴙ 95
36. 6 ⴛ 5 ⴛ 14
37. 11 ⴙ 75 ⴙ 25
38. 8 ⴛ 11 ⴛ 5
39. 19 ⴙ 1 ⴙ 11 ⴙ 39
40. Paul is writing a story for the school newspaper about the landscaping
done by his class. The students planted 15 vines, 12 hedges, 8 trees, and
35 flowering plants. How many plants were used in the project?
41. Earth Science The temperature on Sunday was 58 &deg;F. The temperature is
predicted to rise 4 &deg;F on Monday, then rise 2 &deg;F more on Tuesday, and then rise
another 6 &deg;F by Saturday. What is the predicted temperature on Saturday?
42. Multi-Step Janice wants to
order disks for her computer. She
needs to find the total cost,
including shipping and handling.
If Janice orders 7 disks, what will
her total cost be?
24
Chapter 1 Whole Numbers and Patterns
Description
Computer
Disk
Number
7
Unit Cost
with Tax
Price
\$24.00
Shipping &amp; Handling
Total
\$7.00
Multiply using the Distributive Property.
44. 4 27
43. 9 17
Life Science
45. 11 18
46. 7 51
Determine if each pair of expressions is equivalent. If so, state the property shown.
47. 8 35 and (8 30)(8 5)
48. 14 7 6 and 6 14 7
49. (9 7) 6 and 9 (7 6)
50. 24 6 2 and 6 24 2
51. Life Science Poison-dart frogs can breed underwater, and the females lay
from 4 to 30 eggs. What would be the total number of eggs if four female
poison-dart frogs each laid 27 eggs?
Poison-dart frogs are
members of the
family Dendrobatidae, which includes
Many are brightly
colored.
Use the table for Exercises 52 and 53.
52. Rickie wants to buy 3 garden hoses.
How much will they cost?
Home Center Clearance Sale
53. The boys in Josh’s family are
saving money to buy 4 ceiling
fans. How much will they need
to save?
Table lamp
\$15
Garden hose
\$16
Ceiling fan
\$52
54. Critical Thinking Give a problem that you could simplify using the
Commutative and Associative Properties. Then, show the steps to solve the
problem and label the Commutative and Associative Properties.
55. What’s The Error? A student wrote 5 24 25 6 5 25 24 6 by
the Associative Property. What error did the student make?
56. Write About It Why can you simplify 5( 50 3 ) using the Distributive
Property? Why can’t you simplify 5( 50 ) 3 using the Distributive Property?
57. Challenge Explain how you could find the product of 52 112 using the
Distributive Property. Simplify the expression.
Test Prep and Spiral Review
58. Multiple Choice Which expression does NOT have the same value as 7 ( 4 23 )?
A
7 27
B
( 7 4 ) ( 7 23 )
C
7 4 23
D
28 ( 7 23 )
59. Gridded Response Michelle flew 1,240 miles from Los Angeles to Dallas, and
another 718 miles from Dallas to Atlanta. From Atlanta, she flew 760 miles to
New York City. How many miles did Michelle fly in all?
Estimate each sum or difference by rounding to the nearest thousands place. (Lesson 1-1)
60. 5,237 1,586
61. 915,178 451,836
62. 39,187 24,999
Simplify each expression. (Lesson 1-3)
63. 4 14 12 2
64. 16 42 15 2
65. 62 14 (5 4)
1-4 Properties and Mental Math
25
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