1. Describe the pattern in each sequence.
1. 2, 22, 42, 62, . . . 2. 2.8, 6, 9.2, 12.4, . . .
Write the next three terms of each sequence.
3. 4, 12, 20, 28, . . . 4. 2.1, 2.8, 3.5, 4.2, . . .
5. Every 18 months, National Surveys conducts a population
survey of the United States. If they conducted a survey in
September 2003, when will they conduct the next four
surveys?
6. What is the next term in the sequence 3.2, 12.8, 22.4, 32, …?
Course 2, Lesson 5-3

2. Answers
1. Add 20 to the previous term.
2. Add 3.2 to the previous term.
3. 36, 44, 52
4. 4.9, 5.6, 6.3
5. March 2005, September 2006, March 2008,
September 2009
6. 41.6
Course 2, Lesson 5-3

3. HOW can you use numbers and
symbols to represent mathematical
ideas?
Expressions and Equations
Course 2, Lesson 5-3

4. • 7.EE.1
Apply properties of operations as strategies to add, subtract, factor,
and expand linear expressions with rational coefficients.
• 7.EE.2
Understand that rewriting an expression in different forms in a
problem context can shed light on the problem and how the quantities
in it are related.
Course 2, Lesson 5-3 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and
Council of Chief State School Officers. All rights reserved.
Expressions and Equations

5. Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
7 Look for and make use of structure.
Course 2, Lesson 5-3 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and
Council of Chief State School Officers. All rights reserved.
Expressions and Equations

6. • To identify properties of operations
• To determine if conjectures are true or false
and provide counterexamples for false
conjectures
• To use properties to simplify algebraic
expressions
Course 2, Lesson 5-3
Expressions and Equations

7. Course 2, Lesson 5-3
Expressions and Equations
• Commutative Property
• Associative Property
• property
• Additive Identity Property
• Multiplicative Identity Property
• Multiplicative Property of Zero
• counterexample

8. Course 2, Lesson 5-3
Expressions and Equations
Words The Commutative Property states that the order in which
the numbers are added or multiplied does not change the
sum or product.
Addition Multiplication
Symbols a + b = b + a a • b = b • a
Examples 6 + 1 = 1 + 6 7 • 3 = 3 • 7

9. Course 2, Lesson 5-3
Expressions and Equations
Words The Associative Property states that the way in which
numbers are grouped when they are added or multiplied
does not change the sum or product.
Addition Multiplication
Symbols a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c
Examples 2 + (3 + 8) = (2 + 3) + 8 3 • (4 • 5) = (3 • 4) • 5

10. 1
Need Another Example?
Step-by-Step Example
1. Name the property shown by the statement
2 • (5 • n) = (2 • 5) • n.
The order of the numbers and variable did not change,
but their grouping did. This is the Associative Property
of Multiplication.

11. Answer
Need Another Example?
Name the property shown by the statement
(3 • m) • 2 = 2 • (3 • m).
Commutative Property of Multiplication

12. 1
Need Another Example?
2
Step-by-Step Example
2. State whether the following conjecture is true or false.
If false, provide a counterexample.
Write two division expressions using the Commutative Property.
Division of whole numbers is commutative.
15 ÷ 3 = 3 ÷ 15
5 ≠
State the conjecture.
Divide.
?
The conjecture is false. We found a counterexample. That is,
15 ÷ 3 ≠ 3 ÷ 15. So, division is not commutative.

13. Answer
Need Another Example?
State whether the following conjecture is true
or false. If false, provide a counterexample.
Subtraction of whole numbers is associative.
false; Sample answer: (12 – 5) – 3 ≠ 12 – (5 – 3)

14. 1
Need Another Example?
2
3
4
Step-by-Step Example
3. Alana wants to buy a sweater that costs $38,
sunglasses that costs $14, a pair of jeans that costs
$22, and a T-shirt that costs $16. Use mental math to
find the total cost before tax.
Write an expression for the total cost. You can rearrange the
numbers using the properties of math. Look for sums that are
multiples of ten.
Commutative Property of Addition
Associative Property of Addition
The total cost of the items is $90.
38 + 14 + 22 + 16
= 38 + 22 + 14 + 16
= (38 + 22) + (14 + 16)
Add.
Simplify.
= 60 + 30
= 90

15. Answer
Need Another Example?
In a garden, a decorative pool in the shape
of a box is 2 feet deep, 17 feet long, and
5 feet wide. Use mental math to find the
volume of water in the pool.
170 ft3

16. 1
Need Another Example?
2
3
Step-by-Step Example
4. Simplify (7 + g) + 5. Justify each step.
Commutative Property of Addition
Associative Property of Addition
(7 + g) + 5 = (g + 7) + 5
Simplify.= g + 12
= g + (7 + 5)

17. Answer
Need Another Example?
Simplify 6 + (d + 8). Justify each step.
6 + (d + 8) = 6 + (8 + d) Commutative (+)
= (6 + 8) + d Associative (+)
= 14 + d Simplify.

18. 1
Need Another Example?
2
3
Step-by-Step Example
5. Simplify (m • 11) • m. Justify each step.
Commutative Property of Multiplication
Associative Property of Multiplication
(m • 11) • m
Simplify.= 11m2
= (11 • m) • m
= 11 • (m • m)

19. Answer
Need Another Example?
Simplify a • (9 • b). Justify each step.
a • (9 • b) = (a • 9) • b Associative (×)
= (9 • a) • b Commutative (×)
= 9ab Simplify.

20. How did what you learned
today help you answer the
HOW can you use numbers and symbols
to represent mathematical ideas?
Course 2, Lesson 5-3
Expressions and Equations

21. How did what you learned
today help you answer the
HOW can you use numbers and symbols
to represent mathematical ideas?
Course 2, Lesson 5-3
Expressions and Equations
Sample answers:
• Using numbers and symbols to represent mathematical
properties
• Finding a counterexample to show a conjecture is false
• Applying properties and use mental math when solving
real-world problems

22. Give examples of both
the Associative Property and
the Commutative Property.
Ratios and Proportional RelationshipsExpressions and Equations
Course 2, Lesson 5-3