1. Preview
Warm Up
California Standards
Lesson Presentation

2. Warm Up Evaluate. 1. 42 16 2. |5 – 16| 11 3. –23 –8 4. |3 – 7| 4
Translate each word phrase into a numerical or algebraic expression.
5. The product of 8 and 6
8  6
6. The difference of 10y and 4
10y – 4
Simplify each fraction. 7. 8 8.

3. California
Standards
1.0 Students use properties of numbers to demonstrate whether assertions are true or false Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.

4. Vocabulary
order of operations
terms
like terms
coefficient

5. When an expression contains more than one operation, the order of operations tells you which operation to perform first.

6. Order of Operations First: Second: Third: Fourth:
Perform operations inside grouping symbols.
Second:
Evaluate powers.
Third:
Perform multiplication and division from left to right.
Fourth:
Perform addition and subtraction from left to right.

7. Grouping symbols include parentheses ( ), brackets [ ], and braces { }
Grouping symbols include parentheses ( ), brackets [ ], and braces { }. If an expression contains more than one set of grouping symbols, begin with the innermost set. Follow the order of operations within that set of grouping symbols and then work outward.

8. Helpful Hint
Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.

9. Additional Example 1: Simplifying Numerical Expressions
Simplify each expression.
A. 15 – 2  3 + 1
15 – 2  3 + 1
There are no grouping symbols.
15 – 6 + 1
Multiply.
9 + 1
Subtract. 10
Add.
B ÷ 2
÷ 2
There are no grouping symbols.
÷ 2
Evaluate powers. The exponent applies only to the 3.
Divide. 26
Add.

10. Additional Example 1: Simplifying Numerical Expressions
Simplify each expression. C.
The fraction bar is a grouping symbol.
Evaluate powers. The exponent applies only to the 4.
Multiply above the bar and subtract below the bar.
Add above the bar and then divide.

11. Check It Out! Example 1a
Simplify the expression.
There are no grouping symbols.
Rewrite division as multiplication.
Multiply. 48

12. Check It Out! Example 1b
Simplify the expression.
The square root sign acts as a grouping symbol.
Subtract.
3  7
Take the square root. 21
Multiply.

13. Check It Out! Example 1c
Simplify the expression.
The division bar acts as a grouping symbol.
Add and evaluate the power.
Multiply, subtract and simplify.

14. Additional Example 2: Retail Application
A shop offers gift-wrapping services at three price levels. The amount of money collected for wrapping gifts on a given day can be found using the expression 2B + 4S + 7D. On Friday the shop wrapped 10 basic packages B, 6 super packages S, and 5 deluxe packages D. Use the expression to find the amount of money collected for gift-wrapping on Friday.
2B + 4S +7D
2(10) + 4(6) + 7(5)
Substitute values for variables.
Multiply. 79
Add.
A total of $79 was collected on Friday.

15. Check It Out! Example 2
A formula for a player’s total number of bases is Hits + D + 2T + 3H. Use this expression to find Hank Aaron’s total bases for 1959, when he had 223 hits, 46 doubles, 7 triples, and 39 home runs.
Hits + D + 2T + 3H
(7) + 3(39)
Substitute values for variables.
Multiply.
400
Add.
Hank Aaron’s total number of bases for 1959 was 400.

16. The terms of an expression are the parts to be added or subtracted
The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.
Like terms
Constant
4x – 3x + 2

17. A coefficient is a number multiplied by a variable
A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.
Coefficients
1x2 + 3x

18. Like terms can be combined
Like terms can be combined. To combine like terms, use the Distributive Property.
= 3x
Distributive Property
ax – bx = (a – b)x
Example
7x – 4x = (7 – 4)x
Notice that you can combine like terms by adding or subtracting the coefficients. Keep the variables and exponents the same.

19. Additional Example 3: Combining Like Terms
Simplify the expression by combining like terms.
A. 72p – 25p
72p – 25p
72p and 25p are like terms.
47p
Subtract the coefficients.

20. Additional Example 3: Combining Like Terms
Simplify the expression by combining like terms. B.
A variable without a coefficient has a coefficient of 1.
and are like terms.
Write 1 as .
Add the coefficients.

21. Additional Example 3: Combining Like Terms
Simplify the expression by combining like terms.
C. 0.5m + 2.5n
0.5m + 2.5n
0.5m and 2.5n are not like terms.
0.5m + 2.5n
Do not combine the terms.

22. Caution!
Add or subtract only the coefficients. 6.8y² – y² ≠ 6.8

23. Check It Out! Example 3
Simplify by combining like terms.
a. 16p + 84p
16p + 84p
16p + 84p are like terms.
100p
Add the coefficients.
b. –20t – 8.5t
–20t – 8.5t
20t and 8.5t are like terms.
–28.5t
Subtract the coefficients.
c. 3m2 + m3 – m2
3m2 – m2 + m3
3m2 and – m2 are like terms.
2m2 + m3
Subtract coefficients.

24. Additional Example 4: Simplifying Algebraic Expressions
Use properties and operations to show that 14x + 4(2 + x) simplifies to 18x + 8.
Statements
Reasons 1.
14x + 4(2 + x) 2.
14x + 4(2) + 4(x)
Distributive Property 3.
14x x
Multiply. 4.
14x + 4x + 8
Commutative Property of Addition 5.
(14x + 4x) + 8
Associative Property of Addition 6.
18x + 8
Combine like terms.

25. Check It Out! Example 4
Use properties and operations to show that
6(x – 4) + 9 simplifies to 6x – 15.
Statements
Reasons 1.
6(x – 4) + 9 2.
6x – 6(4) + 9
Distributive Property 3.
6x –
Multiply. 4.
6x – 15
Combine like terms.

26. Lesson Quiz
Simplify each expression. 2.
200 8
3. The volume of a storage box can be found using the expression lw(w + 2). Find the volume of the box if l = 3 feet and w = 2 feet.
24 ft3
Simplify each expression by combining like terms. 4.
5. 14c2 – 9c
14c2 – 9c
6. Use properties and operations to show that
24a + b² + 3a + 2b² simplifies to 27a + 3b².
Check students’ work.