The Power of Algebra

Teacher Guide Exerpt

** NOTE: This is an excerpt from the Power of Algebra Teacher's Guide. The complete 72-page guide is available for purchase.

A. SYNOPSIS

This program will:

Define the order of operations for multiplication, division, addition and subtraction.
Define the order of operations within grouping symbols relative to multiplication, division, addition and subtraction.
Define exponents and the order of operations with exponents and grouping symbols.
Discuss how these rules may be used to simplify algebraic problems.

B. VOCABULARY

FACTOR:: if two numbers a and b are multiplied together to form a product ab, then a and b are called the factors of the product ab.
POWER:: if the number a is used as a factor n times, the product may be written as aⁿ. A product of the form aⁿ is called a power.; In the power aⁿ, a is called the base of the power and n is called the exponent of the power.

Note to the Teacher:

In the power aⁿ, n is called a superscript.
x_n is read “x sub n” where sub stands for subscript and is not the same as the power xn.

OPERATION:: addition, subtraction, multiplication or division of real numbers.

C. FOLLOW-UP ACTIVITY

1. "Please Excuse My Dear Aunt Sally"

This phrase will help students remember the rules for order of operations. Put on the chalkboard:

P
E
M D
A S

Say "Please Excuse My Dear Aunt Sally. This stands for:

Parentheses
Exponents
Multiplication and Division
Addition and Substraction."

Emphasize that multiplication and division have the same priority; that is, they are done from left to right as they occur in the expression. Addition and subtraction also have the same priority. That is why the letters which stand for multiplication/division and addition/subtraction are written on the same line.

2. Show how the addition of parentheses to a problem changes the problem.

2 + 3 · 5 - 8 ÷ 2²

2 + 3 · 5 - 8 ÷ 4

2 + 15 - 2

17 - 2

(2 + 3)5 - (8₂ ÷ 2)²

(5)5 - 4²

(5)5 - 16

25 - 16

3. Explain the difference between -2⁴ and (-2)⁴.

-2⁴ means

(-1)(2)(2)(2)(2)

-16

(-2)⁴ means

(-2)(-2)(-2)(-2)

Note to the Teacher:

You may need to explain all the different ways that the operation of multiplication can be shown in algebra:

2 × 3	2(3)
2 · 3	(2)(3)

4. Review the rules of the Order of Operations.

a) First, perform the exponential operations.

50 - 2² · (2 + 3) ÷ 10
50 - 4 · (2 + 3) ÷ 10

b) Second, perform the operations inside grouping symbols.

50 - 4 · (2 + 3) ÷ 10
50 - 4 · 5 ÷ 10

c) Third, perform multiplication and division as they occur, starting on the left of the expression and moving to the right.

50 - 4 · 5 ÷ 10
50 - 20 ÷ 10
50 - 2

d) Fourth, perform addition and subtraction as they occur, starting on the left of the expression and moving to the right.

50 - 2
48

5. Show that a different result would have been obtained in (4) above if a different order of operations had been applied.

50 - 2² · (2 + 3) ÷ 10	Perform the exponential operations.
50 - 4 · (2 + 3) ÷ 10	Perform the operations inside grouping symbol.
50 - 4 · 5 ÷ 10	Perform multiplication.
50 - 20 ÷ 10	Perform subtraction.
30 ÷ 10	Perform division.
3

6. Solve the equation which appeared in the program.

8 D₂

=

16² 20²

7. In the video, why is X = 3 + 4 · 2 equal to 11 and not equal to 14? Also, why is X = 3 + 4/2 equal to 5 and not equal to 3.5?

D. DISCUSSION QUESTIONS

1. Show how exponents can be used to write a product in simpler form.

2 · 2 · 2 · 5 · 5 · 7 · 7 · 7 · 7 = 2² · 5² · 7⁴

2. Why is 8 - 6X + 5 not equal to 2X + 5?

3. The solution of the equation 2 + (4² + X) + 6X ÷ 3 = 26 is

a) 7
b) 2.68

8

c)

3

d) 106.5

Why?

Note to the Teacher:

aⁿ is read as "a to the nth power." For example, a² is read "a squared," a³ is read "a cubed," and a⁶ is read as "a to the sixth power."
In discussing factors, it should be noted that the product ab is a multiple of a and also a multiple of b. We also call a and b divisors of the number ab.

E. STUDENT WORKSHEETS

Student Worksheet #1
Student Worksheet #2

A. SYNOPSIS

B. VOCABULARY

Note to the Teacher:

C. FOLLOW-UP ACTIVITY

1. "Please Excuse My Dear Aunt Sally"

2. Show how the addition of parentheses to a problem changes the problem.

3. Explain the difference between -24 and (-2)4.

Note to the Teacher:

4. Review the rules of the Order of Operations.

5. Show that a different result would have been obtained in (4) above if a different order of operations had been applied.

6. Solve the equation which appeared in the program.

7. In the video, why is X = 3 + 4 · 2 equal to 11 and not equal to 14? Also, why is X = 3 + 4/2 equal to 5 and not equal to 3.5?

D. DISCUSSION QUESTIONS

1. Show how exponents can be used to write a product in simpler form.

2. Why is 8 - 6X + 5 not equal to 2X + 5?

3. The solution of the equation 2 + (42 + X) + 6X ÷ 3 = 26 is

Note to the Teacher:

E. STUDENT WORKSHEETS

3. Explain the difference between -2⁴ and (-2)⁴.

3. The solution of the equation 2 + (4² + X) + 6X ÷ 3 = 26 is