What you should learn

To recognize and use the properties of identity and equality

To determine the multiplicative inverse of a number

NCTM Curriculm Standards 2, 6-10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Additive Identity

Multiplicative Identity

Multiplicative Inverses

Reciprocals

Reflexive Property of equality

Hypothesis

Conclusion

Symmetric Property of equality

Transitive Property of equality

Substitution Property of equality

 

Introduction: In this lesson we are going to learn about different properties.

Equations such as 20 + a = 20 and 40 + a = 40, can be summarized in algebraic terms. The sum of any number and 0 is equal to that number. Zero is called the additive identity.

Additive Identity Property: For any number a, a + 0 = a

The equations such as 20 * m = 20 and 40 * m = 40, have a solution of one. Since the product of any number and 1 is equal to the number, 1 is called the multiplicative identity.

Mulitiplicative Identity Property: For any number a, a * 1 = 1 * a = a

 

The equation 3 * 0 = p, is an equation where one of the factors is 0 and therefore the value of p is 0. The suggests the following property.

Multiplicative Property of Zero: For any number a, a * 0 = 0 * a = 0

 

Two numbers whose product is 1 are called multiplicative inverses or reciprocals. Zero has no reciprocal because any number times 0 is 0.

Multiplicative Inverse Property: For every nonzero number a/b, where a,b 0, there is exactly one number b/a such that (a/b) * (b/a) = 1

Exercise 1: Name the multiplicative inverse of each number or variable. Assume that no variable equals zero.

a. 5

Since 5 * (1/5) = 1

1/5 is the multiplicative inverse of 5

b. x

c. 2/3

 

At the beginning of algebra class, Ms. Escalante gave each of her studentss a single strip of paper 8 inches long. She instructed the students to divide their paper strips amny way they wished.

Staci left her strip as one 8-inch strip

Amad cut his strip to form a 6-inch strip and a 2-inch strip

Liam cut his strip to form a 5-inch strip and a 3-inch strip.

Using the strips of paper, we know the following to be true.

8 = 8

6 + 2 = 2 + 6

5 + 3 = 3 + 5

The reflexive property of equality says that any quantity is equal to itself.

Reflexive Property of Equality: For any number a, a = a.

 

Many mathematical statements and algebraic properties are written in if-then form. In an if-then statement, the hypothesis is the part following if, and the conclusion isthe part following then.

The symmetric property of equality says that if one quantity equals a second quantity, then the second quantity also equals the first.

Symmetric Property of Equality: For any number a and b, if a = b, then b = a.

 

A third property can also be shown using the paper strips.

If 3 + 5 = 8 and 8 = 6 + 2, then 3+5 = 6 + 2

The transitive property of equality says that if one quantity equals a second quantity and the second quantity equals a thrid quantity, then the first and third quantities are equal.

Transitive Property of Equality: For any numbers a, b, and c, if a = b and b = c, then a = c

We know that 5 + 3 = 6 + 2. Since 5 + 3 is equal to 8, we can substitute 8 for 5 + 3 to get 8 = 6 + 2. The substitution property of equality says that a quantity may be substitued for its equal in any expression.

Substitution Property of Equality: If a = b, then a may be replaced by b in any expression.

You can use the properties of identity and equality to justify each step when evaluating an expression.

Exercise 2: The pep club at Roosevelt High School is selling submariene sandwiches, lemonage, and apples at the district swim meet. Each sandwich costs $2.00 to make and sells for $3.00. Each glass of lemonade costs $0.25 and sells for $1.00. Each apple costs the club $0.25, and the members have decided to sell the apples for $0.25 each. Write an expression that represents the profit for 80 sandwiches, 150 glasses of lemonade, and 40 apples. Evaluate the expression, indicating the property used in each step.

80(3.00 - 2.00) + 150(1.00 - 0.25) + 40(0.25 - 0.25)

= 80 (1) + 150 (o.75) + 40 (0) Substitution

= 80 + 150 (0.75) + 40 (0) Identity *

= 80 + 112.50 + 40 (0) Substitution

= 80 + 112.50 + 0 Multiplicate prop. of 0

= 192.50 + 0 Substitution

= 192.50 Identity +

The club would make a profit of $192.50.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 21 - 45 odd, 46, 47, 49 - 57

Alternative Homework: Enriched: 20 - 44 even, 46 - 57

Extra Practice: Students book page 757 Lesson 1-6

Extra Practice Worksheet: Click Here.

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