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
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
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
a. 5
Since 5 * (1/5) = 1 1/5 is the multiplicative inverse of 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.
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
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 +
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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