What you should learn
To recogize and use the commutative and associative properties to simplify expressions NCTM Curriculm Standards 2, 4, 6 - 10
To recogize and use the commutative and associative properties to simplify expressions
NCTM Curriculm Standards 2, 4, 6 - 10
In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:
Commutative Property Associative Property
Commutative Property
Associative Property
Introduction: In equation 3 + 3 +3 + 2 + 2 + 2 = 2 + 2 + 2 +3 + 3 + 3, the addends are the same, but their order is different. The commutative property says that the order in which you add or multiply two numbers does not change their sum or product.
Commutative Property: For any numbers a and b, a + b = b + a and a * b = b * a
One easy way to find the sum or product of numbers is to group or associate the numbers. The associative property says that the way you group three numbers when adding or multiplying does not change their sum or product.
Associative Property: For any numbers a, b, and c, (a + b) + c = a + (b + c) and (ab)c = a(bc)
You can group 11 + 12 + 7 + 9 + 7 + 6 to make mental addition easy.
11 + 12 + 7 + 9 + 7 + 6 = 11 + 9 + 7 + 6 + 7 + 12 = (11 + 9) + (7 + 6 + 7) + 12 =20 + 20 + 12 =52
11 + 12 + 7 + 9 + 7 + 6 = 11 + 9 + 7 + 6 + 7 + 12
= (11 + 9) + (7 + 6 + 7) + 12 =20 + 20 + 12 =52
= (11 + 9) + (7 + 6 + 7) + 12
=20 + 20 + 12
=52
The commutative and associative properties can be used with the other properties you have studied when evaluating and simplifying expressions.
Exercise 1: Juan Martinez is a product manager for a cereal company. Part of his jobis to determine what size box should be used to package the company's Toasty Oatsies cereal. Juan can choose from among the following sizes: 8" by 11" by 2 1/2", 8 1/2" by 10" by 2", or 7 7/8" by 11" by 3". Juan wants to know which package will hold the most cereal.
The box with the greatest volume will hold the most cereal. To find the volume of each box, multiply the length times the width times the height.
8 * 11 * 2 1/2 = 8 * 2 1/2 * 11 = 20 * 11 = 220 8 1/2 * 10 * 2 = 8 1/2 * 2 * 10 =17 * 10 =170 7 7/8 * 11 * 3 = 7 7/8 * (11 * 3) = 7 7/8 * 33 = 259 7/8
8 * 11 * 2 1/2 = 8 * 2 1/2 * 11
= 20 * 11 = 220
= 20 * 11
= 220
8 1/2 * 10 * 2 = 8 1/2 * 2 * 10
=17 * 10 =170
=17 * 10
=170
7 7/8 * 11 * 3 = 7 7/8 * (11 * 3)
= 7 7/8 * 33 = 259 7/8
= 7 7/8 * 33
= 259 7/8
The box that is 7 7/8" by 11" by 3" has the greatest volume.
The chart below summarizes the properties you can use to simplify expressions.
0 is the identity.
a + 0 = 0 + a = a
1 is the identity.
a * 1 = 1 * a = a
Exercise 2:
a. Write an algebraic expression for the verbal expression the sum of two and the square of t increaced by the sum of t squared and 3. 2 + t + t + 3 b. Then simplify the algebraic expression, indicating all of the properties used.
a. Write an algebraic expression for the verbal expression the sum of two and the square of t increaced by the sum of t squared and 3.
2 + t + t + 3
b. Then simplify the algebraic expression, indicating all of the properties used.
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: 17 - 47 odd, 48 - 56
Alternative Homework: Enriched: 16 - 44 even, 45 - 56
Extra Practice: Students book page 758 Lesson 1-8
Extra Practice Worksheet: Click Here.
Return to Chapter 1