6th Grade Math Textbook, Fundamentals

68 Chapter 3

More Enrichment Topics

1.For all real numbers aand b, a@ b(ab)^2.

a.Find a@ bfor at least five pairs of numbers.

b.Is @ commutative? If so, explain how you know. If not, give a counterexample.

2.For all integers aand b, a# b0 if abis even and a# b1 if abis odd.

a.Find a# bfor at least five pairs of integers.

b.Is # commutative? If so, explain how you know. If not, give a counterexample.

3.Discuss and Write Define an operation that applies to all real numbers.

Make a table showing the result of your operation for at least five pairs

of numbers.

page 75 for exercise sets.

Enrichment:

Define, Substitute, and Compute

Objective To invent arithmetic operations • To explore properties of invented operations

Each of the four basic mathematical operations combines two numbers to get
another number. For example, each basic operation below is combining 12 and 3:

12  3  15 12  3  9 12 • 3  36 12  3  4

You can invent your own operations based on the four

basic operations.

For example, you could define an operation so that for any real

numbers aand b, a★b. So 5 ★ 2 3.5.

The table at the right shows the values of a★bfor more values

of aand b. Note the following about the operation ★.

• The operation ★is commutative.
  For any numbers aand b, a★bb★a.


• For any number a, a★aa.

Consider an operation ❖that is defined for onlyfive whole numbers: 0, 1, 2, 3,

and 4. The result of a❖bis theremainderwhen abis divided by 5.

For example, 4 ❖ 3 2, because 4  3 7 and the remainder when 7

is divided by 5 is 2.

You can use a table to show the result of every possible

combination of numbers. The table shows:

• The operation ❖is commutative.
  For example, 2 ❖ 4  4 ❖ 2 6.


• 0 is the identity element because a❖0 equals a.


• The inverse of a number ais 5 a. That is, a❖(5 a) 0.
  For example, the inverse of 3 is 2, and the inverse of 1 is 4.

7

2

5  2

2

2 a

2

a  a

2

b a

2

a  b

2

a  b

2

❖ 01234

0 01234

1 12340

2 23401

3 34012

4 40123

aba ★b

021

132

222

361.5

 25  33  29

6.4 2.2 4.3