Slide 1Lesson 33

DivisibilityWO.8 Know the basic multiplication facts (products

of one-digit numbers) and the related division facts.

WO.18 Find factors and multiples of whole numbers.

WO.17 Use long division to determine if one number is divisible by another.

WO.20 Find multiple factorizations for composite numbers (e.g., 30 = 3 × 10 = 5 × 6).

Chapter 7

Lesson 33

Slide 2Lesson 33

Objectives

• Understand and apply the definitions of divisibility, multiple, and factor.

• Understand that multiples of a number are divisible by that number.

• Use long division to test for divisibility and find factors of a number.

Slide 3Lesson 33

Remember from Before

• What is a multiple?

• How can you use long division to test if a fraction is a whole number?

Slide 4Lesson 33

Get Your Brain in Gear

1. Use mental math to solve each equation.

a. 4 × a = 36

b. 12 × b = 36

c. 5 × c = 35

d. 3 × d = 72

2. Count by 3’s from 30 to 90.

a = 9

b = 3

c = 7

d = 24

30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90

Slide 5Lesson 33

6 ÷ 2 = 3

6 = 2 × 3

Slide 6Lesson 33

a = 2 × h

Here a is divisible by 2 when h is a whole number.

2 = 2 × 1 4 = 2 × 2 6 = 2 × 3 8 = 2 × 4

Slide 7Lesson 33

The number 5 is NOT divisible by 2.

But the height of this rectangle is 2 + ,which is not a whole number.

121

22 +

Slide 8Lesson 33

Whole numbers that are divisible by 2 are called even numbers.

Whole numbers that are NOT divisible by 2 are called odd numbers.

Slide 9Lesson 33

Check for Understanding

1. Each of the following rectangles is made out of 7 unit squares. The widths are whole numbers but the heights are not. Express the height of each rectangle as a whole number plus a fraction.

h = 3 + h = 2 + h = 1 +

Slide 10Lesson 33

Check for Understanding

2. If you make a rectangle out of 10 unit squares that has a width of 4, what will the height be? Draw a picture to show your answer. Express the height as a whole number plus a fraction.

3. Which of the following numbers are odd? Which are even?

a. 5 b. 7 c. 8

d. 9 f. 0e. 6

The width is 4, the height is 2 + .

odd odd even

odd eveneven

Slide 11Lesson 33

All even numbers (numbers divisible by 2) are described by the following expression:

2 × h

We can use the commutative property of multiplication to rewrite this expression like this:

h × 2

When h = 3, the above expression equals:

3 × 2

2 + 2 + 2

We can also express this using repeated addition:

Slide 12Lesson 33

2 = 2 2 + 2 = 4 2 + 2 + 2 = 62 + 2 + 2 + 2 = 8

Slide 13Lesson 33

Numbers divisible by 3: h × 3 h is a whole number

Slide 14Lesson 33

Check for Understanding

4. Name the first ten multiples of 7 as quickly as you can.

5. Name four multiples of 2 that are also multiples of 3.

6. Name three multiples of 4 that are also multiples of 5.

7. Name three multiples of 3 that are also multiples of 4.

8. What do we call the numbers that are multiples of 1?

0, 7, 14, 21, 28, 35, 42, 49, 56 and 63.

The first four are 0, 6, 12 and 18.

The first three are 0, 20, and 40.

The first three 0, 12, and 24.

Whole numbers

Slide 15Lesson 33

There are many ways to test for divisibility:

• Building rectangles.

• Using the number line.

• Using our knowledge of the basic multiplication facts.

Slide 16Lesson 33

Is 133 divisible by 7?

133 = 7 × 19

Slide 17Lesson 33

Is 3 a factor of 133?

133 = 3 × 44 + 1

We conclude that 133 is NOT divisible by 3. It’s a jump of +1 too big.

Slide 18Lesson 33

Is 133 divisible by 542?

We don’t need to determine what 133 ÷ 542 equals because we know that is less than 1, but greater than 0.

133542

This means 133 ÷ 542 is not a whole number.

From this we can conclude instantly that 133 is NOT divisible by 542.

In general, numbers don’t have factors that are greater than themselves.

The only exception to this is 0.

Zero is divisible by all whole numbers greater than itself.

Slide 19Lesson 33

Check for Understanding

9. Find the smallest whole number (other than 0) that is NOT a factor of 1260.

8 is the smallest whole number that is not a factor of 1260.

Slide 20Lesson 33

Multiple Choice Practice

1. Given the location of on the number line, which statement is true if n and d are whole numbers?

nd

n is greater than d.

n is a multiple of d.

n is divisible by d.

n is a factor of d.

Slide 21Lesson 33

Multiple Choice Practice

2. If m and w are whole numbers and m is a multiple of w, which statement is NOT necessarily true?

Slide 22Lesson 33

A student drew a picture showing that 176 can be divided into 3 equal parts. From this the student claimed that 176 is divisible by 3. What does the student misunderstand about what it means to be divisible? Is 176 really divisible by 3?

Find the Errors

Any number can be broken down into 3 equal parts. To be divisible means that each of those equal parts is a whole number! 176 is not divisible by 3, because when you divide it by 3 you do not get a whole number.