Page 325 -We can also use multiplication to describe this same relationship:
Page 325 -In general, we can describe all numbers that are divisible by 2 using the following equation:
Page 326 -In this form, 2 is being multiplied h times.
Page 327 -From this we see that all multiples of 2 are divisible by 2.
Page 327 -We can generalize this to divisibility by any number.
Page 328 -Long division makes it easy.
Page 328 - Let’s find out using long division: -There is a remainder of 1. We can represent this result as follows:
Lesson 33 Powerpoint
Chapter 7 Lesson 33 Divisibility WO.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 ).
Objectives <ul><li>Understand and apply the definitions of divisibility, multiple, and factor. </li></ul><ul><li>Understand that multiples of a number are divisible by that number. </li></ul><ul><li>Use long division to test for divisibility and find factors of a number. </li></ul>
Remember from Before <ul><li>What is a multiple? </li></ul><ul><li>How can you use long division to test if a fraction is a whole number? </li></ul>
Get Your Brain in Gear <ul><li>1 . Use mental math to solve each equation. </li></ul><ul><li>a. 4 × a = 36 </li></ul><ul><li>b. 12 × b = 36 </li></ul><ul><li>c. 5 × c = 35 </li></ul><ul><li>d. 3 × d = 72 </li></ul><ul><li>2. Count by 3’s from 30 to 90. </li></ul>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
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
<ul><li>The number 5 is NOT divisible by 2. </li></ul>But the height of this rectangle is 2 + ,which is not a whole number. 1 2 1 2 2 +
<ul><li>Whole numbers that are divisible by 2 are called even numbers . </li></ul><ul><li>Whole numbers that are NOT divisible by 2 are called odd numbers . </li></ul>
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 +
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. 0 e. 6 odd odd even odd even even The width is 4, the height is 2 + .
<ul><li>All even numbers (numbers divisible by 2) are described by the following expression: </li></ul>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:
Numbers divisible by 3: h × 3 h is a whole number
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
<ul><li>There are many ways to test for divisibility: </li></ul><ul><ul><ul><li>Building rectangles. </li></ul></ul></ul><ul><ul><ul><li>Using the number line. </li></ul></ul></ul><ul><ul><ul><li>Using our knowledge of the basic multiplication facts. </li></ul></ul></ul>
<ul><li>Is 3 a factor of 133? </li></ul>133 = 3 × 44 + 1 We conclude that 133 is NOT divisible by 3. It’s a jump of +1 too big.
<ul><li>Is 133 divisible by 542? </li></ul>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. We don’t need to determine what 133 ÷ 542 equals because we know that is less than 1, but greater than 0. 133 542
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.
Multiple Choice Practice n is greater than d . n is a multiple of d . n is divisible by d . n is a factor of d . 1. Given the location of on the number line, which statement is true if n and d are whole numbers? n d
Multiple Choice Practice 2. If m and w are whole numbers and m is a multiple of w , which statement is NOT necessarily true?
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.