Download 01-04 Factors of Whole Numbers

Name: _______________
Math 6
Unit 1
Lesson 4
Factors of Whole Numbers
Comparing Ages
Alyssa is twelve years old. She has three cousins.
They are Daron, age 2, Rachel age 3, and Matt age 8.
Alyssa wrote the following mathematical sentences to
show her age:
Alyssa
Age 12
Math 6
Six times as
old as Daron:
12 = 6 x 2
Four times as
old as Rachel:
12 = 4 x 3
%
∆
Daron
Age 2
Rachel
Age 3
One and one half
times as old as Matt:
12 = 11
2 x 8
Matt
Age 8
4-1
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
Reflection
How are the first two examples different from
the last?
Daksha is also 12 years old. He has four cousins. They are Anoop, age 2, Bishnu
age 4, Dayita, age 5 and Rani, age 6. Can you write mathematical sentences for
Daksha’s age like those that Alyssa wrote?
The easy ones should be Anoop, Bishnu and Rani.
Daksha is six times as old as Anoop: 12 = 6 x 2
Daksha is three times as old as Bishnu: 12 = 3 x 4
Daksha is two times as old as Rani: 12 = 2 x 6
These are easy because the numbers 2, 4, and 6 are
factors of the number 12. Since 5 is not a factor of 12 it
is harder to write a description. You cannot find another
whole number to multiply by 5 to get a product of 12.
Daksha
Age 12
Rani
Age 6
Dayita
Age 5
Bishnu
Age 4
Anoop
Age 2
∏
4-2
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
Reflection
Are there any other factors of 12?
Objectives for this Lesson
In this lesson you will explore the following concepts:
•
•
•
•
•
Use arrays to find all the factors of a number
Explain strategies to find factors of a number
Use concrete and visual representations to find factors of a number
Use repeated division to find factors of a number
Solve problems involving factors or multiples
Factors
A whole number that divides evenly into a number is called a factor of the
number. 3 is a factor of 12 because when 3 is multiplied by 4 the product is 12.
12
=
3x4
Product Factors
4 is also a factor of 12 because when 4 is multiplied by 3 the product is 12.
Math 6
4-3
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
Example 1
Identify the factors in the following number sentences:
a. 12 x 7 = 84
b. 15 x 3 = 45
c. 76 x 4 = 304
a. The factors of 84 in this number sentence are 12 and 7.
b. The factors of 45 in this number sentence are 15 and 3.
c. The factors of 304 in this number sentence are 76 and 4.
In some cases you can use your knowledge of multiplication facts to 12 x 12 to
find the factors of a number. You may also need to use some methods when the
numbers are a little harder.
Example 2
Identify the factors of 24.
Start with 1 and list multiplication sentences that have a product of 24:
1 x 24 = 24
2 x 12 = 24
3 x 8 = 24
4 x 6 = 24
At some point the factors will reverse and you will be listing the same factors all
over again:
6 x 4 = 24
8 x 3 = 24
12 x 2 = 24
1 x 24 = 24
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
You may also notice a pattern to listing the factors of a number:
4-4
Factors of 18:
1,
2,
3,
6,
3 x 6 = 18
2 x 9 = 18
1 x 18 = 18
9,
18
Sometimes, this type
of pattern can help
you list your factors.
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
Using Arrays to Find Factors
Just as you used arrays to find the multiples of a number you can also use them
to find factors of a number.
Example 3
Use the grid paper to find all of the arrays that are made using a total of
18 squares. List the factors of 18.
You need to make as many unique rectangles as possible that have a total
of 18 squares:
18
1
9
2
6
3
When you cannot find any more unique rectangles for the number of squares
needed, you have found all the factors of the number.
The factors of 18 are: 1, 2, 3, 6, 9, 18
Some numbers have more factors and some have fewer factors than 18.
Math 6
4-5
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
Reflection
Do you think the size of the number determines the
number of factors? Why or why not?
Let’s Explore
Exploration 1: Arrays to Factors
Materials: Use the graph paper on page 4-12.
For 1 – 5: Use a different coloured pencil crayon for each.
1.Make all possible arrays of 36 squares on your grid paper.
List the factors of 36.
2.Make all possible arrays of 21 squares on your grid paper.
List the factors of 21.
3.Make all possible arrays of 32 squares on your grid paper.
List the factors of 32.
4.Make all possible arrays of 17 squares on your grid paper.
List the factors of 17.
5.Make all possible arrays of 29 squares on your grid paper.
List the factors of 29.
4-6
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
The proper factors of a number are all of the factors less than the number
but greater than 1. The proper factors of 18 are: 2, 3, 6, 9
6. How many proper factors does the number 29 have? _____________________
7.Reflect: What is different about the factors of 29 compared to the
factors of 32? ________________________________________________
8.List the numbers from 1 to 20 that have no proper factors?
__________________________________________________________
Using Repeated Division to Find Factors
You can also use division to identify factors of a number.
The factors of 18 using division:
18 ÷ 1 = 18
18 ÷ 2 = 9
18 ÷ 3 = 6
18 ÷ 6 = 3
18 ÷ 9 = 2
To start finding factors of larger numbers you may want to use division.
A number is always divisible by 1 and itself. To find the factors you may want
to start with a number larger than 1.
Example 4
Zach has 88 baseball cards.
He wants to arrange them in rows
with an equal number of cards in
each row on his wall. What are the
different arrangements he can make
for the cards?
You need to know all of the
multiplication sentences for a
product of 88. You can use upside
down repeated division to find the
factors of 88. Only divide by numbers
that are divisible by 1 and itself.
Math 6
4-7
Math 6
Unit 1
1. D
ivide by a
number and
write the
division upside
down like this:
2 88
Lesson 4: Factors of Whole Numbers
2. N
ow divide
the 2 into 88
and write the
answer beneath
the 88:
2 88
44
3. R
epeat the process going down
the page until you cannot divide
anymore:
2 88
44
22
2 88
44
22
11
The only whole numbers that divide 11 evenly are 1 and itself, so you have
finished dividing.
4. List the factors that you found:
1, 2, 11, 22, 44, 88
5.List all number sentences:
You have the following number sentences that are complete:
1 x 88 = 88
2 x 44 = 88
And you have the following that are incomplete:
__ x 11 = 88
__ x 22 = 88
4-8
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
6.
Zach can make the following arrangements:
1 row of 88 cards 2 rows of 44 cards
4 rows of 22 cards 8 rows of 11 cards
88 rows of 1 card 44 rows of 2 cards
22 rows of 4 cards 11 rows of 8 cards
You can also use factor trees to track the numbers that you have divided into
your number.
One Factor tree of 88:
Another Factor tree of 88:
88
88
2
8
44
2
2
22
2
4
2
11
11
2
Both methods allow you to find several factors of 88. You can then use them to
identify the rest:
1, 2, ____, ____, 11, 22, 44, 88
1, 2, 4, 8, 11, ____, ____, 88
? X 11 = 88
4 x ? = 88
? x 22 = 88
2 x ? = 88
1, 2, 4, 8, 11, 22, 44, 88
Math 6
1, 2, 4, 8, 11, 22, 44, 88
4-9
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
Comparing Factors
You can use Venn diagrams to show the relationship between factors of
two numbers.
Example 5
Complete the Venn diagram with all possible factors of each number.
Factors of 24
Factors of 36
List the factors of each number.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The factors they have in common go in the intersection of the circles.
Factors of 24
Factors of 36
1
8
24
4-10
2
3
4
6
12
9
18
36
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
You can use your Venn diagram to answer questions like:
•
•
•
•
Do the numbers share any factors?
What is the largest factor these two numbers share?
How many factors do the numbers share?
Which number has more factors?
Math 6
4-11
Math 6
Unit 1
4-12
Lesson 4: Factors of Whole Numbers
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
1.
Underline the product in this equation: 4 x 5 = 20
2.
Underline the factors in this equation: 2 x 8 = 16
4-13
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
3.Write three multiplication sentences and identify the factors of
each sentence.
For 4 – 15: Find the factors of each of the following.
4.
46
5.
124
6.
96
7.
65
8.
42
9.
78
10. 108
Math 6
11. 57
4-14
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
12. 225
13. 200
14. 180
15. 140
16.Not including myself and 1, my factors are 2, 3, 4, 6, 8, 12, 16, 24, 32, and
48. What number am I?
17.Alyssa and her family have 72 awards for their sports’ hobbies. She wants
to display them in rows with an equal number of awards. How many
different ways can Alyssa set up the rows?
4-15
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
18.A perfect number is a number whose factors add up to the given number.
The smallest perfect number is six:
Factors of 6: 1, 2, 3, 6
Factors of 6 that are less than 6: 1, 2, 3
The sum of the factors of six: 1 + 2 + 3 = 6
What is the next perfect number?
19. Complete the Venn diagram with all possible factors of each number.
Factors of 30
Factors of 36
20. What is the largest factor of both 30 and 36?
Math 6
4-16
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
21. R
eflect: Describe the method of finding factors that you use most often.
Why do you like this method?
Mixed
Re vi ew
1.
Draw a rectangle.
2.
Draw a square.
4-17
Math 6
Unit 1
Lesson 4: Factors of Whole Numbers
3.
Draw a trapezoid.
4.
Draw a pair of parallel line segments.
5.
Draw a rhombus.
Math 6
4-18
Math 6
Unit 1
4-19
Lesson 4: Factors of Whole Numbers