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Patterns in Multiplication and Division
Factors: numbers you multiply to get a product.
Example:
6 x 4 = 24
Factors
Product
Product: the result of multiplication (answer).
Patterns in Multiplication and Division
Opposites:
using multiplication to solve division
42 ÷ 7 = 6
Dividend
Divisor
quotient: is the result of a division.
Quotient
What 2 multiplication equations can I create from above
1.
2.
Introduction to Fraction Operations
Student Outcome: I will learn why a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 and NOT 0
Divisibility: how can you determine if a number is divisible by
2,3,4,5,6,7,8,9 or 10?
•
•
•
Complete the chart on the next slides and circle all the
numbers divisible by 2,3,4,5,6,7,8,9, and 10.
Then find a pattern with the numbers to figure out
divisibility rules.
Reflect on your findings with your class.
Divisibility Rules for 2, 5, & 10
Student Outcome: I will learn why a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 and NOT 0
Circle the numbers in
the chart that are divisible
by 2 leaving no remainder.
Any patterns?
Can you make a rule?
Can you notice
similarities in the
quotients?
A number is divisible by:
2
If:
The last digit is even (0,2,4,6,8)
Example:
128 is
129 is not
5
The last digit is 0 or 5
175 is
809 is not
10
The number ends in 0
220 is
221 is not
Divisibility Rules for 4, & 8
Circle the numbers in
the chart that are divisible
by 4 leaving no remainder.
Any patterns?
Can you make a rule?
Can you notice
similarities in the
quotients?
A number is divisible by:
4
or
8
or
If:
The last 2 digits are divisible by 4
the last 2 digits divisible by 2 twice
“Double Double”
Example:
1312 is (12÷4=3)
7019 is not
The last three digits are divisible by 8 109816 (816÷8=102) Yes
number is divisible by 2 three times
216302 (302÷8=37 3/4) No
“Triple Double”
Divisibility Rules for 3, 6, & 9
Circle the numbers in
the chart that are divisible
by 3 leaving no remainder.
Any patterns?
Can you make a rule?
Can you notice
similarities in the
quotients?
A number is divisible by:
If:
3
The sum of the digits is divisible by 3
6
The number is divisible by both 2 and 3
9
Example:
381 (3+8+1=12, and 12÷3 = 4) Yes
217 (2+1+7=10, and 10÷3 = 3 1/3)No
114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes
308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No
The sum of the digits is divisible by 9(Note: you can apply this rule to that answer again if you want)
1629 (1+6+2+9=18, and again, 1+8=9) Yes
2013 (2+0+1+3=6) No
Divisibility Rules for 0
Circle the numbers in
the chart that are divisible
by 0 leaving no remainder.
Any patterns?
Can you make a rule?
Can you notice
similarities in the
quotients?
Divisibility Rules
Go to this site for an overall review of the divisibility
rules! (or check your folder for word document)
http://www.mathsisfun.com/divisibility-rules.html
Go to this site for games!
http://www.studystack.com/matching-53156
Divisibility Rules
Assignment
Page 207 - 208 #
3, 22, 24, 25, 26, 28
19
1b,5,15,17,18,23
1b, 3, 9, 11, 15, 23
Student Outcome: Use Divisibility Rules to SORT Numbers
Carroll Diagram
Divisibility
by 6
Venn Diagram
Divisibility
by 9
Not
Divisible
by 9
162
3996
30
31 974
Divisible
by 6 6
162
30
31 9746
Not
Divisible
by 6
23 517
Divisible
by 9 6
39966
23 5176
79
79
Shows how numbers are the
same and different!
Shows relationships between
groups of numbers.
Discuss with you partner why each number belongs where is does.
Student Outcome: Use Divisibility Rules to SORT Numbers
Carroll Diagram
Divisibility
by
Not
Divisible
by
Divisibility
by
Not
Divisible
by
Shows how numbers are the
same and different!
Create a “Carroll Diagram” that
sorts the numbers below
according to divisibility by 3 & 4.
12, 32, 60, 24, 3140,
99
Student Outcome: Use Divisibility Rules to SORT Numbers
Create a “Venn Diagram” that
sorts the numbers below
according to divisibility by 3 & 4.
12, 32, 60, 24,
3140, 99
Venn Diagram
Divisible
by 6
Divisible
by 6
Shows relationships between
groups of numbers.
Student Outcome: Use Divisibility Rules to SORT Numbers
Fill in the Venn diagram with 7 other numbers. There
must be a minimum 2 numbers in each section.
Divisible
by 2 6
Venn Diagram
Divisible
By 5 6
Share your number with the group beside you. Do their numbers work?
Assignment
Page 207 # 7, 8, 13
Practical Quiz #1
Fill in the Venn diagram with these numbers:
Venn Diagram
4, 8, 12, 16, 20, 24, 30, 32, 80
Divisible
By 4 6
Divisible
By 8 6
Factors
Go to this site for showing factors
http://www.harcourtschool.com/activity/elab2004/gr5/9.html
Student Outcome: I will be able to use Divisibility Rules to Determine Factors
Common Factors: a number that two or more numbers are divisible by
OR
numbers you multiply together to get a product
Example: 4 is a common factor of 8 & 12
1x8=8
2x4=8
HOW?
1 x 12 = 12
2 x 6 = 12
3 x 4 = 12
What is the least common factor (LCF) for 8 and 12?
What is the greatest common factor (GCF) for 8 and 12?
How would you describe in your own words (LCF) and (GCF)? Then discuss with
your partner
Student Outcome: I will be able to use Divisibility Rules to Determine Factors
Common Factors: a number that two or more numbers are divisible by
OR
numbers you multiply together to get a product
Example: 3 and 9 are common factors of 18 & 27
1 x 18 = 18
2 x 9 = 18
3 x 6 = 18
HOW?
1 x 27 = 27
3 x 9 = 27
What is the least common factor (LCF) for 18 and 27?
What is the greatest common factor (GCF) for 18 and 27?
How would you describe in your own words (LCF) and (GCF)? Then discuss with
your partner
Student Outcome: I will be able to use Divisibility Rules to Determine Factors
Common Factors: a number that two or more numbers are divisible by.
OR
numbers you multiply together to get a product
List the common factors for the numbers below…
1. 6 & 9
2. 8 & 16
3. 36 & 12
Greatest Common Factor
the greatest number that both numbers are divisible by.
Student Outcome: I will be able to use Divisibility Rules to Determine Factors
Fill in the Venn diagram with factors for 24 and 32.
What factors would go in the middle area?
Factors of
246
Venn Diagram
Factors of
326
Share your numbers with the person beside you. Do their numbers match?
Assignment
Page 207 # 12, 13
Page 208 # 24
Factors
Factor Game
Mr. Bosch will type in a number. You must list all the
factors to get a point. You are playing against your
neighbor. We will play 10 rounds. Person with the most
points wins. Second place person does 15 pushups.
http://www.harcourtschool.com/activity/elab2004/gr5/9.html
Practical Quiz #2
Fill in the Venn diagram with factors for 12 and 30.
What factors would go in the middle area?
Factors of
126
Venn Diagram
Factors of
306
Student Outcome: I will be able to use Divisibility Rules to place fractions in lowest
terms.
Lowest Terms:
when the numerator and denominator of the fraction have no common factors than 1.
Ask Yourself?
÷2
Example:
12 = 6
42 21
What are things you know that will
help with the factoring?
What number can I factor out of
the numerator and denominator?
÷2
Can I use other numbers to make
factoring quicker?
Student Outcome: I will be able to use Divisibility Rules to place fractions in lowest
terms
Place the fractions below into “lowest terms…”
6
10
Share with your neighbor. Did they do more/less/same number of factoring steps?
Student Outcome: I will be able to use Divisibility Rules to place fractions in lowest
terms
Place the fractions below into “lowest terms…”
24
30
Share with your neighbor. Did they do more/less/same number of factoring steps?
Student Outcome: I will be able to use Divisibility Rules to place fractions in lowest
terms
Place the fractions below into “lowest terms…”
9
33
Share with your neighbor. Did they do more/less/same number of factoring steps?
Student Outcome: I will be able to use Divisibility Rules to place fractions in lowest
terms
Place the fractions below into “lowest terms…”
15
35
Share with your neighbor. Did they do more/less/same number of factoring steps?
Student Outcome: I will be able to use Divisibility Rules to place fractions in lowest
terms
Let’s Play a game
http://www.mathplayground.com/fractions_reduce.html
http://www.mathplayground.com/fractions_reduce.html
http://www.jamit.com.au/htmlFolder/app1002.html
http://www.jamit.com.au/htmlFolder/app1002.html
Assignment
Page 207 # 15abc, 16abc
Section 6.3 – Extra Practice Handout
Practical Quiz #3
Place the fractions below into “lowest terms…”
a) 12
16
b) 21
30
c) 32
40
Student Outcome: I will learn how to add fractions with Like denominators
Use Pattern Blocks & Fraction Strips to Model Fractions
They both
represent
ONE WHOLE
1. Using the similar pattern blocks can you make one
whole? How many does it take?
Using Manipulatives to ADD Fractions
How can you divide each whole into equal sections listed in the chart below?
Equal Sections
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
Using Manipulatives to ADD Fractions
Equal Sections
Example: 1 + 1 =
2 2
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
Demo
+
=
Using Manipulatives to ADD Fractions
Equal Sections
Example: 1 + 1 =
3 3
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
Demo
+
=
Using Manipulatives to ADD Fractions
Equal Sections
Example: 1 + 3 =
4 4
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
Demo
+
=
Using Manipulatives to ADD Fractions
Equal Sections
Example: 2 + 3 =
6 6
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
Demo
+
=
Using Manipulatives to ADD Fractions
Equal Sections
Example: Create you own equation
with common denominators
for your partner to solve
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
Demo
+
=
Student Outcome: I will learn how to add fractions with Like denominators
___
1.
2.
3.
4.
+
___
=
____
+
____
=
Name the fractions above…
What if I were to ADD the same fraction to the one above…how
many parts would need to be colored in?
What is the name of our new fraction?
Using other pattern blocks can it be reduced to simplest form?
Student Outcome: I will learn how to add fractions with Like denominators
Using pattern blocks model the following equation. Write the
answer in lowest terms.
2 + 1
6
6
= ___ = __
4 + 1
6
6
= ___ = __
Student Outcome: I will learn how to add fractions with Like denominators
Using pattern blocks model the following equation. Write the
answer in lowest terms.
2 + 1
3
3
= ___ = __
2 + 1
4
4
= ___ = __
Student Outcome: I will learn how to add fractions with Like denominators
Can we add fractions with other denominators other than
“6”? Write the answer in lowest terms.
1 + 1
4
4
= ___ = ___
4 + 1
10
10
= ___ = ___
1 + 5
9
9
= ___ = ___
Assignment
Pages 214-215: 5ab, 6ab, 7ab, 9ab,
10ef, 12, 14
Assignment
6.2 – Add Fractions with like Denominators - Handout
Using Manipulatives to SUBTRACT Fractions
Equal Sections
Example: 2 - 1 =
2 2
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
=
Using Manipulatives to SUBTRACT Fractions
Equal Sections
Example: 2 - 1 =
3 3
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
=
Using Manipulatives to SUBTRACT Fractions
Equal Sections
Example: 3 - 1 =
6 6
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
=
Using Manipulatives to SUBTRACT Fractions
Equal Sections
Example: Create you own equation
with common denominators
for your partner to solve
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
=
Student Outcome: I will learn how to subtract fractions with Like denominators
Using pattern blocks model the following equation. Write the
answer in lowest terms.
5 - 1
6
6
= ___ = __
4 - 2
6
6
= ___ = __
Student Outcome: I will learn how to add fractions with Like denominators
Using pattern blocks model the following equation. Write the
answer in lowest terms.
2 - 1
3
3
= ___ = __
2 - 1
4
4
= ___ = __
Student Outcome: I will learn how to add fractions with Like denominators
Can we subtract fractions with other denominators other
than “6”? Write the answer in lowest terms.
5 - 1
7
7
= ___ = ___
4 - 1
10
10
= ___ = ___
8 - 5
9
9
= ___ = ___
Assignment
Pages 220-221: 5ab, 6ab, 8ab, 10, 11
Assignment
6.3 – Subtract Fractions with like Denominators - Handout
Unit Review Assignment
Chapter Review
Page 222-223
#1-17
Wrap it Up Game
Page 226 “It’s Divisible”
See Smart File
Wrap it Up Assignment
Give handout to students to figure out activities completed during a
24 hour timer frame.
Student Outcome: I will learn how to add fractions with Different denominators
Looking towards the next unit(7)…
Adding and subtracting fractions
with different denominators
Student Outcome: I will learn how to add fractions with Different denominators
Give a fraction for the…
1. Red portion = ____
2. Yellow Portion = ____
3. Green Portion = ____ = ____
4. Blue Portion = ____ = ____
Student Outcome: I will learn how to add fractions with Different denominators
Use Pattern Blocks & Fraction Strips to Model Fractions
They both
represent
ONE WHOLE
1. Using any combination of pattern blocks can you make
one whole? How many of each does it take?
Using Manipulatives to ADD Fractions
Equal Sections
Example: 1 + 3 =
2 6
+
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
=
Using Manipulatives to ADD Fractions
Equal Sections
Example: 1 + 4 =
3 6
+
Color
Fraction
2
Red
1/2
3
Blue
1/3
4
Orange
1/4
6
Green
1/6
=