Download Prime Factor trees and the GCF

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5
prime Factor trees and the GCF
Monitoring progress:
Quiz 1
prime Factor trees and the GCF
How do factor trees help us find the GCF?
Another way to find greatest common factors is to use prime factor
trees for each number.
Here we use prime factor trees to find the GCF for 42 and 36.
Steps for Using Factor trees
Step 1
Find two factors for each number.
42
36
6
7
6
6
Step 2
Continue factoring until only prime factors are left.
36
42
6
2
6
7
3
2
6
3
Step 3
Circle the prime factors.
42
6
2
36
6
7
3
3
2
2
6
3
2
3
Step 4
Find the GCF.
Both 42 and 36 have 2 and 3 as common prime factors. To find the
greatest common factor for 42 and 36, we multiply these two prime
factors: 2 × 3 = 6.
the GCF for 42 and 36 is 6.
Unit 6 • Lesson 5
371
Lesson 5
Let’s look at another example.
example 1
Find the greatest common factor for 14 and 56 using prime
factor trees.
Step 1
Find two factors for each number.
14
2
7
56
8
7
Step 2
Continue factoring until only prime factors are left.
56
14
7
2
8
2
7
4
2
2
Step 3
Circle the prime factors.
14
56
7
2
8
7
4
2
2
2
Step 4
Find the GCF.
We see that 2 and 7 are common prime factors for 14 and 56.
Multiply to find the GCF.
2 × 7 = 14
the GCF of 14 and 56 is 14.
Apply Skills
Turn to Interactive Text,
page 238.
372
Unit 6 • Lesson 5
Monitoring progress
Quiz 1
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Lesson 5
Homework
Activity 1
tell which divisibility rule or rules (2, 3, 5, 6, or 10) can be used to divide
each number.
1. 4,685
2. 1,350
3. 57,912
See Additional Answers below.
4. 45,402
5. 179,031
Activity 2
Find the GCF for each pair of numbers by drawing prime factor trees.
Model 12 and 20
12
20
6
2
2
2×2=4
2
10
3
2
5
GCF = 4
1. 4 and 16
2. 32 and 36
3. 18 and 24
4. 16 and 30
See Additional Answers below.
Activity 3 • Distributed practice
Solve.
1.
33 R2
5,000
 4,999
1
2.
6,978
+ 3,482
10,460
3.
50
50
4.
6q200
64 R6
5. 7q454
2,500
Unit 6 • Lesson 5
373
Lesson
5
Prime Factor trees and the GCF
Monitoring Progress:
Quiz 1
Lesson5 SkillsMaintenance
Name
Skills Maintenance
Finding the GCF, Congruent Shapes
Building Number Concepts:
Date
SkillsMaintenance
FindingtheGCF
Activity1
FindtheGCFforthepairsofnumbers.
1.
What is the GCF of 18 and 20?
2
Prime Factor trees and the GCF
2.
What is the GCF of 24 and 28?
4
3.
What is the GCF of 32 and 36?
4
In this lesson, students use prime factor trees
to determine the GCF for large numbers.
Students learn that when numbers are small,
it is not difficult to examine factor lists. When
numbers are large, we need to come up with a
different strategy for finding the GCF.
4.
What is the GCF of 12 and 48?
12
A prime factor tree is a good tool to use when
we are investigating the common factors,
particularly the GCF, for two large numbers.
When working with large numbers and
complex strategies with many steps, it is
important to work carefully and organize
our work.
CongruentShapes
Activity2
Ineachrowofshapes,thereisoneshapethatiscongruent.Circleit.
1.
Unit 6
Lesson Planner
2.
3.
Objective
Students will use prime factor frees to find
the GCF for large numbers.
4.
Monitoring Progress:
Unit6•Lesson5
237
Quiz 1
Distribute the quiz, and remind students that
the questions involve material covered over
the previous lessons in the unit.
Homework
Students tell which divisibility rules apply
to the given numbers, and use prime factor
trees to find the GCF of pairs of numbers. In
Distributed Practice, students solve problems
using all four of the basic operations.
Skills Maintenance
Finding the GCF, Congruent Shapes
(Interactive Text, page 237)
Activity 1
Students identify the GCF for pairs of numbers. Remind
students to use their knowledge of facts, divisibility
rules, and factor lists to help them.
Activity 2
Students circle shapes that are congruent.
668 Unit 6 • Lesson 5
Lesson
Building Number Concepts:
Prime Factor trees and the GCF
How do factor trees help us find the GCF?
(Student Text, pages 371–372)
Connect to Prior Knowledge
Begin by asking students what the difficulty
could be when they try to find the GCF for 99
and 120, using factor lists.
5
Quiz 1
How do factor trees help us find the GCF?
Another way to find greatest common factors is to use prime factor
trees for each number.
Here we use prime factor trees to find the GCF for 42 and 36.
Steps for Using Factor trees
Step 1
Find two factors for each number.
42
36
6
7
6
2
6
7
3
2
: Use the mBook Teacher Edition
for Student Text, page 371.
Overhead Projector: Reproduce the
trees on a transparency, and modify
as discussed.
Board: Draw the trees on the board,
and modify as discussed.
SteP 1
• Explain that to find the GCF of 42 and 36,
we first look for the prime factors. Start with
the prime factor tree for 42. Break the first
level into the factors 6 × 7. For 36, the first
level is broken into 6 × 6.
6
3
Step 3
Circle the prime factors.
42
2
6
7
3
3
2
36
2
6
3
2
3
Step 4
Find the GCF.
Both 42 and 36 have 2 and 3 as common prime factors. To find the
greatest common factor for 42 and 36, we multiply these two prime
factors: 2 × 3 = 6.
the GCF for 42 and 36 is 6.
Demonstrate
engagement Strategy: teacher Modeling
Demonstrate how we use prime factor trees in
one of the following ways:
6
Step 2
Continue factoring until only prime factors are left.
36
42
6
Link to today’s Concept
Tell students that in today’s lesson, we look at
finding the GCF of larger numbers using prime
factor trees.
Monitoring progress:
prime Factor trees and the GCF
6
Elicit from students that this is a lengthy method
for larger numbers. It is easy to get lost in the
middle and forget to include some of the factors.
prime Factor trees and the GCF
Unit 6 • Lesson 5
371
371
Step 2
• In the next level of the 42 tree, 7 is prime; 6 is
not, so we continue factoring. The next level is 2
× 3. Both numbers are prime. We are done with
42. In the next level of the 36 tree, both 6s can
be factored to 2 × 3. We are done with 36.
Step 3
• Circle all of the prime numbers. The prime
factorization of 42 is 2 × 3 × 7. The prime
factorization of 36 is 2 × 2 × 3 × 3.
Step 4
• Next find the numbers that are common to
both trees. The factors that are common in both
trees are 2 and 3. We multiply 2 × 3 = 6. the GCF is 6.
Unit 6 • Lesson 5 669
Lesson 5
Lesson 5
Let’s look at another example.
How do factor trees help us find the GCF? (continued)
example 1
Find the greatest common factor for 14 and 56 using prime
factor trees.
Step 1
Find two factors for each number.
14
Demonstrate
• Look at example 1 on page 372 of the
Student Text. Take students through the
example as outlined.
2
7
7
8
2
SteP 1
• Find two factors for 14 and two factors for
56. Draw them in the first level of the tree.
80 and 90 (10)
670 Unit 6 • Lesson 5
8
7
4
2
2
Step 4
Find the GCF.
We see that 2 and 7 are common prime factors for 14 and 56.
Multiply to find the GCF.
2 × 7 = 14
the GCF of 14 and 56 is 14.
Apply Skills
Turn to Interactive Text,
page 238.
372
372
120 and 160 (20)
56
7
2
Step 3
• Circle the prime factors. The prime
factorization of 14 is 2 × 7. The prime
factorization of 56 is 2 × 2 × 2 × 7.
Reinforce Understanding
For additional practice, have students draw
prime factor trees to find the GCF of these pairs
of numbers:
2
Step 3
Circle the prime factors.
14
Step 2
• Factor until there are only prime factors left.
Check for Understanding
engagement Strategy: Pair/Share
Put students into pairs and have them find
the GCF of 75 and 100 (25). Have one partner
draw the prime factor tree for 75 and the other
for 100. Then have partners share their trees
and collaborate to find the GCF. Have a pair of
volunteers walk the class through the steps of
finding the GCF.
7
4
2
2
7
Step 2
Continue factoring until only prime factors are left.
56
14
2
Step 4
• Look for common prime numbers in the
trees. Both 2 and 7 are common prime
numbers. Multiply them to get the GCF of 14.
56
8
Unit 6 • Lesson 5
Monitoring progress
Quiz 1
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Lesson5 ApplySkills
Name
Apply Skills
Date
ApplySkills
PrimeFactorTreesandtheGCF
(Interactive Text, page 238)
Activity1
Makeprimefactortreesforthepairsofnumbers,andfindtheGCF.
Have students turn to Interactive Text, page 238,
and complete the activity.
1.
What is the GCF of 66 and 90?
6
66
90
Activity 1
2
Students find the GCF for pairs of large numbers
using the prime factor trees. You might need to
scaffold the steps in this complex process to help
students organize their work. Monitor students’
work as they complete the activity.
2.
9
22
3
10
3
11
What is the GCF of 24 and 36?
3
2
12
24
36
6
2
Watch for:
3.
9
4
3
2
2
What is the GCF of 30 and 25?
3
6
prime factors?
4
3
2
2
5
30
• Can students fully factor each number to its
5
25
5
5
5
• Do students understand how to select the
common prime factors using the method
(e.g., one-to-one correspondence, matching
up the factors from each tree)?
• Do students understand the next step,
ReinforceUnderstanding
Use the mBook Study Guide to review lesson concepts.
238
Unit6•Lesson5
which is to multiply the common prime
factors together?
• Are students able to multiply the string of
prime numbers together to get the GCF?
• Do the students get lost in the procedures of
this multi-step process?
Reinforce Understanding
Remind students that they can review
lesson concepts by accessing the
online mBook Study Guide.
Unit 6 • Lesson 5 671
Unit 6 Quiz1•FormA
Lesson 5
Name
Date
Form A
Monitoring Progress
Common Factors and Number Patterns
Monitoring Progress:
Part 1
Quiz 1
Find the common factors for each set of numbers by writing the
factor lists.
ssess
A
Quiz 1
• Administer Quiz 1 Form A in the Assessment
Book, pages 77–78. (If necessary, retest
students with Quiz 1 Form B from the mBook
Teacher Edition following differentiation.)
3.
Common factors
1, 2, 3, 6
2.
Common factors
18
1, 2, 3, 6, 9, 18
14
1, 2, 7, 14
24
1, 2, 3, 4, 6, 8, 12, 24
21
1, 3, 7, 21
Common factors
1, 2
4.
Common factors
1, 7
1, 5
16
1, 2, 4, 8, 16
15
1, 3, 5, 15
18
1, 2, 3, 6, 9, 18
20
1, 2, 4, 5, 10, 20
20
1, 2, 4, 5, 10, 20
25
1, 5, 25
Unit 6
1.
Part 2
Find the greatest common factor for each set of numbers.
Students
Assess
Differentiate
Day 1
Day 2
1.
Scored
80% or above
Scored
Below 80%
6
2.
Factor lists
24
3.
Extension
GCF
15
20
12
4.
Factor lists
24
5
GCF
1, 3, 5, 15
1, 2, 4, 5, 10, 20
10
Factor lists
1, 2, 3, 4, 6, 12
1, 2, 3, 4, 6, 8, 12, 24
12
GCF
Factor lists
1, 2, 3, 6, 9, 18
1, 2, 3, 4, 6, 8, 12, 24
18
Quiz 1
Form A
All
GCF
1, 2, 5, 10
1, 2, 4, 5, 10, 20
30 1, 2, 3, 5, 6, 10, 15, 30
10
20
Reinforcement
Differentiate
• Review Quiz 1 Form A with class.
Unit6•Quiz1•FormA
Unit 6 Quiz1•FormA
• Identify students for Extension or
Reinforcement.
Monitoring Progress
xtension
e
For those students who score 80 percent or
better, provide the On Track! Activities from
Unit 6, Lessons 1–5, from the mBook Teacher
Edition.
Properties of Shapes, Congruence, and Similarity
Part 3
What properties do these shapes have in common?
einforcement
R
For those students who score below 80
percent, provide additional support in one of
the following ways:
■ Have students access the online
tutorial provided in the mBook
Study Guide.
■
■
672 Answers will vary. Sample answer: They all have curvy edges.
Part 4
Five of the six shapes have a common property. Circle the shape that does
not have this common property.
Have students complete the
Interactive Reinforcement Exercises
for Unit 6, Lessons 1–4, in the mBook
Study Guide.
Provide teacher-directed reteaching
of unit concepts.
Unit 6 • Lesson 5
1
2
3
4
5
6
What property is the shape missing?
Number 4 does not have any sharp edges.
78
Unit6•Quiz1•FormA
77
Unit6 Quiz1•FormB
Unit6 Quiz1•FormB
Name
Date
Name
Form B
MonitoringProgress
MonitoringProgress
CommonFactorsandNumberPatterns
PropertiesofShapes,Congruence,andSimilarity
Part1
Part3
Findthecommonfactorsforeachsetofnumbersbywritingthefactorlists.
1.
3.
Common factors
1, 2, 4, 8
2.
Common factors
Whatpropertiesdotheseshapeshaveincommon?
1, 2, 3, 6
16
1, 2, 4, 8, 16
12
1, 2, 3, 4, 6, 12
24
1, 2, 3, 4, 6, 8, 12, 24
18
1, 2, 3, 6, 9, 18
Common factors
1, 5
4.
Common factors
1, 7
15
1, 3, 5, 15
14
1, 2, 7, 14
30
1, 2, 3, 5, 6, 10, 15, 30
21
1, 3, 7, 21
40
1, 2, 4, 5, 8, 10, 20, 40
28
1, 2, 4, 7, 14, 28
Answers will vary. Sample answer: They all have a straight
edge.
Part2
Findthegreatestcommonfactorforeachsetofnumbers.
1.
GCF
10
2.
GCF
Part4
Factor lists
10
15
20
25
GCF
14
4.
GCF
1, 3, 5, 15
1, 5, 25
1
2
3
2
Factor lists
Factor lists
14
18
1, 2, 7, 14
28 1, 2, 4, 7, 14, 28
Fiveofthesixshapeshaveacommonproperty.Circletheshapethatdoes
nothavethiscommonproperty,thenanswerthequestion.
5
Factor lists
1, 2, 5, 10
1, 2, 4, 5, 10, 20
30 1, 2, 3, 5, 6, 10, 15, 30
3.
Date
4
1, 2, 3, 6, 9, 18
24 1, 2, 3, 4, 6, 8, 12, 24
32 1, 2, 4, 8, 16, 32
5
6
What property is the shape missing?
Number three is not a closed shape.
©2010 Sopris West Educational Services. All rights reserved.
Unit6•Quiz1•FormB
1
©2010 Sopris West Educational Services. All rights reserved.
Unit6•Quiz1•FormB
Unit 6 • Lesson 5 2
673
Lesson 5
Lesson 5
Homework
Activity 1
Homework
Go over the instructions on page 373 of the
Student Text for each part of the homework.
Activity 1
Students tell which divisibility rules apply to the
given numbers.
Activity 2
Students use prime factor trees to find the GCF
for pairs of numbers.
tell which divisibility rule or rules (2, 3, 5, 6, or 10) can be used to divide
each number.
1. 4,685
2. 1,350
3. 57,912
See Additional Answers below.
4. 45,402
5. 179,031
Activity 2
Find the GCF for each pair of numbers by drawing prime factor trees.
Model 12 and 20
12
20
6
2
2
2×2=4
2
10
3
2
5
GCF = 4
1. 4 and 16
2. 32 and 36
3. 18 and 24
4. 16 and 30
See Additional Answers below.
Activity 3 • Distributed practice
Solve.
1.
33 R2
5,000
 4,999
1
2.
6,978
+ 3,482
10,460
3.
50
50
4.
6q200
64 R6
5. 7q454
2,500
Activity 3 • Distributed Practice
Students practice whole number operations
to improve their skills. Remind students to use
mental math whenever possible. Tell them to
use their knowledge of basic facts to solve any
extended facts. Also tell them to use number
sense when they look at the numbers before
solving.
Unit 6 • Lesson 5
In Problem 1, we are subtracting 4,999 from
5,000. Students need to think carefully about
the numbers before they proceed with the
calculation.
Additional Answers
Activity 1
1. The 5 rule.
2. The 2, 3, 5, 6, and 10 rules.
3. The 2, 3, and 6 rules.
4. The 2, 3, and 6 rules.
5. The 3 rule.
(Additional Answers continue on Appendix, page A2.)
674 Unit 6 • Lesson 5
373
373