Download Equivalent Rational Numbers and Percents Unit: 01 Lesson: 01

Identifying Factors
and Multiples
Unit: 02 Lesson: 01
Identifying Factors and Multiples
Unit: 02 Lesson: 01 Table of Contents
1. Lesson Synopsis
2. TEKS
3. Content and Language Objective
4. Vocabulary
5. TAKS Warm-up (one for every day)
6. Engage
7. Explore/Explain (1-5)
8. Elaborate
9. Evaluate
Students begin the lesson by creating factor
pairs using baggies and two-color counters and
then use area models with centimeter tiles and
centimeter grid paper to reinforce prime and
composite numbers. The students will use
prime factors to represent positive integers in
prime factorization form. The students will
investigate factors and multiples of positive
integers and identify the greatest common
factor (GCF) and least common multiple (LCM)
(6.1) Number, operation, and quantitative
reasoning. The student represents and
uses rational numbers in a variety of
equivalent forms. (6.1D) write prime
factorization using exponents.
(6.1E) Identify factors of a positive integer,
common factors and the greatest common
factor of a set of positive integers
(6.1F) Identify multiples of a positive
integer and common multiples and the least
common multiple of a set of positive
integers.
SWBAT use a strategy to represent and write
the prime factorization (with exponents) of
positive integers.
Students will use a strategy to identify and
represent factors, common factors and the
greatest common factor (GCF), and multiples,
common multiples and the least common
multiple (LCM), of a set of positive integers.
-prime number
-composite number
-integers
-factor trees
-exponent
-base number
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Day 1
How many of you have ever had to share something
with a group of friends?
How did you share equally among your friends?
If we have 15 counters, what are all the different
ways the counters may be equally divided into the
baggies? Record your thinking in your math
journal.
Day 1
Use the baggies and counters to model factor pairs
of a set of integers: 1,2,3,4,5,6,7,8,9,10,11 &12.
Record your data in a table in your math journal.
Example of table:
Number Number Counters Multiplication
of
of Bags per bag
Statement
Counters
1
1
1
1x1
Day 1
Day 1
Day 1
How many equal shares did you have for each
number?
What are some patterns you notice about the
products and their factors?
What numbers only have two distinct factors?
2,3,5,7,11
What name do we give these numbers that have
only two distinct factors?
Prime numbers
Day 1
Using these examples, how would you define a
prime number? A number that has ONLY two
distinct factors; 1 and itself
What numbers have more than two distinct factors?
4,6,8,9,10,12
What name do we give these numbers that have
more than two distinct factors? Composite numbers
Using these examples, how would you define a
composite number? Prime numbers
What number is not listed as prime or composite?
Explain.
1—it does not fit the definition of prime or composite
because it only has 1 distinct factor--itself
Day 1
How may we use the square centimeter tiles
to model if the number 7 is prime or
composite?
Day 1
How may we use the square centimeter tiles to
model if the number 7 is prime or composite?
Day 1
How does this model demonstrate the 7 is a prime
number?
Day 1
How may we use the square centimeter tiles to model if
the number 6 is prime or composite?
Day 1
How does this model demonstrate that 6 is a
composite number?
Day 1
How does this model demonstrate 2 is a
composite/prime number? 11? 15? 18?
Day 1
What strategy can you use to list all the
distinct factors for a given product?
Day 1
Homework
Complete Prime and Composite
Summary Sheet
Due TOMORROW
Day 2
What is a pair of
factors for the
product 12?
1 & 12; 2 & 6; 3 & 4
Day 2
Are any of the
factors a prime
number?
Yes, for the factor
pair 2 and 6, 2 is a
prime number.
Day 2
What are two
factors for 12?
3 and 4
Day 2
STAGE 1
4
List a factor pair for the product using two branches
underneath.
Day 2
STAGE 1
4
Are any of the factors a prime number? Composite
number?
Day 2
STAGE 2
4
Circle any prime factors and leave composite factors
alone.
Day 2
STAGE 2
4
What are the two factors of the composite factor?
Day 2
STAGE 3
4
2
2
List a factor pair for the composite factor using two
branches underneath. Bring the circled prime factor
down to the same level.
Day 2
STAGE 3
4
2
2
Are any of the factors a prime number? Composite
number?
Day 2
STAGE 4
4
2
2
Circle any prime factors and leave composite factors
alone.
Day 2
STAGE 4
4
2
2
Are all the factors a prime number?
Day 2
STAGE 4
4
2
2
Write a mathematical sentence using all the prime
numbers.
12= 3 x 2 x 2
Day 2
In pairs, complete
the handout: Factor
Trees.
Day 2
How do you determine what factor pair to begin using
for the factor tree?
Write a factor pair where one of the factors is
a prime number.
How do you know when you have completed the
factor tree?
All the factors are prime numbers and the
product of the prime factors equals the
product at the top of the factor tree.
Day 2
How are the two number sentences: 3+3+3+3=12
and 4 x 3= 12 related?
Both have values of 12 and both use 3 and 4.
In the addition sentence, 3 is written as an
addend 4 times and in the multiplication
sentence the factor 4 indicates the number of
groups to make with 3 in each group; 4
groups of 3
In the number sentence, 4+4+4=12, how many
times is 4 written as an addend?
3 times
Day 2
How would we represent this addition sentence as a
multiplication sentence?
3 groups of 4 is 3x4, the factor 3 indicates the
number of groups to make with 4 in each
group.
How are the two number sentences, 2+2+2=6 and
2x2x2=8, different? Similar?
One involves addition, 2+2+2=6 and has a sum
of 6 and the other, 2x2x2=8, involves
multiplication and has a product of 8. Both
use the number 2, one as an addend and one
as a factor.
Day 2
In math, just as we have a way to represent
2+2+2=6 as 3x2=6, we have another way to
represent 2x2x2=8. We may also represent 2x2x2=8
as 2³=8.
What is the purpose of the 2 in 2³? The little 3
placed above and to the right of the 2?
The 2 indicates what is to be multiplied as a factor
in the multiplication sentence 2x2x2=8. The 3 tells
how many times to use 2 as a factor in the
multiplication sentence.
Day 2
In math, just as we have a way to represent
2+2+2=6 as 3x2=6, we have another way to
represent 2x2x2=8. We may also represent 2x2x2=8
as 2³=8.
According to this pattern, what is another way to
3²
write 3x3=9?
What is the purpose of the 3 in 3²? The little 2
placed above and to the right of the 3?
The 3 indicates what is to be multiplied as a factor
in the multiplication sentence 3x3=9. The 2 tells
how many times to use 3 as a factor in the
multiplication sentence.
Day 2
In math, just as we have a way to represent
2+2+2=6 as 3x2=6, we have another way to
represent 2x2x2=8. We may also represent 2x2x2=8
as 2³=8.
How could we use this pattern to complete the last
column for problems 13 through 24 on the
Refer to the Key
handout: Factor Trees?
If we decompose 2³x3x5², what multiplication
sentence does it represent?
2x2x2x3x5x5=600
Day 2
After students complete the handout: Factor Trees
conduct a class discussion to define bases and
exponents.
In the number sentence, 2³=8, 2 is identified as the
base and 3 is identified as the exponent. According
to this example, how would you define base?
Exponent?
The base indicates what number is to be multiplied
as a factor. The exponent indicates how many
times to use the base as a factor.
Day 3
“I have, Who has” Prime Factorization
Create a factor tree and write the prime
factorization in exponent form for each card after
you have created the card set. Record your
responses on notebook paper or in math journals.
Day 3
“I have, Who has” Prime Factorization
What are the prime factors of this positive integer
using exponents?
Refer to the key
How can we represent the prime factors of the
positive integer using exponents?
Refer to the key
Day 3
Homework
Complete Prime Factorization
Due TOMORROW
Day 4
In groups of four
complete the
handout: Factor
Recording Sheet.
Day 4
20
15
Find the factors of
20 and 15 by
arranging the color
counters.
Day 4
20
15
Choose one color
to represent the
factors of 20 and
choose another
color to represent
the factors of 15.
Record all possible representations
on your handout.
Day 4
20
Day 4
20
Day 4
15
Day 4
15
Day 4
In groups of four
complete the
handout: Factor
Recording Sheet.
Day 4
What are the factors of 20 and 15?
Factors of 20: 1,2,4, 5,10, 20
Factors of 15: 1,3 ,5, 15
What factors do the products 20 and 15
have in common? see above
What is the greatest factor the products 20
and 15 have in common? 5
Use the two color counters to find the factors of 12 and 6.
Day 4
What are the factors of 12 and 6?
Factors of 12: 1,2,3,4,6,12,
Factors of 6: 1,2,3,6
What factors do the products 12 and 6 have
in common? see above
What is the greatest factor the products 12
6
and 6 have in common?
Use the two color counters to find the factors of 4 and 8.
Day 4
What are the factors of 4 and 8?
Factors of 4: 1, 2, 4
Factors of 8: 1, 2, 4, 8
What factors do the products 4 and 8 have
in common? see above
What is the greatest factor the products 4
4
and 8 have in common?
Use the two color counters to find the factors of 4 and 8.
Day 4
Day 4
Day 4
What are the common factors in the array
models for the products 4 and 10?
Factors: 1 and 2
What is the greatest common factor for 4
and 10? 2
What are the common factors in the lists for
12 and 18? 1,2,3,6
What is the greatest common factor for 12
and 18? 6
Day 4
If the greatest factors for each given set of
positive integers are called greatest
common factor, how would you define the
greatest common factor (GCF)?
The largest common factor to a set of numbers
What is the greatest common factor (GCF)
for 12 and 24? GCF=12
Describe how you identified the GCF for 12
and 24?
Day 5
What is the greatest common factor (GCF)?
The largest common factor of a set of positive
integers.
How did we find the greatest common factor (GCF)
of a given set of numbers yesterday?
Listed all factors of the products and identified the
largest factor common to the products.
What is the greatest common factor for 24 and 12?
12
Day 5
Another strategy to identify the greatest common factor
(GCF) of a set of positive integers is to create a factor
tree for each number.
24=2x2x2x3
12=2x2x3
Day 5
What is the product of the common prime factors for
24 and 12?
2x2x3=12
How does the product of the common prime factors
for 24 and 12 compare to the greatest common
factor for 24 and 12?
Both = 12
What is the greatest common factor for 8 and 12?
4
Day 5
Create a factor tree for 8 and 12.
8=2x2x2
12=2x2x3
Day 5
What is the product of the common prime factors for
8 and 12?
2x2=4
How does the product of the common prime factors
for 8 and 12 compare to the greatest common factor
for 8 and 12?
Both = 4
What is the greatest common factor for 18 and 30?
6
Day 5
Create a factor tree for 18 and 30.
30=5x2x3
18=3x2x3
Day 5
What is the product of the common prime factors for
18 and 30?
2x3=6
How does the product of the common prime factors
for 30 and 18 compare to the greatest common
factor for 30 and 18?
Both = 6
Complete the handout: Greatest Common Factor
Practice
Day 6
multiple
2 minutes to
discuss
What do you think the word “multiple” means?
A group of things or objects etc…
How may we define the word “multiple” in
mathematical terms?
Multiples are repeated addition or
multiplication
Day 6
Who can give an example of a multiple?
Is skip counting a way of identifying multiples?
Yes; 2,4,6,8…are all multiples of 2
Is repeated addition a way to identify multiples?
Yes; repeated addition is the same as
multiplying
What are the multiples of 4?
4,8,12,16…
Day 6
The numbers 6,12, 18, 24, 30, 36, 42 etc. are
multiples of what number?
Multiples of 6
What are the first 10 multiples for 5? 7? 9?
In your journals: define the meaning of the
word “multiple” and give two examples.
Day 6
Materials: Handout and 2 Colored Pencils
Day 6
Follow along as I take my colored pencils and mark my number line
What number does our set of counting numbers start
with? 1
If we multiply 4 by 1 and 5 by 1, what numbers do we
have? 4 and 5
Day 6
Follow along as I take my colored pencils and mark my number line
for 4 and 5
Day 6
What is the number where the two colors met? 20
Is 20 a multiple of 4? yes
What did you multiply by 4 to get 20? 5
Is 20 a multiple of 5? yes
What did you multiply by 5 to get 20? 4
Why did the two colors meet on 20?
Because 20 is a common product, multiple, of 4 and 5.
What is the next common multiple of 4 and 5? 40
Are 20 and 40 the only common multiple of 4 and 5?
No
What do you think the next common multiple of 4 and
5 is? 60; 4x15 and 5x12
Day 6
How far does this number line go? 96
Does the number line really stop at 96? No
If the number line continues infinitely, do you think
the common multiple of 4 and 5 continue as well?
yes
How many common multiples of 4 and 5 do you
think exist?
An infinite number of common multiples
What is the least common multiple of 4 and 5?
20
Day 6
With your partner complete the number line using the numbers 4 and
6.
Day 6
What is the number where the two colors met? 12
Is 12 a multiple of 4? yes
What did you multiply by 4 to get 12? 3
Is 12 a multiple of 6? yes
What did you multiply by 6 to get 12? 2
Why did the two colors meet on 12?
Because 12 is a common product, multiple, of 4 and 6.
What is the next common multiple of 4 and 6? 24
Are 12 and 24 the only common multiple of 4 and 6?
No
What do you think the next common multiple of 4 and
6 is? 36
Day 6
What is the operational process used to find the next
multiple in this table? multiplication
What are the common multiples shown in the table?
Based on these examples, how would you define
least common multiple?
Day 7
Identify the least common multiple for:
15
12
What are the common multiples of 15 and 12?
60, 120, 180
What is the least common multiple (LCM) of 15 and
12?
60
Day 7
What are the prime factors of 15 and 12?
What prime factors do 15 and 12 have in common?
3
What is the product of all prime factors of 15 and 12 if
we use the prime factors they have in common only
once? 2 x 2 x 3 x 5= 60
Day 7
What are the prime factors of 15 and 12?
How does this product of prime factors compare to
the least common multiple for 15 and 12? Both = 60
Day 7
What are the prime factors of 12 and 24?
What prime factors do 12 and 24 have in common?
3 and 2, 2
What is the product of all prime factors of 12 and 24 if
we use the prime factors they have in common only
once? 2 x 2 x 2 x 3 = 24
Day 7
What are the prime factors of 12 and 24?
How does this product of prime factors compare to
the least common multiple for 12 and 24?
Both=24
Day 7
Complete the following
handout in pairs.
Turn it in when you are
finished.
Day 8
Complete the following
evaluation on your
own.
Turn it in when you are
finished.
Day 8
Homework
Complete Equivalent Fractions
Due TOMORROW
End of Unit 02 Lesson
01