Download Lesson Notes - The Key Elements To Mathematics Success

Mathematics Success – Level F
T105
LESSON 5: GCF and LCM
[OBJECTIVE]
The student will investigate greatest common factors and least common multiples.
[MATERIALS]
Student pages S39–S48
Transparencies T116, T118, T120, T122, T124, T126, T128
Centimeter cubes – 30 to 40 per pair
Colored pencils – 3 different colors per pair
[ESSENTIAL QUESTIONS]
1. What is the difference between greatest common factor (GCF) and least common
multiple (LCM)?
2. What is the most important word in greatest common factor? Explain.
3. What is the most important word in least common multiple? Explain.
[WORDS FOR WORD WALL]
greatest common factor, least common multiple, factor, multiple
[GROUPING]
Cooperative Pairs (CP), Whole Group (WG), Individual (I)
[LEVELS
OF
TEACHER SUPPORT]
Modeling (M), Guided Practice (GP), Independent Practice (IP)
[MULTIPLE REPRESENTATIONS]
SOLVE, Verbal Description, Concrete Representation, Pictorial Representation, Graphic
organizer
[WARM-UP] (5 minutes – IP, I, WG) S39 (Answers on T115.)
• Have students turn to S39 in their books to begin the Warm-Up. Students will solve
simple multiplication problems and answer questions to review factors, products,
and skip counting (multiples). Monitor students to see if any of them need help
during the Warm-Up. Give students 3 minutes to complete the problems and then
spend 2 minutes reviewing the answers as a class. {Verbal Description}
[HOMEWORK] (5 minutes)
Take time to go over the homework from the previous night.
T106
Mathematics Success – Level F
LESSON 5: GCF and LCM
[LESSON] (50–60 minutes – M, GP, IP, WG, CP, I)
SOLVE Problem
(1 minute – GP, WG) T116, S40
(Answers on T117.)
Have students turn to S40 in their books, and place T116 on the overhead. The
first problem is a SOLVE problem. You are only going to complete the S step with
students at this point. Tell students that during the lesson they will learn how to
find the greatest common factor (GCF) and the least common multiple (LCM) of
numbers. They will use this knowledge to complete this SOLVE problem at the
end of the lesson. {SOLVE}
Greatest Common Factor
(10 minutes – M, GP, IP, WG, CP)
T116, S40 (Answers on T117.)
Ask students what a factor is. (A factor is a number that divides evenly into
another number.) Reference the problems in the Warm-Up on S39 (T115) and
ask students to identify the factors in Problems 1–4. Talk with students about
what the term greatest common factor means. Explain that the word common
means “something that is shared” and that the greatest common factor is the
largest factor shared by two or more numbers.
5 minutes – M, GP, WG: Pass out the centimeter cubes to students. Use the
hundred board on T116 (S40) and the following activity
to model for students a concrete approach to finding the
GCF of two numbers. {Concrete Representation, Graphic
Organizer, Verbal Description}
Mathematics Success – Level F
T107
LESSON 5: GCF and LCM
MODELING
Greatest Common Factor – Hands-On
Step 1: Tell students that they are going to find the greatest common factor (GCF)
of 12 and 18. Explain to students that first they need to find the factors
of 12. Have students put a centimeter cube on each of the factors of 12:
1, 2, 3, 4, 6, and 12.
Step 2: Next, explain to students that they need to find the factors of 18. Have
students leave the cubes on the board and place centimeter cubes on all
the factors of 18: 1, 2, 3, 6, 9, and 18.
Step 3: Have students look at the factors and determine which ones are common
to both 12 and 18, as you point them out on T116. Explain that these
factors will have 2 cubes on them. (They are 1, 2, 3, and 6.) Ask, “Which
is the largest factor 12 and 18 have in common (or share)?” (6) Explain
that six is the greatest common factor of 12 and 18.
Step 4: Record the answer on T116 as students record on S40.
3 minutes – IP, CP:
Have students work in partners with the centimeter
cubes to find the greatest common factor of 14 and
42 and answer the questions on S40. {Concrete
Representation, Verbal Description, Graphic Organizer}
2 minutes – M, WG:
GCF – Pictorial
Use centimeter cubes to model the answer for the
GCF problem on T116 (S40) and go over the answers
to Questions 1–3 as a group.
(6 minutes – M, GP, WG, CP, I) T118, S41
(Answers on T119.)
4 minutes – M, GP, WG: Pass out the colored pencils to students. Have
students turn to S41 in their books, and place T118
on the overhead. Use the following activity to model
for students a pictorial approach to finding the GCF of
two numbers. {Verbal Description, Graphic Organizer,
Pictorial Representation}
T108
Mathematics Success – Level F
LESSON 5: GCF and LCM
MODELING
Greatest Common Factor – Pictorial
Step 1: Tell students that, for Problem 1, they are going to find the greatest
common factor (GCF) of 12 and 30. Have students draw vertical lines with
one colored pencil as shown below to show the factors of 12: 1, 2, 3, 4,
6, and 12.
Step 2: Have students write the factors of 12 in the graphic organizer.
Factors of 12
Factors of 30
Step 3: Next, have students draw diagonal lines as shown below with a different
colored pencil to show the factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30.
Step 4: Have students write the factors of 30 in the graphic organizer.
Factors of 12
Factors of 30
Mathematics Success – Level F
T109
LESSON 5: GCF and LCM
Step 5: Have students look at the common factors of both numbers, 1, 2, 3, and
6, as indicated by the two different colored lines, and circle them on the
hundred board. Then have students circle the common factors in the
graphic organizer.
Step 6: Finally, have students list the common factors on S41 as you do the same
on T118. Ask students to identify the greatest common factor, 6, and
record it.
2 minutes – IP, CP, WG: Have students complete Problem 2 on S41 with a partner
and then come back together as a class to discuss the
answer. {Verbal Description, Pictorial Representation,
Graphic Organizer}
GCF – Abstract
(6 minutes – M, GP, IP, WG, CP) T120, S42 (Answers on T121.)
3 minutes – M, GP, WG: Have students turn to S42 in their books, and place T120
on the overhead. Use the following activity to model for
students an abstract approach to finding the GCF of two
numbers. {Verbal Description, Graphic Organizer}
Greatest Common Factor – Abstract
Step 1: Tell students that, for Problem 1, they are going to find the greatest
common factor (GCF) of 24 and 32. Explain that students will not be
using a hundred board, but will use the graphic organizer they have used
in previous problems to list the factors of 24 and 32. If students are
struggling, have them think about the factor pairs: 1 and 24, 2 and 12, 3
and 8, and so on.
Step 2: Model how to write the factors of 24 in the graphic organizer as students
record on S42.
Factors of 24
T110
Mathematics Success – Level F
LESSON 5: GCF and LCM
Step 3: Model how to write the factors of 32 in the same graphic organizer as
students record.
Factors of 24
Factors of 32
Step 4: Have students circle and then list the common factors. Ask students what
they are and record as students record.
Factors of 24
Factors of 32
Step 5: Ask students what the greatest common factor of 24 and 32 is (8) and
record as students record.
3 minutes – IP, CP, WG: Have students complete Problems 2–4 on S42 with a
partner and then come back together as a class to discuss
the answers. {Verbal Description, Graphic Organizer}
Least Common Multiple
(8 minutes – M, GP, IP, CP, WG) T122, S43
Ask students what a multiple is. (A multiple is a product of two numbers.
The product is a multiple of each number.) Have students count by 5s to 50
to demonstrate multiples of 5. Talk with students about what the term least
common multiple means. Explain that the word common means “something
that is shared” and that the least common multiple is the smallest nonzero number
that is a multiple of two (or more) whole numbers. {Concrete Representation,
Graphic Organizer, Verbal Description}
Mathematics Success – Level F
T111
LESSON 5: GCF and LCM
3 minutes – M, GP, WG: Pass out the centimeter cubes to students. Use the
hundred board on T122 (S43) and the following activity
to model for students a concrete approach to finding the
LCM of two numbers. {Concrete Representation, Graphic
Organizer, Verbal Description}
MODELING
Least Common Multiple – Hands-On
Step 1: Tell students that they are going to find the least common multiple (LCM)
of 4 and 5. Explain to students that first they need to find the multiples
of 4. Have students put a centimeter cube on each of the multiples of 4,
through 24: 4, 8, 12, 16, 20, 24.
Step 2: Next, explain to students that they need to find the multiples of 5. Have
students leave the cubes on the board and place centimeter cubes on all
the multiples of 5, through 25: 5, 10, 15, 20, 25.
Step 3: Have students look at the multiples and determine which ones are
common to both 4 and 5, as you point them out on T122. Explain that
these multiples will have 2 cubes on them. (There is only one. It is 20.)
Ask, “Which is the smallest multiple 4 and 5 have in common (or share)?”
(20) Explain that 20 is the least common multiple of 4 and 5.
Step 4: Record the answer on T122 as students record on S43.
3 minutes – IP, CP: Have students work in partners with the centimeter cubes
to find the least common multiples for 7 and 3, 2 and
8, and 4 and 12. Have students answer the questions at
the bottom of the page. {Concrete Representation, Verbal
Description, Graphic Organizer}
2 minutes – M, WG: Use centimeter cubes to model the answers for the LCM
problems on T122 (S43) and go over the answers to
Questions 1–3 as a group.
LCM – Pictorial
(7 minutes – M, GP, IP, WG, CP)
T124, S44 (Answers on T125.)
5 minutes – M, GP, WG: Pass out the colored pencils to students. Have students
turn to S44 in their books, and place T124 on the overhead.
Use the following activity to model for students a pictorial
approach to finding the LCM of two or more numbers. {Verbal
Description, Graphic Organizer, Pictorial Representation}
T112
Mathematics Success – Level F
LESSON 5: GCF and LCM
MODELING
Least Common Multiple – Pictorial
Step 1: Tell students that, for Problem 1, they are going to find the least common
multiple of 2, 4, and 5. Have students draw vertical lines with one colored
pencil to show the multiples of 2, through 26. Have students write the
multiples of 2 in the graphic organizer.
Multiples of 2
Multiples of 4
Multiples of 5
Step 2: Next, have students draw diagonal lines (from top right to bottom left)
with a different colored pencil to show the multiples of 4, through 24.
Have students write the multiples of 4 in the graphic organizer.
Multiples of 2
Multiples of 4
Multiples of 5
Step 3: Have students draw different diagonal lines (this time from top left to
bottom right) with a different colored pencil to show the multiples of
5, through 25. Have students write the multiples of 5 in the graphic
organizer.
Multiples of 2
Multiples of 4
Multiples of 5
Step 4: Have students circle and then list the multiple that is common to all three
numbers (just 20) and identify the least common multiple of 2, 4, and 5
(20).
Mathematics Success – Level F
T113
LESSON 5: GCF and LCM
2 minutes – IP, CP, WG: Have students complete Problem 2 on S44 with a partner
and then come back together as a class to discuss the
answer. {Verbal Description, Pictorial Representation,
Graphic Organizer}
LCM – Abstract
(7 minutes – M, GP, IP, WG, CP)
T126, S45 (Answers on T127.)
3 minutes – M, GP, WG: Have students turn to S45 in their books, and place T126
on the overhead. Use the following activity to model for
students an abstract approach to finding the LCM of two
numbers. {Verbal Description, Graphic Organizer}
MODELING
Least Common Multiple – Abstract
Step 1: Tell students that, for Problem 1, they are going to find the least common
multiple of 4 and 6. Explain that students will not be using a hundred
board, but will use the graphic organizer they have used in previous
problems to list the multiples of 4 and 6. Ask students what they think
the LCM of 4 and 6 will be and why. (Based on the work they have already
completed, many students may think that the LCM will be 24, because
they simply multiplied 4 and 6.)
Step 2: Model how to write the multiples of 4 in the graphic organizer as students
record on S45. Then model how to write the multiples of 6 in the graphic
organizer as students record.
Step 3: Have students circle the common multiples. Ask students what they are
and record as students record. Ask students to determine what the least
common multiple of 4 and 6 is, and record as students record.
Step 4: Refer to the graphic organizer on T126. Ask students if they needed to
continue to write the multiples once they found the least common multiple.
(No, they can stop when they determine the LCM.).
4 minutes – IP, CP, WG: Have students complete Problems 2–4 on S45 with a
partner and then come back together as a class to discuss
the answers. {Verbal Description, Graphic Organizer}
T114
Mathematics Success – Level F
LESSON 5: GCF and LCM
SOLVE Problem
(3 minutes – GP, WG)
T128, S46 (Answers on T129.)
Remind students that the SOLVE problem is the same one from the beginning
of the lesson. Complete the SOLVE problem with your students. Ask them for
possible connections from the SOLVE problem to the lesson. (Students will be
using least common multiples to solve the problem. {SOLVE, Verbal Description}
If time permits…
(10 minutes – IP, I) S47 (Answers on T130.)
Have students solve Problems 1–5 on S47 independently. Review the answers
as a class.
[CLOSURE] (2 minutes)
To wrap up the lesson, go back to the essential questions and discuss them with
students.
• What is the difference between greatest common factor (GCF) and least common
multiple (LCM)? (The greatest common factor is the largest number that is a
factor of 2 (or more) whole numbers. The least common multiple is the smallest
number that is a multiple of two (or more) whole numbers. It can not be 0.)
• What is the most important word in greatest common factor? Explain. (“Greatest”
is the most important word because we are looking for the largest number that
is a factor of 2 (or more) whole numbers.)
• What is the most important word in least common multiple? Explain. (“Least” is
the most important word because we are looking for the smallest number that is
a multiple of two (or more) whole numbers – except for 0.)
[HOMEWORK] Assign S48 for homework. (Answers on T131.)
[QUIZ ANSWERS] T132
The quiz can be used at any time as extra homework or to see how students did on
working with greatest common factors and least common multiples.
T116
Mathematics Success – Level F
LESSON 5: GCF and LCM
TRANSPARENCY MASTER
Directions: Complete the following SOLVE problem with your teacher. You will only
complete the S step.
Maddie and Elli are selling tickets to the school play. The tickets cost $3.50 each.
They sold tickets for several hours over the weekend. If Maddie sold 5 tickets
per hour and Elli sold 8 tickets per hour, what is the least number of tickets they
would need to sell to have sold exactly the same amount of tickets?
S Underline the question.
This problem is asking me to find ________________________________.
Directions: Complete the rest of this page with your teacher.
GCF of 12 and 18 =
GCF of 14 and 42 =
1.What is a factor?
2.What does common mean?
3.What does greatest common factor mean?
Mathematics Success – Level F
T117
LESSON 5: GCF and LCM
Here is the key to S40.
Directions: Complete the following SOLVE problem with your teacher. You will only
complete the S step.
Maddie and Elli are selling tickets to the school play. The tickets cost $3.50 each.
They sold tickets for several hours over the weekend. If Maddie sold 5 tickets
per hour and Elli sold 8 tickets per hour, what is the least number of tickets they
would need to sell to have sold exactly the same amount of tickets?
S Underline the question.
This problem is asking me to find the number of tickets Maddie and Elli have to sell to have sold the same amount of tickets.
Directions: Complete the rest of this page with your teacher.
GCF of 12 and 18 = 6
GCF of 14 and 42 = 14
1.What is a factor? a number that divides evenly into another number; a
number that can be multiplied by another number to equal a product
2.What does common mean? shared
3.What does greatest common factor mean? the largest factor shared by two or more numbers
T118
Mathematics Success – Level F
LESSON 5: GCF and LCM
TRANSPARENCY MASTER
Directions: Complete this page with your teacher.
1. Find the GCF of 12 and 30.
Factors of 12
Factors of 30
Common Factors of 12 and 30:
Greatest Common Factor:
2. Find the GCF of 21 and 42.
Factors of 21
Factors of 42
Common Factors:
Greatest Common Factor:
Mathematics Success – Level F
T119
LESSON 5: GCF and LCM
Here is the key to S41.
Directions: Complete this page with your teacher.
1. Find the GCF of 12 and 30.
Factors of 12
Factors of 30
Common Factors of 12 and 30: 1, 2, 3, 6
Greatest Common Factor: 6
2. Find the GCF of 21 and 42.
Factors of 21
Factors of 42
Common Factors: 1, 3, 7, 21
Greatest Common Factor: 21
T122
Mathematics Success – Level F
LESSON 5: GCF and LCM
TRANSPARENCY MASTER
Directions: Complete this page with your teacher.
LCM: 4 and 5 =
2 and 8 =
7 and 3 =
4 and 12 =
1.What is a multiple?
2.What does common mean?
3.What does least common multiple mean?
Mathematics Success – Level F
T123
LESSON 5: GCF and LCM
Here is the key to S43.
Directions: Complete this page with your teacher.
LCM: 4 and 5 = 20
7 and 3 = 21
4 and 12 = 12
2 and 8 = 8
1.What is a multiple? The product of two numbers is a multiple of each
number.
2.What does common mean? something that is shared
3.What does least common multiple mean? the least number that is a multiple
of two or more numbers
T124
Mathematics Success – Level F
LESSON 5: GCF and LCM
TRANSPARENCY MASTER
Directions: Complete this page with your teacher.
1. Find the LCM of 2, 4, and 5.
Multiples of 2
Multiples of 4
Multiples of 5
Least Common Multiple: Common Multiples:
2. Find the LCM of 6 and 8.
Multiples of 6
Multiples of 8
Common Multiples:
Least Common Multiple:
Mathematics Success – Level F
T125
LESSON 5: GCF and LCM
Here is the key to S44.
Directions: Complete this page with your teacher.
1. Find the LCM of 2, 4, and 5
Multiples of 2
Multiples of 4
Multiples of 5
Common Multiples: 20
Least Common Multiple: 20
2. Find the LCM of 6 and 8.
Multiples of 6
Multiples of 8
Common Multiples: 24
Least Common Multiple: 24