Factors and Multiples – Grade Six
Ohio Standards
Connection
Number, Number Sense and
Operation Standard
Benchmark G
Apply and explain the use of
prime factorizations, common
factors, and common multiples
in problem situations.
Indicators 2
Find and use the prime
factorization of composite
numbers. For example:
a. Use the prime factorization to
recognize the greatest
common factor (GCF).
b. Use the prime factorization to
recognize the least common
multiple (LCM).
c. Apply the prime factorization
to solve problems and
explain situations.
Mathematical Processes
Benchmark H
Use representations to organize
and communicate mathematical
thinking and problem solutions.
Lesson Summary:
In this lesson, students solve problem situations using greatest
common factors (GCF) and least common multiples (LCM).
Estimated Duration: Two and one-half hours
Commentary:
Students need to see the relevance in the mathematics they
learn. This lesson builds on the concepts of least common
multiple (LCM) and greatest common factor (GCF) and
presents real-world problem situations for students to solve.
A strong understanding of these concepts and skills benefits
students as they develop algebraic thinking. Understanding
that the same set of numbers can answer multiple questions
related to a situation and applying different algorithms to the
same set of numbers in multiple situations develops
fundamental problem solving skills.
Pre-Assessment:
• Direct each student to fold a sheet of notebook paper in
half. Ask students to label one side of the folded paper,
“Factors” and the other, “Multiples.”
• Direct the students to write what they know about factors
and multiples on their paper. Check the information on the
lists as students write and assist as necessary.
• Have students share their responses with the class. Record
information about factors and multiples on the board while
students check off the information they already have. If a
classmate presents new information, students may add it
to their papers. Record all responses shared by students.
• Collect the sheets and informally assess prior knowledge
and determine readiness for lesson.
Instructional Tips:
• Elicit examples and vocabulary as students share
responses.
• Use leading questions to assist students who share
incorrect responses. Correct any misconceptions.
Scoring Guidelines:
Students should know the difference between factors and
multiples. They may associate factors with the numbers in a
multiplication fact or multiples with skip counting. The
responses may include examples of each concept. Record
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Factors and Multiples – Grade Six
the names of students who have little or no previous knowledge of factors and multiples, as well
as students who have the terms confused. Informally assess the class as a whole during the
discussion part of the pre-assessment.
Post-Assessment:
Use any of the following strategies as a post-assessment:
• Collect and assess student work completed during the lesson.
• Direct students to choose one of each type of problem used in class (GCF and LCM) and
explain their solutions.
• Direct students to complete Attachment A, GCF and LCM Post-Assessment.
Scoring Guidelines:
Assess students’ progress in using greatest common factor and least common multiple in
problem situations. Students should use the appropriate concept, accurately solve the problem
and explain their reasoning using strategies and mathematical thinking.
Adequate
Understanding
Partial
Understanding
Limited
Understanding
Uses greatest common factor and least common multiple appropriate to the
context and provides accurate solutions. Explanation includes clear strategies
with mathematical thinking.
Provides accurate solutions but omits or provides and incomplete explanation
or
Provides incorrect solutions with explanations that show some understanding.
Provides an inaccurate factorization of some or all of the numbers.
Confuses the use of GCF and LCM for the contexts.
Instructional Procedures:
Part One – Greatest Common Factor
1. Present the following problem to students:
Ms. Smith is planning a scavenger hunt for the field trip to the science museum. To be fair,
she wants each team to have the same number of students. She also wants to keep the grade
levels separate. There are 48 fifth graders and 60 sixth graders attending the trip.
Determine possible group sizes and how many groups for each size. What is the largest
number of students in groups of equal size Ms. Smith can make?
2. Ask students to work in small groups to solve the problem, recording their work on chart
paper. Students may use chips or square tiles to model the situation.
Instructional Tip:
Distribute different colored markers or crayons to each group, and direct each group member to
choose a different color to record his/her contribution. This shows at a glance who was
responsible for each part of the group’s work. In addition, the mistakes can be caught as the
group works because they cannot erase their errors. A record of every group’s work and errors
helps inform instruction.
3. Circulate as students work, give guidance and ask leading questions to correct
misconceptions when needed.
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Factors and Multiples – Grade Six
4. Select groups to share their work and solution with the class.
5. Lead a discussion on all the possible group sizes. Elicit responses and list them on the board,
in order from least to greatest. Direct questions to help students see that this is a list of the
common factors of 48 and 60 and that the solution to the problem (12) is the greatest
common factor, GCF, of 48 and 60.
• What is the largest possible group size?
• How do you know this group size will work for both the fifth and sixth-graders?
• What are some other group sizes?
• What do you notice about the list of possible group sizes?
6. Provide additional information regarding the problem situation.
Ms. Smith decides to invite the seventh grade students to the field trip. There are 54 seventh
graders. Determine possible group sizes and how many groups for each size. What is the
largest number of students in groups of equal size Ms. Smith can make?
7. Circulate as students work, give guidance and ask leading questions to correct
misconceptions when needed.
8. Select groups to share their work and solution with the class.
• What is the largest possible group size?
• How do you know this group size will work for all three grades?
• What are some other group sizes?
• What do you notice about the list of possible group sizes?
9. Direct students to complete Attachment B, Field Trip Fever.
10. Ask the class to share their strategies for finding the GCF.
11. Close part one of the lesson by using the strategies shared by the students to create a common
strategy for finding the GCF and solve a problem situation. Have students record the strategy
in their journal and use it to solve a problem situation.
Part Two – Least Common Multiple
12. Present the following problem to students:
Every morning at breakfast, Mindy reads the cereal box. She notices that each box contains
12 servings. Mindy reads the label on the gallon of milk and discovers that it contains 16
servings. If she uses exactly one serving of milk and one serving of cereal each day, how
many days would it take until Mindy emptied a box of cereal and a gallon of milk on the
same day? Continue to have students use different colors to record their work.
13. Ask students to work in small groups to solve the problem, recording their work on chart
paper. Students may use chips or square tiles to model the situation.
14. Circulate as students work and give guidance and leading questions to correct
misconceptions when needed
• What would happen if Mindy used different sizes of cereal boxes and/or milk containers?
Instructional Tip:
If students have difficulty in determining a starting point, suggest that they begin with simpler
numbers to discover the process.
15. Ask each group to share their work and solution with the class.
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Factors and Multiples – Grade Six
16. Ask students to give responses for the number of boxes of cereal and the number of gallons
of milk.
17. Create a list for each category using student the responses. Each list will be a multiple list.
• How many days does it take to use a box of cereal? (12)
• When will the next box of cereal be empty? (12 days later on day 24)
18. Use questions to help students see that the first number that is common for both lists (48) is
the least common multiple (LCM) of 12 and 16.
19. Have students solve similar problems, but with different numbers, such as 9 and 12.
20. Ask the class to share strategies for finding the LCM.
21. Provide different breakfast contexts such as eggs to bacon, or toast to servings of jelly.
Clarify strategies used for finding the least common multiple.
22. Have students complete Attachment C, Breakfast Brainteaser.
23. Ask the class to share their strategies for finding the LCM.
24. Close part two of the lesson by using the strategies shared by the students to create a
common strategy for finding the LCM and solve a problem situation. Have students record
the strategy in their journal and use it to solve a problem situation.
Two buses leave from the same station at 7:00 a.m. The Island Express leaves the station
every 18 minutes. The Bay Express leaves the station every 15 minutes. At what times during
the day will both buses depart the station at the same time?
Differentiated Instructional Support:
Instruction is differentiated according to learner needs, to help all learners either meet the intent
of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified
indicator(s).
• Allow students to use counters or square tiles to model the problem situations.
• Vary the difficulty of the numbers used in the problems.
• Help students develop the concept of factors and multiples by developing slogans, such as,
“When you multiply factors by factors, you get multiples.”
• Direct the students to illustrate a factor/multiple machine. When you put in two factors, a
multiple comes out.
Extensions:
• Challenge students to find the LCM and GCF for three or more numbers. Present scenarios
such as:
We have been asked to make 100 treat bags containing a piece of chocolate, a sucker and a
stick of gum. There are 36 pieces of chocolate in a package. Suckers come in packs of 24 and
gum in packs of 15 sticks. What is the minimum number of packages of each type of candy
that should be purchased to make enough treat bags? How many treat bags will it make?
Will there be any left over? If so, how many?
• Introduce students to the connection between LCM and common denominators and GCF and
simplifying fractions.
• Ask students to predict whether the LCM of two numbers is less than, greater than or equal to
the product of the two numbers.
• Direct the students to develop/explore rules for divisibility.
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Factors and Multiples – Grade Six
Home Connections:
• Have students identify real-world situations that involve GCF and LCM.
• Have students practice skip counting (multiple counting).
• Have students practice making all possible factor pairs for a number.
Materials and Resources:
The inclusion of a specific resource in any lesson formulated by the Ohio Department of
Education should not be interpreted as an endorsement of that particular resource, or any of its
contents, by the Ohio Department of Education. The Ohio Department of Education does not
endorse any particular resource. The Web addresses listed are for a given site’s main page,
therefore, it may be necessary to search within that site to find the specific information required
for a given lesson. Please note that information published on the Internet changes over time,
therefore the links provided may no longer contain the specific information related to a given
lesson. Teachers are advised to preview all sites before using them with students.
For the teacher: chart paper
For the students: markers or crayons for each group, a sheet of chart paper for each group,
counters or square tiles, and calculators (optional)
Vocabulary:
• factor
• greatest common factor (GCF)
• least common multiple (LCM)
• multiple
Technology Connections:
• Use a spreadsheet to develop a multiple list for large numbers.
• Use calculators to generate multiples.
• Use calculators to check if a number is a factor of another number.
Research Connections:
Cawletti, Gordon. Handbook of Research on Improving Student Achievement. Arlington, Va.:
Educational Research Service, 1999.
Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works:
Research-Based Strategies for Increasing Student Achievement, Alexandria, Va.: Association for
Supervision and Curriculum Development, 2001.
Attachments:
Attachment A, GCF/LCM Post-Assessment
Attachment B, Field Trip Fever
Attachment C, Breakfast Brainteaser
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Factors and Multiples – Grade Six
Attachment A
GCF/LCM Post-Assessment
Name_______________________________
Date__________________________
Directions: Complete the problems below.
1. Emily’s family just purchased two new cars. As they were reading the owner’s manuals,
they realized that one of the cars needs an oil change every 3000 miles. The other car needs
an oil change every 4000 miles. How many miles will be on each car when they can both be
taken in for an oil change together for the first time? How many miles will be on each car
when this happens again? Describe the pattern.
2. It is Greg’s turn to bring snacks and juice for the soccer team. At the grocery store, he found
some juice boxes and crackers that were on sale. The juice boxes come in packs of six, while
the crackers come in packs of eight. If he wants an equal number of juice boxes and
crackers, how many of packages of each should he buy?
If there are 12 members on the team, how many crackers and juice boxes would each team
member receive?
3. How do you know whether to use the GCF or the LCM to solve a problem?
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Factors and Multiples – Grade Six
Attachment B
Field Trip Fever
Name_______________________________
Date__________________________
A. Find the best group size that meets all of Ms. Smith’s conditions (equal group size, grades do
not mix, largest group possible).
1. 75 fifth graders and 100 sixth graders
2. 12 fifth graders, 36 sixth graders, 18 seventh graders
3. 24 fifth graders and 42 sixth graders
4. 60 fifth graders, 75 sixth graders, 45 seventh graders
B. Is it possible to create a situation where the “best group size” is 1? Is it possible for the GCF
of two numbers to be 1? If so, explain how. If not, explain why not.
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Factors and Multiples – Grade Six
Attachment C
Breakfast Brainteaser
Name_______________________________
Date__________________________
A. Tell how many days it would take to empty a cereal box and a carton of milk on the same
day. How many boxes of cereal and cartons of milk would be used?
1. 14 servings of cereal and 8 servings of milk
2. 18 servings of cereal and 6 servings of milk
3. 16 servings of cereal and 14 servings of milk
4. 20 servings of cereal, 12 servings of milk, 6 servings of eggs
5. 24 servings of cereal, 16 servings of milk, 10 servings of bacon
B. Jeri thinks there is s shortcut to figuring out the LCM. She says that if you multiply the two
numbers, you will always get the LCM. Is Jeri right? Does her method work every time?
Sometimes? Never? Explain your reasoning.
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