In this Warm-up, students explore the concept of least common multiples using rhythm.
Launch
Tell students they will be establishing a steady beat as a class. Use a visual hand motion or display a visual timer to set the beat. Tell one half of the class to clap on every other beat, and tell the other half to stomp their feet on every third beat. If time allows, repeat this activity for the following intervals: every 3rd and 4th beat, every other and every 4th beat, every 4th and 6th beat. Follow with a whole-class discussion.
Activity
Student Lesson in Spanish
None
Your teacher will give you instructions to create a class rhythm. As you participate, think about these questions:
When will the two sounds happen at the same time?
How does this game relate to common factors or common multiples?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to share their thinking on how this activity is related to common multiples or common factors. Invite students to explain what they noticed. If not mentioned by students, make sure they understand that students clapping on every other beat is like finding multiples of 2, and students stomping on every 3rd beat is like finding multiples of 3. When the two things happen at the same time, it is like a common multiple.
18.2
Activity
Standards Alignment
Building On
Addressing
6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express as .
In this activity, students work in pairs to solve problems that involve thinking about factors and multiples, including the greatest common factor and least common multiple. Students reflect on what type of mathematical work was used in each problem and begin to notice similarities in problems that involve factors and and those that involve multiples. Students must make sense and persevere as they decide how the problems relate to common factors and common multiples (MP1).
Launch
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms, or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norm supported the mathematical community during the activity.
Arrange students in groups of 2. Give students 15–20 minutes of work time, and follow with a whole-class discussion.
MLR2 Collect and Display. Collect the language that students use to solve problems involving common multiples and common factors. Display words and phrases, such as “multiples,” “factors,” and “multiply,” or “divide evenly.” During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed. Advances: Conversing, Reading
Activity
None
Work with your partner to solve the following problems.
Party. Elena is buying cups and plates for her party. Cups are sold in packs of 8, and plates are sold in packs of 6. She wants to have the same number of plates and cups.
Find the smallest number of plates and cups that meets her requirement.
How many packs of each supply will she need to buy to get that number?
Name two other quantities of plates and cups she could get to meet her requirement.
Tiles. A restaurant owner is creating a mural on one wall made entirely with square tiles. The tiles will be laid side-by-side to cover the entire wall with no gaps, and none of the tiles can be cut. The wall is a rectangle that measures 18 feet by 12 feet.
What is the side length, in feet, of the largest possible tile she could use? Explain or show your reasoning.
How many of these largest size tiles are needed?
Name the side lengths, in feet, of two other tile sizes that can be used.
Stickers. To celebrate the first day of spring, Lin is putting stickers on some of the 100 pages in her math notebook. She puts a skateboard sticker on every 4th page, and a kite sticker on every 5th page.
Find three page numbers that will get both stickers.
Will the 30th page have no stickers, 1 sticker, or 2 stickers? Explain or show your reasoning.
Kits. The school nurse is assembling first-aid kits for the teachers. She has 75 small bandages and 90 large bandages. All the kits must have the same number of each size bandage, and all bandages must be used.
What is the largest number of kits the nurse can make?
How many small bandages and large bandages will be in each kit?
Put a checkmark in the appropriate column to show what kind of mathematical work was involved in each problem situation.
problem
finding multiples
finding least
common multiple
finding factors
finding greatest
common factor
Party
Tiles
Stickers
Kits
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for students to express what kinds of problems have to do with least common multiples and what kinds have to do with greatest common factors. For each problem, ask students to indicate whether they think the problem had to do with common multiples or common factors, and invite a few to share their reasoning. Select students to explain their reasoning about how they solved the problems as time allows.
Math Community
Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:
“I noticed our norm ‘’ in action today, and it really helped me/my group because .”
Engagement: Provide Access by Recruiting Interest. Invite students to generate a list of additional examples of situations that involve greatest common factor or least common multiple that connect to their personal backgrounds and interests. Supports accessibility for: Conceptual Processing, Attention
18.3
Activity
Optional
Standards Alignment
Building On
Addressing
6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express as .
In this activity, students work in groups to predict whether a problem involves common factors or common multiples. Groups then solve one assigned problem, create a visual display to represent their work, and prepare a brief presentation. While analyzing only pairs of numbers is the expectation for this course, this activity asks students to think about the greatest common factor and least common multiple for sets of three whole numbers.
Launch
Arrange students in groups of 4. Provide access to tools for creating a visual display. Give students 5 minutes to read through each problem and discuss whether its solution has to do with finding common factors or common multiples. Then assign each group one of the problems. Give students 15–20 minutes of work time to solve their problem, create their visual display, and prepare a short presentation.
Activity
None
Read and discuss each problem with your group. Without solving, predict whether each problem involves finding common multiples or finding common factors. Circle one or more options to show your prediction.
Then your teacher will assign one problem to your group. Work with your group to solve the problem. Then create a visual display that explains your group’s mathematical thinking while solving the problem. Your display may include a diagram, lists, tables, equations, descriptions, and math vocabulary.
Soccer. Diego and Andre are both in a summer soccer league. During the month of August, Diego has a game every 3rd day, starting August 3rd, and Andre has a game every 4th day, starting August 4th.
common multiples
least common multiple
common factors
greatest common factor
What is the first date that both Diego and Andre will have a game?
How many of their games fall on the same date in the month of August?
Performances. During a performing arts festival, students from elementary and middle schools will be grouped together for various performances. There are 32 elementary students and 40 middle-school students. The arts director wants identical groups for the performances, with students from both schools in each group. Each student must be in a group and can be a part of only one group.
common multiples
least common multiple
common factors
greatest common factor
Name all the possible groupings of elementary- and middle-school students.
What is the largest number of groups that can be formed? How many elementary-school students and how many middle-school students will be in each group?
Lights. A string of lights has red, gold, and blue lights. The red lights are set to blink every 12 seconds, the gold lights are set to blink every 8 seconds, and the blue lights are set to blink every 6 seconds. The lights are on an automatic timer that starts each day at 7:00 p.m. and stops at midnight.
common multiples
least common multiple
common factors
greatest common factor
How often will all 3 colors of lights blink at the exact same time?
How many total times will this happen in one day?
Banners. Noah is making identical square banners for students to hold during the Opening Day game. He has two square pieces of cloth—one is 72 inches wide, and the other is 90 inches wide. He wants to use up all the cloth and make the largest square banners possible.
common multiples
least common multiple
common factors
greatest common factor
How wide should he cut the banners?
How many banners can he cut?
Dancers. Elena is part of a recital where 48 dancers perform in the dark. All the dancers enter the stage in a straight line wearing glow-in-the-dark accessories. Every 3rd dancer wears a glow-in-the-dark headband, every 5th dancer wears a glow-in-the-dark belt, and every 9th dancer wears a set of glow-in-the-dark gloves.
common multiples
least common multiple
common factors
greatest common factor
If Elena is the 30th dancer, what accessories will she wear?
Will any of the dancers wear all 3 accessories? If so, which one(s)?
How many of each accessory will the dance teacher need to order?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is for groups to share their visual displays and explain their reasoning. Give each group the opportunity to briefly present their visual display and approach to solving their problem. If time allows, highlight the different ways in which students used diagrams, equations, and vocabulary to represent their work.
18.4
Activity
Optional
Standards Alignment
Building On
Addressing
6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express as .
In this activity, students play bingo to review factors and multiples. There are two versions of the game:
Version A: 10 Anywhere
The calling cards are mixed, and one is randomly selected.
The statement on the card is read out loud and recorded for all to see.
Students cover all numbers that fit the statement. If a number has already been covered, it gets a second chip stacked on top.
10 markers anywhere on the board make a bingo.
Version B: 4 in a Row
The calling cards are mixed, and one is randomly selected.
The statement on the card is read out loud and recorded for all to see.
Students cover all numbers that fit the statement. If a number has already been covered, it does not get a second chip stacked on top.
4 markers in a row make a bingo.
Launch
Arrange students in groups of 2. Distribute bingo chips and 1 bingo board to each group. Explain the rules for the version of bingo that will be played. Play as many rounds as time allows before a whole-class discussion.
Activity
None
Your teacher will explain the directions for a bingo game.
Share one bingo board and some bingo chips with a partner.
Your teacher will read some statements out loud. Work with your partner to decide which numbers fit each statement.
For each number you cover with a chip, be prepared to identify which statement it corresponds to and to share your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to reflect on any strategies used during the game. After playing a few rounds, discuss:
“Is this a game of luck, strategy, or a combination of both? Explain your reasoning.” (Luck is part of the game, but students must also be familiar with the math vocabulary words and must also be accurate in identifying all the factors and multiples of various numbers.)
“Are some numbers better to have on a game board than others? Explain your reasoning.” (Yes, the number zero is not a good number to have because not many statements fit this value. Larger composite numbers are more likely to get covered because they have many factors and could also be multiples of the smaller numbers.)
Lesson Synthesis
The goal of this discussion is for students to summarize their understanding of factors and multiples. Here are some questions for discussion:
“What is the greatest common factor of two whole numbers? How can it be determined?” (The greatest common factor of 2 whole numbers is the greatest number that evenly divides both numbers, without a remainder. One way to find the greatest common factor is to list all factors for both numbers and choose the largest number that appears in both lists.)
“What types of situations involve finding the greatest common factor?” (situations that involve having to divide two different numbers into equal groups, with no remainders)
“What is the least common multiple of two whole numbers? How can it be determined?” (The least common multiple of two whole numbers is the smallest number that is a multiple of both numbers. One way to find the least common multiple is to list multiples of each number in order and choose the first number that appears on both lists.)
“What types of situations involve finding the least common multiple?” (situations that involve different numbers that need to be multiplied to make the same number)
Student Lesson Summary
If a problem requires dividing two whole numbers by the same whole number, solving it involves looking for a common factor. If it requires finding the largest number that can divide into the two whole numbers, we are looking for the greatest common factor.
Suppose we have 12 bagels and 18 muffins and want to make bags so that each bag has the same combination of bagels and muffins. The common factors of 12 and 18 tell us possible number of bags that can be made.
The common factors of 12 and 18 are 1, 2, 3, and 6. For these numbers of bags, here are the number of bagels and muffins per bag.
1 bag: 12 bagels and 18 muffins
2 bags: 6 bagels and 9 muffins
3 bags: 4 bagels and 6 muffins
6 bags: 2 bagels and 3 muffins
We can see that the largest number of bags that can be made, 6, is the greatest common factor.
If a problem requires finding a number that is a multiple of two given numbers, solving it involves looking for a common multiple. If it requires finding the first instance the two numbers share a multiple, we are looking for the least common multiple.
Suppose forks are sold in boxes of 9 and spoons are sold in boxes of 15, and we want to buy an equal number of each. The multiples of 9 tell us how many forks we could buy, and the multiples of 15 tell us how many spoons we could buy.
Forks: 9, 18, 27, 36, 45, 54, 63, 72, 90,. . .
Spoons: 15, 30, 45, 60, 75, 90,. . .
If we want as many forks as spoons, our options are 45, 90, 135, and so on, but the smallest number of each utensil that we could buy is 45, the least common multiple. This means we would buy 5 boxes of forks () and 3 boxes of spoons ().
Standards Alignment
Building On
Addressing
6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express as .
