Author:
Angela Vanderbloom
Subject:
Mathematics
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Tags:
  • 6.NS.B.4 Least Common Multiple Greatest Common Factor
  • License:
    Creative Commons Attribution Non-Commercial
    Language:
    English

    Education Standards

    6.NS.B.4 Lesson 3

    6.NS.B.4 Lesson 3

    Overview

    In this lesson, students apply what they have learned about factors and multiples to solve a variety of problems. In the first activity, students to use what they have learned about common factors and common multiples to solve less structured problems in context (MP1).

    Warm Up - Two Numbers

    Students will be given clues about two numbers in which they determine those two numbers.

    Two numbers can be described with the information below:

    • Both numbers are less than 20.
    • The greatest common factor of the two numbers is 2.
    • The least common multiple of the two numbers is 36.

    What are the two numbers?

    Factors and Multiples - Problem 1

    In this activity, students work in pairs to solve problems that involve thinking about factors and multiples as well as the greatest common factor and least common multiple. After solving the problems, they reflect on what type of mathematical work was part of each problem and then record this information into a table. Students begin to notice similarities in the types of problems that involve factors and and in 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

    Arrange students in groups of 2. Give students 15 minutes of work time followed by whole-class discussion.

    Support for Students with Disabilities

    Executive Functioning: Eliminate Barriers. Chunk this task into more manageable parts (e.g., presenting one question at a time), which will aid students who benefit from support with organizational skills in problem solving.

    Receptive/Expressive Language: Processing Time. Students who benefit from extra processing time would also be aided by MLR 6 (Three Reads).

    Social-Emotional Functioning: Peer Tutors. Pair students with their previously identified peer tutors.

    Student Response

    1. Answers vary. Sample response: Elena could have 24 cups and 24 plates.
    2. Answers vary. Sample response: Elena could get 3 packages of cups and 4 packages of plates to get 24 of each.
    3. Answers vary. Sample response: She could get 48 of each if she gets 6 packages of cups and 8 packages of plates. She could also get 72 of each if she gets 9 packages of cups and 12 packages of plates.

    Work with your partner to solve the following problem.

    1. 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.

      1. Find a number of plates and cups that meets her requirement.
      2. How many packs of each supply will she need to buy to get that number?
      3. Name two other quantities of plates and cups she could get to meet her requirement.

    Factors and Multiples - Problem 2

    Same as noted within Factors and Multiples Problem 1.

    Student Response

    1. The largest square that will fit has a length of 6 ft.
    2. 12 tiles are needed to cover the floor.
    3. She could also use tiles of length 1, 2, or 3 ft.

    Work with your partner to solve the following problem.

    2.  Tiles. A restaurant owner is replacing the restaurant’s bathroom floor with square tiles. The tiles will be laid side-by-side to cover the entire bathroom with no gaps, and none of the tiles can be cut. The floor is a rectangle that measures 24 feet by 18 feet.

    1. What is the largest possible tile size she could use? Write the side length in feet. Explain how you know it’s the largest possible tile.
    2. How many of these largest size tiles are needed?
    3. Name more tile sizes that are whole number of feet that she could use to cover the bathroom floor. Write the side lengths (in feet) of the square tiles.

    Factors and Multiples - Problem 3

    Same as noted within Factors and Multiples Problem 1.

    Student Response

    1. Answers vary. Any 3 lockers from 20, 40, 60, 80, and 100 are valid responses. The least common multiple of 4 and 5 is 20. All common multiples of 4 and 5 are multiples of 20.
    2. Since 30 is not a multiple of 4, the 30th locker will not get a flower sticker. It will only get 1 sticker.

    Work with your partner to solve the following problem.

    3.  Stickers. To celebrate the first day of spring, Lin is putting stickers on some of the 100 lockers along one side of her middle school’s hallway. She puts a skateboard sticker on every 4th locker (starting with locker 4), and a kite sticker on every 5th locker (starting with locker 5).

    1. Name three lockers that will get both stickers.
    2. After Lin makes her way down the hall, will the 30th locker have no stickers, 1 sticker, or 2 stickers? Explain how you know.

    Factors and Multiples - Problem 4

    Same as noted within Factors and Multiples Problem 1.

    Student Response

    1. The greatest amount of kits that can be made is 15.
    2. Each kit will have 5 bandages and 6 lozenges.

    Work with your partner to solve the following problem.

    4.  Kits. The school nurse is assembling first-aid kits for the teachers. She has 75 bandages and 90 throat lozenges. All the kits must have the same number of each supply, and all supplies must be used.

    1. What is the largest number of kits the nurse can make?
    2. How many bandages and lozenges will be in each kit?

    Factors and Multiples - Problem 5

    Same as noted within Factors and Multiples Problem 1.

    Student Response

    problemfinding multiplesfinding least
    common multiple
    finding factorsfinding greatest
    common factor
    Partyxx  
    Tilesx xx
    Stickersxx  
    Kitsx xx

    What kind of mathematical work was involved in each of the previous problems? Put a checkmark to show what the questions were about.

    problemfinding multiplesfinding least
    common multiple
    finding factorsfinding greatest
    common factor
    Party    
    Tiles    
    Stickers    
    Kits    

    Cool Down: What Kind of Problem?

    Student Response

      1. Greatest common factor
      2. Least common multiple
      3. Least common multiple
      4. Greatest common factor
    1. Answers vary. Sample responses:
      • A problem is about finding the greatest common factor when it involves looking for the greatest number of equal sets that can be made from a certain number of items (with no left over), or looking for the greatest length of a rectangle given a certain number of square units.
      • A problem is about finding the greatest common factor when it involves finding the largest possible number that divides into two whole numbers without a remainder.

     

    1.  For each problem, tell whether finding the answer requires finding a greatest common factor or a least common multiple. You do not need to solve the problems.

    a.  Elena has 20 apples and 35 crackers for making snack bags. She wants to make as many snack bags as possible and wants each bag to have the same combination of apples and crackers. What is the largest number of snack bags she could make?

    b.  A string of holiday lights at a store have three colors that flash at different times. Red lights flash every fifth second. Blue lights flash every third seconds. Green light flashes every four seconds. The store owner turns on the lights.  After how many seconds will all three lights flash at the same time for the first time?

    c.  A florist orders sunflowers every 6 days, starting from the sixth day of the year, and daisies every 4 days, starting from the fourth day of the year. When (or on which day) will she orders both kinds of flowers on the same day?

    d.  Noah has 12 yellow square cards and 18 green ones. All the cards are the same size. He would like to arrange the square cards into two rectangles—one of each color. He wants both the yellow and green rectangles to have the same height and to be as tall as possible. What is the tallest possible height for the two rectangles?

     

    2.   Explain how you know which problem(s) involves finding the greatest common factor.