Author:
Angela Vanderbloom
Subject:
Numbers and Operations
Material Type:
Lesson Plan
Level:
Middle School
Grade:
6
Tags:
  • 6th Grade Mathematics
  • Greatest Common Factor
  • Models
    License:
    Creative Commons Attribution Non-Commercial
    Language:
    English
    Media Formats:
    Interactive, Text/HTML

    Education Standards

    6.NS.B.4 Intro Lesson 2

    6.NS.B.4 Intro Lesson 2

    Overview

    Lesson Overview

    Students use a geometric model to investigate common multiples and the least common multiple of two numbers.

    Key Concepts

    A geometric model can be used to investigate common multiples. When congruent rectangular cards with whole-number lengths are arranged to form a square, the length of the square is a common multiple of the side lengths of the cards. The least common multiple is the smallest square that can be formed with those cards.

    For example, using six 4 × 6 rectangles, a 12 × 12 square can be formed. So, 12 is a common multiple of both 4 and 6. Since the 12 × 12 square is the smallest square that can be formed, 12 is the least common multiple of 4 and 6.

    Common multiples are multiples that are shared by two or more numbers. The least common multiple (LCM) is the smallest multiple shared by two or more numbers.

    Goals and Learning Objectives

    • Use a geometric model to understand least common multiples.
    • Find the least common multiple of two whole numbers equal to or less than 12.

    From Rectangles to Squares

    Lesson Guide

    Begin the class by projecting the 2 × 3 card from the interactive on the board. Show students how to make copies of the card and then give them a few minutes to make and describe squares.

    The purpose of this discussion is to ensure that all students understand how to make squares with the cards. When you choose student responses to share with the class, pick responses that make this concept clear.

     

    • Six cards can make this square: The next larger square would have side lengths of 12 units.
      These errors are likely to surface: These shapes are not squares. You might suggest that students keep in mind the question, “Why squares?”
    • Answers will vary.

    Opening

    From Rectangles to Squares

    Use the 2 × 3 rectangle to do the following.

    • Build at least two squares.
    • Describe the squares you built.

    INTERACTIVE: Build Squares with Rectangles

    Math Mission

    Lesson Guide

    Discuss the Math Mission. Students will find common multiples and the least common multiple of two numbers using a geometric model.

    Opening

    Investigate how to find the common multiples of two numbers, and identify the least common multiple.

    Build Squares

    Lesson Guide

    Have students build squares individually.

    Most likely, students will build a 12-unit square and a 24-unit square, although some students may mention squares with sides of 36 or 48 units.

    Answers

    • Squares will vary.

    Work Time

    Build Squares

    Build squares using the 4 × 6 rectangle.

    INTERACTIVE: Build Squares with Rectangles

    Hint:

    • Start with two rectangles side by side to make a 4×12 rectangle.
    • Add another row of two rectangles to make an 8×12rectangle.
    • Continue adding rows of two rectangles until you make a square.

    The Smallest Square

    Mathematical Practices

    Mathematical Practice 4: Model with mathematics.

    Students use a geometric model to identify common multiples and the least common multiple of two numbers.

    Mathematical Practice 6: Attend to precision.

    Watch for students who attend to precision when building their squares and when recording their results. Students who do not create squares or who do not record their results precisely will not get a correct solution.

    Mathematical Practice 8: Look for and express regularity in repeated reasoning.

    Identify students who find relationships between the side lengths of the rectangles and the side lengths of the squares, and apply reasoning to determine common multiples and the least common multiple of two numbers.

    Interventions

    Student has an incorrect solution.

    • Have you checked your work?
    • How can you describe the lengths of the sides of a square? Is this shape a square? [Point to the student’s square.] Explain why or why not.
    • If it is not a square, what do you need to do to make it a square?
    • How can you represent the length of the square as a multiplication problem? How can you represent the width of the square as a multiplication problem?
    • What do the side lengths of the squares represent?

    Student has a solution.

    • Describe how to use multiplication to describe the length and width of a square.
    • What do the side lengths of the squares represent? Explain your thinking.
    • How did you determine the smallest possible square?
    • What does the smallest possible square represent?
    • How could you find the least common multiple without drawing rectangles?

    Possible Answers

    • The height of the smallest square made by 4 × 6 rectangles is 3 rectangles high.
    • The width of the smallest square made by 4 × 6 rectangles is 2 rectangles wide.
    • One side of a rectangle is shorter than the other side. The sides of a square need to be the same length, so you need more rectangles along one direction to equal the length of the rectangles along the other direction.
    • The side length of the smallest square made by 4 × 6 rectangles is 12 units.
    • A 12 × 12 square is the smallest square that can be made with 4 × 6 rectangles, because 12 is the first multiple of 4 that is also a multiple of 6.
    • The two equal sides of the square represent the same number that both the length and the width of the rectangle must divide evenly.

    Challenge Problem

    Answers

    • The smallest square made from 2 × 5 rectangles is a 10 × 10 square.
    • Students build the 10 × 10 square to see if their prediction is correct.

    Work Time

    The Smallest Square

    Think about the smallest square you made using the 4 x 6 rectangles.

    • How many rectangles high is the square?
    • How many rectangles wide is the square?
    • Why did you use more rectangles in one direction than in the other direction?
    • How long is one side of the square?
    • How do you know that the square is the smallest square you can make?
    • The side lengths of all of the squares are the common multiples of 4 and 6. The side length of the smallest square is the least common multiple (LCM) of 4 and 6. Why would you build squares rather than rectangles to find the least common multiple?

    Challenge Problem

    • Start with the 2 × 5 rectangle. Predict what size the smallest possible square would be that you could build using the rectangle.
    • Build the square to see if your prediction is correct.

    INTERACTIVE: Build Squares with Rectangles