- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Interactive, Text/HTML

# Rectangular Grid

# Reviewing The Greatest Common Factor

## Overview

Students use a geometric model to investigate common factors and the greatest common factor of two numbers.

# Key Concepts

A geometric model can be used to investigate common factors. When congruent squares fit exactly along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid. The greatest common factor (GCF) is the largest square that fits exactly along both the length and the width of the rectangular grid. For example, given a 6-centimeter × 8-centimeter rectangular grid, four 2-centimeter squares will fit exactly along the length without any gaps or overlaps. So, 2 is a factor of 8. Three 2-centimeter squares will fit exactly along the width, so 2 is a factor of 6. Since the 2-centimeter square is the largest square that will fit along both the length and the width exactly, 2 is the greatest common factor of 6 and 8.

Common factors are all of the factors that are shared by two or more numbers.

The greatest common factor is the greatest number that is a factor shared by two or more numbers.

# Goals and Learning Objectives

- Use a geometric model to understand greatest common factor.
- Find the greatest common factor of two whole numbers equal to or less than 100.

# Which Tiles Will Exactly Cover the Grid?

# Lesson Guide

Start the lesson by projecting the 12-unit × 18-unit rectangular grid from the interactive and having partners discuss the two questions.

The purpose of this discussion is to ensure that all students understand what it means to cover the grid with congruent square tiles without any gaps or overlaps. When you choose student responses to share with the class, pick responses that make this concept clear. The 1-unit squares can cover the grid. Twelve of the square tiles fit along one side of the grid, and 18 of them fit along the other side. The 5-unit square tiles do not work. There is a 2-unit gap along the 12-unit side of the grid and a 3-unit gap along the 18-unit side.

## Opening

# Which Tiles Will Exactly Cover the Grid?

Suppose you want to cover a 12-unit by 18-unit rectangular grid exactly, without any gaps or overlaps. Can you do this using square tile stamps in the following unit sizes?

- 1-unit square tiles? Explain why or why not.
- 5-unit square tiles? Explain why or why not.

INTERACTIVE: Rectangular Grid

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will find common factors and the greatest common factor of two numbers using a geometric model.

## Opening

Investigate how to find the common factors of two numbers.

# Square Tiles on the Grid

# Lesson Guide

No product is required for the first problem; it is simply a chance to explore. However, students do use the results of the exploration for the second and third problems.

After a few minutes of exploration, direct students to record their findings on a separate sheet of paper. Have students work on their presentations individually or in pairs.

As students work, look for work that:

- Correctly identifies the 1-unit, 2-unit, 3-unit, and 6-unit squares as exactly covering the 12 × 18 rectangular grid.
- Misidentifies the other unit squares as exactly covering the 12 × 18 rectangular square grid.

It is not necessary to wait until all students have completed this portion of the work. As soon as about half the class is ready, start the Ways of Thinking discussion.

# Mathematical Practices

**Mathematical Practice 4: Model with mathematics.**

Students use a geometric model to identify common factors and the greatest common factor of two numbers. Finding squares that fit along each edge of the rectangular grid is similar to dividing the edge length of the rectangular grid by the edge length of the square tile.

**Mathematical Practice 6: Attend to precision.**

Watch for students who attend to precision when aligning square tiles along the length and the width of a rectangular grid and when recording their results in a table. Students who leave gaps between tiles or have overlaps will not get the correct solution.

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

Identify students who find relationships in their recordings and apply reasoning to determine the common factors and the greatest common factor of two numbers.

# Interventions

**Student has an incorrect solution.**

- Have you checked your work?
- Show me how you got [answer] as a side length of a square tile for both 12 and 18. Does your answer make sense? What happens if you try the tile along the [width, length]?
- Use your 3-unit square tile. How many tiles fit exactly along the 12-unit side? What is 12 divided by 3? Which operation are you modeling when you are finding squares that fit along the side of the rectangular grid?
- What does the side length of the square tile that fits exactly along the side of the rectangular grid represent?

**Student has a solution.**

- What do the side lengths of squares that fit along the 12-unit side represent?
- What do the side lengths of squares that fit along the 18-unit side represent?
- What do the side lengths of the square tiles that exactly cover the rectangular grid represent?
- What does the square tile with the largest side length that will exactly cover the rectangular grid represent?
- How could you find the greatest common factor without drawing rectangular grids?
- Do you think a 1-unit square would fit exactly along the side of any rectangular grid with whole-number side lengths? What can you conclude about the number 1?

# Answers

- These are the side lengths of square tiles that will exactly cover the 12 × 18 rectangular grid:

1 unit, 2 unit, 3 unit, and 6 unit. - The square tiles that will exactly cover the 12 × 18 rectangular grid are factors of both 12 and 18.
- The largest square that will exactly cover the 12 × 18 rectangular gird is a 6-unit square.

## Work Time

# Square Tiles on the Grid

- Explore the interactive by trying square tiles of different sizes.

- Answer these questions:
- Which square tiles will exactly cover a 12 × 18 rectangular grid?
- What do you notice about the square tiles that will exactly cover the 12 × 18 rectangular grid?
- What is the largest square tile that will exactly cover the 12 × 18 rectangular grid?

INTERACTIVE: Rectangular Grid

## Hint:

The numbers 12 and 18 are both even. Do you think 2-unit squares will exactly cover this rectangle?

# Prepare a Presentation

# Preparing for Ways of Thinking

As students work, look for work that:

- Correctly identifies the 1-unit, 2-unit, 3-unit, and 6-unit squares as exactly covering the 12 × 18 rectangular grid.
- Misidentifies the 4-unit, 12-unit, or 18-unit squares (or others) as exactly covering the 12 × 18 rectangular square grid.

# Challenge Problem

# Answer

- Squares with side lengths of 1, 2, 3, 4, 6, and 12 units will exactly cover a 36 × 12 rectangular grid. The 12-unit square is the largest possible square.

## Work Time

# Prepare a Presentation

List all the tile sizes you were able to use to exactly cover the 12 × 18 rectangle. Explain what your findings tell you.

# Challenge Problem

- Predict what is the largest square you think will cover a 36 x 12 rectangular grid exactly. Then, test your prediction to see if it is correct.

# Make Connections

# Lesson Guide

During this discussion, help students understand that finding squares that fit along each edge of the rectangular grid is a geometric modeling of dividing the edge lengths of the rectangular grid by the edge lengths of the square.

- When you are finding squares that fit along an edge of the rectangular grid, what are you doing? Think about the relationship between the edge length of the rectangular grid and the edge length of the square.
- What are you finding when squares fit exactly along both the width and the length of the rectangular grid?
- Why does a 1-unit square fit exactly along the side of any rectangular grid with whole-number side lengths?
- What does the length of the largest square that fits exactly along both the width and the length of the rectangular grid represent? What can you reason from this model?

Explain that students are using squares to cover the rectangular grid because they are looking for factors that are common to both 12 and 18. The edge length of any square that exactly fits along both the width and the length of the 12 × 18 rectangular grid is a factor of both 12 and 18.

The 1-unit square will exactly cover any rectangular grid with whole-number dimensions, and the number 1 is a factor of all whole numbers.

The edge length of the largest square that will cover a rectangular grid is the greatest common factor of the length and width of the rectangular grid.

If you ask students how they would find the greatest common factor without sketching rectangular grids, some students might suggest listing the factors of both numbers in a table and then finding the greatest factor that is common to both. Other students might suggest listing the factors of one number, and then dividing the other number by those factors to see if there is a remainder. The greatest factor that divides evenly is the greatest common factor.

## Performance Task

# Ways of Thinking: Make Connections

Takes notes about your classmates’ approaches to finding the tile sizes that will cover a grid exactly.

## Hint:

As your classmates present, ask questions such as:

- How did you decide which squares to use to try to cover the grid?
- Were there any square sizes that you knew right away would not cover the grid?
- What are you finding when squares fit exactly along both the width and the length of the rectangular grid?
- What does the largest square tile that covers the grid represent?

# Greatest Common Factor

# Interventions

Ask questions such as the following as students are working:

- What strategy did you use to find the greatest common factor?
- Can you use a model to find the greatest common factor? Show me.
- What are other common factors of the two numbers? How do you know?

# Answers

- The greatest common factor of 8 and 12 is 4.
- The greatest common factor of 18 and 24 is 6.
- The largest square tile that will cover a 27 × 63 rectangle is a 9-unit square.
- The greatest common factor of 54 and 126 is 18.

## Work Time

# Greatest Common Factor

Common factors are factors that are shared by two numbers. The greatest common factor of two numbers is the greatest number that is a factor of the two numbers.

- Find the greatest common factor of 8 and 12.
- Find the greatest common factor of 18 and 24.
- Find the largest square tile that will cover a 27 × 63 rectangle.
- What is the greatest common factor of 54 and 126?

## Hint:

- Find numbers that will divide evenly into both numbers. Which is the largest?
- Find all the factors of both numbers. Look for the common factors.

# All About Factors

# Mathematics

- Have pairs discuss factors, common factors, and greatest common factors.
- As student pairs discuss, listen for students who may still not understand how to use the model to find factors, common factors, and the greatest common factor. Make a note to clarify any misconceptions during the class discussion.
- Then discuss the Summary as a class. Be sure to highlight these points:
- Common factors are factors that are shared by two numbers. For example, 2 is a common factor of 4 and 8. The greatest common factor of two numbers is the greatest number that is a factor of the two numbers. So the greatest common factor of 4 and 8 is 4, not 2.
- When you can exactly fit congruent squares along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid.
- Common factors are all of the factors that are shared by two or more numbers.
- The greatest common factor is the greatest factor shared by two or more numbers.

ELL: Be prepared to assist students with explaining the subject matter in problems. Keep in mind that the content and vocabulary may be unfamiliar to some students. Make sure students know and understand the word *congruent* and understand how congruent squares can be used to find the greatest common factor of two numbers in a rectangular grid.

SWD: To ensure that all students make the connection between the multiplication table, factors, and multiples, highlight the multiplication table with different colors on the outside (factors) and inside (multiples).

## Formative Assessment

# All About Factors

**Read and Discuss**

- A factor of a number is a whole number that divides a given quantity without a remainder. For example, 5 is a factor of 10 but not a factor of 8.
- Common factors are factors that are shared by two numbers. For example, 2 is a common factor of 4 and 8.
- The greatest common factor of two numbers is the greatest number that is a factor of the two numbers. So, the greatest common factor of 4 and 8 is 4.
- When you can exactly fit congruent squares along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid.

## Hint:

Can you:

- Describe the relationship between the side length of the square and the side lengths of the rectangular grid?
- Identify what the largest square that exactly covers the rectangular grid represents?
- Use the terms
*factor*,*common factor*, and*greatest common factor*appropriately?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to find out what students learned about common factors and greatest common factors.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**What I learned about common factors and greatest common factors is …**