## From Rectangles to Squares

## 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

Use the 2 × 3 rectangle to do the following.

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

INTERACTIVE: Build Squares with Rectangles

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

Build squares using the 4 × 6 rectangle.

INTERACTIVE: Build Squares with Rectangles

- 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.

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?

INTERACTIVE: Build Squares with Rectangles

Prepare a presentation that shows your work and supports your conclusion about how you know you made the smallest square possible.

- 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

What do you notice about the length of the square compared to the length of either side of the rectangle?

Take notes about your classmates’ thinking as to how they know the square they built is the smallest square possible.

As your classmates present, ask questions such as:

- What strategy did you use to make your squares?
- How can you determine the length of your squares?
- Explain how you know that you found the smallest possible square.
- What does the length of a square represent?
- What does the length of the smallest square represent?
- Where do you see a multiple represented in your model?
- Where do you see a common multiple represented in your model?
- Where do you see the least common multiple represented in your model?

Write the least common multiple of each set of numbers.

- 4, 6
- 18, 36
- 20, 36
- 12, 20, 30

Generate and list the multiples of each number. Look for multiples in common. Then find the smallest one.

**Read and Discuss**

- A common multiple of two or more whole numbers is a number that is a multiple of all the numbers. For example, 12 is a common multiple of 2 and 3 because 2 × 6 = 12 and 3 × 4 = 12.
- The smallest of all the common multiples of any two or more natural numbers is called the
*least common multiple (LCM)*. For example, the least common multiple of 2 and 6 is 6, because 2 × 3 = 6 and 1 × 6 = 6. (Multiples of 2 that are less than 6 are 2 and 4; those numbers are not multiples of 6. 6 is the smallest multiple of 6.) - When you build a square using rectangles that have edges with whole-number lengths, the edge length of the square is a common multiple of the length and width of the rectangle.

Can you:

- Describe the relationship between the side lengths of a rectangle and the side length of a square that was built from the rectangle?
- Use the terms
*multiple*,*common multiple*, and*least common multiple*?

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 multiples and least common multiples is …**