Lesson 13Rectangles with Fractional Side Lengths

Learning Goal

Let’s explore rectangles that have fractional measurements.

Learning Targets

  • I can use division and multiplication to solve problems involving areas of rectangles with fractional side lengths.

Warm Up: Areas of Squares

Problem 1

Three squares. The first square is labeled with side length 1 inch on the vertical side and 1 inch on the horizontal side. The second square is labeled with side length one half inch on the vertical side and one half inch on the horizontal side. The third square is labeled with side length 2 inches on the vertical side and 2 inches on the horizontal side.

  1. What do you notice about the areas of the squares? Write your observations.

  2. Consider the statement: “A square with side lengths of inch has an area of square inches.” Do you agree or disagree with the statement? Explain or show your reasoning.

Activity 1: Areas of Squares and Rectangles

Your teacher will give you graph paper and a ruler.

Problem 1

On the graph paper, draw a square with side lengths of 1 inch. Inside this square, draw another square with side lengths of inch.

Use your drawing to answer the questions.

  1. How many squares with side lengths of inch can fit in a square with side lengths of 1 inch?

  2. What is the area of a square with side lengths of inch? Explain or show your reasoning.

Problem 2

On the graph paper, draw a rectangle that is inches by inches.

For each question, write a division expression and then find the answer.

  1. How many -inch segments are in a length of inches?

  2. How many -inch segments are in a length of inches?

Problem 3

Use your drawing to show that a rectangle that is inches by inches has an area of square inches.

Activity 2: Areas of Rectangles

Problem 1

Each of these multiplication expressions represents the area of a rectangle.

  1. All regions shaded in light blue have the same area. Match each diagram to the expression that you think represents its area. Be prepared to explain your reasoning.

    1. Vertically oriented rectangle that has a horizontal dotted line about three quarters of the way down its width. The top portion is shaded blue.
    2. Vertically oriented rectangle that has a horizontal dotted line about three quarters of the way down its width and a vertical dotted line about three quarters of the way to the right of its length. The top left portion of the rectangle is shaded blue.
    3. Vertically oriented rectangle that is shaded blue.
    4. Vertically oriented rectangle that has a vertical dotted line about three quarters of the way to the right of its length. The left portion of the rectangle is shaded blue.

  2. Use the diagram that matches to show that the value of is .

Are you ready for more?

Problem 1

The following rectangles are composed of squares, and each rectangle is constructed using the previous rectangle. The side length of the first square is 1 unit.

Five pictures of rectangles start with a square then adding more squares in each picture.

  1. Draw the next four rectangles that are constructed in the same way. Then complete the table with the side lengths of the rectangle and the fraction of the longer side over the shorter side.

    short side

    long side

  2. Describe the values of the fraction of the longer side over the shorter side. What happens to the fraction as the pattern continues?

Activity 3: How Many Would It Take? (Part 2)

Problem 1

Noah would like to cover a rectangular tray with rectangular tiles. The tray has a width of inches and an area of square inches.

  1. Find the length of the tray in inches.

  2. If the tiles are inch by inch, how many would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning.

  3. Draw a diagram to show how Noah could lay the tiles. Your diagram should show how many tiles would be needed to cover the length and width of the tray, but does not need to show every tile.

Lesson Summary

If a rectangle has side lengths units and units, the area is square units. For example, if we have a rectangle with -inch side lengths, its area is or square inches.

A large square is divided into 4 equal squares. The large square has bottom horizontal side length labeled 1 inch. Of the four smaller squares, the top left square is shaded blue. It has side lengths labeled one half inch.

This means that if we know the area and one side length of a rectangle, we can divide to find the other side length.

A rectangle with the horizontal side labeled 10 and one half inches and the vertical side labeled with a question mark. In the center of the rectangle, 89 and one fourth square inches is indicated.

If one side length of a rectangle is in and its area is in², we can write this equation to show their relationship: Then, we can find the other side length, in inches, using division: