Rectangles with Fractional Side Lengths

Warmup

Activity #1

Explore the Effect of Side Lengths of a Rectangle on the Area.

• Observe the changes to the rectangle as you adjust the “length” and “width” sliders. Each square in the grid is 1cm by 1cm.

• Answer the questions below.

(1.) Adjust the sliders so l = 5 and w = 3. How many squares fit inside the rectangle?

(2.) When we say that the area of the rectangle is “32 square cm” or “32 cm squared”, what do we really mean by that?

(3.) What measurements of the rectangle would give you an area of 32 square cm?

(4.) What is the formula for the area of a rectangle? Your answer should be in terms of l and w.

Activity #2

Explore Rectangles with Fractional Dimensions.

• In the applet below, draw a square with side lengths of 1 inch. 
• Inside this square, draw another square with side lengths of ¼ inch.

• Answer the following questions;

(1.) From the graph, 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.

In the app below, draw a rectangle that is 3 ½ inches by 2 ¼ inches.

• In the applet below, draw a rectangle that is 3 ½ inches by 2 ¼ inches. 

• Use your drawing to answer the questions. 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

(3.) Each of these multiplication expressions below represents the area of a rectangle. All regions shaded in light blue have the same area.

• Match each diagram to the expression that you think represents its area.

(4.) Explain your reasoning for the matches above.

Activity #3

Complete a Table with the Ratio of Side Lengths of a Rectangle.

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

• Draw the next four rectangles that are constructed in the same way.

• Complete the table with the side lengths of the rectangle and the fraction of the longer side over the shorter side.

(1.) Describe the values of the fraction of the longer side over the shorter side.

(2.) What happens to the fraction as the pattern continues?

Challenge #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 your reasoning.

Challenge #2

Find the unknown side length of the rectangle below.

Challenge #3

Select all the equations that represent the relationship of the side lengths and area of the television below.