Lesson 10Rectangles and Triangles with Fractional Lengths

Learning Goal

Let’s explore rectangles and triangles that have fractional measurements.

Learning Targets

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

Warm Up: Areas of Squares

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.

What do you notice about the areas of the squares? Write your observations.
Consider the statement: “A square with side lengths of $\frac{1}{3}$ inch has an area of $\frac{1}{3}$ square inches.” Do you agree or disagree with the statement? Explain or show your reasoning.

Activity 1: How Many Would It Take? (Part 2)

Noah would like to cover a rectangular tray with rectangular tiles. The tray has a width of $11 \frac{1}{4}$ inches and an area of $50 \frac{5}{8}$ square inches.

Find the length of the tray in inches.
If the tiles are $\frac{3}{4}$ inch by $\frac{9}{16}$ inch, how many would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning.
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.

Activity 2: Bases and Heights of Triangles

Problem 1

The area of Triangle B is 8 square units. Find the length of $b$ . Show your reasoning.

A triangle labeled B has a horizontal side on the bottom of the triangle and a vertex above the horizontal side. A dashed line from the vertex to the horizontal side is drawn and a right angle symbol is indicated. The horizontal side is labeled b and the dashed line is labeled eight thirds.

Problem 2

The area of Triangle C is $\frac{54}{5}$ square units. What is the length of $h$ ? Show your reasoning.

A triangle labeled C has a horizontal side at the top of the triangle and a vertex below the horizontal side and to the left. A horizontal line extends from the horizontal side and to the left. A dashed line is drawn from the bottom vertex to the extended horizontal line and a right angle symbol is indicated. The dashed line is labeled h and the horizontal side of the triangle is labeled 3 and three fifths.

Lesson Summary

If a rectangle has side lengths $a$ units and $b$ units, the area is $a \cdot b$ square units. For example, if we have a rectangle with $\frac{1}{2}$ -inch side lengths, its area is $\frac{1}{2} \cdot \frac{1}{2}$ or $\frac{1}{4}$ 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 $10 \frac{1}{2}$ in and its area is $89 \frac{1}{4}$ in $^{2}$ , we can write this equation to show their relationship: $10 \frac{1}{2} \cdot ? = 89 \frac{1}{4}$

Then, we can find the other side length, in inches, using division: $89 \frac{1}{4} \div 10 \frac{1}{2} = ?$