Lesson 5A New Way to Interpret a over b

Learning Goal

Let’s investigate what a fraction means when the numerator and denominator are not whole numbers.

Learning Targets

I understand the meaning of a fraction made up of fractions or decimals, like $\frac{2.1}{0.07}$ or $\frac{\frac{4}{5}}{\frac{3}{2}}$ .
When I see an equation, I can make up a story that the equation might represent, explain what the variable represents in the story, and solve the equation.

Lesson Terms

coefficient
solution to an equation
variable

Warm Up: Recalling Ways of Solving

Solve each equation. Be prepared to explain your reasoning.

$0.07 = 10 m$
$10.1 = t + 7.2$
$x - 4.4 = 19$
$\frac{x}{2.3} = 8$

Activity 1: Interpreting $\frac{a}{b}$

Solve each equation.

$35 = 7 x$
$35 = 11 x$
$7 x = 7.7$
$0.3 x = 2.1$
$\frac{2}{5} = \frac{1}{2} x$

Are you ready for more?

Solve the equation. Try to find some shortcuts.

$\frac{1}{6} \cdot \frac{3}{20} \cdot \frac{5}{42} \cdot \frac{7}{72} \cdot x = \frac{1}{384}$

Activity 2: Storytime Again

Take turns with your partner telling a story that might be represented by each equation. Then, for each equation, choose one story, state what quantity $x$ describes, and solve the equation. If you get stuck, consider drawing a diagram.

$0.7 + x = 12$
$\frac{1}{4} x = \frac{3}{2}$

Lesson Summary

In the past, you learned that a fraction such as $\frac{4}{5}$ can be thought of in a few ways.

$\frac{4}{5}$ is a number you can locate on the number line by dividing the section between 0 and 1 into 5 equal parts and then counting 4 of those parts to the right of 0.
$\frac{4}{5}$ is the share that each person would have if 4 wholes were shared equally among 5 people. This means that $\frac{4}{5}$ is the result of dividing 4 by 5.

We can extend this meaning of a fraction as a quotient to fractions whose numerators and denominators are not whole numbers. For example, we can represent 4.5 pounds of rice divided into portions that each weigh 1.5 pounds as: $\frac{4.5}{1.5} = 4.5 \div 1.5 = 3$ . In other words, $\frac{4.5}{1.5} = 3$ because the quotient of 4.5 and 1.5 is 3.

Fractions that involve non-whole numbers can also be used when we solve equations.

Suppose a road under construction is $\frac{3}{8}$ finished and the length of the completed part is $\frac{4}{3}$ miles. How long will the road be when completed?

We can write the equation $\frac{3}{8} x = \frac{4}{3}$ to represent the situation and solve the equation.

The completed road will be $3 \frac{5}{9}$ or about 3.6 miles long.

$\begin{aligned} \frac{3}{8} x & = \frac{4}{3} \\ x & = \frac{\frac{4}{3}}{\frac{3}{8}} \\ x & = \frac{4}{3} \cdot \frac{8}{3} \\ x & = \frac{32}{9} = 3 \frac{5}{9} \end{aligned}$

Lesson 5A New Way to Interpret a over b

Learning Goal

Learning Targets

Lesson Terms

Warm Up: Recalling Ways of Solving

Problem 1

Activity 1: Interpreting ab

Problem 1

Are you ready for more?

Problem 1

Activity 2: Storytime Again

Problem 1

Lesson Summary

Activity 1: Interpreting $\frac{a}{b}$