Lesson 5Say It with Decimals

Learning Goal

Let’s use decimals to describe increases and decreases.

Learning Targets

I can use the distributive property to rewrite an equation like $x + 0.5 x = 1.5 x$ .
I can write fractions as decimals.
I understand that “half as much again” and “multiply by 1.5” mean the same thing.

Lesson Terms

long division
percentage
rational number
repeating decimal
tape diagram
unit rate

Warm Up: Notice and Wonder: Fractions to Decimals

A calculator gives the following decimal representations for some unit fractions:

$\frac{1}{2} = 0.5$

$\frac{1}{3} = 0.3333333$

$\frac{1}{4} = 0.25$

$\frac{1}{5} = 0.2$

$\frac{1}{6} = 0.1666667$

$\frac{1}{7} = 0.142857143$

$\frac{1}{8} = 0.125$

$\frac{1}{9} = 0.1111111$

$\frac{1}{10} = 0.1$

$\frac{1}{11} = 0.0909091$

What do you notice? What do you wonder?

Activity 1: Repeating Decimals

Problem 1

Use long division to express each fraction as a decimal.

$\frac{9}{25}$
$\frac{11}{30}$
$\frac{4}{11}$

Problem 2

What is similar about your answers to the previous question? What is different?

Problem 3

Use the decimal representations to decide which of these fractions has the greatest value. Explain your reasoning.

Are you ready for more?

One common approximation for $π$ is $\frac{22}{7}$ . Express this fraction as a decimal. How does this approximation compare to 3.14?

Activity 2: More and Less with Decimals

Problem 1

Match each diagram with a description and an equation.

Diagrams:

Tape diagram A has 4 green segments labeled "x" and three of those together labeled "y". Diagram B has 4 green segments labeled "x" and one white segment and all together are labeled "y"

Descriptions:

An increase by $\frac{1}{4}$

An increase by $\frac{1}{3}$

An increase by $\frac{2}{3}$

A decrease by $\frac{1}{5}$

A decrease by $\frac{1}{4}$

Equations:

$y = 1. \overset{―}{6} x$

$y = 1. \overset{―}{3} x$

$y = 0.75 x$

$y = 0.4 x$

$y = 1.25 x$

Problem 2

Draw a diagram for one of the unmatched equations.

Activity 3: Card Sort: More Representations

Your teacher will give you a set of cards that have proportional relationships represented 2 different ways: as descriptions and equations. Mix up the cards and place them all face-up.

Take turns with a partner to match a description with an equation.

For each match you find, explain to your partner how you know it’s a match.
For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking.
When you have agreed on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches.

Lesson Summary

Long division gives us a way of finding decimal representations for fractions.

For example, to find a decimal representation for $\frac{9}{8}$ , we can divide 9 by 8.

So $\frac{9}{8} = 1.125$ .

$\begin{array}{r} 1.125 \\ 8 9.000 \\ \underset{―}{8} \\ 1 0 \\ \underset{―}{8} \\ 20 \\ \underset{―}{16} \\ 40 \\ \underset{―}{40} \\ 0 \end{array}$

Sometimes it is easier to work with the decimal representation of a number, and sometimes it is easier to work with its fraction representation. It is important to be able to work with both. For example, consider the following pair of problems:

Priya earned $x$ dollars doing chores, and Kiran earned $\frac{6}{5}$ as much as Priya. How much did Kiran earn?

Priya earned $x$ dollars doing chores, and Kiran earned 1.2 times as much as Priya. How much did Kiran earn?

Since $\frac{6}{5} = 1.2$ , these are both exactly the same problem, and the answer is $\frac{6}{5} x$ or $1.2 x$ .

When we work with percentages in later lessons, the decimal representation will come in especially handy.