Lesson 5: Say It with Decimals

Let's use decimals to describe increases and decreases.

5.1: Notice and Wonder: Fractions to Decimals

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

$\frac12 = 0.5$

$\frac13 =0.3333333$

$\frac14 = 0.25$

$\frac15 = 0.2$

$\frac16 = 0.1666667$

$\frac17 = 0.142857143$

$\frac18 = 0.125$

$\frac19 = 0.1111111$

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

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

What do you notice? What do you wonder?

 

5.2: Repeating Decimals

  1. Use long division to express each fraction as a decimal.

    $\frac{9}{25}$

    $\frac{11}{30}$

    $\frac{4}{11}$

  2. What is similar about your answers to the previous question? What is different?
  3. Use the decimal representations to decide which of these fractions has the greatest value. Explain your reasoning.

5.3: More and Less with Decimals

  1. Match each diagram with a description and an equation.

    Diagrams:

    Descriptions:

    An increase by $\frac14$

    An increase by $\frac13$

    An increase by $\frac23$

    A decrease by $\frac15$

    A decrease by $\frac14$

    Equations:

    $y=1.\overline{6}x$

    $y=1.\overline{3}x$

    $y=0.75x$

    $y=0.4x$

    $y=1.25x$

  2. Draw a diagram for one of the unmatched equations.

5.4: 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.

  1. For each match you find, explain to your partner how you know it’s a match.
  2. For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking.
  3. 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.

Summary

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

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

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

\(\require{enclose} \begin{array}{r} 1.125 \\[-3pt] 8 \enclose{longdiv}{9.000}\kern-.2ex \\[-3pt] \underline{8{\phantom{.0}}} \phantom{00} \\[-3pt] 1\phantom{.}0\phantom{00} \\[-3pt] \underline{8\phantom{0}}\phantom{0} \\[-3pt] 20\phantom{0} \\[-3pt] \underline{16\phantom{0}} \\[-3pt] \phantom{0} 40 \\[-3pt] \underline{40} \\[-3pt] 0 \\ \end{array}\)

Sometimes it is easier to work with the decimal expansion 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 $1.2x$.

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

Practice Problems ▶

Glossary

repeating decimal

repeating decimal

A repeating decimal is an infinite decimal expansion that eventually repeats the same sequence of digits over and over again. The repeated sequence is indicated by a line above it.