Lesson 14Decimal Representations of Rational Numbers
Learning Goal
Let’s learn more about how rational numbers can be represented.
Learning Targets
I can write a fraction as a repeating decimal.
I understand that every number has a decimal expansion.
Lesson Terms
- repeating decimal
Warm Up: Notice and Wonder: Shaded Bars
Problem 1
What do you notice? What do you wonder?
Activity 1: Halving the Length
Problem 1
Here is a number line from 0 to 1.
Mark the midpoint between 0 and 1. What is the decimal representation of that number?
Mark the midpoint between 0 and the newest point. What is the decimal representation of that number?
Repeat step two. How did you find the value of this number?
Describe how the value of the midpoints you have added to the number line keep changing as you find more. How do the decimal representations change?
Activity 2: Recalculating Rational Numbers
Problem 1
Rational numbers are fractions and their opposites. All of these numbers are rational numbers. Show that they are rational by writing them in the form
0.2
0.333
-1.000001
Problem 2
All rational numbers have decimal representations, too. Find the decimal representation of each of these rational numbers.
Activity 3: Zooming In On
Problem 1
On the topmost number line, label the tick marks. Next, find the first decimal place of
using long division and estimate where should be placed on the top number line. Label the tick marks of the second number line. Find the next decimal place of
by continuing the long division and estimate where should be placed on the second number line. Add arrows from the second to the third number line to zoom in on the location of . Repeat the earlier step for the remaining number lines.
What do you think the decimal expansion of
is?
Are you ready for more?
Problem 1
Let
For each of the following questions, first decide whether the fraction or decimal representations of the numbers are more helpful to answer the question, and then find the answer.
Lesson Summary
We learned earlier that rational numbers are a fraction or the opposite of a fraction. For example,
Rational numbers can also be written using decimal notation. Some have finite decimal expansions, like 0.75, -2.5, or -0.5. Other rational numbers have infinite decimal expansions, like 0.7434343 … where the 43s repeat forever. To avoid writing the repeating part over and over, we use the notation
A decimal expansion of a number helps us plot it accurately on a number line divided into tenths. For example,