Illustrative Mathematics Grade 8, Unit 8, Lesson 14: Decimal Representations of Rational Numbers

Learning Targets:

  • I can write a fraction as a repeating decimal.
  • I understand that every number has a decimal expansion.

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Illustrative Math
Grade 8

Lesson 14: Decimal Representations of Rational Numbers

Let’s learn more about how rational numbers can be represented.

Illustrative Math Unit 8.8, Lesson 14 (printable worksheets)

Lesson 14 Summary

The following diagram gives some examples of decimal representations of rational numbers.
Decimal Representations of Rational Numbers

Lesson 14.1 Notice and Wonder: Shaded Bars

What do you notice? What do you wonder?

Lesson 14.2 Halving the Length

Here is a number line from 0 to 1.

  1. Mark the midpoint between 0 and 1. What is the decimal representation of that number?
  2. Mark the midpoint between 0 and the newest point. What is the decimal representation of that number?
  3. 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?

Lesson 14.3 Recalculating Rational Numbers

  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 a/b or -a/b.
  2. All rational numbers have decimal representations, too. Find the decimal representation of each of these rational numbers.

Lesson 14.4 Zooming In On 2/11

  1. On the topmost number line, label the tick marks. Next, find the first decimal place of 2/11 using long division and estimate where 2/11 should be placed on the top number line.
  2. Label the tick marks of the second number line. Find the next decimal place of 2/11 by continuing the long division and estimate where 2/11 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 2/11.
  3. Repeat the earlier step for the remaining number lines.
  4. What do you think the decimal expansion of 2/11 is?

Are you ready for more?

Let x = 25/11 = 2.272727… and y = 58/33 = 1.75757575…
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.

  • Which of x or y is closer to 2?
  • Find x2.
    • Show Answers

      The decimal representation is better when deciding which of x or y is closer to 2.
      y is closer to 2.

      The fraction representation when finding x2.
      x2 = 252/112 = 625/121

Lesson 14 Practice Problems

  1. Andre and Jada are discussing how to write 17/20 as a decimal. Andre says he can use long division to divide 17 by 20 to get the decimal. Jada says she can write an equivalent fraction with a denominator of 100 by multiplying by 5/5, then writing the number of hundredths as a decimal.
    a. Do both of these strategies work?
    b. Which strategy do you prefer? Explain your reasoning.
    c. Write 17/20 as a decimal. Explain or show your reasoning.
  2. Write each fraction as a decimal.
  3. Write each decimal as a fraction.
  4. Find the positive solution to each equation. If the solution is irrational, write the solution using square root or cube root notation.
  5. Here is a right square pyramid.
    a. What is the measurement of the slant height l of the triangular face of the pyramid? If you get stuck, use a cross section of the pyramid.
    b. What is the surface area of the pyramid?

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

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