In the last two lessons in this unit, students explore decimal representations of rational and irrational numbers. The zooming number line representation used in these lessons supports students' understanding of place value and helps them form mental images of the two different ways a decimal expansion may go on forever (depending on whether the number is rational or irrational).
This first lesson explores the different forms of rational numbers. The warmup reviews the idea of rational numbers as fractions of the form \(\frac{a}{b}\) using tape diagrams. The first classroom activity, which is optional, continues with the same fractions by writing them as decimals.
In the second classroom activity students work with a variety of rational numbers written in different forms, including fractions, decimals and square roots. They see that it is not the symbols used to write a number that makes it rational but rather the fact that it can be rewritten in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers, e.g. \(\sqrt\frac19=\frac13\).
In the last activity students explore the decimal expansion of \(\frac2{11}\). They use long division with repeated reasoning (MP8) to find that \(\frac2{11}=0.1818….\). Students realize that they could easily keep zooming in on \(\frac2{11}\) because of the pattern of alternating between the intervals \(\frac1{10}\frac2{10}\) and \(\frac8{10}\frac9{10}\) of the previous line. The goal is for students to notice and appreciate the predictability of repeating decimals and see how that connects with the \(\frac{a}{b}\) structure.
By the end of this lesson students have seen that rational numbers can have decimal representations that terminate or that eventually repeat. This begs the question if there are numbers with nonterminating decimal representations that do not repeat. This leads into the next lesson.
Lesson overview
 14.1 Warmup: Notice and Wonder: Shaded Bars (5 minutes)
 14.2 Optional Activity: Halving the Length (10 minutes)
 14.3 Activity: Recalculating Rational Numbers (20 minutes)

14.4 Activity: Zooming In On \(\frac2{11}\) (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 14.5 Cooldown: An Unknown Rational Number (5 minutes)
Learning goals:
 Comprehend that a rational number is a fraction or its opposite, and that a rational number can be represented with a decimal expansion that “repeats” or “terminates”.
 Represent rational numbers as equivalent decimals and fractions, and explain (orally) the solution method.
Learning goals (student facing):
 Let’s learn more about how rational numbers can be represented.
Learning targets (student facing):
 I can write a fraction as a repeating decimal.
 I understand that every number has a decimal expansion.
Glossary:
 repeating decimal  A repeating decimal has digits that keep going in the same pattern over and over. The repeating digits are marked with a line above them. For example, the decimal representation for \(\frac13\)is \(0.\overline{3}\), which means 0.3333333 . . . The decimal representation for \(\frac{25}{22}\) is \(1.1\overline{36}\) which means 1.136363636 . . .
 Access the complete Grade 8 glossary.
Standards:
 This lesson builds on the standard: CCSS.7.NS.A.2.dMS.7.NS.2d
 This lesson builds towards the standard: CCSS.8.NS.A MS.8.NS
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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