Lesson 11

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 warm-up 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. \(\sqrt{\frac{1}{9}} = \frac{1}{3}\).

In the last activity students explore the decimal expansion of \(\frac{2}{11}\). They use long division with repeated reasoning (MP8) to find that \(\frac{2}{11}=0.1818\dots\). Students realize that they could easily keep zooming in on \(\frac{2}{11}\) because of the pattern of alternating between the intervals \(\frac{1}{10}\)–\(\frac{2}{10}\) and \(\frac{8}{10}\)–\(\frac{9}{10}\) 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 non-terminating decimal representations that do not repeat. This leads into the next lesson.