# Lesson 11

Decimal Representations of Rational Numbers

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

• 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.

### Student Facing

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

Building On

Building Towards

### Glossary Entries

• cube root

The cube root of a number $$n$$ is the number whose cube is $$n$$. It is also the edge length of a cube with a volume of $$n$$. We write the cube root of $$n$$ as $$\sqrt[3]{n}$$.

For example, the cube root of 64, written as $$\sqrt[3]{64}$$, is 4 because $$4^3$$ is 64. $$\sqrt[3]{64}$$ is also the edge length of a cube that has a volume of 64.

• 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 . . .