Lesson 15
Infinite Decimal Expansions
Lesson Narrative
In this lesson, students further explore finding decimal expansions of rational numbers as well as irrational numbers. In the warmup, students find the decimal expansion of \(\frac37\), which starts to repeat as late as the seventh decimal place. However, once the first repeating digit shows up, repeated reasoning allows the students to stop the longdivision process (MP8). The discussion of the warmup is a good place to introduce students to the overline notation for repeating decimal expansions.
In the first classroom activity, students learn how to take a repeating decimal expansion and rewrite it in fraction form. The activity uses cards with the steps and explanations of the process and asks students to put these cards in order. Once they have the correct order, they use the same steps on different decimal expansions. While the numbers are different, the structure of the method is the same (MP7).
In the last activity of this lesson, and of this unit, students investigate how to approximate decimal expansions of irrational numbers. In an earlier lesson, students learned that \(\sqrt{2}\) cannot be written as a fraction and they estimated its location on the number line. Now they use “successive approximation,” a process of zooming in on the number line to find more and more digits of the decimal expansion of \(\sqrt{2}\). They also use given circumference and diameter values to find more precise approximations of \(\pi\), another irrational number students know. In contrast to the previous lesson, students see that there is no easy was to keep zooming in on these irrational numbers. They are not predictable like a repeating decimal. Because it is not possible to write out the complete decimal expansion of an irrational number we use symbols to name them. However, in practice we use approximations that are good enough for a given purpose.
Learning Goals
Teacher Facing
 Compare and contrast (orally) decimal expansions for rational and irrational numbers.
 Coordinate (orally and in writing) repeating decimal expansions and rational numbers that represent the same number.
Student Facing
Let’s think about infinite decimals.
Required Materials
Required Preparation
Prepare enough copies of the Some Numbers are Rational blackline master for each group of 2 to have a set of 6 cards.
Learning Targets
Student Facing
 I can write a repeating decimal as a fraction.
 I understand that every number has a decimal expansion.
CCSS Standards
Glossary Entries

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 . . .
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