Lesson 11

The purpose of this warm-up is for students to review what rational numbers are—fractions and their opposites. In earlier lessons, students explored square and cube roots and now return to strictly rational numbers. While students do not need to be able to recite a definition for rational numbers, this type of diagram is one students are likely familiar with and gives students an opportunity to use related language such as part, whole, halves, 1 out of 8 parts, one eighth, etc.  After students work with different representations of rational numbers in upcoming lessons, they will expand their definition and understanding of irrational numbers. 

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. 

The purpose of this optional activity is for students to see what the figure in the warm up didn’t explore: as 1 is successively divided in half, the number of digits needed to accurately describe the resulting value increases. Students likely have memorized that a half is 0.5 and a fourth is 0.25, but what is half of 0.25? How can you calculate it? Use this activity if you think your students need a reminder about place value or long division with decimals.

Do not provide access to calculators. Give students 2 minutes quiet work time followed by partner then whole-class discussion.

Discuss:

• “How can you use long division to answer the third problem?”
• “What are these values when written in fraction notation? How do you know?”

The purpose of this task is for students to rewrite rational numbers with terminating decimal expansions in fraction form and fractions with terminating decimal expansions as decimals. This activity is the first of a series of three in which students rewrite numbers in different ways, supporting their understanding of what rational and irrational numbers are and how they can be represented.

Monitor for students who write \(\frac15\) and \(\frac{2}{10}\) for 0.2.

Arrange students in groups of 2. Do not provide access to calculators. 

Remind students that all rational numbers are fractions and their opposites, and they can all be written in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers (with \(b\neq 0\)). In fact, there are many equivalent fractions that represent a single rational number. For example, 5 is equivalent to \(\frac{10}{2}\) and \(\frac{15}{3}\). 

Students should complete the problems individually, then compare with their partners and come to resolutions over any differences. Repeat this process for the second problem.

Some students may write that \(0.333=\frac13\). If so, ask them to check their work by calculating the decimal representation of \(\frac13\).

The purpose of this discussion is to highlight different strategies for rewriting rational numbers in different forms. Select previously identified students to share their solutions, including one student who wrote \(\frac15\) for 0.2 and another who wrote \(\frac{2}{10}\). For the problems with roots, the values were purposefully chosen to emphasize to students that just because a number is written with a square or cube root does not mean it is not rational.

For 0.2, draw a number line with the numbers 0, 1, and 2 with plenty of space between the integers. Subdivide the segment from 0 to 1 into 5 equal pieces, and then ask where to plot \(\frac15\). Then label \(\frac15\) and ask how we can see that this is \(\frac{2}{10}\). (Subdivide each fifth into two equal pieces—now each one is \(\frac{1}{10}\).) Then label the point \(\frac{2}{10}\) and 0.2.

Remind students that the point corresponds to a rational number, and we have a lot of different ways we can represent that number.

This activity continues with the work from the previous activity by examining a decimal representation of a rational number that, when written as a decimal, repeats forever. The purpose of this activity is for students to use repeated reasoning with division to justify to themselves that \(0.1818 . . .\) repeats the digits 1 and 8 forever (MP8).

Arrange students in groups of 2. Do not provide access to calculators. Tell students that now we are going to think about the decimal representation of \(\frac{2}{11}\). Remind students that they should be prepared to explain their reasoning for each step in the activity.

If you think students need a reminder of how the zooming number lines work, which were used in an earlier unit, demonstrate how to show where \(\frac{1}{8}\) is using 3 number lines, starting with one from 0 to 1.

Students in groups of 2. Give 2 minutes for students to begin individually and then ask students to discuss their work with their partner and resolve any differences. Follow with a whole-class discussion.

The purpose of this discussion is to explicitly state the repeated reasoning and successive approximation used to calculate each digit after the decimal point for \(\frac{2}{11}\). Select one or two students to share their reasoning about the decimal representation of \(\frac{2}{11}\).

Tell students that we often find that the decimal representation of a rational number repeats like this, and we have a special notation to represent it. Then write

\(\displaystyle 0.18181818181818181818 . . . = 0.\overline{18}\)