In this activity, students review the idea that a rational number can be rewritten as a positive or negative fraction, and they rewrite rational numbers with finite decimal expansions in fraction form. Students will study rational numbers with infinite decimal forms in a following activity.

For the problems with roots, values were purposefully chosen to emphasize that not all numbers written using square root or cube root notation are irrational.

Monitor for students who write these different representations of 0.2 for the first problem:

The focus here is to see that there are many ways to express a rational number.

The goal of this discussion is to highlight how rational numbers can be represented in different ways. Begin by asking students what they noticed about the values used in the Task Statement (Some are positive and some are negative. Some are written as decimals and some written as square roots or cube roots.) Explain that while these numbers are all represented in different ways, they are all rational numbers because they can all be expressed as a positive or negative fraction.

Display answers for 0.2 from previously selected students for all to see.

Use Compare and Connect to help students compare, contrast, and connect the different approaches and representations. Draw a number line with the numbers 0, 1, and 2 with plenty of space between the integers for all to see. Here are some questions for discussion:

• “How do we plot ? on this number line?” (Subdivide the segment from 0 to 1 into 5 equal pieces and  is at the first mark after 0.) Then label the point .
• “How can we see that this is the same as ?” (Subdivide each fifth into two equal pieces—now each piece is .) Then label the point  and 0.2.
• “How do these different representations show the same information?” (One representation is partitioned into twice as many parts, and we count over twice as many of them.)

Emphasize the idea that any rational number can be plotted on the number line this way when written as a fraction —by subdividing the interval into  parts and counting over  of them.

In this activity, students rewrite fractions with finite decimal expansions as decimals. This activity deepens students’ understanding of what rational numbers are and how different representations highlight different features of a rational number.

The goal of this discussion is to highlight that all rational numbers have a decimal representation in addition to a fractional representation.

Begin by inviting students to share how they rewrote  as a decimal representation (0.375). Demonstrate the steps for long division, or have a student demonstrate the steps for all to see

Ask students how they would go about plotting the fraction  on a number line. (Divide the interval between 0 and 1 into 8 equal parts and count 3 parts to the right from 0.

Then ask students how they would go about plotting the decimal 0.375 on a number line. Explain that while 0.375 can be plotted at the same location as , the same strategy does not make sense when plotting a number represented as a decimal since there is no denominator. In a decimal such as 0.375, the digits are referring to place value instead of equal parts.

Then display the number lines from the Warm-up.

Tell students that the three points on each number line all represent the same number: 0.375. Explain how the first number line shows the value of the number between 0.3 and 0.4, but closer to 0.4. In order to get a more precise idea of the number’s location, the second number line is “zoomed in” to the interval between 0.3 and 0.4, and now we can tell that the number is between 0.37 and 0.38. By zooming in one more time, we see that the number lands exactly on one of the tick marks: 0.375.

If time allows, repeat this explanation with .

Conclude the discussion by telling students that all rational numbers have a decimal representation. Rational numbers that eventually come to fall exactly on one of the tick marks—such as 0.375 did on the third number line—have a finite decimal expansion and are sometimes called a "terminating decimal." Eventually, all the digits after a certain number are just 0, so typically they are not written. For example, we can write  as 0.375, 0.3750, or 0.3750000000.

Students will investigate repeating decimals in a following lesson, so there is no need to address them here.

In this activity, students rewrite a fraction with an infinite decimal expansion as a decimal. Students use repeated reasoning with division to justify to themselves that 0.1818 . . . repeats the digits 1 and 8 forever (MP8). Students learn that this is an example of a repeating decimal because it has digits that keep going in the same pattern over and over.

In later activities, students will deepen their understanding of rational numbers with infinite decimal expansions by learning how to rewrite numbers in that form as fractions

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 . Invite one or two students to share their number lines or display this image for all to see.

Ask students to share things they notice and wonder (I notice that the point for keeps jumping from left to right. I notice that the pattern seems to be repeating itself. I wonder if this pattern will continue forever.) Now ask students to share what they think the decimal representation of  is, and record responses for all to see.

Tell students that sometimes the decimal representation of a rational number repeats like this forever. Unlike 0.375 where the decimal expansion eventually falls exactly on a tick mark since all the digits after the 5 are 0, the decimal expansion of  will never land exactly on a tick mark when we zoom in by powers of 10, but will continue this pattern forever. Fractions like have an infinite decimal expansion and are sometimes called repeating decimals.

Tell students that there is a special notation to represent repeating decimals, and display the notation for all to see:

Emphasize that even though  goes on forever, it is still a rational number because it repeats and is the decimal representation of the fraction .