This lesson explores different representations of rational numbers. Students begin by considering an image of three number lines, each showing a more precise location of 0.375.

Next, students rewrite rational numbers written as decimals, square roots, and cube roots as numbers in fraction form. They see that it is not the symbols used to write a number that make it rational, but rather the fact that it can be rewritten in the form , where  and  are integers and .

Reversing their thinking, students then find the decimal representation of fractions with finite decimal expansions.

In the last activity students use long division with repeated reasoning to find that  (MP8). Students plot this decimal expansion on a series of zooming number lines and observe an alternating pattern between the intervals that will continue forever.

This zooming number line representation supports students' understanding of place value and helps to form mental images of the two different ways a decimal expansion can represent a rational number—both as a terminating decimal when the decimal expansion eventually lands exactly on a tick mark and as a repeating decimal when the decimal expansion repeats forever but in a predictable way.

In the next lesson, students continue to work with rational numbers that have infinite decimal expansions and contrast the decimal representations of irrational numbers.