In this activity, students practice using long division to calculate the decimal form of a rational number. Students are expected to notice a repeating pattern (MP8).

The goal of this discussion is to make sure students understand that all rational numbers have a decimal expansion that eventually repeats. Ask students to share the next 4 digits and record them on the long division calculation for all to see. Discuss:

• “Without calculating, what number do you think will be next? Why?” (I think 2 will be the next digit because I can see the starting pattern has begun again.)

Continue the calculation and verify that 2 comes next and continue until reaching 4 again. Point out that this cycle will continue indefinitely—we can predict what will happen at each step because it is exactly like what happened 6 steps ago.

Tell students that all rational numbers have a decimal expansion that eventually repeats. Sometimes they eventually repeat 0s, like in . Sometimes they repeat several digits like in . If necessary, remind students that in overline notation, the line goes over the digits that repeat.

Be careful in the use of the word “pattern,” as it can be ambiguous. For example, there is a pattern to the digits of the number 0.12112111211112 . . . , but the number is not rational.

In this activity, students learn and practice a strategy for rewriting repeating decimals as a fraction. Students begin by arranging cards in an order that shows how a strategy is used to show . Next, students use the strategy to calculate the fractional representations of two other values.

The extension problem asks students to repeat the task for decimal expansions of and . It may be counter-intuitive for students to conclude that . Students might insist that must be strictly less than 1, which can prompt an interesting discussion. Invite students to make their argument more precise.

In this activity, students revisit irrational numbers to emphasize the point that no fractional representation exists for these numbers. For all irrational numbers, long division doesn't work because there are no two integers to divide, and different methods to approximate the value need to be discussed.

Students will approximate the value of using successive approximations and discuss how they might figure out a value for using measurements of circles. They will also plot these values on number lines to reinforce the idea that irrational numbers are still numbers—they have a place on the number line even if they cannot be written as a positive or negative fraction.