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.

The goal of this discussion is to reinforce student understanding of the strategy introduced to rewrite a repeating decimal as a fraction. Begin the discussion by selecting 2–3 students to share their work for the second problem, displaying each step for all to see. Ask if anyone completed the problem in a different way and, if so, have those students also share.

If no students notice it, point out that when rewriting , we can multiply by 100, but multiplying by 10 also works since the part that repeats is only 1 digit long.

Conclude the discussion by asking students to rewrite  using this strategy.

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.

The purpose of this discussion is to deepen students' understanding that irrational numbers cannot be written as fractions, but they are still numbers with a location on the number line. As such, their values are approximated using different methods than for rational numbers. Discuss with students:

• “How long do you think you could keep using the method in the first problem to find more digits of the decimal representation of ?” (My calculator only shows 9 digits to the right of the decimal, so that's as far as I could go.)
• “Would this method work for any root?” (Yes, I could find the two perfect squares that the square of the number we are taking the square root of is between and then keep approximating the value of the root from there.)

Tell students that the strategy they used in the first problem is called “successive approximation.” It takes time, but successive approximation works for finding more and more precise approximations of irrational numbers, as long as there is a clear value to them check against. In the case of , since , there is a clear value to test whether approximations are too high or too low.

Lastly, select 2–3 students to share their reasoning to the last part of the second problem. Make sure students understand that since measuring has limitations of accuracy, any calculation of using measurement, such as with the beaker and the fuel tank, will have limited accuracy.