Lesson 15

The purpose of this warm-up is to give students practice rewriting numbers in different forms. Students have re-written rational numbers with terminating decimals in fraction form and the reverse, as well as calculated the decimal form of of \(\frac{2}{11}\), which has an infinitely repeating two-digit pattern. Students are only asked to get to the first digit that repeats, which for \(\frac37\) is the 7th digit after the decimal point.

Since students are expected to notice a repeating pattern, they should not use a calculator for this activity.

Remind students of the previous activity where the decimal expansion of \(\frac{2}{11}\) was shown to be \(0.1818 . . .\) using long division and repeated reasoning.

Students in groups of 2. 1 minute of quiet work time followed by partner and whole-class discussion. 

Ask students what the next 4 digits are, and record them on the calculation for all to see. Ask students, “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. Repeat this process until reaching 4 again. Point out that this cycle will continue indefinitely, because we can completely predict what will happen at each step because it is exactly like what happened 7 steps ago.

Define a repeating decimal and show students overline notation again. State that all rational numbers have a decimal expansion that eventually repeats. Sometimes it eventually repeats 0's like \(\frac12 = 0.5000000 . . .\), and we call this terminating decimal. 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.

The purpose of this activity is for students to learn and practice a strategy for rewriting rational numbers with decimal representations that repeat eventually into their fraction representations. Students begin by arranging cards in order that show how the strategy was used to show \(0.4\overline{85}=\frac{481}{990}\). Next, students use the strategy to calculate the fraction representations of two other values.

The extension problem asks students to repeat the task for decimal expansions with a single repeating digit, such as \(0.\overline{3}\) and \(0.\overline{9}\).  While simpler to process algebraically, it may be counter-intuitive for students to conclude that \(0.\overline{9}=1\). The fact that one number can have two different decimal expansions is often surprising, and hints at the fact that the number line has some quite subtle aspects to it.  Students might insist that \(0.\overline{9}\) must be strictly less than 1, which can invite an interesting discussion as to the nature of infinite decimal expansions.  If such a discussion arises, invite students to try to make their argument precise, and to try to explain where they feel there is a flaw in their argument that \(0.\overline{9}=1\).  Discussion points to help resolve lingering dissonance include:

• Asking whether any numbers on the number line could be between \(0.\overline{9}\) and \(1\).
• Whether expressions like \(0.\overline{9}5\) make any sense.  (They do not, since the notation \(0.\overline{9}\) notation means the pattern continues forever.)
• A second argument for those who accept that \(\frac{1}{3}=0.\overline{3}\) is to multiply both sides by 3.

Demonstrate the algorithm with an example such as converting \(0.\overline{12} = \frac{12}{99}\).

Arrange students in groups of 2. Do not provide access to calculators. Distribute a set of the slips cut out from the blackline master to each group. Tell students that once they have the cards arranged, they should work on the second problem individually and then compare their work with their partner. Finish with a whole class discussion.

The goal of this discussion is to help students build a more robust understanding of how the strategy works. 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 \(0.7\overline{88}\), we can multiply by 100, but multiplying by 10 also works since the part that repeats is only 1 digit long.

End the discussion by asking students to rewrite \(0.3\overline{0}\) using this strategy. This is like using a sledgehammer for a nail, but it works and is reflective of the work they did in an earlier activity.

Students first encountered irrational numbers at the start of this unit as a way to denote the side lengths of squares. They also spent time attempting to find a number of the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers, that is equal to \(\sqrt{2}\) only to have it revealed that \(\sqrt{2}\) is irrational. Now that students have recently done a lot of thinking about decimal representations of rational numbers and how to convert the infinitely repeating decimal representation of a number into a fraction representation, we revisit irrationals to further the point that no such fraction representation exists for these numbers. For many irrationals, long division doesn't work as a tool to calculate a decimal approximation because there are no two integers to divide. It can be said that a square of area 2 has a side length of \(\sqrt{2}\) or that \(\sqrt{2}\) is a number that, when squared, has a value of 2 (with \(\text-\sqrt{2}\) being the other number). All this means is that different methods to approximate the value need to be discussed.

In this activity, students will approximate the value of \(\sqrt{2}\) using successive approximation and discuss how they might figure out a value for \(\pi\) using measurements of circles. Students will also plot these values on number lines accurately to the thousandth place to reinforce the idea that irrational numbers are numbers. Therefore, irrational numbers have a place on the number line even if they cannot be written as a fraction of integers.

Calculators are okay, but students should not use the root button or the \(\pi\) button. Before beginning, remind students that we have previously found decimal representations for fractions, and that knowing these representations made it easier to plot numbers on a number line. Today we are going to do the same thing with irrational numbers. Ask students “Earlier, we used long division to find decimal representations of numbers. Why can't we do that today with irrational numbers?” (Irrational numbers are ones that cannot be written as fractions.)

Follow work time with a whole-class discussion.

The purpose of this discussion is to deepen students understanding that irrational numbers are not fractions. As such, their values are approximated using different methods than for rational numbers. Begin the discussion by asking:

• “How long do you think you could keep using the method in the first problem to find more digits of the decimal representation of \(\sqrt{2}\)?” (My calculator only shows 9 digits to the left 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 the square of the root is between and then start approximating the value of the root from there.)

Tell students that what they did in the first problem is a strategy called “successive approximation.” It takes time, but successive approximation works for finding more and more precise approximations of irrational numbers so long as you have a clear value to check against. In the case of \(\sqrt{2}\), since \(\sqrt{2}^2=2\), there was a clear value to test approximations against whether they are too high or too low.

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