standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Activity
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.
Launch
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.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner, detailing the steps they took to calculate the fraction representation of the repeating decimals. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “Next I . . .”, and “Finally, in order to solve, I _____ because . . . .”
Supports accessibility for: Language; Social-emotional skills
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to revise and refine their response to “Use Noah’s method to calculate the fraction representation of 0.1
\(\overline{86}\)
and 0.7
\(\overline{88}\)
”. Ask each student to meet with 2–3 other partners for feedback. Display prompts for feedback that will help students strengthen their ideas and clarify their language. For example, “Can you explain how...?” and “You should expand on....”, “Can you say that another way?”, etc. Give students 1–2 minutes to revise their writing based on the feedback they received.​
Design Principle(s): Optimize output (for generalization)
Student Facing
Your teacher will give your group a set of cards. Each card will have a calculations side and an explanation side.
The cards show Noah’s work calculating the fraction representation of
\(0.4\overline{85}\)
. Arrange these in order to see how he figured out that
\(0.4\overline{85} = \frac{481}{990}\)
without needing a calculator.
Use Noah’s method to calculate the fraction representation of:
\(0.1\overline{86}\)
\(0.7\overline{88}\)
Student Response
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Student Facing
Are you ready for more?
Use this technique to find fractional representations for
\(0.\overline{3}\)
and
\(0.\overline{9}\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
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.


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