Category Archives: Precalculus

0.9 Repeating

I got to teach one of my favorite lessons in a Precalculus class this week, which I developed several years ago from a paper by Thomas Osler, Fun with 0.999…

We started with a Quick Poll. Students could select as many or as few choices as they wanted.

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I shared their responses separated

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and grouped together.

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In the first class, one student selected all three choices.

In the second class, 5 students selected all three choices.

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I set the timer for a few minutes and asked students to think individually about how they could argue their selection(s).

Then I asked them to talk together about their ideas.

I walked around and listened. These are the conversations I heard:

A: 1/3 is 0.3 repeating. 2/3 is 0.6 repeating. If we add 1/3 and 2/3, we get 1. If we add 0.3 repeating and 0.6 repeating, we get 0.9 repeating.

B: 1/9 is 0.1 repeating. If we multiply 1/9 by 1, we get 1. If we multiply 0.1 repeating by 9, we get 0.9 repeating.

C: 1/3 is 0.3 repeating. If we add 1/3 three times, we get 1. If we add 0.3 repeating three times, we get 0.9 repeating.

D: If x=0.9 repeating, then 10x=9.9 repeating. (It was clear that a few students had seen Vi Hart talk about 0.9 repeating. Even so, this was all they had for now.)

E: I think this is like Zeno’s Paradox. To walk across the room, you have to walk halfway, and halfway again, and halfway again.

This was the perfect opportunity to deliberately sequence the students’ thinking and let them make connections between their arguments (5 Practices style). With which conversation would you start?

We started with argument C. More than one person shook their head in disbelief, even though they agreed that the argument was convincing.

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Next we moved to argument A, which was very similar to argument C.

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Next we moved to argument B.

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I had a few suggestions of what to do, based on the article from the AMATYC Review. We went to one of those next that the students hadn’t thought of: If x=0.9 repeating, what happens when you divide the equation by 3?

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A student shared their work differently in each class, showing that x=1.

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We moved next to argument D. Again, students shared their thinking differently in each class.

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No one thought about Zeno’s Paradox in the first class. So I asked them how we could express 0.9 repeating as a sum.

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And then I sent a Quick Poll to collect their responses.

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In the second class, I asked the students with argument E to share their thoughts. They got at the infinite sum idea, so without decomposing 0.9 repeating as a class, I sent the Quick Poll. Lots of students came up with a sum that equaled 1. Only one of those was clearly 0.9+0.99 +0.999+…

(I didn’t show them the responses equal to 1 in green when I showed them their results.)

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So we practiced look for and make use of structure together. How can we decompose 0.9 repeating into a sum?

I sent the poll again.

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We concluded the lesson by polling the first question again. In the first class, 4 additional students believed only that 0.9 repeating = 1 at the end.

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In the second class the number of students selecting only choice A changed from 6 to 13.

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Our #AskDontTell journey continues, one lesson at a time …


Posted by on May 2, 2015 in Precalculus


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Odd Functions

The students in a class that meets in my room were working on a few problems at the beginning of their class period. I overhead one tell the other that an odd function goes through the origin.

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I suggested to the teacher that she might try sending a True/False question to start the class discussion for the day: An odd function must go through the origin.

And so she did.

And the results were the following. (Note that the teacher deselected “Show Correct Answer” before she displayed the results for the class.)

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There was a lot of talk about an odd function having a graph with symmetry about the origin. It took a long time for someone to find a counterexample to the statement. They were in the family of polynomials – and rightly so, the name of their unit was Polynomials. I kept waiting for someone to go back to their trigonometric functions.

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But they didn’t. Finally someone asked about f(x)=1/x. A hyperbola. That doesn’t go through the origin. But that is an odd function. How do we know? We can show that –f(-x)=f(x), and the graph is symmetric about the origin.

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What type of odd functions must go through the origin? Will every polynomial odd function go through the origin?

And a possible variation: How would the conversation have played out if the teacher had shown the correct answer and told students to determine an odd function that doesn’t go through the origin?

And so the journey continues, searching for questions that push students’ thinking and probe for misconceptions. And every once in a while, we find one that we need to share with others.



Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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Posted by on March 14, 2014 in Precalculus


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