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.

I shared their responses separated

and grouped together.

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

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

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.

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

Next we moved to argument B.

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?

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

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

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.

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

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.)

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.

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.

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

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