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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Connecting Factors and Zeros

NCTM’s Principles to Actions suggests Mathematics Teaching Practices for teachers. Two of those are the following.

MTP 1 Establish mathematics goals to focus learning

MTP 6 Build procedural fluency from conceptual understanding

If the goal for students is to use the factors of a quadratic function to determine its zeros, what concepts must students understand to meet that learning goal?

Our team wrote this leveled learning progression for our lesson.

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros.

Level 2: I can expand the product of two binomials.

Level 1: I can solve an equation in one variable.

Level 1: I can determine the zero(s) of a function from the graph of a function.

We decided to first ensure that students know what a zero is, and we checked this is more than one way on the opener for the day. (See this source for similar Level 1 problems.)

Students had to place a point at the zero of the function.

Almost all students were able to note that the point of interest is where the graph intersects the x-axis.

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Students had to name the coordinates of the zero of the function, which about half could do.

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And then students had to answer a question about a zero in context. A few more than half could do this.

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We decided that students also need to be able to solve an equation in one variable.

Which they could easily do.

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And we also decided that if students are going to meet the learning goal, they are also going to have to be able to multiply binomials. Which you can tell from the results that they could not easily do (Q8 and Q9).

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In the lesson, we started with the zeros of a linear function.

What do you notice?

If I give you a similar equation, can you tell me the zero?

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What do you notice on this page?

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If I give you a similar equation, can you tell me the zero?

We checked in with students using some Quick Polls.

What do you notice about the answers for this first poll?

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(I noticed that not all are x-intercepts.)

Students showed some improvement as we continued.

The answers are all x-intercepts.

We asked questions like …

How can we tell that (-6,0) is the correct choice using the equation?

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We spent a long time on linear functions. Some might think we spent too long.

Then we looked at a quadratic function.

And we related the linear factors to the quadratic visually.

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This is part of a Math Nspired activity called Zeros of a Quadratic Function, where there is a lot more flexibility in changing the factors.

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Our leveled learning progression for the second lesson changed a little:

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros, and I can use the zeros of a quadratic function to determine its factors.

Level 2: I can rewrite a quadratic function given in factored form to standard form.

Level 1: I can determine the zero(s) of a quadratic function from the graph of a function.

When we checked for student understanding during the opener of the second lesson, we saw that students were able to determine the zero(s) of a quadratic function from the graph of a function.

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Lots of students were at Level 1, determine the zeros when given the graph and the equation.

Not as many were at the target – but definitely more than had reached it the day before.

We have worked to build procedural knowledge from conceptual knowledge in our unit on Zeros and Factors. Our standards say that we want students to “Factor a quadratic expression to reveal the zeros of the function it defines”. The standards don’t say that we want students to factor a quadratic expression just for the sake of factoring.

What opportunities are you providing your students to concentrate on relationships rather than just results?


Posted by on April 27, 2015 in Algebra 1


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The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

  1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

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It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.


Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?


What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

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So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

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And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

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Here are the results of the questions that they worked.

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What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

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And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

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How can we graph the circle, limited to functions?

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How can we tell which points are correct?

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I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

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54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

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I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

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We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

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And so the journey continues, one #AskDontTell learning episode at a time.


Posted by on April 19, 2015 in Circles, Coordinate Geometry, Geometry


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SMP6: Attend to Precision #LL2LU

We want every learner in our care to be able to say

I can attend to precision.


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But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?


Level 4:
I can distinguish between necessary and sufficient conditions for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.


How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).

He had written


And then he wrote:

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He absolutely knows what he means: y=-4+3(x-2).

But that’s not what he wrote.


Which student responses show attention to precision for the domain and range of y=(x-3)2+4? Are there others that you and your students would accept?

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How often do our students notice that we model attend to precision? How often to our students notice when we don’t model attend to precision?

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Attend to precision isn’t just about numerical precision. Attend to precision is also about the language that we use to communicate mathematically: the distance between a point and a line isn’t just “straight” – it’s the length of the segment that is perpendicular from the point to the line. (How many times have you told your Euclidean geometry students “all lines are straight”?)

But it’s also about learning to communicate mathematically together – and not just expecting students to read and record the correct vocabulary from a textbook.

[Cross posted on Experiments in Learning by Doing]


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Classifying Triangles

We look specifically at 45-45-90 triangles on the first day of our Right Triangles unit. I’ve already written specifically about what the 45-45-90 exploration looked like, but I wanted to note a conversation that we had before that exploration.

Jill and I had recently talked about introducing new learning by drawing on what students already know. I’ve always started 45-45-90 triangles by having students think about what they already know about these triangles (even though many have never called them 45-45-90 triangles before). After hearing about one of Jill’s classes, though, I started by asking students to make a column for triangles, right triangles, and equilateral triangles, noting what they know to always be true for each. This short exercise gave students the opportunity to attend to precision with their vocabulary.

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It occurred to me while we were talking that having students draw a Venn Diagram to organize triangles, right triangles, and equilateral triangles might be an interesting exercise. How would you draw a Venn Diagram to show the relationship between triangles, right triangles, and equilateral triangles?

In my seconds of anticipating student responses, I expected one visual but got something very different.

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What does it mean for an object to be in the intersection of two sets? Or the intersection of three sets? Or in the part of the set that doesn’t intersect with the other sets?

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Then we thought specifically about 45-45-90 triangles. What do you already know? Students practiced look for and make use of structure.

One student suggested that the legs are half the length of the hypotenuse. Instead of saying that wouldn’t work or not writing it on our list, I added it to the list and then later asked what would be the hypotenuse for a triangle with legs that are 5.


I wrote 10 on the hypotenuse and waited.

But that’s not a triangle?


5-5-10 doesn’t make a triangle.

Why not?

It would collapse (students have a visual image for a triangle collapsing from our previous work on the Triangle Inequality Theorem).

Does the Pythagorean Theorem work for 5-5-10?

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Students reflected the triangles about the legs and hypotenuse to compose the 45-45-90 triangle into squares and rectangles.

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And they constructed an altitude to the hypotenuse to decompose the 45-45-90 triangle into more 45-45-90 triangles.

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And then we focused on the relationship between the legs and the hypotenuse using the Math Nspired activity Special Right Triangles.

And so the journey continues … listening to and learning alongside my students.


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Special Right Triangles: 45-45-90

I gave my students our learning progression for SMP 8 a few weeks ago as we started a unit on Right Triangles and had a lesson specifically on 45-45-90 Special Right Triangles.


The Geometry Nspired Activity Special Right Triangles contains an Action-Consequence document that focuses students attention on what changes and what stays the same. The big idea is this: students take some kind of action on an object (like grabbing and dragging a point or a graph). Then they pay attention to what happens. What changes? What stays the same? Through reflection and conversation, students make connections between multiple representations of the mathematics to make sense of the mathematics.

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Students start with what they know – the Pythagorean Theorem.

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Looking at the side lengths in a chart helps students notice and note what changes and what stays the same:

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The legs of the triangle are always the same length.

As the legs increase, the hypotenuse increases.

The hypotenuse is always the longest side.


Students begin to identify and describe patterns and regularities:

All of the hypotenuses have √2.

The ratio of the hypotenuse to the leg is √2.


Students practice look for and express regularity in repeated reasoning as they generalize what is true:

To get from the leg to the hypotenuse, multiply by √2.

To get from the hypotenuse to the leg, divide by √2.

hypotenuse = leg * √2

Teachers and students have to be careful with look for and express regularity in repeated reasoning. Are we providing students an opportunity to work with diagrams and measurements that make us attend to precision as we express the regularity in repeated reasoning that we notice?

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In a Math Practice journal, Kaci writes about “look for regularity in repeated reasoning”. We figured out that half of a square is a 45-45-90 triangle, and students were trying to determine the other two sides of the triangle given one side length of the triangle. She says “To find the length of the hypotenuse, you take the length of a side and multiply by √2. The √2 will always be in the hypotenuse even though it may not be seen like √2. In her examples, the triangle to the left has √2 shown in the hypotenuse, but the triangle to the right has √2 in the answer even though it isn’t shown, since 3√2√2 is not in lowest form. She says, “I looked for regularity in repeated reasoning and found an interesting answer.”

What opportunities can we provide our students this week to look for and express regularity in repeated reasoning and find out something interesting?


Posted by on April 7, 2015 in Uncategorized


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Visual: SMP-8 Look for and Express Regularity in Repeated Reasoning #LL2LU

Many students would struggle much less in school if, before we presented new material for them to learn, we took the time to help them acquire background knowledge and skills that will help them learn. (Jackson, 18 pag.)

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.


But…what if I can’t? What if I have no idea what to look for, notice, take note of, or attempt to generalize?

Investing time in teaching students how to learn is never wasted; in doing so, you deepen their understanding of the upcoming content and better equip them for future success. (Jackson, 19 pag.)

Are we teaching for a solution, or are we teaching strategy to express patterns? What if we facilitate experiences where both are considered essential to learn?

We want more students to experience the burst of energy that comes from asking questions that lead to making new connections, feel a greater sense of urgency to seek answers to questions on their own, and reap the satisfaction of actually understanding more deeply the subject matter as a result of the questions they asked.  (Rothstein and Santana, 151 pag.)

What if we collaboratively plan questions that guide learners to think, notice, and question for themselves?

What do you notice? What changes? What stays the same?

Indeed, sharing high-quality questions may be the most significant thing we can do to improve the quality of student learning. (Wiliam, 104 pag.)

How might we design for, expect, and offer feedback on procedural fluency and conceptual understanding?

Level 4
I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3
I can look for and express regularity in repeated reasoning.

Level 2
I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1
I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

If we are to harness the power of feedback to increase student learning, then we need to ensure that feedback causes a cognitive rather than an emotional reaction—in other words, feedback should cause thinking. It should be focused; it should relate to the learning goals that have been shared with the students; and it should be more work for the recipient than the donor. (Wiliam, 130 pag.)

[Cross posted on Experiments in Learning by Doing]

Jackson, Robyn R. (2010-07-27). How to Support Struggling Students (Mastering the Principles of Great Teaching series) (Pages 18-19). Association for Supervision & Curriculum Development. Kindle Edition.

Rothstein, Dan, and Luz Santana. Make Just One Change: Teach Students to Ask Their Own Questions. Cambridge, MA: Harvard Education, 2011. Print.

Wiliam, Dylan (2011-05-01). Embedded Formative Assessment (Kindle Locations 2679-2681). Ingram Distribution. Kindle Edition.


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