Tag Archives: construct a viable argument and critique the reasoning of others

Squares on a Coordinate Grid

I was excited to find a new Illustrative Mathematics task using coordinate geometry.

CCSS-M G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

In the picture below a square is outlined whose vertices lie on the coordinate grid points:

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The area of this particular square is 16 square units. For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.

As a precursor to the task, I included the following statements on the bell work for students to discuss in their groups before we had a brief class discussion.

The three sides of a right triangle can all be even.

The three sides of a right triangle can all be odd.

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Last year, I heard Linda Griffith talk about giving a part of this task to some students in Arkansas. I began with my students the way she began with hers. Each student had a sheet of graph centimeter graph paper and a straightedge. Near the top left corner, draw a square with an area of 1 square centimeter.

Challenge accepted, although some students drew their square in the top right corner.

Next, I want you to choose a point, which can be above your square on even on your square, and I want you to dilate your square by a scale factor of two.

This took longer, but it was a good reminder of what we need for a dilation.

What happened? What can you tell me about your image?

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It has an area of 4 square centimeters.

How do you know?

I counted the squares.

Someone else noted that the similarity ratio is 1:2, so the ratio of the areas is 1:4.

What will happen if you dilate your original square by a scale factor of 3.

We will get a square with an area of 9.

And so they did.

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Now. Here is our goal for this lesson: For each whole number n between 1 and 10, find a square with vertices on the coordinate grid whose area is n square units or show that there is no such square.

So far, we have 1, 4, and 9. What do you know about those numbers?

They’re perfect squares.

Yes. So now I want you to draw a square with an area of 2 square centimeters. I’d like for you to work by yourself for 2 minutes, and then you can share what you’ve found with your group.

I watched while they worked. I saw many students approximating √2 on their calculator. I saw several students who had made a rectangle with an area of 2 square centimeters. I saw one student who had immediately thought of 45-45-90 triangles and had drawn a square with an exact area of 2. Everyone was doing something, even if they were using approximations.

After two minutes, I told students they could work together now, and that I had two reminders: I have asked you to draw a square. And I want it to have an exact area of 2 square centimeters.

I heard great conversation. I asked a few of those who had approximated the side length of their square how they knew the side was √2. Linda Griffith told a great story last year about some of her students: they decided to put “not drawn to scale” next to their diagram, as they had seen on one too many of the diagrams from their geometry class. Several others made the 45-45-90-connection for an isosceles right triangle with a leg of length 1 cm to get the desired square. I listened to one group who realized they had confused whether a square is always a rectangle or a rectangle is always a square take their rectangle and compose its parts a different way to get a square.

I decided to have them share first. It occurred to me after they started talking that I should video their explanation. I caught part of it.

I love that these two took their rectangle of area 2 and rearranged it to make a square of area 2.

Next I asked the student who had immediately thought of 45-45-90 to explain her thinking.

She related her work to the Pythagorean Theorem.

And finally one other student shared who had composed his square differently from the girls with the rectangle.

Now that we have a square with an area of 2, what other square areas can we easily get?

Of course a dilation by a scale factor of 2 will give us a square with an area of 8.

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What side length does that square have? 2√2.

So what is next? We still need squares with areas of 3, 5, 6, 7, and 10.

What could we do to get 5?

Several students simultaneously thought about 3-4-5 right triangles. So what does that give us? An area of 25, which we can get with oblique side lengths from the 3-4-5 triangle or horizontal/vertical side lengths of 5 cm.

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It isn’t really 5 we need. What can we do to get √5 for a side length?

Students continued working, many coming up with a 1-2-√5 triangle from which to draw a square with an area of 5.

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If we can get 5, can we get 10?

I was expecting to hear 12+32=10, and I did hear that. But I also heard (√5)2+(√5)2=10, which I didn’t hear as loudly because I wasn’t expecting to hear it. You would think I’d have learned by now to pay closer attention to what my students actually say. What I am learning, though, is that it takes time to process student thinking for a task that isn’t “cookie cutter”, and I don’t always do that quickly in class, especially when the bell is about to ring. We ended with a discussion of more than one way to get a square with an area of 10 – and I left 3, 6, and 7 for the students to finish exploring outside of class.

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So I would have liked to talk about why 3, 6, and 7 don’t work. We didn’t get there this year.

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But we did make it farther than last year as the journey continues …


Posted by on March 17, 2014 in Coordinate Geometry, Geometry


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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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Mastering Circles

We had a “mastering” lesson on Circles recently. I wanted to share three quick stories.

HJ wanted to talk about problem 23.

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She was having a hard time describing a particular region about which she had a question, so she asked if we could label the points and angles. Of course!

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I have read and watched Dan Meyer talk about providing students an opportunity to recognize a need for labeling, a need for organizing by a coordinate grid, etc. I cringe to think about how much time I used to spend in geometry on basic notation and labeling … valuable time that we could have spent doing more interesting tasks. What I’ve found is that the notation will follow when we model it in our everyday use. And when it’s not there, students might even ask for it!


I was surprised by the student responses to number 20.

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Not surprised that some students got it wrong, but what was is that so many students did to get 50π+100 instead of 50π+96?

FJ talked us through a solution to get 50π+96, and the students agreed with his reasoning. Will someone who got 50π+100 tell us what you did?

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They had the same 12-16-20 right triangle as the others. But instead of calculating the area using the two legs of the right triangle as the base and height, they calculated the area of the triangle using the hypotenuse as the base and drawing in the height. Incorrect assumption? The height coincides with a radius of the circle, and will be 10.

RE wrote a reflection on this problem, which was a good opportunity to construct viable arguments and critique the reasoning of others.

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And #10. Three tennis balls are packaged in a pressurized can, one on top of the other. Is the height of the can or its circumference greater? Justify your answer.

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I’ve talked about this before…the importance of monitoring student responses before showing them to the class. I deselected “show correct answer” before I displayed the results. And then I asked them to find someone at a different table who answered differently. Critique their reasoning as to why they chose what they chose. Do you agree or disagree?

And I sent the poll again.

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And then HR shared her argument with the whole class before we moved to the next problem.

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I am convinced more every day of the value of my response system. The feedback helps me make decisions about what to do next. The feedback makes the Standards for Mathematical Practice a reality in my classroom. The feedback helps my students make decisions about what to do next. Am I leveling up to the learning goals for this unit? Or do I need some type of intervention before the summative assessment?

I’m still reading Transformative Assessment in Action by James Popham.

And so the journey continues …

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Posted by on March 7, 2014 in Circles, Geometry


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Pythagorean Relationships: The Question That Probes

I don’t have the exact quote, but I read in Embedded Formative Assessment (Dylan Wiliam) that teachers having time to plan the questions that they will ask in a lesson to push and probe student thinking is important. We should ask questions to push students’ thinking forward and to uncover student thinking (and misconceptions).

I have read more about this idea of selecting questions ahead of time in Transformative Assessment (James Popham) and Transformative Assessment in Action (James Popham). Popham suggests that teachers not only need to plan what questions they will ask for formative assessment but also how they will respond when all students answer the question correctly versus the majority of students versus half of the students versus few or none of the students. I think that is ideal. I haven’t decided yet that it is practical for every lesson I teach.

In our dilations unit, we did a lesson on Pythagorean Triples and Pythagorean Relationships. We used part of the linked Math Nspired activities by the same name. The main purpose was to provide students an opportunity to make connection between a primitive Pythagorean Triple and the resulting triangles that can be dilated from that triangle. But at the recommendation of my upperclassman who have already taken the ACT and SAT, we also spent a bit of time providing students an opportunity to determine whether a triangle was acute, right, or obtuse given its 3 side lengths. While I think this concept could be implied from CCSS 7.G.A.2, it will be 2-3 more years before we have high school students who have been through CCSS Grade 7.

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

I started the lesson by asking students what question they might ask if they knew the side lengths of a triangle were 8, 16, and 17. Many asked about classifying the triangle, but a few asked what the area of the triangle would be. The question provided a nice problem solving points opportunity for those who wanted to learn about calculating the area of a triangle given the three side lengths.

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I exported the questions students asked and have included below:

Is it a right triangle? 6

What is the area of a triangle with side lengths of 8, 16, and 17?          1

Does it have a right angle?   2

What type of triangle do these 3 sides make?       2

is it a right triangle   1

The question would be what type of triangle is this        1

is it scalene?   1

what is the question that goes with this information?     1

what kind of triangle is it     1

classify the triangle? 1

what type of triangle is it     2

Is it a pythagorean triple?    2

what kind of triangle is it?   1

is it a pythagorean triple?    1

What type of triangle is it?   3

is it an acute triangle?          1

with these given measurements is there a pythagrian triple?    1

which side is the base?         1

what type of triangle does this three sides make 1

Is it obtuse?   1

is it a right triangle?  8

is this a right triangle?          2

Is the set of numbers a Pythagorean Triple?        1

can we find the area 1

what is the height of the triangle?  1

is it a pythagorean triple      1

What is the height?   1

Which are the legs, and which is the hypotenuse?           1

what is the area?       1

Can the area be found?        1

how do you know what the base is 1

is this a right triangle           1

whats is the base      1

what is the area        1

is it a right triangle ? 1

what type of triangle is it?   2

what is the height? to find area.      1

As students interact with the Pythagorean Relationships TNS document, they record whether the given side lengths form a triangle that is acute, right, obtuse, or nonexistent. Then students look for regularity in repeated reasoning to describe a relationship that is true about the squares of the sides of the triangles.

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For all of the given triangles, a≤b≤c. Some of my students wrote that a triangle is acute when a2+b2>c2. Others wrote that the triangle is acute when c2< a2+b2. Students already knew that a triangle is right when a2+b2=c2 or c2= a2+b2. Students also determined that a triangle is obtuse when a2+b2<c2 or c2>a2+b2. Students already knew that for 3 lengths to form a triangle, a+b>c, a+c>b, and b+c>a.

I sent a Quick Poll to assess student understanding. Our geometry team has a set of pre-prepared Quick Polls for each lesson. Teachers send them as needed in their classrooms. We don’t use every one of them, and we don’t necessarily send the same polls each time we teach a lesson. We practice formative assessment during our lessons to decide which questions to send and use the results to adjust the lesson.

I didn’t want to send three side lengths that formed a right triangle. Nor did I want to send three lengths that did not form a triangle. The first one I came upon happened to be an obtuse triangle. I sent the poll.

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And I was surprised by the results. The students had determined that for a triangle to be obtuse, a2+b2<c2. Why did one-third of them miss the question? I had to think fast. I could have shown them the correct answer. And then I could have worked the problem correctly. Or I could have asked what misconception the students who marked acute had. Would everyone have paid attention to that?

What I did instead was to show the students the results without displaying the correct answer.

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I asked students to find another student in the room and construct a viable argument and critique the reasoning of others. I walked around and listened to their arguments. And I sent the poll again.

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At that point, the students shared what happened to those students who had marked acute the first time. They had only observed that 8+15>18 was true instead of also noting that 82+152>182 was not true. And so it struck me at that moment that I had gotten lucky. Without realizing it ahead of time, I had chosen the right problem to send students to uncover their misconceptions. Had I sent another of the prepared Quick Polls instead that asked students to classify a triangle with side lengths 16, 48, and 50, all of the students would have gotten it correct, but they would have gotten it correct for the wrong reason. For that triangle, both 16+48>50 and 162+482>502 are true, and so the students would have chosen acute even if they had incorrectly used the Triangle Inequality Theorem to decide that.

I am glad that I sometimes get lucky as the journey continues …


Posted by on January 12, 2014 in Geometry, Right Triangles


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