Tag Archives: special right triangles

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?


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Team Sorting – Right Triangles & Circles

We’ve been sorting students into teams on the first day of each unit this year. After the team sort for Dilations, we were motivated to again make the cards challenging and interesting.

We went to the cafeteria so that students would have plenty of room to find their teams. I told them that we have 6 teams and 30 students.

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I overhead the following:

  • I’ve just got two sides.
  • Am I supposed to calculate the side or the angle?
  • I’m noticing we both have 60-30 triangles.
  • You would have to find the hypotenuse, and you would have to find the leg.
  • Do you have 33 degrees?
  • We think we have a team: two sides and an x.
  • Who has an x, an angle, and a hypotenuse?
  • Do you have the included leg or the opposite leg?
  • Is everybody here today?

Eventually, all teams sorted themselves.

From the beginning, students began to notice when we can use the Pythagorean Theorem and when we can’t.

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Students began to notice the difference between the opposite leg of the acute angle of a right triangle and the included leg.

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Students began to notice when we were looking for the angle of a right triangle instead of one of the sides.

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Students began to notice special right triangles.

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Yesterday, we sorted into teams for our circles unit.

Students began to notice the difference between central angles and inscribed angles (even though they don’t know the names of them yet).

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Students began to notice the difference between diagrams with two chords and two secants (even though they don’t know the definition of a chord versus a secant).

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Students began to notice the difference between diagrams with tangents only and diagrams with a secant and a tangent.

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Just in case you want to use or edit, you can find all of our team sorting cards here.

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Posted by on January 28, 2015 in Circles, Geometry, Right Triangles


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Hopewell Geometry – Misconceptions

Last year I posted student work from the Mathematics Assessment Project task Hopewell Triangles. This year I want to talk about the conversations that we had.

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Students worked on the task for 15-20 minutes by themselves. For those who didn’t need that long, I had two other tasks from Illustrative Mathematics on the back of the handout: Seven Circles I and Shortest Line Segment from a Point P to a Line L.

After students worked by themselves, I collected their responses in the TNS document I had sent. They started talking about their results with their groups. Apparently I didn’t make it clear to the whole class that I would be collecting their responses, because everyone had not keyed in their responses, but I looked at the responses that I did get to monitor student progress while they were talking with their groups. I noticed that one student had answered question 4 incorrectly.

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So I went to his group and listened to them as they constructed a viable argument and critiqued the reasoning of others. Alex explained how he had arrived at his wrong answer. Can anyone figure out Alex’s misconception?

In our whole class discussion, I made sure that Alex shared his thinking. We all have something to learn from other’s misconceptions and we have created a classroom where it is okay to learn from incorrect thinking. It turns out that Alex used inverse tangent to calculate the acute angle measures. He then concluded that the triangle was right because it had a right angle (since the two acute angles were complementary), not realizing his circular logic in assuming that the triangle was right to use right triangle trigonometry for his acute angle calculations.

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Can we show that Alex’s thinking doesn’t work in conjunction with what we know about the other triangles?

Triangle 3 is a 45°-45°-90° triangle, and so its acute angles are 45°. At least half of the students recognized that they had already calculated one of the acute angles for Triangle 1 in problem #2. Then we can show by the Angle Addition Postulate that we have a problem when the shaded triangle is right since 53°, 90°, and 45° don’t sum to 180°.

Most students used the Pythagorean Theorem to show that the shaded triangle was not right, which was our next whole class conversation. One student told us that the hypotenuse of Triangle 3 was 5√2, which he knew because of our work with Special Right Triangles. Another student used the Pythagorean Theorem on Triangle 2 to get its hypotenuse of 7√5. And then another showed us that (5√2)2+(7√5)2≠(15)2. This would have been the end of the conversation in my class when I first started teaching. But because our focus is not only on correct answers but also on logical arguments and understanding, a student stopped us from continuing by asking a question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3? What a good question. Why is the hypotenuse of Triangle 2 7√5 instead of 7√3?

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How did you get 7√3? The triangle has side lengths 7 and 14, so I assumed the triangle was a 30°-60°-90° triangle. Another student cleared up that misconception. Side lengths of x and 2x are for a 30°-60°-90° triangle only when 2x is the length of the hypotenuse.

The last whole class conversation happened because I overhead a group talking about how they knew Triangle 1 was similar to Triangle A. I had asked them to remember that part of the conversation because I was going to ask them to discuss it with the whole class later. One girl said that she knew the triangles were similar because she saw that 3-4-5 was proportional to 9-12-15. But she was having a hard time visualizing how the triangles were similar. How could we show that the two triangles are similar to each other?

How do we define similarity? Two figures are similar if there is a dilation (and if needed, a sequence of rigid motions) that will map one figure onto the other. Can we show that to be true for Triangle A and Triangle 1?

We need several rigid motions along with the dilation to show that Triangle A and Triangle 1 are similar:

Reflect Triangle A about the side that is 3 units.

Rotate Triangle A’ 270° about the vertex of the angle opposite the side that is 3 units. Let’s call that point X’.

Translate Triangle A’’ by a vector that goes from X’’ to the left vertex of the given rectangle.

Dilate Triangle A’’’ about point X’’’ by a scale factor of 3.

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I was amazed at the mathematics in this lesson and the opportunity for students to make connections. What a great task! If I had written the task, it wouldn’t have occurred to me to include a triangle that students might mistake for a 30°-60°-90° triangle. Instead, the students were given the opportunity to construct a viable argument and critique the reasoning of others. It might not have occurred to me to include a calculation in an early question (#2) that students could use again in the final question (#4) – I would have likely changed the angle measures to have them perform two different calculations. Instead, the students were given the opportunity to look for regularity in repeated reasoning. If I had written the task and included a triangle similar to one of the others, I doubt it would have occurred to me to make one of them have a different orientation than the other. Instead, the students were given the opportunity to look for and make use of structure.

And so the journey continues, with great tasks like this one to make student misconceptions evident and to correct those misconceptions …


Posted by on February 2, 2014 in Geometry, Right Triangles


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Special Right Triangles

I blogged last year about providing students the opportunity to make sense of special right triangles. This year I just want to make a few observations. Over the last year, I have gone through and thrown out old binders of notes and transparencies that I used to use. I took a picture of how I used to make sure students could solve special right triangles, just as a reminder of how far we’ve come.

Special Right Triangles

In our lesson on 45°-45°-90° triangles, students use the Pythagorean Theorem to look for regularity in repeated reasoning, reason abstractly and quantitatively, and look for and make use of structure.


Students did well recognizing that a 45°-45°-90° triangle is half of a square divided by a diagonal. And they did well on the formative assessment Quick Polls that I sent to assess their progress.

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Note: These two questions were not on the same side of the page on the student handout.

But something happened when I asked them to calculate the perimeter of a non-familiar polygon.

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I monitored their progress after I sent the poll. After two minutes, I saw the following results. 4 students had a correct response, and 15 students had an incorrect response.

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After another minute, 5 students had a correct response, and 21 students had an incorrect response.

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I stopped the poll.

What would you do next?

I unchecked “Show Correct Answer” when I showed the results. And I asked a student who got 15 to explain his thinking (construct a viable argument). He counted 15 “pieces of segments” in the figure. Then I asked the class to critique his reasoning. Is every piece congruent? Another student asked to come to the board so that she could show how she used the practice look for and make use of structure.

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Of course everyone understood the mistake in measuring after the second student drew the auxiliary line. But I wonder why more students didn’t connect what we had been learning to this diagram on their own?

What would have happened if we had started class with the perimeter of the polygon? Whether I had asked them to answer it then or not, would it have made a different in what they saw later?

Not unrelated, I recently asked my 9-year old daughter to load the dishwasher. Several hours later, I opened the dishwasher and found a big surprise.


I’m not sure if you can tell or not, but among other problems, the coffee mugs still have on their lids. It’s hard to believe that AKW has ever unloaded the dishwasher, much less on many occasions.

We talked about how the dishwasher works – and why we should turn dishes towards the water. This was take 2, a week later.

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I’m still a bit flabbergasted at how difficult learning how to load the dishwasher has been without direct instruction. But I wonder what would have happened if I had asked her a different question to get her to think about how the dishwasher works. I wonder what would have happened if I had specifically asked her what she noticed when she was unloading the dishwasher, causing her to look for regularity in repeated reasoning. My daughter is still learning how to think and problem solve.

As all learners are.

And I am still learning the questions to ask, to promote thinking and problem solving, to uncover misconceptions.

As all Learners are.

And so the journey continues …

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Posted by on January 28, 2014 in Geometry, Right Triangles


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