Tag Archives: 45-45-90 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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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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