We started a unit on Geometric Measure and Dimension a few weeks ago. During the first lesson I asked whether students had made sense of the formulas for the areas of a triangle, parallelogram, and trapezoid, or whether they had just memorized them without understanding. Since they said the latter, we took a few minutes to make sense of the formulas.

We used Area Formulas from Geometry Nspired for demonstration.

We moved B and H. What do you notice about the given rectangle and parallelogram?

Their bases have the same length.

They have the same height.

What is true about their areas?

They are equal.

How do you know?

What is the formula for the area of a parallelogram?

We moved B and H. What do you notice about the given parallelogram and triangle?

Their bases have the same length.

They have the same height.

What is true about their areas?

They are not equal.

What relationship does the area of the parallelogram have to the area of the triangle?

It is twice the area of the triangle.

The area of the triangle is ½ the area of the parallelogram.

How do you know?

So what about the area of a trapezoid?

I didn’t show them the next page in the TNS document.

I showed them a general trapezoid and asked them to work for a few minutes alone to make sense of how to calculate the are of the trapezoid.

I walked around and watched.

Whose would you choose for a whole class discussion?

I asked DT to show us her picture first. She had decomposed the trapezoid into two triangles using a diagonal. We were able to use the distributive property to show that the area of the two triangles was the same as the textbook trapezoid formula.

As you can see in the bottom right, another student had decomposed the trapezoid into three triangles. We didn’t take the time to show that the area of the three triangles was the same as the textbook trapezoid formula.

I asked BA to show us her work next. She had decomposed the trapezoid into a parallelogram and a triangle. Again, we were able to show how the area she calculated was the same as the textbook trapezoid formula.

I asked MA to show us her work next. She had recomposed the trapezoid into a rectangle that had a base equal to the median of the trapezoid and a height the same as the trapezoid. Again, we were able to show how the area she calculated was the same as the textbook trapezoid formula.

I asked BE, whose work I somehow missed photographing, to explain what he did last. Unknown to him, his work was like that shown in the TNS document.

I teach a high school geometry class. Every day I wonder whether I waste my students’ time going back and making sense of concepts that they’ve been using for a while mathematically. I struggle to know what is our best use of class time. Surely there is a way that my students could have done this outside of class … would it have been effective? We were at five ways and counting to make sense of the formula. I’m not sure that is trivial. My students were reasoning abstractly and quantitatively, even if the math content was not an explicit part of our standards. And so the journey continues …

Travis

April 28, 2014 at 3:37 am

…never a waste of time…’why?’ is powerful…harness that curiosity and perplexity. Students benefit with multiple passes, much like layers of lacquer. When we think they ‘got it’, then it is our job to find another tack and see if that sheds more light. [mixed metaphors, yum]

I had to use shearing to think through the equal areas of the triangle with medians–from your previous post–that concept must continually be re-visited for the the students.

Dinner conversation tonight about your tangerine peels and a calc teacher reminded me of the deriv. connection with volume and S.A. for the sphere–cool. (it was actually a thank you dinner for CREW and I happened to start chatting with him and he teaches at another school. And oh, he had never heard of Nspire! I was shocked. He uses 89 and 92. I felt like it was Acts ‘we have never heard of another baptism other than John’s’ or the road to Emmaus ‘are you the only one who has not heard of such things?!’ I tried not to let my jaw drop too much.

suevanhattum

April 28, 2014 at 1:42 pm

>Every day I wonder whether I waste my students’ time going back and making sense of concepts that they’ve been using for a while mathematically.

Not at all. I teach college, and most of my students can’t imagine making up their own formulas – which they must. What is the surface area for a mug (no handle)? Simple. Just take the top off that cylinder. But they can’t do it, because they just memorized.

jwilson828

April 28, 2014 at 1:50 pm

Thank you both for your comments. I regularly ask students to give me feedback on whether lessons are helpful, and they all seem to agree that they’ve never thought about the topics we discuss in-depth. But I still want to be sure that I’m giving them what they need content-wise. The SA of the mug example is helpful. Pat commented on the Volume Formulas post about checking for understanding by asking students to calculate surface area for a compound shape like a cone on top of a cylinder, which is also helpful. I can at least go back through my lessons to be sure I’m using those types of examples.

suevanhattum

April 28, 2014 at 1:53 pm

Actually, it’s beter if you leave some out of your lessons, and ask the students if they could do those on their own.

jwilson828

April 28, 2014 at 2:07 pm

Good point. Thank you for making me use my language correctly … I will be sure I am providing my students the opportunity to solve those kinds of problems. We definitely don’t follow the “I work – you mimic” in our course. Thanks again for your comment. I found your blog this weekend (I think through someone’s tweet) & look forward to reading.

Mr-Butler

April 28, 2014 at 2:22 pm

This is good stuff. Just starting to read your blog and liking it already. I’m completely on the same page for process with formulas like these. I teach HS math (Integrated Math 1 primarily) and in my classes we did some similar things. Here’s a couple of Geogebra Applets I had my students play with.

Box:http://www.geogebratube.org/student/m105529

Half Box: http://www.geogebratube.org/student/m112429

We categorized rectangles/parallelograms and triangles/trapezoids into box and 1/2 box respectively. I’m fortunate to have 1:1 Chromebooks so the students can play with the applets as well.

I’ve been trying to use more of this visual play to develop conceptual understanding. If you have more of this stuff keep posting it. I’m creating/curating such items over at transformulas.org

Thanks again for contributing for us to see.

-@mathbutler

jwilson828

April 28, 2014 at 3:38 pm

Thanks for reading. I’ll keep posting … and I look forward to keeping up with your blog and the transformulas site as well. It is good to be able to learn from each other.