Length & Area – Sliding Triangles

We started our unit on Geometric Measure & Dimension with a Mathematics Assessment Project formative assessment lesson on Evaluating Statements about Length and Area.

I chose the card that each group would explore ahead of time. I only gave each group one card, and I had another task ready for them if they finished early. They didn’t. Students chose whether they wanted to explore on paper or using technology.

Each card had a hint, which I held until I felt like the group might need it. While students were working, I monitored their progress.

The second group had Sliding Triangles.

If you slide the top corner of a triangle from left to right, its area stays the same.

If you slide the top corner of a triangle from left to right, its perimeter changes.

They read through their card and drew a few diagrams, but they decided to spend most of their time building the scenario using our dynamic geometry software. This group did not need the hint card.

When it was time for this group to present their work, we sent the Quick Polls so that we would know what students instinctually thought, even though they had not all had time to explore the statements in depth. I am learning to make use of the TI-Nspire Navigator allowing me to send more than one Quick Poll at a time.

I showed the results, but I deselected Show Correct Answer before doing so. I wanted the group to know what their peers thought before they just told them the results.

I used the Live Presenter feature of Navigator to make one of the student’s calculators live on the projector at the front of the room. They grabbed and moved the point that changed the top vertex of the triangle, noting with the square they had constructed off the base that the height of the triangle remained constant.

The area of the triangle will always be the same.

What about the perimeter? The perimeter changes.

Does the perimeter always change? Sometimes the perimeter is the same.

When is the perimeter the same? When one triangle is a reflection of the other about the perpendicular bisector of the base.

And so the journey continues …

 

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Length & Area: Cutting Shapes

We started our unit on Geometric Measure & Dimension with a Mathematics Assessment Project formative assessment lesson on Evaluating Statements about Length and Area.

I chose the card that each group would explore ahead of time. I only gave each group one card, and I had another task ready for them if they finished early. They didn’t. Students chose whether they wanted to explore on paper or using technology.

Each card had a hint, which I held until I felt like the group might need it. While students were working, I monitored their progress.

The first group had Cutting Shapes.

When you cut a piece off a shape you reduce its area.

When you cut a piece off a shape you reduce its perimeter.

They thought Always for both statements.

Most of them were taking a quadrilateral and cutting off a triangle on the corner.

How could I get them to figure out it was sometimes besides just telling them? I also didn’t want to give this group the hint card because I felt like it gave too much away.

What if you cut off something besides a triangle? Someone cut off a rectangular corner.

What just happened to the area? It got smaller.

What just happened to the perimeter? It stayed the same!

How do you know?

What if you cut somewhere besides a corner?

I went to another group after I asked this question.

By the time I got back to them, they had a great explanation as to why both statements were not always.

I’m sure there was a better way to do this, which might include having students evaluate the statements by themselves before coming to class for the lesson. We could have spent a week on these six cards. But we didn’t.

Before each group presented their work, we sent out a Quick Poll to see what the rest of the class instinctively thought about the statements.

I showed the results, but I deselected Show Correct Answer before doing so. I wanted the group to know what their peers thought before they just told them the results.

The group presented their work, not just giving us their results, instead talking us through their thinking about the statements, and how they arrived at their conclusion.

And so the journey continues …