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 third group had Rectangles.
Draw a diagonal of a rectangle and mark any point on it as P. Draw lines through P, parallel to the sides of the rectangle. The two shaded rectangles have equal areas.
Draw a diagonal of a rectangle and mark any point on it as P. Draw lines through P, parallel to the sides of the rectangle. The two shaded rectangles have equal perimeters.
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.
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 point on the diagonal.
What do you see? A lot of rectangles.
What is true about the rectangles? Some have a diagonal.
What does the diagonal buy us? Triangles. Congruent triangles.
So the shaded areas are always equal.
What about their perimeters? When are they equal? When P is the midpoint of the diagonal, we can show that all four rectangles are congruent.
The fourth group had Medians of a Triangle.
If you join each vertex of a triangle to the midpoint of the opposite side, the six triangles you get all have the same area.
This group spent most of their time building the scenario using our dynamic geometry software.
The students used technology to see that the statement was true. Then they talked about why. Technology helps us see things that we might not see at first glance. Technology shows us that something is true (or not). Even though we still have to make sense of why it is true (or not). Technology makes more problems accessible to more of my students.
What did the class think about this group’s statement?
I didn’t get to hear this group’s presentation. Another teacher in my department had my students on this day. But this group spent a lot of time looking for and making use of structure.
How do we know that the brown triangle has the same area as the yellow triangle?
How do we know that the triangle formed by the green, orange, and brown triangles is equal to the triangle formed by the blue, pink, and yellow triangles?
How do we know that the triangle formed by the green, orange, and blue triangles is equal to the triangle formed by the brown, pink, and yellow triangles?
The fifth group had Square and Circle.
If a square and a circle have the same perimeter, the circle has the smallest area.
This group mostly reasoned quantitatively. They didn’t use the technology, but we looked at a square and circle that had been built with the same perimeter so that we could have that dynamic feel to make sense of their results.
The last group had Midpoints of a Quadrilateral.
If you join the midpoints of the sides of a quadrilateral, you get a parallelogram with one half the area of the original quadrilateral.
We’ve talked before about the result of joining the midpoints of the sides as a parallelogram, but we had never discussed the areas of the figures.
I enjoyed looking at the work this group did using their technology to make sense of the statement.
Why is this statement always true?
Last year a teacher in our department had several students wonder if they would do cards like these again in class. She asked them why they wanted to do more. The students answered, “Because they were like puzzles. It was fun to figure them out.”
And so the journey continues …