Trying Triangles

Over the summer I reorganized my geometry curriculum. Matt put together a great sheet of triangles to try and I reformatted it slightly. It worked great, except I have toolkits with short rulers so I wish that some of the side lengths were smaller (plus more examples would fit on a page that way). No one was able to figure out the area or perimeter ones, but I think they're worth keeping as challenges and thought experiments. (Even if you can't draw it, do you think it exists?)

Trying Triangles

We used pencil and paper, Matt mentions GeoGebra but we have only used it once so far and I think physically manipulating rulers works well for convincing yourself something just won't reach.

When students had drawn as many triangles as they could and made their own conjectures, I used the hint cards I saw in a video of a Japanese classroom. Propped up against the white board were ten index cards numbered 1-5. Each number had a different question on it which prompted students to make particular types of conjectures. I made two copies of each card since some students wanted to bring the card to their seats last time I used hint cards. The idea is, students should be able to form conjectures on their own, but if they need a hint, the cards are available to look at. No one is allowed to claim they are "done" until they have checked all 5 cards. These are the questions I would ask if I was circulating the room and noticed a student was stuck, but this way they don't have to wait for me to arrive.

Hint Card 1: What angles work to make a triangle?
Hint Card 2: What side lengths work to make a triangle?
Hint Card 3:  What can you say about the lengths of the sides of a triangle if you know the measures of its angles?
Hint Card 4:  What can you say about the measures of the angles of a triangle if you know the lengths of its sides?
Hint Card 5: What do you need to know to prove two triangles are the same?

Friday we shared out conjectures in two of my classes and ended up with:

The triangle will be scalene if we're given 3 different sides or 3 different angles.
If two angles are the same, then the triangle will be isosceles.
If the sides are equal, then the angles will be equal.
The angles add up to 180 degrees.
The sides work when they are close, but not too close
          45, 3, 4 fails           3, 4, 10 fails           3, 4, 5 works
          3, 4, 6 works          3, 4, 7 flattened triangle

One class was just what they came up with on their own, the other class we had enough time to try some extra examples which supplemented their ideas. Next up, working to verify this list, which will remain a work in progress throughout our unit on triangles.