First, draw what you consider to be the most beautiful triangle. Measure and record all of the side lengths and angles for your triangle. Be as precise as you can.
Second, you have just created what you consider to be the most beautiful triangle ever created. You are so excited about your triangle that you want to share it with the world. You go to your Twitter account to send a tweet and realize that the small number of characters you can send may be a problem.
Undaunted you decide to see what the fewest number of sides and angles of your triangle that you can tweet and still have followers be able to re-draw your beautiful triangle exactly as you have drawn it. Your task is to determine the fewest number of sides and angles of your triangle that you can tweet. Think about this problem and decide what you would do.
I gave students a couple of minutes and then have students pair up and see if their method works. I walk around to see what they come up with. I had students share out their findings, which follow:
- give 2 angles and 2 sides
- an equilateral triangle with the side length
- right or isosceles triangle and two side lengths
- 2 congruent side lengths and 1 angle
Many students had elected to draw isosceles or right triangles. As I walked around, one group had concluded that you just need 3 pieces of information about the triangle, deciding that AAA and SSA would also work.
I decided to tackle this by having the one group state that just three parts of the triangle were needed. I wrote this on the board. I focused on the right or isosceles triangle and two side lengths first. I didn't want to tackle right triangles yet, so I asked students to focus on the isosceles triangles.
I asked the class to re-create an isosceles triangle I was thinking of, the congruent side lengths were 5". After pondering this, students realized that there wasn't enough information. Without the angle formed by the congruent sides, they could not replicate the triangle. So, we basically needed to know the two sides and the angle formed by those two sides.
Does this work for any triangle? I asked students to try this out. There was still some question about the process working, mostly due to students' poor use of protractors and rulers (although they are much better than they were). I asked them to all draw a triangle that had side lengths of 3" and 4" with the angle formed by these sides being 70o. When they compared their drawings they realized that they had drawn congruent triangles. I wrote under the three parts side-angle-side and the abbreviation SAS. At the end of the line I wrote works.
I asked if the order given for the two sides and angle mattered. What would happen if you were told that a side of 3" joined a side of 4" and at the other end of the 4" side there was a 70o angle? Students determined that this didn't work. I wrote side-side-angle SSA and then didn't work on the board.
I next used the equilateral triangle as a springboard. In this situation, you are essentially communicating the lengths of all three sides. Would the idea of communicating all three side lengths work for any triangle? I gave them triangle side lengths of 3", 4", and 6" and asked them to draw the triangle to see if they had the same triangle. After the typical struggles and my referencing back to what was going on when they copies angles using a compass and straight-edge, the class decided that providing three sides worked. I wrote side-side-side SSS and works after it.
We were running out of time so I asked the class to consider the analogous situations of ASA and AAS, which flipped the SAS and SSA relationships. I'll use next class to work through these and also bring to attention of the entire class the use of AAA.
I asked the class to re-create an isosceles triangle I was thinking of, the congruent side lengths were 5". After pondering this, students realized that there wasn't enough information. Without the angle formed by the congruent sides, they could not replicate the triangle. So, we basically needed to know the two sides and the angle formed by those two sides.
Does this work for any triangle? I asked students to try this out. There was still some question about the process working, mostly due to students' poor use of protractors and rulers (although they are much better than they were). I asked them to all draw a triangle that had side lengths of 3" and 4" with the angle formed by these sides being 70o. When they compared their drawings they realized that they had drawn congruent triangles. I wrote under the three parts side-angle-side and the abbreviation SAS. At the end of the line I wrote works.
I asked if the order given for the two sides and angle mattered. What would happen if you were told that a side of 3" joined a side of 4" and at the other end of the 4" side there was a 70o angle? Students determined that this didn't work. I wrote side-side-angle SSA and then didn't work on the board.
I next used the equilateral triangle as a springboard. In this situation, you are essentially communicating the lengths of all three sides. Would the idea of communicating all three side lengths work for any triangle? I gave them triangle side lengths of 3", 4", and 6" and asked them to draw the triangle to see if they had the same triangle. After the typical struggles and my referencing back to what was going on when they copies angles using a compass and straight-edge, the class decided that providing three sides worked. I wrote side-side-side SSS and works after it.
We were running out of time so I asked the class to consider the analogous situations of ASA and AAS, which flipped the SAS and SSA relationships. I'll use next class to work through these and also bring to attention of the entire class the use of AAA.
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