Ok, it's time to actually start covering material that might be new to students. I started by stressing the purpose of my curriculum because many of the students have noticed that I'm doing everything differently than the other geometry teachers in school. We're doing things differently so that everything is built from stuff we've learned. Unit 3 was all about squares, so we're going to start Unit 4 from there and go forward.
Major point to stress in the "review" is the diagonal to side length ratio. Looking over the U3 Assessment, many students still count diagonal spaces on a graph in the exact same fashion as horizontal & vertical spacings.
So with a diagonal drawn in a square, we very obviously have two triangles. Coming from a square, we know the sides are equal and there are 90 degree angles in the "corners." Intro the vocab that 2 equal sides --> isosceles, so we can call this triangle an "Isosceles Right Triangle" (IRT).
Also coming from the unit on squares, we know that the diagonal acts as an angle bisector. Half of 90 is 45 degrees, so each acute angle in the IRT must be 45 deg. 90+45+45 = 180 which conforms with expectations since the four 90 deg angles in the square summed to 360, and we cut the square in half.
I'm trying to keep everything very explicit as we build the unit from the ground up. I make a point to stress that we only know the angles of an IRT sum to 180 because we only knew the 4 angles of a square sum to 360. We can branch out later and verify that the idea holds true for all triangles.
I then tasked the class with a challenge. Draw a right triangle that is NOT isosceles on graph paper, keeping the sides to integral lengths. Determine the area by counting boxes (remembering that we defined area in U3 with squares) and try to find a connection between the lengths of the legs of the right triangle and the area.
I really thought kids were going to just start with the assumption that A = 1/2 b*h and claim they were done, but they actually didn't see the connection, presumably because I never used the terms 'base' and 'height.'
We'll recap the challenge at the start of class tomorrow and start work on U4 WS1: Area & Perimeter of Right Triangles.