Lesson 7-4: Overlapping Triangles (Day 43)

Today I subbed at a middle school. It was mostly for sixth and seventh grade English, but during the conference period, I also covered a special education math class for seventh graders.

They were working on adding rational numbers. This time, they had to add numbers that already had like denominators -- but since this is seventh grade, these were signed rational numbers. The trickiest problems for students were ones where they had to keep track of the sign, reduce the fraction, and convert it to a mixed number all in the same problem. There were also a few questions where the students had to explain their answers for Common Core. We already mentioned last week that special ed students often struggle with these.

I know that I shouldn't still be discussing this from yesterday, but I was thinking about what to do when faced with a student who refuses to put squared and cubed units for areas and volumes -- especially considering some of the silly reasons that teachers deduct points for. We want him to see that putting squared and cubed is crucial, not silly. Well, I was thinking about the old trick of turning a dollar into a penny due to carelessness with units:

$1 = 100c = (10c)^2 = ($.10)^2 = $0.01 = 1c

If we want to emphasize the connection to measurement, use meters and centimeters, or even yards and inches, instead of dollars and cents.

Here's another trick that might work to convince students to do extra things to help them in math -- in this case Algebra I -- without making it seem that we're taking off points for correct answers. In Algebra I, we often want students to draw a line through the equal sign to make sure that they're not getting the two sides of the equation confused. But some students can already solve equations and get correct answers without drawing the line. Now here's a possible solution that I may implement if I ever teach Algebra I -- when solving equations in class, I make sure that I always draw the line. Then if I ever forget to draw the line, I give the student who catches my error an extra point. That way, the only person I'm requiring to draw the line is myself -- yet the weaker students see the importance of drawing the line.

Lesson 7-4 of the U of Chicago text covers more proofs. These proofs are trickier, since they involve overlapping triangles.

Last year, I tried to post an activity about overlapping triangles, but this was still in the middle of my scanner not working, and the activity didn't come out too well. So now I'm making this into a regular lesson and rewriting it from scratch.

Once again, I had to be careful not to include any questions that require theorems that we have yet to prove, such as the Isosceles Triangle Theorem, that we're still waiting until next week for. Because the triangles overlap, it may appear that, in Question 4, we need to show that Triangle SUA is isosceles. But as it turns out, we actually don't need to show this to complete the proof.

The bonus question is somewhat interesting here. It asks whether there is a valid congruence theorem for quadrilaterals, SSASS. Last year I tried to solve it, but got confused, so I want to take the time to set the record straight.

As it turns out, SSASS is not a valid congruence theorem for quadrilaterals. A counterexample for SSASS is closely related to a counterexample to SSA for triangles -- we start with two triangles that satisfy SSA yet aren't congruent -- one of these will be acute, the other obtuse. Then we reflect each triangle over the congruent side that is adjacent to the congruent angle. Each triangle becomes a kite -- as the original triangles aren't congruent, the kites can't be congruent either, yet they satisfy SSASS (with the A twice as large as the A of the original triangles).

I tried to prove SSASS by dividing each quadrilaterals into two triangles, then using SAS on the first pair and SSS on the second. The problem with this is that that division doesn't produce two triangles unless the quadrilateral is known to be convex. With our two kites, notice that the acute triangle becomes a convex kite, while the obtuse triangle becomes a nonconvex (or concave) kite -- which is also known as a dart. If both quadrilaterals are already known to be convex, then my proof of SSASS is valid.

One congruence theorem that actually is valid for quadrilaterals is SASAS. We can prove it the same way that we proved SAS for triangles. We put one of the sides -- in this case the congruent side that's between the other two congruent sides -- on the reflecting line. Then we can prove that the two far vertices are on the correct ray, the correct distance from the two vertices on the reflecting line -- this works whether the quadrilateral is convex or concave. We can also prove SASAS by dividing the quadrilateral into triangles. There are separate cases for convex and concave quadrilaterals, but all of them work out.

Other congruence theorems for quadrilaterals are ASASA and AASAS. Another congruence theorem, AAASS, is also valid, but it's similar to AAS in that there's a trivial proof based on the angle-sum that reduces it to ASASA (just as AAS reduces to ASA), only in Euclidean geometry. A neutral proof of AAASS exists, but it's more complicated.