One of my classes is at an impasse. They all agree that having proportional sides does not guarantee similarity in a quadrilateral (yay for the students who thought to grab a rhombus and a square from the tub of shapes to prove their point!). Yet we’re stuck on triangles. Half of the students are convinced of side-side-side similarity in triangles, the other half are not. They have been unable to get anyone to change their minds.
I believe they are waiting for me to tell them who is correct.
So, I’m going to try this.
- Measure each side and record.
- Determine if the sides are proportional (they are).
- Determine the scale factor (left as an exercise to the reader).
- Then carefully cut out each segment (kinda worried about this part).
- Next create a triangle out of each set (one triangle will use the letter only sides, one the primed letters).
- Measure the angles. (thinking they may need to tape the triangle together)
- Determine if SSS guarantees similarity in triangles.
I tried it. It seemed doable.
We’ll see how it goes. Any suggestions, advice, insights, criticism is/are of course welcome.
(There are two more sets in the .pdf -bonus, one set is special.)
When I went through this with my kids, I must have timed it just right. I haven’t done a lot of construction this year, but I introduced SSS right after they’d grasped the concept that the compass is not about drawing circles, but about creating congruent lengths.
I had them draw a line segment, and then two full circles centered on each end. It was easy to see that there were only two possible triangles, and that they wee congruent. I challenged them to follow the process to try to find some lengths for which it didn’t work out, and they were all fairly quickly convinced.
If only all of my other lessons this year had gone that well.
Good for the skeptical for being skeptical! You have done well by these young ones, Obi Wan.
Do they know right triangle trig? sin cos and tan (inverse, actually) wouldn’t work unless a side proportion guaranteed an angle.
Your exercise involves compelling inductive reasoning, but your skeptics might still object that it’s not a proof – how do they know there isn’t some other triangle somewhere with the same proportions with different angles?
Sorry I don’t have any better ideas…let us know how it goes.
Mr. K. Glad to hear your lesson went well – we don’t do constructions in this course. Now I’m wondering if we have the time to delve into them.
Kate Nope, we haven’t done right angle trig yet – that’s coming up in a week or so (freshman course). We’re going for an intuitive approach at the moment, the proofs will come later this week. Which is why I’m okay leaving it open ended for the moment. I’d rather have them intuitively understand then go for the formal proof.
I’m with Kate on the potential for persistent skepticism. Could your students think that you chose 2 sets of segments that just happened to work? If so, you could have each student make their own set of segments (in two different colors or some other way to distinguish them). They would choose their scale factors (ensuring variety), assemble the triangles, then measure the angles. Everyone could report their findings. Perhaps that would be even more convincing?
Have you considered letting each student choose three lengths to make a triangle (sneak in a little triangle inequality while you are at it), then decide on their own scale factor and calculate the lengths of the resulting sides. They can then use dry spaghetti to create the segments and piece their two triangles together? If you have 30 kids you will have 30 different examples of SSS similarity. Even the skeptic should allow this inductive proof to convince them.
Oh yeah – I forgot.
I also had them cut up straws and thread them on string to make a triangle.
Part A involved them cut a straw into two unequal pieces, and then doing the same to the shorter piece. (Demonstration of the triangle inequality.)
Part B involved them cutting two congruent sets of 3, and trying to make different triangles out of them.
I can’t remember whether I did this before or after the construction. If I were to do it again, I’d do it before.
I take it by “similar” you mean “equiangular”? Otherwise I’d say you’re begging the question here.
Shorter version: Start with two triangles with sides proportional (call them “A” and “B”). Pick one side of triangle B and construct on it a new triangle (“C”) similar to triangle A (copy the two base angles, and argue that the third angle is uniquely specified because they have to add up to 180 degrees). Since C is similar to A, its sides are proportional (the converse of what you’re proving here). Then it’s all down to comparing proportionalities to show that C and B are congruent.
Any sufficiently skeptical student should not be convinced by any of these ideas for constructing one or another example. Yes, these two triangles with proportional sides are similar, but we’re making a statement about all such pairs of triangles.
I was about to suggest Mr K’s drinking straw method. 1) Cut out straws to match the lengths of the line segments. 2) Use bendable pipe cleaners to secure the sides together. 3) Once the triangles are constructed the students can physically match the angles, no measurement needed.
To help with the generality of the statement, have each student then create their own example to ‘verify’ their results.
I didn’t read all of the comments to see if this was already there, so I apologize if it is. Get some SteelTec or Erector Set pieces and make triangles with them. Tighten the joints down at first, but then loosen them up. The kids will hopefully start to understand the idea when they realize that they can’t change the triangle even when it’s loosened. As an aside, this is why triangles are so common in building/support. Since triangles can’t change their angles without changing their sides, they are inherently the strongest shape (even a circle will oblate under stress.)
Hi. The CMP curriculum has Polystrips for this very purpose:
They work great. Here are some virtual ones:
I’m glad we didn’t get to this activity today. Thanks for all of your comments – I’ll respond more in more detail after the craziness of the next few days are over. You have no idea how much I appreciate all of your input!
I used a geogebra file to try to convince kids of SSS congruence. Perhaps if I had had students use geogebra to convince me of SSS congruence we’d have gone further. Anyway, that’s a link to the file that you might find helpful in visualizing.
I haven’t come in a while… somehow I left you out of my reader… Sorry to have missed so much.
This was a nice question.
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this is stupid! I didn’t even learn what I needed to!