Not sure how everyone else is doing, but as we approach the end of unit 1 and begin looking at the assessment, we are feeling the need for a little clarity. For confidentiality, I will not talk directly about test items, but I do want to focus you on a few ideas that need to be coming out as you wrap up this unit. You may get the most out of this post if you sit with a copy of the assessment next to you and have a little scavenger hunt through the document to find how the various math topics I discuss come up in the assessment. Then check out some of the resources I have at the end, and see what you can use to revisit what you missed on the first time around.
Ready? Here we go. . .
Are two shapes Congruent?
In lessons 1-13, you dug deeply into congruence. This was not a brand new, unconnected topic. The understanding of congruence is built on the rigid transformations we learned about in the first half of the unit. Justifying congruence is the WHY of those lessons. The anchor chart above makes some of those connections.
But if you are like me, and have taught lots of years, this is probably not how you have spent your career thinking about congruence. At the high school level, congruence implied formal proofs based on matching corresponding parts. At the middle school level, it was an more informal set of thoughts around “exactly the same size and shape” and “fit perfectly on top of one another”. It’s this fits perfectly thing we are messing with here. If we are sitting together at the same table and you want to tell me two pattern blocks or puzzle pieces are congruent, you can pick one up and put it onto the other. If they are pictures on a piece of paper, you can trace one, move the tracing paper, and lay it right on the other to demonstrate that congruence.
But when you don’t have the person you are trying to convince right next to you, you need to write directions for how they need to move the figure so they stack up. Our precise mathematical language helps us write clear descriptions of that movement so they can follow those steps and be convinced.
Anyone done a puzzle with a toddler recently? If you haven’t, here’s an adorable 15 month old puzzle pro:
Notice his favorite transformation is translation. He slides the puzzle piece back and forth until they drop in. At around 1:25 seconds he has one that is not oriented correctly and his mother says “rotate counter-clockwise 180 degrees.” Well actually she says “ Flip it around.” He’s 15 months after all.
How would “flip it around” have worked for you? If you were blindfolded and trying to do that puzzle, would you have understood what she meant?
Now that our students are teenagers instead of toddlers, we are looking for better, more precise communication. But the idea is straight forward. Convince me they are congruent by helping me put the shapes on top of each other. Strong, well defined mathematical language is what makes that happen.
Sometimes the diagram they are working from lacks landmarks and that makes description difficult. Your students need to learn to be resourceful and add those landmarks as needed. That could mean drawing in and naming a line of reflection. That could mean adding point names to important vertices. Lesson 13 practice problem 4 is a great chance to practice adding what is needed to make their communication clear. Take a little time to make that a piece of conversation. What did you add? Why? In lesson 15 practice problem 4 it comes up again. Notice that both of these are on a grid, and that might not be true of all the problems your students are about to see (scavenger hunt time). Think about how you will prepare your students to address that situation.
But what if they are not congruent?
What if? How does that play out when we are sitting next to each other at the table with cut out shapes or pattern blocks? I imagine stacking them up and saying, “See? This angle is wider on the top shape,” or “See? The left side is longer on this one.”
That is how it works here as well. Show me what doesn’t match. That probably means you need to name, highlight, or circle the non congruent corresponding parts. That is practiced in Lesson 7 practice problems number 1 and 3.
Lesson 12 makes great practice on this topic. Consider introducing this anchor chart there.
“So there is some geometry vocabulary from previous grade levels that my students just don’t know.” (scavenger hunt time)
If that’s not true in your classroom, yours may be the only one in the world like that. News flash- kids don’t remember everything from previous grades. Shocking, I know. ( Aside: I just read an awesome article, Addressing Unfinished Learning in the Context of Grade Level Work, if you are interested in how you could address that in your classroom.)
So if you find yourself wondering, “Do my students actually know what a rhombus is?”, optional activity 4 from lesson 12 is a great place to bring some of those words up in the context of congruence. Don’t make it too tricky for yourself. We used plastic place value sticks of 10 because that’s what we had. Toothpicks would have worked great. For those of you who have too much money and don’t know what to spend it on, Ang-legs would be awesome for this. We gave each student six, and then asked them to build things. Make a triangle. Can you build one not congruent to your partner’s? Someone thought of doubling the sides, and so I asked if that also doubled to angles. I used the phrase “Convince me” liberally as we worked through building square, rectangle, parallelogram, and rhombus. In both classes students build a regular hexagon while trying to make a parallelogram different from their partner’s. All of a sudden I knew we needed to talk more about what a parallelogram was. We brought up that ALL the corresponding parts had to be the same for the shapes to be congruent. Just equal sides ( square non-square rhombus) and just equal angles ( square and non-square rectangle) did not guarantee congruence.
It was fun, it was quick, and we got a chance to work a bit on some unfinished learning from previous grade levels that the kids really needed. It would fit great as part of a review day.
So define design . . .
Check out activity 13.4 again. This is a place to bring out that even though all the corresponding parts are equal, they are not arranged in the same way, and so the entire right face is not congruent to the entire left one. (scavenger hunt time) Also check out the extra diagram in your lesson synthesis for lesson 13. It reiterates this point in a way you could use as you reviewed if you didn’t use it before:
If you did, here’s another sample that you can use to review this idea:
There is not a single point of rotation that the entire first figure can be rotated around to give the second figure. Each piece of the first figure was rotated, but that did not create a rigid transformation of the entire figure.
More Review Resources:
So for most of us it’s our first year, and you might not have got it perfect the first time around. In case you are wanting to revisit some things, here are a few extra review activities to choose from: