Day 103: Recap & Practice

The first part of the class was part review (snow day yesterday), and part organizing our notes. On Tuesday, we made a Venn Diagram of the parallelograms, but didn't list the conclusions directly (just categorized the shapes). So I spent ~15 minutes making a list of everything that was true for each shape. 

Obviously, we started with parallelograms in general and wrote the definition and the 4 conclusions we found. Then we branched off to rectangles, and from there we actually moved down to squares, since we never formally discovered the major properties of rhombi. I liked the way it shaped up because we had a list of things that were true for squares, and some of them weren't true for rectangles. So logically, properties such as diagonals being perpendicular and bisecting their angles must have come from rhombi. 

I insisted to the students that any justification for their work MUST come from the conclusions that were now cleanly listed on the board. 

Students were then given the remainder of the hour to work on U7 WS1 regarding Parallelograms & Rhombi. 

Picture of the notes

Day 102: The Venn Diagram

Continuing where we left off with the discovery of the parallelogram yesterday, I began class by building a Venn diagram of "4 sided shapes" (didn't want to burden them with the Q-word just yet). So if all squares must fit inside the rectangle category, and all rectangles must fit inside the parallelogram category, if we look at the major properties (squares have equal sides AND 90 deg angles, while rectangles only have the 90 deg angles), that must mean there's another class of parallelograms that has equal sides. I tried to create a genetic/inherited trait metaphor and use the idea that squares are the child of rectangles and this new shape. Most students knew the name rhombus, but the power of the metaphor was helpful in "guessing" the properties that would hold true for it without the investigation. 

For example, if the diagonals of squares are angle bisectors, but is NOT true for rectangles, then where did squares get that trait from? Must have been the rhombus!

What was incredibly frustrating was that as I was creating the diagram with circles (as one generally does with a Venn diagram), I had a lot of students puzzled as to why I was pointing to a circle and labeling it "rectangles." Seriously. I couldn't make this up. My hope was that once I finished, they would see the Venn diagram and recognize it from the rest of the world and all would be right again. Nope. It wasn't until I used the name 'Venn' that any connections were made. *sigh*

Unit 7 Worksheet 1 was also assigned in preparation for the expected snow day tomorrow. 

Day 101: Unit 7 - Quadrilaterals

Well, it finally happened. Last summer, when I was writing this new curriculum, I got overviews of how to sequence the first 6 units done, figuring I'd just finish the rest when I had time (xmas break, mid winter break, etc). Did I? Of course not. So I had to spend the weekend prepping Unit 7 and thinking about what I wanted to do when and how. 

Keeping with my theme of building up from specific ideas into more general ones, we spent the day reviewing properties we knew about rectangles (from Unit 6) and I reminded everyone that we developed those properties by first starting with squares (from Unit 3). Now we're going to alter the rectangle, create a new shape, and test to see what still holds true. 

From here, we were able to show that the opposite sides are still parallel, which gives us our definition (and name). We then tested things like congruent opposite sides (distance formula), and 90 angles at the vertices (slope), as well as perpendicular diagonals, congruent diagonals, and bisecting diagonals. Tomorrow we'll wrap up the discussion, transition quickly into the rhombus, show how all these sets of quadrilaterals can be related and start the practice. 

Day 100; Unit 6 Assessment

This test was unlike most others I'd been writing this year for this new curriculum. It was very much adapted from questions of the worksheets an quizzes. Not because I was consciously trying to make it "easy," but I'm not sure how many different ways there are to see if students can classify triangles and use SSS/SAS/ASA/AAS. 

On the upside, the grades were very good across the board. Which now means I'll have to fight complacency going forward, as students will take the good grade and assumes it means they're a genius and no longer need to put forth any effort. 

Day 99: Review

I allowed the students to work on the proofs activity, prepare a note sheet for tomorrow's test, or simply ask questions regarding topics they were still struggling with. 

Again, not everyone takes uses their time wisely, but I stressed the structure of the class and that I was deliberately pushing the responsibility of preparing for the test on them, so hopefully they at least can identify who's really in charge of their success. 

Day 98: Intro to Proofs

Part of the revamped curriculum I created was to find a better way to teach/use proofs. In the textbook my school uses, proofs are started in Ch. 3 amid a random assortment of other information (such as solving a system of linear equations algebraically), and they're not heavily emphasized as the book progresses. I don't see the point in starting them out so early, because there are not a lot of applicable skills so early in the course, so any proofs would be so short to be almost pointless. 

All I really want out of proofs is for my students to get the idea that they need to be able to justify their reasoning with other concepts we've leaned in the course. Hearing a student say "that angle is 90 degrees because it looks like it is" is one of those things that makes me die a little each time I hear it. 

With the short week and the Unit 6 Assessment on Friday, I wasn't sure what else to do, so I created a handout with 4 very basic proofs involving triangle congruence. In my now 5 attempts at teaching proofs, this was probably the best go at it I've had. Students really did seem to appreciate what was possible with proofs as well as the structure and sequence of it. 

My hope is that moving forward, I can really push hard on using proofs to test true understanding of the content. Instead of memorizing that parallelograms have congruent opposite angles, can you prove WHY that must be true? In my opinion, the former is a MUCH better test of learning than the latter. 

Day 97: Online Review

Students were given the class period to work in the computer lab as they saw fit, under the direction that we would have a quiz on Wednesday (no school Monday or Tuesday for President's Day). I directed them to the Khan Academy page for Congruent Triangles as a starting point both to watch tutorials if they were struggling or to practice if they wanted additional work beyond the worksheet. 

As expected, some students really do take advantage of opportunities like this, but far too many do not. Even if I enjoyed nagging (which I do not), the students in question don't respond. They're either falsely confident that they don't need to study, or they're confidence that studying won't do any good. The kids who spend their time wisely aren't necessarily the smartest, they're simply the most self-aware of their skill set and understand how to make gains.