Thursday, February 28, 2013

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

Tuesday, February 26, 2013

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. 

Monday, February 25, 2013

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. 

Friday, February 22, 2013

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. 

Thursday, February 21, 2013

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. 

Wednesday, February 20, 2013

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. 

Friday, February 15, 2013

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. 

Thursday, February 14, 2013

Day 96; Practice

Students were given the class period to practice working on U6 Worksheet #3 which involves identifying given information and deciding which triangle congruence postulate/theorem to use (SSS, SAS, ASA, AAS). I chose to omit Hypotenuse-leg to avoid confusion. If it comes up, I'd rather see them use the 3rd Angles Thm in conjunction with something else to complete the proof. 

Wednesday, February 13, 2013

Day 95: ASA & AAS

Between not wanting to wrestle with the straws and pipe cleaners again, and not thinking that activity would continue to be as useful, I opted instead for a more direct discovery. Random Student #1 picks a number between 1 & 10. #2 picks a number between 20 and 70. #3 picks another number between 20 and 70. Now we can draw a line of length (#1 cm) in between a (#2) angle and a (#3) angle. Complete the triangle, measure everything and show that everyone's triangles are congruent (ASA).

For AAS, I show the basic outline of the proof that involves the 3rd Angles Thm with ASA rather than go through any specific discovery. 

Tuesday, February 12, 2013

Day 94: SAS Congruence

My original plan here was to have everyone use half of the "bundle" we used yesterday - the long straw, the middle straw, and the class would agree on an angle. You can then set the two sides to the given angle with the pipe cleaner, and trace out what you've created. Then, complete the triangle by drawing in the 3rd side. Once the triangle is complete, measure EVERYTHING you don't already know (3rd side and two angles). Idea being that if everyone started with two congruent sides and a congruent included angle, everyone should end up with the same (congruent) triangles. 


  • Holding the created angle steady while tracing was problematic
  • Students are not very skilled with either the ruler or the protractor
  • Some students trace the inside of the angle, some the outside
Overall, the conclusion comes across, but it might help to encourage rounding to the nearest cm to avoid measurement errors clouding the idea. 

Monday, February 11, 2013

Day 93: Proving Triangles Congruent

This section of Unit 6 was a major focus of mine while I wrote this curriculum last summer. I hate teaching proofs in the traditional style presented by most textbooks because of how formal it's presumed that proofs need to be. I just want students to be able to justify their reasoning and think sequentially, which has been the underlying focus of this entire venture. 

My school's geometry textbook introduces proofs in Chapter 3, which would be fine if it relied on them heavily throughout the remainder of the book. Instead, they're introduced at a very basic level (which is necessitated by starting them so early) and basically abandoned after that. 

My thoughts are along the lines of "let's not deal with proofs until we're well equipped" (read: 2nd semester). Once the students have the basic skills down pat (such as types of angles), we can attack proofs that have more substance. And maybe we could even use proofs to test/discover new ideas. 

So to start out, I made "bundles" which consisted of 3 pieces of drinking straw and 3 pipe cleaners. Each bundle has the same 4 inch, 5 inch, and 6 inch piece of drinking straw, so students can use the pipe cleaners as angles (inserted in the straws) to make triangles. Even though the supplies were cheap, I only made enough to have students work in pairs and simply asked each group to build a triangle, trace its outline onto paper, then measure all three angles and sides. We collected the data and the hope was to show that everyone's triangle was the same, thus demonstrating the SSS postulate. 

The major snag is that too many students cannot use a ruler or protractor reliably, so there's a much larger spread in the data that you'd like. I still think it was a useful activity though. 

Thursday, February 7, 2013

Day 92: Discussion

Class review of yesterday's worksheet. After checking the work for completion, upwards of 85% of the students made a solid effort. Again, that might be because it was easy, but I'll take it. 

Rather than deal with WB's, I used my tablet and alternated between randomly calling on students for their answers, and in some cases allowing students to draw their work on the tablet. They love playing with the toy, but like any other snazzy tech educators try to incorporate, it loses it's appeal VERY quickly if relied on too heavily. 

As important as routine is, I do see an advantage to "mixing it up" in terms of how I'll present or discuss the material. I might use the document camera, the tablet, or go "old school" and simply use the whiteboard at the front of class. 

Wednesday, February 6, 2013

Day 91: Practice

Students were given the class period to work on U6 Worksheet #1 which deals with both Area & Perimeter of Rectangles as well as the classification and naming of triangles. 

This is the first worksheet being checked for a grade under SBG, and it tentatively appears as if more students are making an attempt at completion. That could also be connected to the fact that this is incredibly easy work. 

Tuesday, February 5, 2013

Day 90: Classification of Triangles

First, a quiz on the basics of rectangles and the angles formed when a transversal intersects a pair of parallel lines. Then, moving on to the idea of how triangles are classified and how we identify corresponding pieces.

Surprisingly, a lot of students remember the different types of classification schema. It always amazes me which piece of random trivia from a class will stick with students from year to year (and conversely, which won't). 

Explaining how to name triangles, and stressing why it's important was a little harder to deal with. It's one of those things that appears silly and more about math being harder than it needs to be to the students. Of course I try to use an example from the future of when this will be relevant, but many students have pre-decided that even the example is stupid, so there's no point in being good at the skill we're practicing. Actually, that last sentence describes 95% of the struggles I face as a teacher. 

Monday, February 4, 2013

Day 89: Discussion

Still a a loss for what to do with my 5th & 6th hour. 2nd hour whiteboarded the worksheet and got through a discussion of every problem with time to spare. I knew a traditional WB presentation wouldn't work with my later classes, so I opted for the "WB Parade" instead, and even that didn't work. Students are off task and loud, simply choosing to do absolutely nothing. 

In the end, the problems to all 14 problems were posted and displayed, and students had ample opportunity to look them over, but I only had time to talk about a couple in a whole class setting, which I'm predicting will be used against me tomorrow during the quiz (the old "you never showed us how to do this" defense). 

I'm considering keeping a blank checklist roster along with my HW list and my Class Monitor list and simply having students initial on days that they're choosing to not work. I'll have a basic 'pledge' at the top of the page that will be something along the lines of "I'm choosing to not participate in the class work today and accept full responsibility for the grade that I will receive."

I hate resorting to guilt trips like that, but it seems to be the only type of reinforcement that works, and most students themselves would agree. 

Friday, February 1, 2013

Day 88: Looking for patterns

I began class with a derivation of the angle relationships that appear when a transversal intersects a pair of parallel lines. We started with a new, horizontally drawn rectangle on a graph so that the distances would be easier to eyeball. Just as we did when initally making conclusions about rectangles, we solved for the measure of an acute angle formed by the diagonal using a tangent ratio. From there, we can complete the right triangle to solve for a third angle. Then use the angle addition postulate (which still provides a perplexing amount of trouble for students) to solve another angle, and continue to solve for angles using linear pairs and vertical angles where needed.

When it's all done, the picture is a mess and students are stressing that they'll have to replicate all 15 (or however many) steps there were for every problem. "Wouldn't it be great if we had some shortcuts to follow?" I asked. "Do you notice any commonalities that we could look for again in the future?" Some angle pairs jump right out, but even the others aren't hard to see with a little effort. 

So I created 4 copies of the framework of the picture (just the parallel lines and the transversal) and identify one pair at a time, naming them as I go along. 

Students then had ~20 minutes to work on U6 WS1 and were told that it would be checked (for a grade now) on Monday.