Solving Systems Review by ezmoreldo on Scribd

# Don't Panic, The Answer is 42

A journal on Teaching Math and my only hope for Professional Development

## Monday, November 13, 2017

### Solving systems of linear equations review for algebra 2

I whipped up this worksheet out of a balance problem worksheet I found online years ago and some desmos activities on systems. I really wanted to do some of the desmos activites themselves, but I tried one and my students couldn't stay on-task. They kept cutting and pasting quotes from the communist manifesto and advertisements for bitcoin into the short response boxes. Sigh. This activity that I mashed together actually went really well. We did most of the exercises as think-pair-shares and they remembered enough from algebra 1 for it to be a quick, easy and intuitive review of systems solving. I did steal everything so I take no credit except in terms of the presentation.

## Sunday, October 15, 2017

### NWMC 2017

Holy Cosines! A lot has been happening in the math world in the three years since I had my first child and my life was consumed by this entropy machine. I just attended my first professional conference in 4 or 5 years and my mind is abuzz. I need to write it all down before I forget.

First I attended Tom Reardon's "Problem Solving: All-Time Favorite Mathematically Rich Precalculus Activities, Individualized- with Complete Solutions". Here are some things I want to remember:

Then I did Dan Meyer's "Charge Up Your Classes with Free Desmos Technology." I have to insert another Holy Cosines here! Dan sat next to me at one point. I was too star dazzled to say a single word to him, but I was 2 feet away! I actually haven't always been a big three act fan. I've followed Dan's blog for a long time, but I was too overworked to implement any of his ideas. I'm not tech saavy and when he was first posting his three act videos I thought they were super cool, but I couldn't see how I could make any, didn't have the technology in my classroom to even show them, and I also didn't think I could spare the class time to really do them justice. Also I was teaching 14 preps so really didn't have the planning time either (yes, it really was 14

Then I was tired and the next day we had staff development for my school so I played hookie and didn't go to any of the Friday sessions. Which was a huge bummer.

Then I did a 7:30 AM (ON A SATURDAY!) breakfast keynote with Fawn Nguyen titled, "What if We've Been Teaching Mathematics All Wrong." WOW. There is so much from this that I want to remember that I wish I'd video taped it. But I've already forgotten so much of it.

Then I did Andrew Stadel's "Lessons that Make Math Stick." Again, he may have made me a groupie for life. Things I want to remember:

Finally I did Jeff Crawford's "Visual Algebra: Current Research and Practical Applications." Oh my gcf! He had us looking at visual patterns in such cool ways. Blocks are sooooo cool.

First I attended Tom Reardon's "Problem Solving: All-Time Favorite Mathematically Rich Precalculus Activities, Individualized- with Complete Solutions". Here are some things I want to remember:

- We started with "the great applied problem" which involved a cylindrical tank lying on it's side partially filled with water. The goal is to figure out how much water is in the tank, and how much water is needed to finish filling up the tank. He asked us first to ask him for the information we'd need to solve the problem. I never do this in class because I'm always in such a hurry. What a super important first step when tackling any problem. This was consistent through all the activities he showed us.
- Then we did a bunch of other fun problems and we played with graphing calculators a lot. My classroom has a set of donated calculators I scrounged up from the app Next Door this summer, so I don't know how much of this fun my kids will get to have, but it reminded me of how powerful they are, and how intimidating they are. He had us doing stuff with them I never new was possible. But remembering how to do all the little steps and where all the different buttons were was hard, and I use TI-84s every day. I almost asked him if it was worth it teaching students this way or if we should just switch over to Desmos, screw the standardized tests, but I was too chicken.

Then I did Dan Meyer's "Charge Up Your Classes with Free Desmos Technology." I have to insert another Holy Cosines here! Dan sat next to me at one point. I was too star dazzled to say a single word to him, but I was 2 feet away! I actually haven't always been a big three act fan. I've followed Dan's blog for a long time, but I was too overworked to implement any of his ideas. I'm not tech saavy and when he was first posting his three act videos I thought they were super cool, but I couldn't see how I could make any, didn't have the technology in my classroom to even show them, and I also didn't think I could spare the class time to really do them justice. Also I was teaching 14 preps so really didn't have the planning time either (yes, it really was 14

*preps.*Isn't that insane?) But in the few years I've been away he's made Desmos into a lean, mean three act machine. Some things I don't want to forget.- He did this super cool thing where we made a list of values from -5 to 5, then he defined points like (L, L) or (L, -L) or (L, L^2) and had us predict what the graph would look like. What a super super super cool way to finally cement the idea that when we graph a function like y=x^2, the ordered pairs will be: (x, x^2). I've had such a hard time getting my students to understand that (x,y) means the same thing as (x, f(x)) which means the same thing as (x, whatever f(x) is defined as, like x^2, or x^2 -2x+7)
- I MUST go back and finish the desmos scavenger hunt then use desmos all the time.
- Dan said something during the presentation that rubbed me the wrong way and I'm still trying to make sense of it. He had this shtick of playing super dead-pan and skeptical of anyone's answer. He said he worked hard to make students doubt any answer they presented. His point that we want students to justify their thinking, to be open to alternate solutions, and to really own their answers even in the face of doubt is well taken, and maybe if I'd been taught with someone like him as a teacher, someone I trusted, I would have learned more confidence. But I have too many experiences of being ignored or doubted when I was right in less trusting environments to be comfortable with this teaching style.

Then I was tired and the next day we had staff development for my school so I played hookie and didn't go to any of the Friday sessions. Which was a huge bummer.

Then I did a 7:30 AM (ON A SATURDAY!) breakfast keynote with Fawn Nguyen titled, "What if We've Been Teaching Mathematics All Wrong." WOW. There is so much from this that I want to remember that I wish I'd video taped it. But I've already forgotten so much of it.

- First, I want a poster that has three rules on it. Rule #1: Never give up. Rule #2: Never give someone else an answer. Rule #3: Love being stuck.
- She used visual patterns in ways I'd never thought about before. Of course I can't remember them anymore! She did a cool paper folding activity where you take a strip of paper and fold it in half. Then unfold and count the creases. Then fold it back in half and then in half again. Count the creases. Repeat. I've seen this activity before, but I'd forgotten about it so hopefully this blog post will make me remember.
- I need to play WAAAAAY more with visual patterns and between 2 numbers and everything she's ever done ever.

Then I did Andrew Stadel's "Lessons that Make Math Stick." Again, he may have made me a groupie for life. Things I want to remember:

- Give every activity the 3-C test; each one should be conceptual, spark curiosity, and should connect students to real experiences. He had a great video of a girl on a see-saw. She put a milk crate on the other side of the see-saw and started filling it with bricks. Each brick weighed 5 lbs. The obvious questions was how many bricks it took to balance with the girl. What a really cool way to model division and multiplication. He had so many of these great videos that made me see how truly important it is to spark curiosity.
- He talked about how baseball players who practice three kinds of hitting in blocks learn less than those who do mixed practice. Then he advocated for Steve Leinwand's 2-4-2 homework model. Here's a link to a blog post talking about it. I really really really want to try this with my pre-calc class.
- Finally he talked about meaningful feedback- whether feedback should be immediate or delayed. I post all my answer keys online for students to look at. I need to give this question a LOT more thought.

Finally I did Jeff Crawford's "Visual Algebra: Current Research and Practical Applications." Oh my gcf! He had us looking at visual patterns in such cool ways. Blocks are sooooo cool.

- I want to investigate proofs without words, Jo Boaler videos and youcubed (which I'd never heard of before!) and open-up resources which I'd already started playing around with for my 6th grade math class.
- He talked about Finger gnosia which is crazy and I want to experiment on my toddler with.
- He showed us all the different ways our brains see patterns and how while all those different ways can be distilled into the same algebra expressions, they are beautiful in their uniqueness and the different ways we see them can lead to some understandings of the function's that are more useful than others. (Particularly in tracing the patterns backwards)
- The key to functions is identifying what stays the same and what changes. Then when it changes,
*how*it changes. - I need to have students prove why sqrt(a^2+b^2) IS NOT a+b with blocks. Because they KEEP making this mistake.
- I need to read PEAK.

Whew. I'm pooped. Maybe later I'll come back and fill out more details but I've got the heart of the sessions distilled for future me to enjoy with tea and biscuits.

## Tuesday, October 3, 2017

### Inverse Function Activity

Here's a quick activity to develop the idea of inverse functions for my algebra 2 students. We just learned composition and domain and range. I hope it'll be fun.

Update: We just did this activity. The game was GREAT. The worksheet was terrible. I need to tweak it. I'll upload a new version once I think it'll work better.

Update update: Here's a new version of the worksheet. I'm actually kinda proud of this one. I want to go back in time and kick myself. Or at least hand this over to past me and make me teach it this way instead.

Inverse Functions Activity by ezmoreldo on Scribd

Here's a word version: inverse functions gameUpdate: We just did this activity. The game was GREAT. The worksheet was terrible. I need to tweak it. I'll upload a new version once I think it'll work better.

Update update: Here's a new version of the worksheet. I'm actually kinda proud of this one. I want to go back in time and kick myself. Or at least hand this over to past me and make me teach it this way instead.

Inverse Functions Activity2 by ezmoreldo on Scribd

And here's a word version
## Thursday, September 14, 2017

### Sunday Funday: Classroom Organization

I guess I'm behind the times as the Sunday Funday Blog challenge prompt I want to respond to is already a little old, but it inspired me to write a post, so here I am.

I want to write about classroom organization, specifically what to do if you don't have a classroom. There are some of us nomadic teachers out there, shlepping stuff from classroom to classroom, trying to figure out how to connect to the digital projector with the wrong cords when our laptops only have hdmi ports and the projector only has vga. There are some interesting and unique challenges you face when you don't have your own space.

Challenge #1: Collecting homework. If you have to move from one classroom to another, to another without much of a break in between, and you need a teeny bit of time for set-up, you don't have time to go stash homework somewhere. So if you collect homework in one class, you have to shlep it to another classroom, then if you collect homework in that classroom, your homework shlepping multiplies. It's impossible to do things like notebook or binder checks, unless you try to check it during class while the kids are occupied, but our periods are only 50 minutes which makes that strategy tricky too.

Solution: I don't collect homework. I've come up with two ways of assigning homework without needing to collect it.

- Way 1: I assign the homework, show them the answer key in class and while they're grading their own work, I go around and give them a stamp for having it done on time. I use alphabet stamps and go through the alphabet so at the end of the unit, it's easy for me to see when an assignment is missing. Pros: the kids grade their own work so are more cognizant of their mistakes. They can get their questions addressed way more quickly than if I collected work and they ask deeper questions because they can see where their work deviated from mine and catch misconceptions they didn't even know they had. Cons: Takes time away from instruction.
- Way 2: I post the answer keys to the homework on my class website and the kids check their work on their own and come to class with questions. Then in class, I choose a problem from the homework and display it. Then I hand out blank note cards and the kids solve the problem from the homework without their homework or notes in front of them. If they did the homework and checked their work thoughtfully, recreating the solution should be a breeze. Pros: They have access to the answer key as they're working through their homework so can be more thoughtful and can come to class with really specific questions. Having to then reproduce the work in class the next day really reveals if they understood it or not. Cons: cheating is a possibility but as only the note cards are graded it won't help their score. I also see a smaller sample of their work so I have less info on their understanding.

Solution: Our school does have a digital projector in each room so I bought myself a document camera and with that, my laptop, and a vga to hdmi converter I can reliably use the projectors. I write a notes template for the lesson that day and have students take notes from that. Then I can scan in the template each night and they'll always have access to the notes!!! I don't have to ever argue with students anymore about notes. If they didn't take them- go check the website! If they're absent, go check the website! Teaching this way also makes displaying student work a breeze. We all had a great time when I had my algebra 2 students solve a quadratic formula problem on note cards. One by one I showed the cards via document camera and every answer was different! There was some laughter and a lot of sheepish "I guess you were right in telling us to be more careful"s.

Challenge #3: Classroom management has always been a struggle for me, and I thought I'd finally come to grips with it a few years ago when I was teaching in California. Teaching without a classroom makes me feel less valid. I can't control my space, I'm always puffing from one place to another frantically trying to set things up. I lose authority this way. I don't have a solution for this one. It's just an interesting observation. I do my best, I try to pretend, but without feeling like I own my space I also don't feel as in command of my students (not that I want to command them- gently trick them into doing what I want without them noticing?)

I'm definitely not as good a teacher when I have to teach this way. But I'm pleased with the grading system and I'm really enjoying the ethical conversations we're having about how to use answer keys responsibly. Students have so many resources at their fingers, but don't know how to properly use them. Even with the answer key sitting in front of him a student today couldn't scan through his paper and compare it to the key. He kept skipping around or missing details. There are so many hidden executive functioning skills/deficiencies that are revealed when kids have to grade their own work.

## Friday, September 1, 2017

### I'm Back! Again?

So two years ago I posted an "I'm back" post. But I hadn't actually started teaching again so I had nothing to say. Now I'm really back. Back to teaching and hopefully, back to blogging. I want to spend this first post reflecting on why I stopped blogging in the first place because I think it speaks to some of the issues our students struggle with.

First, I stopped following teacher blogs. The good ideas have been so helpful and so inspiring, but every good idea I didn't have the chance to use made me feel bad, every boring lesson that I didn't spice up made me feel bad, and then all those good ideas made me feel bad about my own paltry ones. So then I stopped posting my own blogs.

It's so silly to fall into that comparison trap. To think that because there are so many amazing things happening out there, so many things that I could never have dreamed of, that means my ideas are worthless. That I have nothing to contribute. But that's also not what blogging is about. It's not an arena where the best ideas have to pin the good or mediocre ideas down and hold them down for a count of 10. I did feel like I was "losing" at some game and I'd never be smart enough to win.

I've felt this way about math too. I often didn't have the insight fast enough, or wasn't able to chug the numbers competently enough to shine in class. I always thought my contributions were worth less than other peoples'. It took me until grad school to realize that the people speaking up in class often were wrong, or were bsing, or were questioning. They weren't better at math than me, they were better at participating in a mathematical community than I was. My math partner who talked a lot of jargon and had a deeper pool of knowledge than I did often missed the key insights our proofs needed and I usually saw them. I was quieter about it, and more tentative. But I could see them. I fight this crippling insecurity every day. I know where it came from- a string of sexist math teachers and an older brother incredibly gifted at math- I don't know how to conquer it other than being very aware of it and fighting against it.

So I'm fighting now by resuming this blog. And I'm going to spend some time thinking about how to get my students to fight too because I know that a lot of them also feel like they're losing the game. They can't collect enough points, or see the ideas fast enough, or be smart enough. There are so many ways to help all students feel valued, but at the same time they're constantly inundated with messages about achievement that are divorced from real learning and from the real contributions they can make to their mathematical community. It's not about winning, it's about participation and I want to figure out how to weave this message into every aspect of my classroom culture. That's my mission for this return to teaching and this return to blogging. Wish me luck!

First, I stopped following teacher blogs. The good ideas have been so helpful and so inspiring, but every good idea I didn't have the chance to use made me feel bad, every boring lesson that I didn't spice up made me feel bad, and then all those good ideas made me feel bad about my own paltry ones. So then I stopped posting my own blogs.

It's so silly to fall into that comparison trap. To think that because there are so many amazing things happening out there, so many things that I could never have dreamed of, that means my ideas are worthless. That I have nothing to contribute. But that's also not what blogging is about. It's not an arena where the best ideas have to pin the good or mediocre ideas down and hold them down for a count of 10. I did feel like I was "losing" at some game and I'd never be smart enough to win.

I've felt this way about math too. I often didn't have the insight fast enough, or wasn't able to chug the numbers competently enough to shine in class. I always thought my contributions were worth less than other peoples'. It took me until grad school to realize that the people speaking up in class often were wrong, or were bsing, or were questioning. They weren't better at math than me, they were better at participating in a mathematical community than I was. My math partner who talked a lot of jargon and had a deeper pool of knowledge than I did often missed the key insights our proofs needed and I usually saw them. I was quieter about it, and more tentative. But I could see them. I fight this crippling insecurity every day. I know where it came from- a string of sexist math teachers and an older brother incredibly gifted at math- I don't know how to conquer it other than being very aware of it and fighting against it.

So I'm fighting now by resuming this blog. And I'm going to spend some time thinking about how to get my students to fight too because I know that a lot of them also feel like they're losing the game. They can't collect enough points, or see the ideas fast enough, or be smart enough. There are so many ways to help all students feel valued, but at the same time they're constantly inundated with messages about achievement that are divorced from real learning and from the real contributions they can make to their mathematical community. It's not about winning, it's about participation and I want to figure out how to weave this message into every aspect of my classroom culture. That's my mission for this return to teaching and this return to blogging. Wish me luck!

## Monday, July 20, 2015

### Teaching Again

I've taken the last year off to have my son.

It's been quite a ride.

People have been asking me which is easier, parenting or teaching. Since he's only 11 months old, I don't really know yet what parenting consists of but I can say that this has been the most relaxing year of my adult life. What does that say about being a teacher?

I was able to take some classes for myself (abstract algebra, graph theory 1 and graph theory 2) and I reconnected with what I love and hate about being a student. It was really helpful for me to remember what not knowing math feels like and what a different persona I adopt as a student (super quiet, shy and uncertain) vs. who I am as a teacher (gregarious, adventurous, unashamed of making mistakes.) I spent a lot of time observing the other women in the classes (only about a quarter of the graduate students were women) and how much they participated compared to the men (about 90% of the comments made in class were by men.) None of the students were black or Latino. I'm still processing how these observations should influence my teaching but for now, it's clear that I need to do more for my female and minority students. Why don't women participate? Why don't I participate? My personal reasons are related to fear that at some point, I will hit a wall mathematically and just won't be able to understand something (even though I've overcome every wall so far), inherent shyness and introvertedness, fear of being wrong, math being so tied to my identity that I don't want to be revealed as a fraud (which I do feel like sometimes. What right do I have to be telling other people how to do math when I'm unsure I could have pursued math seriously.) I did have some sexist math teachers. I never felt encouraged in math. But these are my reasons. Does every woman in math share these misgivings? Or do we all have our own individual insecurities reinforced by our cultural context? My sample size was really tiny. And my shyness prevented me from sharing my observations with other women in the class.

Anyway. I'm going back to teaching. Algebra 1 and Japanese for next year. I'm excited and scared to go back but I'm looking forward to catching up with what everyone's been doing on the MTBoS while I've been away. I hope I can start contributing again and I'm so grateful I have this community to lean on when I'm in need of inspiration, which I always am! I hope someday I can contribute something useful in exchange for all this community has given me.

It's been quite a ride.

People have been asking me which is easier, parenting or teaching. Since he's only 11 months old, I don't really know yet what parenting consists of but I can say that this has been the most relaxing year of my adult life. What does that say about being a teacher?

I was able to take some classes for myself (abstract algebra, graph theory 1 and graph theory 2) and I reconnected with what I love and hate about being a student. It was really helpful for me to remember what not knowing math feels like and what a different persona I adopt as a student (super quiet, shy and uncertain) vs. who I am as a teacher (gregarious, adventurous, unashamed of making mistakes.) I spent a lot of time observing the other women in the classes (only about a quarter of the graduate students were women) and how much they participated compared to the men (about 90% of the comments made in class were by men.) None of the students were black or Latino. I'm still processing how these observations should influence my teaching but for now, it's clear that I need to do more for my female and minority students. Why don't women participate? Why don't I participate? My personal reasons are related to fear that at some point, I will hit a wall mathematically and just won't be able to understand something (even though I've overcome every wall so far), inherent shyness and introvertedness, fear of being wrong, math being so tied to my identity that I don't want to be revealed as a fraud (which I do feel like sometimes. What right do I have to be telling other people how to do math when I'm unsure I could have pursued math seriously.) I did have some sexist math teachers. I never felt encouraged in math. But these are my reasons. Does every woman in math share these misgivings? Or do we all have our own individual insecurities reinforced by our cultural context? My sample size was really tiny. And my shyness prevented me from sharing my observations with other women in the class.

Anyway. I'm going back to teaching. Algebra 1 and Japanese for next year. I'm excited and scared to go back but I'm looking forward to catching up with what everyone's been doing on the MTBoS while I've been away. I hope I can start contributing again and I'm so grateful I have this community to lean on when I'm in need of inspiration, which I always am! I hope someday I can contribute something useful in exchange for all this community has given me.

## Tuesday, July 15, 2014

### Intro to Proofs in Geometry

I wanted to blog about this a looooong loooong time ago but the school year got in the way along with moving across the country twice because of family health dramas (NY to California in the fall, now California to Oregon. I know Cali to Oregon doesn't seem that far, but it is over 1,000 miles from San Diego to Portland. California is freekishly big.) So though I know posting lesson plans in the summer is kind of silly, I want to get it out of my system before I forget what I did.

The school in which I taught this past year was a high poverty school where 30% of our students had IEPs. It is a charter school so it's pretty small meaning I got to work very closely with my students, colleagues and parents but we did lack funding and our students were weak in a lot of basic skills. In fact, most of my geometry students this past year

So here is the description of how the unit flowed. It's definitely a more traditional approach to proofs and I stuck to two-column proofs. I tried to transition the students to paragraph proofs, but their skills and confidence were too low; they liked the organized nature of two-column proofs. First, I didn't nix the logic unit. Even though logic is not in Common Core anymore, I think that the reasoning done in the logic unit helps prepare students for proofs.

Logic Unit Lesson 1: Intro to conditional statements.

The school in which I taught this past year was a high poverty school where 30% of our students had IEPs. It is a charter school so it's pretty small meaning I got to work very closely with my students, colleagues and parents but we did lack funding and our students were weak in a lot of basic skills. In fact, most of my geometry students this past year

*hadn't even passed algebra 1 yet.*The previous algebra teacher found them so lacking in basic skills that she gave the entire algebra 1 class "incompletes" because they didn't finish the algebra 1 curriculum. My principal decided to enroll all these students in geometry anyway because she figured (rightly I think) that they needed a bit of a break from algebra and if they saw some algebra in a geometrical context it might make going back to algebra more meaningful (which it did. Every time algebra popped up in geometry the students were actually excited because it was familiar and wasn't too difficult. They really mastered equation solving, writing expressions and equations of lines by studying these topics through geometry.) All of this meant that when proofs came up I was super freaked out. I've always struggled with teaching them and I feel like I've done a very poor job in the past. I put a lot of thought into how to build proofs into our curriculum this past year and I feel like what I did was relatively successful. My students weren't scared of proofs for the first time in my teaching career. When they came up, the students knew at least where to start and always attempted them. So I want to lay down what I did just so I don't forget.So here is the description of how the unit flowed. It's definitely a more traditional approach to proofs and I stuck to two-column proofs. I tried to transition the students to paragraph proofs, but their skills and confidence were too low; they liked the organized nature of two-column proofs. First, I didn't nix the logic unit. Even though logic is not in Common Core anymore, I think that the reasoning done in the logic unit helps prepare students for proofs.

Logic Unit Lesson 1: Intro to conditional statements.

- First I did Sam Shah's lesson introducing conditional statements. I did the drawing activity and posted all their pictures on the wall. It went really well- I was surprised at how much trouble some students had following the directions precisely. A lot of them didn't know the geometric vocabulary (like what an isosceles triangle is) or were hesitant drawing so it was a great activity to do towards the beginning of the class.
- Then I just did a mini-lecture on the notation of conditional statements, Euler diagrams and what a negation is.
- Then I gave them this assignment: Logic Unit Lesson 1: Conditional Statements Intro

Logic Unit Lesson 2: Manipulations of Conditional statements

- I did the second part of Sam Shah's lesson on conditional statements
- I did a mini lecture on the vocab: converse, inverse, contrapositive and negatin
- Finally I gave them this assignment: Logic Unit Lesson 2: Manipulations of Conditional Statements.

Logic Unit Lesson 3: Word Proofs

- First we did a syllogism activity where I just cut the cards apart and had them put the syllogism in the correct order and a mini lecture on syllogisms
- Then we did word proofs. This is one of the most successful lessons I've ever taught on proofs. I totally stole it from another blogger, and of course forgot to save their name in the name of the file I downloaded like I normally do. When I figure out who made it I'll update this post. I reformatted the file I stole from that other blogger and did the lesson in kind of a workshop style. I did one or two of the word proofs on the board to demonstrate how to do it, then I gave them time to work on their own and then we compared answers. I had them come up and show different solutions they'd discovered and we talked about the fact that there's more than one way to do a proof correctly. Like I mentioned before, this was actually a pretty bright class but one lacking in discipline and both basic math and study skills. I had to
*hold the kids back.*They were chomping at the bit to do more and more and more puzzles. I made the last two pages of the lesson optional and almost all students did them anyway. Whoever designed this lesson was brilliant because it really hooked students who are usually disengaged with math. You can't not want to solve one of these puzzles when they're presented to you. - There are two pages in the lesson to have the students make their own puzzles and switch papers with each other. We had to skip this part because we ran out of time.
- Here's the lesson: Logic Unit Lesson 3: Word Proofs

- This lesson was a little less fun than the last one, but it was very effective. First we went over basic angle terminology: complementary and supplementary angles, vertical angles and linear pairs (they'd learned these before), and we also talked about the algebra properties of equality, transitivity and the substitution property.
- Then I had them do the lesson below in pairs. It's structured exactly like the word proofs from the last lesson with 4 or so "rules" and space for them to use the rules to go from the given to the prove. The students were able to stumble their way through these proofs
*without me doing any examples on the board*based on what they did in the last lesson. - Here's the lesson: Logic Unit Lesson 4: Angle Proofs
- Finally we concluded with a Scrambled Proofs Activity

Congruent Triangles Lesson 1: Intro to Congruence

- For this lesson I just did a standard lecture over what congruence is, the notation for congruence and examples of using congruence to find missing parts. Nothing really exciting. The only thing about this lesson that I like is my warm-up. We define congruence as "identical in every way" and then I ask students if the two identical twins are congruent which leads to a great discussion of what "corresponding parts" means:
- Also, I always get a laugh out of my student by choosing another teacher who's the same height as me (at this school I used the principal) and I use that teacher to discuss the difference between the "equal" symbol and the "congruence" symbol. Our heights can be equated, but if you accidentally use the congruence symbol you're saying I and this other teacher are identical in every way.
- Here's the assignment for that lesson: Congruent Triangle Lesson 1: Intro to Congruence

Congruent Triangles Lesson 2: Triangle Congruence Theorems

- I used this cool illuminations app on congruent triangle theorems along with this worksheet that I wrote to introduce SSS, SAS, ASA and AAS. I've done this activity twice, the first time I let the students pair up and use their own computers. A lot of the students wouldn't or couldn't follow the directions so I spent the whole period frantically running around trouble shooting. The more motivated students in the class were able to make the connections I wanted but the other students left class mostly confused. So the second time, we did each activity first as a whole class with computers closed, then I let them open their laptops and play with the simulation to confirm the results for themselves. This worked much better. All the students were successful on the homework without need for more instruction.
- Then I totally stole a few worksheets here and here from Math Teacher Mambo for the students to work on independently. Here's my mash-up of her brilliance: Congruent Triangle Lesson 2: Congruent Triangle Theorems

Congruent Triangles Lesson 3: Using Theorems in Proofs

- This is a lesson I created that I'm exceptionally proud of. It was super boring though. I realized that students weren't really getting what theorems were for or how to use them. They were still struggling with problems like: if angle A and angle B are a linear pair and angle A measures 40 degrees, what is the measure of angle B. They weren't thinking about what "linear pair" means and how to connect that meaning to the problem. Even if they got this far they didn't understand that in a "proving" situation they needed to state how they know that angle A and angle B add to 180 (that they need to say: by def of linear pair or state the theorem).
- So I made this worksheet: Congruent Triangles Lesson 3: Using Theorems to Make Deductions.
- In pairs I had them go through their notes and fill in the blanks for all the theorems we've covered. This was the boring part. I gave them a time limit though which helped keep them focused and this activity also reinforced the importance of taking notes. I refused to tell them any answers. If they didn't have it in their notes they needed to find someone who did.
- The "classwork" part of the lesson is where it all really paid off. I did the first few examples with them- how to cite the correct theorem that had been used in each situation. As soon as I started doing these problems on the board, a bunch of "ohhhh so that's why we needed those stupid theorems" exclamations went through the room. It was especially rewarding to watch them do the last page where they have to think backwards- find the theorem that applies to the situation then figure out what deduction can be made.
- Students guarded the list of theorems they made as the first part of the lesson fiercely and insisted that I do a similar "fill in the blank theorem review" at the end of every unit.

- This lesson went quite well although it's a very traditional lesson. I just did a few example proofs with them and talked about how to set up a two-column proof table (I know... but these students liked structure. I tried to show them a paragraph proof and their eyes all crossed and they started throwing paper.)
- I gave them this packet of problems from letspracticegeometry.com (there are a lot of typos in this worksheet though. I would like to rewrite it) but without the first two pages.
- Then I gave them this proof template worksheet thing that I created. It has the students choose which proofs they want to try, the harder proofs being worth more points. They have to reach a certain number of points to get full credit.
- This small spin on a worksheet created a night and day difference in students' attitudes about proofs. Every other time I've taught proofs students have been super whinny about them and would give up quickly. But when I handed out the above assignment I saw at least half the students immediately turning to the last page to do the harder proofs. A lot of them struggled on the proofs through a good chunk of the period without finishing and I kept suggesting they just go do more of the easier proofs, or build up to the harder ones but they said that they wanted to do hardest ones. The fact that there was a choice between easy and hard involved made them want to prove to themselves that they could do the hard. Students with less confidence started with the easy ones and were able to advance to the harder ones pretty smoothly. Everyone was engaged and no one was complaining that I was making them do proofs.

Congruent Triangles Unit Lesson 5: CPCTC theorem proofs

- I taught this lesson the same as the last one. Examples then a "choose your own problems" proof worksheet. Here's the packet from letspracticegeometry.com and here's the proof template worksheet I gave them.
- I've had trouble in the past with students using CPCTC inappropriately so I put the following message up on the projector in giant letters and made them recite it in unison a few times. I kept it up as they worked on the proofs and I didn't have students misusing CPCTC!

To use CPCTC you MUST

FIRST: Prove triangles congruent

THEN: Say parts are congruent with CPCTC

CPCTC says that:

IF two triangles are congruent THEN their corresponding parts are congruent.

Prove the "IF" first, Only then can you use the "THEN"

- Again, through this class period students were working on proofs
*without complaint*and without giving up. If they started to have trouble they could persevere or choose a new problem and this flexibility eliminated a lot of the griping I've experienced in the past with proof practice. Boring but effective.

WHEW. That's all. I just wanted to catalog what I'd done because this was my most successful proof teaching experience so far. It still needs a lot of work though. And I know that under Common Core, it may not even be relevant anymore because I didn't work in any proving congruence with transformations.

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