Basic Triangle Congruence Proof Tools

I'll be honest- I wasn't quite sure how the start of triangle proofs would go with my Geo with Lab students this year. Not because they aren't capable, but because they were so scared! With students who already have some math anxiety, the anticipation of starting a famously formidable topic was hanging ominously over the classroom our very first day. They had heard things from their older sibling and friends and judged proofs before they ever even gave them a chance. So far, so good, though. No tears. A few "why did everyone make such a big deal of this?" comments. I count all of those in the W column. 

I've been scaffolding proofs slowly, a little more each day. We've been using this super basic graphic organizer to plan before writing anything, making a constant analogy to how your English teacher asks you to outline a paper before you write the final draft. Knowing where your headed before you ever start makes the destination easier to get to, for sure. I printed about 8 of these to a page and have them in sheet protectors in my room so kids can write on them with dry erase markers with no commitment as they plan. I also have a bunch of extras that they can pick up for homework or use on quizzes/tests if it helps them. 

Here's how it looked in action on our first day:

Here's a link if you're interested in using or adapting: 

Proof Planning Graphic Organizer

I also designed this really basic sorting activity for triangle congruence. I know there are a ton of these out there already, but I wanted something targeted for my type of kiddos, who need to build some confidence and hadn't been exposed to everything yet. I didn't even use this with my standard level class because they were ready for more open-ended activities more quickly. There are 3 proofs and all the statements and reasons scrambled for each. My students at this point hadn't seen HL or CPCTC yet, so these are really just getting them to engage with the very first steps of proof. It went well with my kiddos and made them feel some success, which is really half the battle in a lab class. 

Triangle Congruence Proofs Sort

Steal away if this would be helpful to you! And as always, feel free to offer suggestions if you see room for improvement! Happy proving!  

WODB- Differentiation Techniques

We just finished implicit differentiation last class and I wanted to get my students thinking about when to use which differentiation technique as we come to the close of this unit. I made this quick WODB to have students analyze differences among structures. This isn't as true to WODB form as I'd like, but it will still get them talking and thinking. 

Comment suggestions if you have any! 

A: Only one that uses implicit differentiation, only one where point is given (not just x coordinate)

B- Only one asking for general derivative, not derivative at a point

C- Only one that requires product rule, only one not in terms of y (in function notation with g(x)), only one with trig function

D- Only one that does not equal 1/2 (equals -1/2), only one from a table, only one that requires quotient rule

Limit Definition of the Derivative- Multiple Representation Reference Sheet

I pinned this picture years ago when I started teaching calculus and it's stuck with me every year. It's so simple and clean, but shows exactly where we came from and where we're going. Math is so elegant sometimes *sighhhhhhh*. 

I really want my students this year to make concrete connections between representations as often as possible. The limit definition of the derivative is such a foundational and, at its core, simple concept. It's slope. We know slope. 

I made this reference sheet for my kids to get them talking, thinking, and connecting between representations of the derivative. I have them start with their Algebra I definition of slope, then progress through notational changes to derive the formula for slope of the tangent line. I've included the general formula on the front and the definition at a point on the back. 

I'm  giving this as a warm up in class tomorrow so I know I'll be tweaking, but if you see anything you think should be tweaked leave a comment! 

Limit Definition of Derivative- Multiple Representation Reference 

UPDATE: Here is my filled in key from class! I really liked this and will incorporate it again next year for sure! 

Limit Definition of Derivatives KEY

Intermediate Value Theorem Justifications Gallery Walk

One of my favorite parts of AP Calculus is seeing my students' ability to make mathematical arguments develop. This is a long and drawn out process that takes the length of the course to complete. One thing that I found really useful last year was having students critique others' arguments. 

To get students analyzing arguments for IVT, I used this gallery walk and a whole lot of post its. 

IVT Gallery Walk 

Students were asked to critique arguments around the room, offering feedback and identifying one they thought made the best argument. They started to give specific feedback on how things could be improved and identify which was really "true dat." 

We will do more like this throughout the year, but it's still one of my favorite ways to get into justifying! 

Intro to Auxiliary Lines Warm Up & Activity

As we wrap up our first geometry unit this week, one thing we begin to discuss are auxiliary lines. My students already have so much background knowledge on parallel lines cut by a transversal from their middle school days (and our first few days of the year), so I wanted a short task to get them connecting what they've done before with where we are headed next. 

Auxiliary Lines Warm Up

I designed this task as a way to get students thinking independently about how auxiliary lines might help us. The above diagram was given to them, along with a bunch of questions about angle pairs and measures. I had them work on this individually after they finished a quiz, then find a high five partner around the room with whom they could discuss their answers. For many this was a "duh" kind of activity. It transitioned well into more complicated diagrams, giving us a foundation to which we could refer. 

Here's the actual Word file:

Auxiliary Lines Warm Up

Two Truths and a Lie

I was inspired by this one from Math=Love Two Truths and a Lie Activity Template post, in which she had students create their own version of Two Truths and a Lie on a particular topic. This brief activity had students decide which statement was a lie and then explain why. They worked in partners and we debriefed as a class. In my more talkative classes it sparked a lot of debate. In my less talkative it involved a LOT of creeping on students' papers and calling on the right kids to share their answers (which some did not appreciate, especially on a 90 degree day during first block). Overall, though, I liked it and would the format again for other topics! 

Again, the Word file: 

Two Truths and a Lie- Auxiliary Lines

Nothing groundbreaking but if anyone can use it, please steal! 

Storyboarding to Introduce Limits

I had the rare opportunity this year to start from scratch with my AP Calc students when introducing limits. Due to adjustment to anew curriculum, our Pre-Calc PLC just didn't get there last year so we are doing more thorough coverage in class than I've done since my Pre-Calc teaching days. I wanted to try something new this year to get my kids thinking about the concept of a limit. 

I was planning to put this slide on the board:

I'd then asked for 4 volunteers, 2 to be a part of Scene #1 and 2 to be a part of Scene #3. Scene #2 was a mystery that would come back later. I'd give each group a set of a secret directions only they got to see and about 30 seconds to plan, then had to pose to represent their "scene" for the class. 

Scene #1: 

Two friends are having a frisbee toss. Friend #1 is ready to toss frisbee to friend #2, who is wearing sunglasses.

Scene #3: 

Friend #2 is holding broken sunglasses, having clearly gotten hit in the face with a frisbee. Friend #1 is reacting to having just hit his/her friend. 

The class has to study the scenes and make a prediction as to what Scene #2 would look like. Most would go for the obvious choice. A few would go for something crazy (a bird flew in his eye, he got punched in the face, etc), which is great. We start to discuss what it looks like should have happened in that scenario, given the information we were given. We also discuss the fact that this predicted scenario doesn't necessarily have to be what actually happened. The kids grasp on to this pretty easily and it becomes an easy framework through which we can view a limit, especially a limit where the defined value of the function and the limit are not equal. And it's a goofy, fun way to get seniors out of their seat! 

But, of course, I forgot we had a senior assembly this morning and I would lose 45 minutes of my 2nd period.....teacher problems for sure. It remains hypothetical, but I have a lot of hope for it. Sharing if it can be a fun and goofy thinking activity for anyone else and storing it away for a rainy day for myself!  

The First Week: Building a Triad of Responsibility

First Week 2015

First Week 2009

I have a history of very interesting first weeks of school.  There have been good ones, meh ones, and downright weird ones. My very first first day of school ended abruptly in an early dismissal and subsequent closing of school for days due to a massive flood. Two years ago, I sliced the tip of my finger off trying to make pickled radishes and went to school in full middle finger bandage....basically flipping off my classes for the entire first week. (Luckily, my husband got to see silver nitrate in action when I got my finger cauterized and had a cool story to tell his chemistry classes that year...)

NYC Math Lab Triad of Responsibility

This week was, by far, my favorite first week I've had in my 7 years. I loved all of my classes, felt like I got to know more than usual about my kids, and it's been the most perfect September weather here in Upstate NY. I tried a bunch of new things and brought out some old favorites. I celebrated with coworkers and went to bed by 8:30 pm on Friday. 

The biggest difference maker for me this year was the marriage of 2 different activities I learned about this summer. First, I had my classes participate in Sara Van Der Werf's 1-100 Task as one of our first day activities. I decided to fuse this with one of my experiences from NYC Math Lab this summer. In Math Lab, we used the Triad of Responsibility to help set norms for the students. I loved this idea and thought it was a natural fit for the first day. 

Before we started the activity, I had my students individually brainstorm their responsibilities to themselves as students. It was amazing to see the difference between answers in a Geometry with Lab class (1.5 times the class minutes of a typical Geometry class) and an AP Calculus class. 

Side Note: The Calc ones almost made me sad....so grade oriented, no examination of yourself as a whole person. It would have been how I answered in high school and it's part of the reason I love doing this particular work with these kids. I hope they all begin to see that they are valuable for more than just a number. 

Next, I had the students perform the 1-100 task as described in Sara's blog. Overall, we did it 3 times and discussed group norms or "Responsibilities to My Group" in reference to the activity. 

Lastly, we discussed "Responsibilities to our Classroom Community" in reference to our whole group discussion. Each class had common themes, but brought many unique answers. Even if lot of math teachers aren't into them, it was a warm, fuzzy day and I ate it up. 

Here are what my kids generated from our conversations:

These are nice to hang in the classroom, sure, and I've already found myself referring back to them in our first 3 days of instruction. But you cannot imagine the difference this short activity made with my students.  I have seen the type of group work I normally don't see until November or December during the 3rd day of class. I have seen students checking in with each other to make sure everyone in the group understands. I have seen communication from every student in a group. Students are taking small risks, testing the reactions of their classmates and seeing that it is okay to be wrong. And they're asking questions! I'm very interested to see how this initial sense of community carries on through the year and I feel a huge responsibility to live up to the expectations of a safe environment that I've set from the first day. Anyone have any favorite ways to keep that sense of community and safety going all year long? 

Best of all, the posters got approval from a very harsh and deciphering academic who only wags her tail at her favorite first day strategies.