Tuesday, February 27, 2018

Which Ratio? Activity for Right Triangle Trig

One of the things my geometry kids consistently struggle with each year is deciding just which ratio to use for each problem. They can reliably go from there, but when you get stuck on the set up, it can be a SINE that you're lost!

I tried something new this week to combat this issue and so far, it's been really effective. My lab students are especially benefiting as I see them consistently being able to identify the ratio and therefore gain an entry point into the problem.

Pam Wilson wrote a great blog post about an activity she did with her students- read it here- and I stole the document she created to adapt to my kiddos. I put each one of these on a slide and students are asked to analyze (think-pair-share) style which ratio is represented in each picture.

My favorite part is I gave each student a set of these paddles to participate. They were quick to make....run an exacto knife through some cardboard and hot glue popsicle sticks to the back of each (how most people spend their Sunday nights, right?!).  Instant feedback for me, fun for them....an all around win. I've been just leaving these on the tables as we work and I'll ask the kids to periodically show me which ratio they think we should use before we attempt a problem. This has been especially helpful with word problems because I do a quick analysis before moving on to make sure we've "got it" and know how to approach it. 

Also, I know this could totally be done in a Kahoot or another way, but something about the hands on "auction" style get the kids super engaged and I love that I can continue to use them throughout the unit. A nice tool to keep around whenever necessary, not just on "Kahoot" day.

A small thing but so far, especially for me struggling learners, it's made a big difference! 

Thursday, February 8, 2018

Differential Equations CSI

Differential Equations are a topic with so many applications in the real world. Knowing how rough the Potato Question was for AP Calculus AB students last year (49% of students earned 0/9 points), I wanted to bring some more of these applications to my kiddos within our unit. Luckily, I'm married to a scientist and he's overflowing with ideas to help me out! 

First, we looked at rate laws in Chemistry. First order rate laws are a direct application of the law of exponential change. Honestly, these have more to explore when we get into definite integrals, but I wanted students seeing them for the first time now so we can continue to work with them when we start definite integrals. Since about half of my class is enrolled in AP Chem, this was an easy connection for them to make. We also got to discuss why they're called Integrated Rate Laws (oh, THAT'S what the chem teacher was talking about!) and most were able to easily tie this to the derivation we always do for this. 

After this and some more examples, I handed out the lab for the day-

 Determining Time of Death with Newton's Law of Cooling.

I modified it from an old Houghton Mifflin activity, adding in the information about forensic science. I used this article on using ODE's to determine time of death as a resource for what I added, if you're interested. I felt like I'd scaffolded it enough and was excited to get my kids working.

When they walked in, I had the room set up like a crime scene. My goal was that they'd solve the murder before they left. I had a "breaking news" picture of the murderer poised and ready....it was obviously the Night King from Game of Thrones. But I'll be honest....I should have left more time for this. It felt rushed, so we didn't finish it. My students this year struggle a lot more with fundamentals than in years past, so properties of logarithms that should be 2nd nature were taking way more explanation than the time I'd allotted. We got through the first page- the separate and solving for C. We analyzed what C represented. The rest we will save for next class.

I still love the idea and the intrigue it created for the kids- walking into a "crime scene" and being responsible for solving it. Next year, hopefully with less snow days, we will leave ourselves more time to make sure our algebra skills are on point and we have time to really solve the mystery! Next year I also want to tie in temperature probes and have students actually gather their own data, instead of using the data provided in the problem. 

Sunday, February 4, 2018

Hands On Slope Fields

Slope fields have always been a weird topic for me...I never loved my approach to them. They weren't hard for my kids, but they weren't particularly engaging and I always saw my student getting frustrated by the "stakes" of doing them on paper and feeling like they were doing it wrong before they got a real feel for it. I wanted to have my kids be able to experiment with it in a hands on way before having them commit anything to paper, so I've been tweaking this approach for the past few years and felt like this week it finally went the way I'd like it to go! 
I started the class by using Rebecka Peteron's Slope Field Activity, where each student is given a coordinate point and then asked to come up to the board and draw a small segment with that slope. We used these discussion questions pictured to talk about isoclines and beginning to see patterns beyond just "plug and chug" methods of determining information from the field itself.

From there, we did some low-stakes practice with the interactive slope fields I built. To make them, I went to Staples and bought washi tape, brass paper fasteners, and a roll of packing cardboard. I cut out 1 ft x 1 ft squares of cardboard, used washi tape to make the axis, and put the fasteners through the back. In class, I showed the students a differential equation and they modelled the slope field for me. We were able to work out kinks and make sure we all understand what they should look like before going any further. Then, I would ask the students to trace a particular solution using their Twizzlers through a point. We were able to discuss the general vs particular solution, the patterns we saw, and more. All of this was followed by a small group exploration where students played with dependence on x vs y, determining particular solutions, and relating differential equations and slope fields in different representations. 

I still want to work on the type of practice we do after the exploration and relate it to some more applications, but this approach was way effective for my kids than normal. I have a group this year who really benefit from more visual and hands on approaches anyway. 

Do you have any other approaches to slope fields that you love? Share in the comments! 

Tuesday, January 16, 2018

Learning to Live-Stream: Review Sessions from my Couch

This year has been a hot mess of delays, snow days, and general schedule interruptions that have gotten me a little jumpy about my AP pacing. Yes, I know the exam is a week later this year. Yes, I know I had 5-6 weeks of review the last 2 years. Yes, I know we started behind since our Pre-Calc courses didn't get to cover the final unit as thoroughly as in years past. But like most teachers, I have my "checkpoints" of where I think I should be at certain points in the year and midterm week has gotten me feeling a little spooked. 

My kids wanted a bit more review that I just didn't have the time to give them for our last test, so we came to a compromise....I would figure out how to do a live stream review if they would watch it. We'd try for an hour the day before the test and see how that went. 

Anyone who has been around my blog for a while knows I'm a huge devotee of Notability and knew I wanted a way to live stream my iPad so  I could write by hand. After doing some research, I came across AirServer,  which took my normal airplay capabilites and allowed me to tie in live streaming directly to YouTube. Students saw my screen and a small image of me in the corner....good for the Italian in me who needs to talk with my hands constantly. 

I knew I should have majored in interpretive dance of ladders falling down a wall at a constant rate. I truly seemed to have missed my calling. 

My favorite feature of doing this was that students could interact with me live, so it wasn't just another video they were watching....they could actually chat. I gave them option of either using the chat on the live stream or if they didn't want everyone to see their questions they could send them on Remind. Questions looked mostly like this:

But occasionally like this:

A nice part of having the questions come in Remind meant that I had the power to answer whichever I chose....ignoring the horse sized duck I was about to take down until a more appropriate time in the conversation. 

The kids seemed to enjoy it and I saw big payoffs on their test. I am going to be trying it again this week for my midterm review with younger students...we will see if it's as beneficial for non-AP students. Though more than 2/3 of my class watched and interacted on the live stream, only 6 filled out my feedback form (THANKS, GUYS). Here's what they had to say:

And my final favorite feature?

YouTube archives the live stream so it can be accessed immediately by students who weren't able to watch live or who didn't want to watch live. They can speed it up or slow it down or just skip to the parts they need. 

I will definitely be tweaking for next time, but overall I thought it was a positive first attempt and reached students in a really accessible way for them on a day when they otherwise would have been unable to have the chance to review with the help of a teacher. 

Sunday, January 14, 2018

Dilation Constructions Mini-Project

As you may know from wherever you are currently reading this, it's been quite a winter so far in the great Northeast. We've already used 2 snow days and have had a 2 hour delay for cold, along with a fair number of cancelled after school activity days. Not to mention we've had a ton of stomach bugs and the flu bouncing from student to student like a pinball game. Needless to say, the best laid plans of math teachers have gone astray once or twice.

Since we run on an A/B day schedule and almost all of our time lost has been on B days, one of my geometry with lab classes inevitably zoomed in front of the other. In an effort to get them back on track, I designed this mini-project for my class that was light years ahead. I liked that it was creative, gave them some choice, and had them really practice the skills of constructing a dilation. Understanding the construction so strongly ties to the conceptual understanding of both dilations and similarity, so I knew this wouldn't be time wasted. 

You don't need much for the project. To prep, I printed tons of tiny images of famous characters. I went cartoons, but you could have kids bring in their own or design their own "logo." I had students put a dot at the center of the top of a sheet of printer paper, then glue down the original. This would be our pre-image and our center. From there, we started constructing! Students identified significant points and then created a "connect the dots" to help them draw the dilated figure. Many experimented with scale factor, trying to find the one that would be the largest without falling off of the page. 

After completing their dilations, students wrote a reflection relating what they'd done that day to the Desmos activity (Working with Dilations) we'd been working on prior to the activity. They needed to discuss center, projection lines, scale factor, and ratios. I've seen this play out in a deeper understanding of dilations in the remainder of our unit and a stronger ability to work with centers off the origin and dilations off the coordinate plane in general. 

Here are a few of my favorites! 

Thursday, January 4, 2018

Perimeter & Area of Similar Figures

We're in the heart of our dilations and similarity unit in Geometry and my students have started being surprised when we aren't using the computers that day. So thankful for our recently expanded access to more devices for our kiddos. 

I love starting my unit with this amazing Working with Dilations activity. It builds the concept of similarity from similarity transformations, which I love. It also helps students visualize how the center of dilation and the k value influence the image. It made the compass and straight-edge construction for a dilation a "duh" kind of moment....OF COURSE that's how you do it! 

Students doodling on pattern
block mats to show their thinking. 

I designed the next activity so students could discover the relationship between scale factor and ratios of perimeter and area. Students use a combo of interactive shapes and pattern block mats to discover the pattern! I took the idea from a Big Ideas Math activity and adapted it:

Perimeter and Area of Similar Shapes Desmos Activity

Overall, it went well. My students this year are struggling a bit more with abstraction that they have in the past, but they enjoyed the activity and once one person in the group saw that pattern they were excited to see it for themselves.

Feel free to steal, adapt, or offer suggestions for improvement!

Monday, November 20, 2017

WODB for First Derivative Test

Using this as a warm up after our first day of curve sketching. Feel free to use or offer feedback, if you have any!