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! 

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. 

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. 

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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!