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.
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.