Math Chat: Routes, Rates, and Spooky Skeletons

These past few weeks, my kids and I have been on the hunt for Halloween decorations, specifically skeletons. So to aid in this endeavor, we have been playing our favorite Halloween songs (a classic like Chumbala Cachumbala is a must!) and using our commute to scout out decorations. Since we have a bit of a drive to and from our weekday locations, we find new routes as an excuse to snoop on other families’ yard decorations. 

My kids, creatures of habit that they are, noticed that these route changes meant we were arriving home at different times. This simple observation opened the door to discuss a fundamental mathematical concept: rates of change.

In our spooky scenario, this led us to a conversation about why our travel time was increasing. I asked, “Well, why do you think it’s taking longer?” 

Their answers were pretty insightful: “Because you slowed down to look at people’s decorations” (they weren’t wrong!), “Because the streets we’re taking have a lower speed limit”, and “We’re probably just going a longer distance.” 

These were all excellent observations, and naturally led into discussing how a rate works. In this case, the rate concerned how much distance we covered in a certain amount of time. Now, I didn’t use the word “rate” in this conversation with them (though I was tempted!), since my kids are in elementary school and the concept is a bit advanced. But even at their young ages, they know that things change, and that these changes can be related to other changes. They could see our arrival time was changing, that the distance itself was changing, or that my speed (which is a relationship between distance and time) was changing.

And this conversation is more accessible than you might think! While we often associate rates with speed, but we can think about them in tons of other contexts. Like the classic Halloween scenario: candy per kid (can you tell that Halloween consumes my thoughts this time of year?). Imagine you have a bowl of candy and tell everyone they can take 2 pieces. What happens if some kids sneak an extra? If most kids take 3 pieces, we might run out of candy! Or, consider picking up Legos or Magnetiles. If one person only picks up 5 pieces when they were supposed to pick up 10, how does that affect the number of pieces the other group members have to pick up? 

Throughout all of time, only one constant remains: things change. It’s one of the primary reasons why mathematics exists! We needed a way to understand the relationships between our ever-changing world, in a format flexible enough to track these changes while they change.

So, while my kids are learning about changing distances and times, I’m teaching my calculus students about how rates of change themselves can change. It’s a topic that arises in both the classroom and in life, and it can start with a simple conversation about some fall fun.