“You have a choice of which version of the game to play,” I said, as I placed out four different gameboards and spinners for our focus game.

Tyler’s eyes narrowed. “But I don’t *want* to play the game,” he protested. “I’ve already played it.”

I wanted to roll my eyes, but I have enough teaching experience to know better. “Yes,” I continued. Everyone has. But today, you get to work with the Stage 4 gameboard. It’s different, and there’s something extra challenging about it! What do you notice about it?”

Everything about Tyler’s body — his pursed lips, his tense shoulders — said that he had no interest in playing this game, and perhaps less interest in talking to me. He continued to lament: “I already have it all memorized.”

“This isn’t really a memorization game,” I said, even though I knew it was time to walk away. The game did involve a lot of reasoning about quantities and strategy about moves to make. Tyler had done well with the first stage of the game, but I wouldn’t have said even that seemed “too easy,” as he claimed it was. “You can just memorize it in two hours.”

It didn’t matter. Tyler was done. We had entered into a power struggle, and there was no way he was going to engage mathematically at this point. I was also nervous that his negative attitude might infect other students who were quite cheerfully playing the game. They were noticing patterns and making connections. Meanwhile, Tyler had convinced his friend Dylan to sulk with him in the corner.

____

Engagement is a tricky thing. Some students may find a prompt motivating, pushing them to go deeper and deeper with the mathematics. Another student may look at it and think, “that’s easy and a waste of my time,” even though the student has not even broken the surface of possibility.

Some materials look deceptively easy. My interactions with Tyler this morning reminded me of a sign I had seen on the highway earlier this week.* (photo at left)*

It looked like such a simple doubling pattern. **But what do these numbers even represent?** Also: the car I was riding in was traveling 18 mph when I took the photo. **Should I expect that we will continue at a roughly constant? Go faster?**

There are a few ways I thought to approach this dilemma of speed. I decided to focus on the first distance — to get from the spot we were located to the intersection with Rt 95, a 7 mile trip predicted to take 14 minutes.

That meant that it would take me 2 minutes to travel 1 mile, on average. (Of course, highway traffic on a rainy Friday is anything but predictable, but the stop and go should even out over the 7 mile trip.)

Or I could say that, in 1 minute, I travel an average of 1/2 mile.

Notice that we’ve now identified two different unit rates: 1/2 miles per minute and 2 minutes per mile. It’s true that we can identify two different unit rates to describe the relationship between any two quantities in a ratio. Even students that proficiently determine unit rates may not have wrapped their heads around that yet. (Look at those lovely reciprocals that emerge, too, taking us deeper into the connection between ratios and fractions!)

To determine the standard measure of motor speed in the US — *miles per hours, or mph* — I would then need to find the rate of miles per 60 minutes of driving. Both of these unit rates are pretty helpful in calculating the mph. I could use a scale factor of 30 on the minutes per mile (2 minutes per mile –> 60 minutes per 30 miles) or a scale factor of 60 on the minutes per mile (0.5 miles per minute –> 30 miles per 60 minutes).

This is all well and good if I use the data about the time it should take to travel me the 7 miles to Rt 95. What if I want to go the 13 miles to Rt 125? This sign suggests it will take me 25 minutes to travel the 13 miles, which seems like a faster rate than the first leg. I might be able to shave an entire minute (maybe even two!) off of my average rate for the first 7 miles.

I decided to use the ratio of minutes to miles. If it takes 25 minutes to go 13 miles, it will take 5 minutes to go 13/5 miles (2.6 mi). An hour is 12 times longer than that, so the distance I can travel in an hour is 2.6 • 12, or 31.2 mi.

The numbers are decidedly less pretty using the ratio of miles to minutes.

Hideous! I do not want to calculate anything with thirteenths on a Friday afternoon!

##### This then starts to get me thinking about how our speed is only representative of a single moment in time. And how big is a moment in time? This links back to the first chapter in Ben Orlin’s latest, *Change is the Only Constant *(October 2019). I highly recommend the book. (Yes, I’m biased because I am Ben’s sister, but it’s also fantastic, funny, and thought provoking.)

There is a great lesson from *Illustrative Mathematics 6-8 Math, vIII* about precisely this idea of twin unit rates. It’s Grade 6, Unit 3, Lesson 6: Interpreting Rates. (Link goes to Kendall Hunt instance of the curriculum, which is freely available) Students are asked to calculate the two unit rates for a series of several quantities.

Grade 6, Unit 3, Lesson 6: Interpreting Rates

(*IM 6-8 Math™ v. III certified by Illustrative Mathematics*)

In fact, I was looking at this lesson yesterday with a team of sixth grade teachers in Colorado. An experienced teacher told me that some of her students struggled with the milk problem (problem #2) last year because of the fractions getting in the way of reality.

If Mai’s family drinks 10 gallons of milk every 6 weeks, that means that they drink 10/6 gallons of milk every week. That seems reasonable. I can picture 1 2/3 gallons of milk quite easily.

The trouble sets in with how many weeks does it take for a family to consume 1 gallon of milk.

What is 6/10 (or 3/5) of a week? I think that’s a fun dilemma to think about. Well, 3/5 of 7 days is 4.2 days. 0.2 days is 4.8 hours, which is 4 hours and 48 minutes. (So we arrive at the hyper specific amount of 4 days, 4 hours, and 48 minutes to drink one gallon of milk. This is where the calculus starts to sneak back up. I believe that the family buys 10 gallons of milk over a 6 week period, but are they drinking at a constant rate? Derivatives…)

So the team thought that this might be fun for us, as a group of nerdy math teachers, to explore, but that it would be excruciating for some of their sixth grade students. Fair enough. So I decided to help them change the numbers so that it that we wouldn’t have to deal with fractional days.

“Oh! If we want it to be divisible by 7 for the 7 days, let’s do 10 gallons of milk in 7 weeks,” I suggested. Problem solved!

…at least, the problem was solved until I actually solve the problem, and realized that this did not result in a nice, whole number of days.

I realized my error immediately. If I want the number of weeks to be chunked into 7ths, that means we need to multiply by a scale factor of 1/7 starting at the number of gallons of milk, not the number of weeks. We switched it to 7 gallons of milk over a 6 week period.

This resulted in the completely reasonable 7/6 gallons of milk per week, and 1 gallon of milk every 6 days.

In order to engineer the problem to our design, I needed to understand the mathematics much deeper.

________

I suppose this is my fantasy response to Tyler, the fifth grader who grumbled about being “bored” by the idea of re-playing a game ripe with opportunities to mathematical experimentation. The mildly vindictive part of me would love to share all of this with him. This is what it feels like to go to deep into the math! This is what it feels like to connect and add onto understandings! This is what it feels like to journey through twists and turns, emerging at some new ideas about a familiar topic!

Unfortunately, this would never convince Tyler. I accepted his feedback, and told him that we *would* be introducing another game next week, but that being a mathematician involved a lot more. It involves looking for patterns, and making connections, and comparing and contrasting others’ ideas.

“I don’t want to be a mathematician,” Tyler said, “I just want you to give me hard math. Math that’s not boring.”

“What’s hard math?” I asked, doing my best to be earnest, but inevitably betraying some of the skepticism that coursed through my veins.

“You know. Like big numbers. Algebra. Did you know that I can solve algebra? I don’t want to play games about division and fractions!”

As frustrated as I get when I think a student just wants math that *looks* hard, rather than math that *is* hard, I do need to take into consideration what Tyler is feeling here. At the root of this, there is something about his identity. It doesn’t matter to him that other students are working on algebra and *also* willing to engage in deep learning about fractions. He isn’t interesting in consuming the kind of math that other people find interesting. What is it about solving simple algebraic equations that’s more satisfying for him? How can I acknowledge how he feels and also help him change it?

More succinctly: how can we hook Tyler on mathematical thinking for real? …and regularly? That’s a question I need to dive into — even if I would rather play with more rates and ratios on this cold and gray afternoon.

So thought-provoking! I don’t have a solution, but I am in the same situation to a certain extent with one of my students… Have you shared on your blog the games that you refer to here? I’d love to see these “games about division and fractions.” 🙂

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I will PM you about the game. 🙂

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