# The In-N-Out Problem

I did “The In-N-Out Problem with my Algebra Support classes this week. See “The In-N-Out Problem” here.

This conversation introduced our lesson:

Me: “So, have you guys ever been to In-N-Out before?”

5 minutes of discussion, which inevitably includes one kid screaming across the room to another kid, “WHAT?! YOU’VE NEVER BEEN TO IN-N-OUT BEFORE?!”

Me: “Great. Have you ever heard of the ‘secret menu’?”

5 more minutes of discussion, which inevitably includes half the class yelling that they’ve never heard of a secret menu and demanding to know how you can see the secret menu, and the other half of the class still yelling about how much they love In-N-Out.

I introduced the idea of the 3×3, and 4×4, and showed them the picture of the 20×20 (see link above).

5 more minutes of discussion, which inevitably includes someone thinking they are HILARIOUS when they yell out that the picture looks like diabetes.

Me: “No, diabetes is caused by too much sugar. This is heart disease.”

We did noticing and wondering with the 20×20 picture, which always results in these statements/questions (usually in this order):

1. How do you eat that?!
2. That looks gross.
3. You can order that?!
4. Who would order that?!
5. How much does that cost?! [fantastic – this is the one I wanted to hear]
6. How many calories are in that?! [this one’s good too, but the cost one is easier to answer]
7. Miss, I’m going to order that the next time I go to In-N-Out.
8. Miss, I’m going to bring that to class tomorrow! [this kid is usually trying to outdo the one who said #7, and I have to make it very clear that I DO NOT want that in my classroom]
9. How long does it take to eat?
10. How much does it weigh?
11. How long is it?

I ask them which questions they can answer during class today, and we agree that the cost and calories are the only ones. Then I ask which one they want to answer, and (fortunately) they generally agree on the cost question.

I randomly assigned students to groups of 3, gave them big (2ft by 2ft) whiteboards, and told them to figure out how much a 3×3 costs.

## The Work

One or two groups figured out right away that they needed to isolate the meat and cheese, since that is what’s getting added to get the bigger burgers. The rest of the groups (about 14 total between my 3 classes) were split pretty evenly between adding the cost of a single and a double-double, and multiplying the single by 3.

I talked with each group about their approach. We had lots of conversations about how 3 cheeseburgers, or the double-double plus single results in too many buns and toppings. Some groups picked up on this difference faster than others. (I got a lot of blank looks at first. In some groups I actually forced them to write out a list of components in a single, a double, and a 3×3. Then I asked them what was different.) Some groups figured out that subtracting to find the change in cost was helpful. Some got stuck on the fact that the cheese is \$0.25, and couldn’t figure out the meat AND cheese. All of them eventually reached the point where they figured out the rate of change, or the cost of 1 layer of meat and cheese. I made them write that down on their boards.

They were super paranoid about other groups copying their work, which I thought was funny. (I actually felt like they were more paranoid about it when they had wrong answers on their boards. Maybe they didn’t have the time or extra brain power to worry about it when they were on the right track.) I told them to leave the boards where they were repeatedly. In one class, someone yelled across the room at me that A IS CHEATING MISS HE’S LOOKING AT OUR BOARD! and I turned around and said, “A, way to use your resources,” and gave him a high-five. The entire class stared at me, before making a mad dash to look at everyone else’s boards. I did tell them I didn’t want them talking outside their groups, but looking is fine.

So I had each group do a 3×3 and explain their process. They had to explain whether they were right or wrong. I tried really hard not to ever tell them if they were right or wrong, and just make them explain until either we found a hole in their reasoning, or their reasoning held up under examination. Then I told them to do a 4×4. This is easy if you’ve done a 3×3.

Then I told them to do a 20×20, like the picture they’d seen earlier. Their eyes bugged out of their heads. They looked at each other and exclaimed, “20 BY 20?!” I walked away, partly so they could get to work without me being involved, and partly so they wouldn’t see me grinning.

There were 2 main approaches to the 20×20. 1: Multiply the rate of change by 20, then add the cost of the 4×4. 2: Calculate a 5×5, 6×6, 7×7, etc. until you reach 20.

If students chose Option 1, I had to talk them through the logic again. Ok, so you’ve got 20 slices here, and you’ve got 4 here. If you add those two together, how many slices do you have altogether? Yeah, 24 is too many. Then I’d walk away again.

If students chose Option 2, I’d usually cruise by their workspace at some point during the process, and I’d make some offhanded comment like, “Hmm, you’re really going to regret doing it this way when I tell you have to do a 50×50 next.” Their eyes would bug out of their heads, they stare at each other, and exclaim, “50 BY 50?!” And I’d walk away so they couldn’t see me laughing.

Anyway, at some point the groups who chose Option 1 would figure out they needed to do a 4×4 plus 16 extra slices instead of 20, and the groups who chose Option 2 would finish their calculations (mostly correctly).

After getting them to explain their reasoning again, I’d drop the bomb. “100 by 100.”

I feel like I should stop here and explain something. One of my favorite parts of my job is “scaring” my kids. I love to assign them something that seems impossible and watch them figure out how to make it possible. I love seeing them overcome challenges, and try new methods, and persevere until they get it. Their incredibly melodramatic reactions are just icing on the cake.

Anyway, I tell them they need to do 100×100 (the kids who thought they were going to a 50×50 next feel betrayed so I shrug and tell them I lied) and they freak. It’s hilarious. And I walk away. (Again, mostly so they can’t see that I’m laughing. Maniacally.)

One group insisted on adding the rate of change 80 more times to get to 100. It was a horribly painful process, partly because I kept trying to guide them to a shorter method but B, who was the group’s de facto leader, was being stubborn, and partly because they did it wrong. They got the wrong answer, and since B kept erasing her work because she kept running out of space (80 addition problems using the standard algorithm take up a lot of space), I have no idea what happened. All I know is that she finished, yelled across the room at me that she was done, and I took one look and yelled back “No.” She was off by about \$4. I explained that this was why I had been trying to get them to use a different method – adding something 80 times introduces A LOT of opportunity for error. (Especially after I took away her calculator.)

Most groups calculated a 20×20 with 80 extra patties and cheese. A few groups calculated a 4×4 with 96 extra patties and cheese. One group calculated a single with 99 extra patties and cheese. (This was after quite a few failed attempts on their part, so I think they had just gone back to the drawing board completely to come up with a new method, which is why they didn’t start with the 20×20.)

## What did I see?

Perseverance. I saw 2 class periods of students who did whatever it took (albeit with a fair amount of scaffolding in some situations) to answer the question. (1 class period didn’t get to finish the problem on a second day because of some behavior issues during their work.)

CCSS Math Practice 1 says “Make sense of problems and persevere in solving them”. This is the full description, with my annotations:

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. [I scaffolded this through my introduction to the problem, and my instructions to do 3×3, 4×4, 20×20 before the 100×100.] They analyze givens, constraints, relationships, and goals. [We discussed what the burger includes, what changes, how the price changes, etc.] They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. [One kid gave a really cool estimate for what the 100×100 would cost and explained his reasoning. He got his estimate written on the board. Most of the time, however, the kids did tend to just jump into solving. I could work more on discussing their plan before they begin.] They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. [Through my scaffolding, they did this.] They monitor and evaluate their progress and change course if necessary. [I forced this by checking their answers to each part of the question – 3×3, 4×4, etc.] … Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. [We didn’t use enough diagrams, but we did make lists of burger components sometimes. Every group did eventually find the trend that was the rate of change – one patty and slice of cheese.] … Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” [Again, I scaffolded this by asking them to explain their reasoning and then questioning them through the gaps in their reasoning.] They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. [We didn’t really do a very good debrief after the problem, so I didn’t see this. I could have done better at this.]

So, next time: Have groups discuss their plan (maybe write down their strategy?) before they begin to calculate. Encourage the use of tables or graphs or other representations to organize their data. Debrief at the end of class to see different approaches to the problem and identify similarities in the methods.