## December 19, 2014

I came across this activity way back when - during student teaching. I've modified it a few times and while this year was an improvement I'm still not 100% satisfied. I had to do a ton of hand holding today. So this post is a stream of consciousness as I work through what happened and how the activity should adjust to reflect that. Help me out?

Your class needs to raise \$100 to go on a field trip.  You decide to sell cups of iced tea and lemonade after school.  At the stand, iced tea costs \$0.50 per cup and lemonade costs \$0.80 per cup.

We started noticing and wondering with just the situation, no prompts. This was a particularly engaging context for some of my students since they had run a business in their math class last year. I clarified a few things based on their commentary (ex: assume all the supplies were donated so you have 100% profit) and told them that I wondered "How many cups of lemonade and iced tea will we need to sell to get \$100?"

Then I handed out the photocopy and told them to find 3 possibilities. Then check in with me and I'd give them graph paper. Everyone was able to start, they grabbed a calculator and tried something. A majority of them could figure out 200 cups of iced tea was a solution. Not everyone recognized it as (200 ice tea, 0 lemonade). Of those, some went directly to lemonade, others I gave minimal prompting got them to (0 ice tea, 125 lemonade). Everyone who had started with one or the other needed a lot of guidance to get a third point. I'm thinking that two points might be enough, and I'll just let them assume the situation is linear because we're studying lines. They can pull some other points off the graph later. So the pressing issue in (after?) #1 is getting students to see the solutions as ordered pairs. Many students who found another solution, say, (40 ice tea, 100 lemonade) recorded the final answer as 140 cups.

Ideas:
give them a table (that's more structure than I want in the first question, so perhaps #2 becomes a table)
give them some sentence frames (I was pushing to get them to write "40 cups of ice tea is \$20, 100 cups of lemonade is \$80") so that they clarify their thinking
What about a 5 column table? Cups ice tea | Profit from ice tea | Cups lemonade | Profit from lemonade | Total Profit
That's too much structure to have anywhere on the paper while they're pondering - but I could leave the situation and the question on the board, then after students have done some work in their notebook they could have the handout which would start with that five column table... I might like that best because that will also assist with the equation writing process.

OMG, making a coordinate plane on blank graph paper was such an ordeal for several kids! I told them to count by 10's so the graph would fit on the page and a few of them put 10 after one box, then skipped a line and put 20 after two boxes. This is a conversation worth having (If the first box is 10 units, all the boxes are 10 units on that axis.) so no changes here. Just a reminder to myself to watch for this error and make sure to watch when the kid restarts so they don't have to do it three times (sorry kid I walked away from!).

More than one kid remarked on the fact that the line was decreasing as soon as they drew it. If they're noticing it, let's capitalize on this. Ask if it's increasing or decreasing and what the slope is. Then ask what the slope means. For my fundamentals kids I think a fill in the blank sentence is the way to go here. I want them to say "As they sell more cups of ice tea, they sell less cups of lemonade to maintain a constant profit." So we'll start with "As x increases, y _____." Then put it in context and explain why this makes sense?

Now they are well equipped to find a variety of other points. They need to check that each point actually works to give a profit of \$100 (especially since the scale is so... big? small? uh... especially since each box is worth 10). So let's go back to that table from before and fill in some more rows.

By now they've done enough repeated calculation that they're ready to write an equation. In class today I had to show them how to write out their one point that wasn't an intercept in a single equation, and then they were able to substitute the variables in to generalize. Actually, let's make the last row of the table x | ___ | y | ___ | ___. Then they can pull the equation almost directly from there. Nice.

Then let's capitalize on the fact that they're bound to have found a point that requires selling a fraction of a cup. I think the color coding questions are okay. I like that there are a variety of answers that don't make sense (fractions, negatives).

The last part is to rewrite in y=mx+b form and recognize the parts. I am not sure if this is worthwhile. It is a nice aha moment when they solve the equation and then see that b matches the y-intercept and m matches the slope. So maybe it's helpful for making connections to one of the ways we've written equations in the past?

Okay, I put some of those thoughts into a word doc. Here is the newest draft, eagerly awaiting your ideas. Remember, they will notice and wonder, then work on finding at least two solutions before they get the handout. Oh, and if you're interested, here's the version I used 5 years ago when I last taught Algebra.

1. Thanks for taking the time to share all of this. I think you've documented some great observations and ideas. Hopefully I can revisit some, but I'll stick to the beginning of the task.

1) I really like how you started with the situation and no prompts. What would you think about subtracting the first sentence?
Poll the students about what a reasonable dollar amount would be for this sale and their field trip goal. I bet their answers will be all over the place, so maybe use some average or mode. I think the important discussion to have is getting students to argue why \$50 might be too little and not worth their time or \$1000 might be too high and selling refreshments might not be the best vehicle.

2) Say you take the median or coax them toward \$100 as the goal, would student guesses help? Could we simplify this to a quick number sense opportunity: if they sell only 100 iced teas, will they reach their goal? Convince me.
If they sell only 100 lemonades, will they reach their goal?
Convince me.
What if they sell 100 of both?
What if they sell 100 iced teas, what's the fewest lemonades they could sell?
What if they sell 100 lemonades, what's the fewest iced teas they could sell?
* I'm rambling.

3) For the kids who struggled with giving a third solution (combination), what about asking them to forget the \$100 and use something extremely small like \$4.50?

1. I'll echo Andrew's praise and suggest an extension along his lines. With smaller values such as his \$4.50 suggestion it is probably easy enough for the kiddos to come up with a variety of sales combos that match the target value. Having multiple solutions might nudge them along the path you hoped for with the \$100 goal. You can also have them talk about maximum and minimum values for the number of drinks to sell (and prepare) to meet these goals. Lay the groundwork for optimization problems later on.

2. I think your line of questioning in 2 is exactly what we need. It's a bit more structured (instead of pick any number, we ask what if we sell 100?) and will get those students who were stuck on open endedness moving.

We had some conversations about the reasonableness of pricing, goal and potential timeframes even with that first sentence included!

2. By the way, I also want to know more about the blank paper coordinate plane ordeal.

1. Tina: "You have earned your graph paper. To make the whole graph fit on one page you won't be able to count by ones. I think counting by 10's will be fine"
Student: "Okay!" Heads back to seat and writes |0 |10 |(blank) |20 |(blank) |30... Runs out of space before reaching 200. Stops.
Tina: "Oh no! Look what happened: this first box is 10 units. But then it takes you two boxes to make 10 units the rest of the time. All the boxes have to be the same amount on the x-axis. That means that every box has to be worth 10 for it to fit."

I think it's because cm graph paper is small so writing every multiple of 10 on every line is a bit squishy. So kids knew from the past that it was okay to skip numbers and skip lines. The skipping lines combined with the one box=10 units meant that things got mixed up and they were inconsistent.

2. Gotcha.
Did Desmos make an appearance at anytime?
The part I struggle with is when to use Desmos in the sequence of an activity like this.
Would I want my students to graph by hand? Yes.
Can I afford the time if they make mistakes graphing? Not always.
If I use Desmos too early on, does it become a crutch? Possibly.
Will the visual representation of Desmos help with future tasks similar to this? Probably.