# Prices, Proportions, Percents

I was in Garden Grove with my son on Sunday, and he insisted that I try this smoothie place called Tastea. With Jamba Juice and Blenders and all the other juice bars around town, I was skeptical that this joint’s concoctions would be anything different. He ordered a taro milk tea and I got a Thai tea, both with boba. Just one sip and I said to him, “Let’s order another round! It’s a long drive home!” Soooo delicious.

While there I saw a math lesson brewing, so I picked up their menu with prices. This is the lesson with my two classes of 6th graders.

Me: (I tell them about Tastea and how I wish it were closer.) Okay, let’s start with something you might be more familiar with, Starbucks. I love that now I live within walking distance from one! Do you know the different sizes that they have there?

Class: (When I refer to “class,” I don’t mean the whole class, of course, but somebody in the class joins in on the conversation.) Tall, grande, venti.

M: Do you know exactly how much liquid each size holds? (They make various guesses. I bring out the 3 sizes that I got from Starbucks so they have a visual.) I normally order a tall mocha frappuccino, let’s say the price is \$3.50. Do you think the venti, which is twice the volume of the tall, would cost twice as much, or \$7.00?

C: No.

M: Why not?

C: You normally get a better deal with a bigger size.

M: What do you mean by a “better deal”?

C: (All their answers show me that they understand the idea of more bang for your buck. Then finally someone says…) Lowest unit price!

M: Right! That’s why so many families go to Costco. Buying in bulk normally saves us money because the item has the best unit price. Well, we’re talking about Starbucks now, so buying more is a better deal, but drinking more is not so good for our body. Let’s fill in this sheet. (I pass out this handout Prices_proportions_percents.) How do we calculate unit price? What place value should we round it to? How do you write thirty-one-cents-per-ounce?

M: Tastea has three sizes: mini, gigantic, and even more. Their teas can also be purchased by the “partea jug,” which holds a gallon. I’ve given you the prices of the 10-ounce minis for the three different types of drinks, your job is to figure out the prices of the other sizes. You’ll work in small groups to figure out these out. So, do you think the gigantic will cost twice as much as the mini because it holds twice as much?

C: No. It’ll cost less.

M: How much less? Well, that’s your group’s job to come up with the best estimate. We have Starbucks’ prices for their three sizes, you could look at how they price their drinks. But here’s the sweet deal for you. You and your group mates do the math that you need to, then write down your first estimation right here in this column. Bring your paper up to me (only the “captain’s” paper), give me a few seconds to figure out the percentage that your estimation is off by, and I’ll write it in this column and give you back your paper. What percent do you want to see, large or small? What if your estimation were the actual price — what percent would I write there?

I tell them that they could figure out the actual price of the drink if they knew how I calculated the percentage of error. So, work work work. Think think think. What makes sense? Oh, I remind them that the percentage does not indicate if their estimation is too high or too low. So, again, what makes sense?

We also note that prices generally end in a 0, 5, or 9. So, even if the calculation tells them the price should be \$4.23, they might want to change that to \$4.25 or \$4.20.

When they bring up their paper again with the second estimation, all I do is write their estimation again in pen and circle it — this is so they can’t change their answer and I know that I’ve seen it. I do NOT fill in the “Actual Price” column at this time because the groups are working at different rates, and in a crowded room, it’s easy for kids to see each other’s papers, even inadvertently, and the game of estimation is over if they saw the actual price beforehand.

They simply move on to the next size to make a first estimation again. We repeat the process.

When all groups are done with estimations for the first type of juice — smoothies — I tell them what the actual prices are.

Now, it’s their turn to figure out the percentage of error. I give the groups about 10 minutes to do so without help from me. At the end of the 10 minutes, either there’s at least one group that knows how to do so and can show it to the class, or no group knows how, then I’ll walk them through the calculation by asking them questions to figure this out.

They continue in the same manner for the Slushy Freeze and Specialteas on page 2. This time hopefully they’ll be able to work backward from the error percentage that I give them after their first estimation.

Reasons I’m proud of this lesson:

1. It’s about proportions, but many priced items in real life are not directly proportional. The kids knew this coming in because they’ve been consumers.
2. We get to talk about business strategies that entice people to buy the larger sizes while still make a profit. (Starbucks calls it “tall” because it rhymes with “small,” but clearly the word tall naturally elongates the imagination.)
3. Students get to make estimations throughout, but they know these aren’t “wild-ass guesses.” They start with the calculation of proportions and adjust the prices accordingly. They get to critique and argue with their group mates to come up with the best estimations.
4. I get kids to think about percentage in a context that they can wrap their heads around. And they want to know how because their second estimation could be dead on if they knew.
5. It’s fun that the error percentage does not indicate if their estimation is too high or too low. A few groups do go farther in the wrong direction. Oh, well — good to learn that now.
6. It’d be fun for me to get Starbucks or Jamba Juice for the group with the lowest total in percentage errors.

[Updated 04/05/14]

I got some thoughtful reflections on this lesson, I’ll just share two:

I learned how to work backwards with percentages and try to get the number spot on. I also learned how business would price things by dropping the price by the perfect amount. My number sense got a lot better from all the multiplying, dividing, and reasoning. It was very difficult, which I’m very happy about. The teamwork was probably the hardest part of the project. M and I are very competitive, and we got different answers a lot. I learned how to work together with others a lot better, and it doesn’t move your team along to place blame and argue. I’m really grateful we did this project because it was very hard and worthwhile. It was a great use of three days!

I learned how to use different data to get answers. Also, we have to see a pattern. This Tastea assignment was really fun. I enjoyed it and look forward to another. Teamwork is really important even though people can’t agree, you got to support it. If your group gets it wrong, but your answer was right, you can’t blame someone or put them down because probably they will get some right for you. So always stay positive to your teammates and encourage them.