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Lessons From the Classroom
Discount and Sales Tax
Instead of doing the above exercises in our textbook, I had my 6th graders do this:
Shopping Contest at Target
Let’s pretend Target has a contest. The contest is for shoppers to find merchandise from their online store.
Contest rules
You must choose at least 10 different items.
The items must come from at least 5 different departments, such as footwear, kitchen, clothing, toys, etc.
You may buy more than 1 same item — you can buy 2 or more packs of athletic socks.
The winner
The winner is the shopper whose merchandise receipt totals exactly $500 or closest to it — without going over. They get to keep all the merchandise!
Your task
Phase 1: Go to Target’s website and find some items that you’ll want to get for yourself and your family. List the items and their original prices in the table provided separately.
Phase 2: The store manager (really, it’s your teacher) then announces the percent discounts for different categories of merchandise. Write these down below. Apply these discounts to your items and calculate the new sale prices.
Phase 3: The current tax rate for our city is 8%, so you must add this to your total. In this phase, you get to add or remove items on your shopping list to reach the target goal of $500 without going over.
I only did Phase 1 with the kids, my sub supposedly did Phases 2 and 3 with them. (I wanted to do the rest of the lesson with them when I return from NCTM — and after spring break — but they voted to continue the lesson with the sub as they were really into the task. I'll find out tomorrow when class resumes.)
Why this task
You can change everything about it. Shop somewhere else instead of Target.
Change the rules depending on time available and access to computers. Are there enough computers for each kid or do they need to work in groups of 2 or 3? My kids were in pairs. (We have enough laptops, but at least 30% of them have issues.)
I like the idea of not going over a certain amount — $500 in this case — instead of "whoever is closest to the target price" because I think it keeps the kids more reserved in their shopping spree. Students understand that if there were only 2 contestants, then the one with a final receipt total of $154 would win over the other with a $501 total.
Kids don't know what the exact discounts are until Phase 2, so this makes it a fun temporary secret. But they know to go over budget in Phase 1 because there will be discounts — not all departments have to have discounts either. Their totals in Phase 1 were in the $600 to $700 range because they also know that it'd be easier (faster) to remove items than add them later in Phase 3. They were also told that the discounts would be somewhat realistic, meaning Electronics will get a smaller percent discount, if any, than Clothing.
They'll become more aware of how much things cost — and how quickly they add up in the shopping cart.
Handout Discount and Sales Tax
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. And here’s the key.) 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:
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.
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.)
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.
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.
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.
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.
"Working Together" Problem
Heya, back-to-back post about a problem from Five Triangles mathematics.
When I tweeted how much I love this problem, a few people did not feel the same at all. Here are my reasons for appreciating this problem:
It's a notched up "working together" problem that I have not seen before.
It has percentages and fractions.
I can use rectangles to solve this. (I was asked on Twitter how I would solve this using rectangles, hence this post.)
I had to work on this problem. This is a big reason for me. We should assume that if we're teaching a particular math subject — Geometry, Statistics, or Calculus — that we're able to easily do all the exercises in the textbook. A set of exercises allows us to practice a particular skill. But a problem should require us to think. I hope I've encouraged problem-solving enough with my students that they value a problem more when they have to struggle with it, when they don't know immediately how to start it, when they get stuck and become frustrated, when they seek others for help, when they can leave the problem and come back to it another day.
While I'm at it, I also love the site Five Triangles in general for a couple of reasons:
The Geometry problems are simply stated and interesting. They make me pause and think, very few have been automatic gimmes.
The solutions are not posted. I really appreciate this because if they were, we might be tempted (mainly due to lack of time) to check the answers too early before we allow ourselves a chance to work through the problem and perhaps struggle with it. "Anticipating" is the first of 5 Practices that gives us insight on how students might solve the problem.
I did, however, retype the question above so it's easier to read and track information. I also numbered the paragraphs for quicker reference.
How we worked through this problem. Colors and all.
Draw a rectangle to represent the task. It has an area of 80 square units because that's the LCD of the three fractions in the problem.
Because this grid represents the task, we use it to fill in the amount of work done. Paragraph [3] is the first concrete piece of information that allows us to do this.
We continue to fill in the work done as described in paragraph [4].
Paragraph [5] is the first piece of information that allows us to figure out C's rate. Knowing that C can do 16 boxes in 8 hours means C can do the task — 80 boxes — in 40 hours.
With C's rate, we can now take on paragraph [2]. We know from the last step that C's hourly rate working alone is 2 boxes per hour or 10 boxes in 5 hours. But when working with A, C's rate is 40% faster, therefore instead of getting just 10 boxes done, C can get 14 boxes done in 5 hours when working with A.
From picture above in green, we know A and C did 24 boxes in 5 hours, and since C was responsible for 14 of those, the remaining 10 boxes were done by A.
Then A's hourly rate when working with C is 2 boxes per hour. Because this hourly rate represents a 20% increase than if A were to work alone, the math we need to do is 2 boxes divided by 1.2 to get 5/3 boxes. Solving for x in the proportion below gives us the answer that A completes the task in 48 hours.
Lastly we use paragraph [1] to figure out rate for B. We know A's alone rate is 5/3 boxes per hour, but when working with B, A's rate is 40% faster. Thus we multiply 5/3 by 1.4 to get 7/3. If A can do 7/3 in 1 hour, then A can do 35/3 in 5 hours when working with B.
The yellow boxes show that A and B can do 25 boxes in 5 hours, so subtracting 35/3 from 25, we see that B did 40/3.
To get B's alone rate, we divide 40/3 by 1.2 (because B is 20% faster when working with A) to get 100/9. Solving the proportion below gives us the answer of B completing the task in 36 hours.
