# Punch-A-Bunch: A 3-Act Math Lesson on Probability

Welcome back math geeks! I need your help making a 3-Act math lesson on probability better.

I love Price is Right because many of the games require contestants to make predictions. This often involves estimating prices of products. But sometimes contestants have to make choices of a different nature, and these choices are ripe opportunities to think about probability and expected value. And I love when a fruitful 3-Act Math opportunity presents itself. (I’ve written about one before here.)

The example I want to share now doesn’t seem to fit a 3-Act format. Maybe that’s because it’s not truly a 3-Act Math lesson. But I don’t know what else to call it. I’m curious about your thoughts on how to make it better.

Some questions I’m asking:

- Is it too clunky?
- What grade levels will find this lesson useful?
- What concepts/standards does it best target?
- What opportunities did I miss?
- What extensions can be made?

I’m inviting your feedback in the comment section. Thanks for helping me get better!

## Quick Overview

Here’s a quick sketch of the game. A woman has an opportunity to win money. By guessing the prices of 3 products (not shown in any of the videos), she earns three opportunities to win money. She chooses three cards at random. She does not know the value of these cards.

Here’s the catch! She has to choose to keep the money on a card, or throw away the card and choose the next one. In other words, should she keep the money or hope for more? These are interesting choices that I think students and teachers would enjoy exploring.

One last thing: On the game board, you’ll see a table that shows the different amounts and how frequent those amounts occur (out of 50). This table is really important to the math in this lesson and the tool students will need to use to make arguments about her choices.

## Act 1

Play this video.

After explaining the rules of the game, these are some possible teacher questions.

- What do you notice?
- What do you wonder?
- What information is missing?
- What do we know?
- What can she expect to win?
- What is the “average value” of the cards?
- How likely will she win $25,000?
**Key Question: How much do you think she’s going to win?**

## Act 2a

When students are ready, play this video. My hunch is that students will need a lot of think and talk time in Act 1. Students need to know that she does not get to keep the sum of all three cards. She can only keep one card and once she throws a card away, she can’t undo that choice.

Should she keep the $250? Or try for more? How can math help us make this choice?

When student thinking has maximized, show the next video so students can see what she did.

## Act 2b

Then play this video.

Should she keep the $5000? Or try for more? How can math help us make this choice?

When student thinking has maximized, show the next video so students can see what she did.

## Act 3

Before showing the reveal, consider the following questions. As always, it’s best if students can come up with these questions themselves.

- How much money do we think is on the last card?
- Did she make the right choices?
- How can math help us make some predictions?
- How confident are we?
- What are the odds she made a bad decision?

Possible extension questions:

- What if the second card was $2500, what should she do?
- Does her expected value for winning change if she has 1 punch? 2 punches? 5 punches?
- How many punches would she need so that winning $25,000 is
*likely*?

## Closing thoughts

Some questions I’m asking:

- Is it too clunky?
- What grade levels will find this lesson useful?
- What concepts/standards does it best target?
- What opportunities did I miss?
- What extensions can be made?

Would love to here your thoughts on these questions or anything that you think might be useful.

My first reaction is that if you’re looking to tie the task to probability/stats standards in K-5, you won’t find any. It’s not a reason not to do it in elementary, but just saying. I agree it’s a little clunky. There’s a lot of starting and stopping which might cause the task to lose momentum. Maybe combining all the act 2s with act 1 would help? We could see her getting the $250 and the $5000 and then the question is what should she do? Keep the $5,000 or get the third card? Doing that might obviate the necessity to explain how it works at the beginning.

You’re right it’s not really a 3-act because there’s no information missing, at least that I can see. You’d have to hide the number of each amount that’s on the board. Either way the kids would need that information so it would be helpful to have it printed out, or you could give it to them orally ala Peter Liljedahl (on a VNPS of course.) It would be cool to see what the kids would do with that info. Maybe grade 4? But not sure if there’s an elem CC standard that fits.

I forgot to say that I like it! And I think the kids would be very excited to see what happens. They could even set up the same scenario and try it over and over again and record the results.

Glad you enjoy! I agree with you and think you’re on point. I’m not sure how accessible this could be. I have some more pondering to do about the matter. I do wish she had gotten the $2500 second and made the choice more interesting. I believe the mean value of the cards is about $2000, but that is certainly not the median which leads to an interesting discussion (again, probably beyond 5th perhaps). Thanks for taking the time and chiming in and helping to make it better! I do appreciate! c

Yeah, this is a blast. It has one of my favorite features of modeling tasks (3Act or otherwise) in that it allows for the

iterationof a student’s mental model. Because there are three opportunities here.It’s super helpful that the first card punched is an extreme value. Feels obvious to bail on it. Was keeping $5,000 the right move? A little trickier.

Here’s where I want to ask students to

changethe problem scenario so that they’d switch their intuition. If a student says, “yeah, keep the $5,000,” what about the problem scenario (the image at the top of the post) would have to change to make you change your mind. This is, again, an effort at helping her develop aninformalmodel of expected value. Same if the student said, “no, ditch the $5,000.” Eventually we use all of these cognitive resources to help students create aformalmodel of expected value.I also want to create a simulator so students can play the game and test their model. You want to partner up w/ Desmos on this one? Create a fun lesson everyone can use?

Thanks Dan! That’s some insightful input. I appreciate it. This problem gets more interesting as I think more deeply about it. I wish the second card was $2500 (or better yet, $1000) because a perplexing thing about this game is that the mean (expected value?) is about $2000. But the median is $1000…a very different value for the measure of center. I’m wondering how it looks in Desmos and would certainly love to flush it out more. I like the “change the problem scenario to they’d switch their intuition.” What would have to change in the image to make $5000 worth discarding? One variable that does not come up in this problem is: What if she had 5 cards to look at…should she ditch the $5000? 10 cards? 20 cards? She (contestant) doesn’t get that choice. I’m wondering how to leverage this perplexity and curious how you see Desmos making this happen. Happy to help.

This looks like a lot of fun. I’d use it with my junior high kids. However, I’d save the full video to the very end, if I use it at all. The quality of the lesson is on the punch board itself, so I just want that image. I agree with Joe, the stopping and starting of the videos are tiresome for me, especially when her actual choices are pretty easy to predict. I would NOT show the Act 1 pic that you have only because I’d be really interested in the students coming up with the numbers in the orange circles themselves. I’d leave them with the different values and the total amount (I think it’s $103,000) and ask them to figure out how many of each amount should there be. (I’m always trying to strip away information to see what the kids come up with.) Have kids do this individually, then get them into groups of 3, and now they must agree with on ONE set of answers. Then share this with the whole class. Now, I’d reveal what the actual game show has. Now, play the game as a class by making these into cards and just shuffling them. Draw 3 cards, and reveal each card, and after each reveal, take a tally of how many students would toss or keep the card. Play this like 10 rounds, and each kid can keep a score of his/her own success. I like repetitions and lots of samples if we’re gonna talk about probability. I just see a whole lot of playing and discussion without ever playing the video itself because unfortunately, the video is rather anti-climatic (her choices are predictable) and like you said, “clunky.” Thank you for sharing, Chase. I love the premise.

Thanks Fawn! You and Dan are encouraging me to think more deeply about the simulation aspect of this lesson. One thing that strikes me is that the mean of the cards is just over $2000. (I agree with you on the $103,000…and divided by 50 it’s $2060.) But the median is $1000. I wish she was faced with a $1500 card and had to choose. Makes me think of a “Would You Rather” scenario.

You’re right…there’s a lot more to strip away. So you’re envision students doing most of the math thinking before showing more of the videos? Have students explore the theoretical before diving in and watching one experimental outcome?

Thanks for taking the time to geek out!

So much going on here that I like. I think many points were brought up in other comments including Fawn mentioning that the choices were fairly obvious. That being said, I think that actually gives you an opportunity to streamline this. Instead of spending lots of time on this particular problem, you can go through it fairly quickly to get students to the point where you can then ask them some version of “How would you advise them to adjust the game to make the decisions more challenging?” Let’s assume the quantity of choices is fixed. You could potentially also do something like have them keep the mean win amount the same. Then you could focus only how altering the amounts (such as making a max of $20,000 but making other amounts higher or alternatively making it $2000 instead of $2500 and raising the grand prize).

Also, in terms of standards, there are many that could work but this one comes to mind: http://www.corestandards.org/Math/Content/HSS/MD/B/5/a/