In this meeting on our third anniversary, I tried out a 3-Act Math Task I’ve been planning for a while. It’s got to do with the 5% bonus the MTA gives us on non-unlimited MetroCards and the odd remainders I often find when I run out of funds on the card.

### Materials:

- Lesson Plan: Mark’s Metrocard – A 3-Act Math Task
- Act One Video: http://bit.ly/marksmetrocard
- MTA fare information – March 2017 (abridged)
- Act Three – pre-valued MTA cards
- Act Three Video: http://bit.ly/marksmetrocardact3

### Act One

We started by watching a short video. We didn’t have access to a projector so we huddled around a tablet. The small size of the video made it difficult to see some of the numbers, which created some challenges later when the group found itself in disagreement about the value on the Metrocard. In the end, we watched the video three times.

Video: Mark’s Metrocard

#### Write down the first question that comes to your mind.

*Share your question with a partner. See if it is the same question or a different question.*

- What is the valuable info? What’s the catch?
**How much was left on the card after all the swipes?****How many swipes were there? How many trips for $20?**- Was it coming or going?
- Was he actually traveling?
- Why was he charged $21 when he chose $20?
- Was it the same person?
- Does he get a bonus for $20? What is it?
- Is he really getting $21 on the card?
- How is he getting the extra dollar?
- Does he get 6 or 7 rides?
- Does he get a bonus ride?

Participants noticed that Mark chose $20 in value to put on the card, was charged $21 because of a new card fee, but got $21 in value on his card, which was proved by the fact that he had $18.25 on the card after one swipe. However, one group explained this by saying that the first trip cost only $1.75. There was a lot of discussion about how much value Mark had on his card when he left the MetroCard machine and went to the turnstile the first time.

### Act Two

#### What information do you need from me in order to answer our question/s?

*I asked the group to focus on the questions in bold to start. Before trying to answer them, though, I asked what information they wanted from me.*

- What is the ratio for the bonus? (I answered this by saying that there is a 5% bonus for all amounts put on the card over $5.50.)
- What dollar amount bonus do I get for $19? (I didn’t answer this question.)
- Is the bonus growing linearly or exponentially? (I didn’t answer this question either.)

I made a decision not to hand out the MTA fare info sheet yet because it seemed that the group had all the information they needed to answer the current questions.

*I then gave the group 7 minutes of individual problem-solving time. *

### Act Three

*When we came back together, Emerald explained her method of answering the questions.*

Emerald knew she was starting with $21 because 5% of $20 is $1, which is added to the value of the card. She could also confirm that it was $21 at the start since after taking away the first 2.75, there was 18.25 left over. If you subtract 2.75 from 21 seven times, you have 1.75 left over. She then counted the number of times she subtracted 2.75 to find there were 7 trips.

I think this could be the end of our work on this problem, but I had a concern I wanted to share with the group. I asked the group what other questions they had. And shared my concern. This led to a new…

### Act One (B)

*What about that $1.75? That doesn’t bother you?*

- How much are you actually paying per swipe? With the bonus, with the new card fee…
- Are we really getting a bonus? Since they give 5% on $20, then take it away with the new card fee…
- Is it possible to break even? There was skepticism about whether it is even possible to add money so that you don’t have value left over on the card.

I asked the group to focus on these two questions:

*What amount should Mark have put on his card originally?*

*OR*

*What amount should Mark add to $1.75 on his card?*

### Act Two (B)

I gave out an abridged MTA fare information sheet that explains the 5% bonus, new card fee, and allowable increments of new value ($.05).

### Act Three (B)

#### Problem-Solving

*I asked the group to work individually for 10 minutes on these questions, then to work in pairs.*

We were starting to run out of time, but the group did get to a couple solutions:

Q: How much should Mark have put on the card originally? A: $55. (Looking for when the 5% bonus gives us exactly one extra ride. 55*.05 = 2.75. 55/2.75 = 20. $55 plus bonus results in exactly 21 trips.)

— Eric Appleton (@eappleton) November 9, 2017

Q: What amount should Mark add to the $1.75 on his card? A: $1.00 (Forget the bonus. Who cares about losing a nickel?) I’m not sure if I’m satisfied by this…

— Eric Appleton (@eappleton) November 9, 2017

Open questions:

- Is there an amount smaller than $55 that can be put on a MetroCard so that it doesn’t result in leftover money after all the swipes?
- Is there an amount larger than $1 that can be added to Mark’s MetroCard value of $1.75 so that there wouldn’t be leftover money the next time?

#### Solutions

At the end I showed a short video that demonstrates the MTA’s MetroCard Calculator. I also shared cards with information about the MTA’s pre-valued cards with value amounts that swipe to zero, accounting for the 5% bonus.

- Act 3 video – Mark’s Metrocard (http://bit.ly/marksmetrocardact3)
- MTA’s pre-valued card info

In preparation for this meeting, I spent the last few weeks asking for feedback and solutions from friends outside of CAMI. Open the MetroCard Problem Google Doc to see my attempted solution of the MetroCard Problem and interesting approaches from other people, including use of graphing calculators and spreadsheets. Feel free to comment with your own solution strategies.

- Extensions to the MetroCard Problem
- What if you had a card with ___ in value on it? How much money could you add so that none would be wasted?
- Imagine this situation: When Mark gets down to $1.75 on his card, he adds another $20 in value. How much is on the card now? If I swipe, swipe, swipe until all trips are gone, is there a remainder? If so, add another $20 in value. How long will it take to get to an empty card?
- My friend David adds $40 each time he sees “insufficient funds” when he swipes at the turnstile. If he starts by spending $40 on a new card and adds $40 every time he runs out, how long (how much money, how many refills) will it take him to get to 0 value on the card?
- Why do you think the MTA created a system with such messy remainders? Can you think of a better system to collect money and give bonus for larger purchases?

In attendance: Maggie, Maryam, David, Deneise, Emerald, Eric

Programs represented: CUNY Adult Literacy/HSE Program, Queens EOC, RiseBoro, York College

From a friend:

“Eric, this may be useful for a lesson in self-advocacy if not a word problem in math. I had two metrocards, and both, super annoyingly, had $2.45 remaining on them. (How and why is that?!) No time to dither at the ticket slot machine, I asked the attendant in the booth if she could combine the fares onto one card, and to my great surprise, she did it. I thanked her about three times.

So I now had $4.90 on one card and zero on the other, and passed the gate.

I am not sure what this means for the leftover on the one card with a balance, and how to possibly make it pay down the line, but at least I’m on that train.”