Becky Blessing was one of my substitute teachers last year. She re-introduced herself at the start of my UC Berkeley presentation and halfway through my WCYDWT? thesisie. “capture anything that interests you and present it to your kids in the most compelling way possible.” she called me over and showed me this gem, which she captured for Dor Abrahamson’s problem-solving class.

Download high quality here. See the pilot for instructions.

This meter is different with how you actually get a deal depending on how you pay. I would have students tell me the best deals. Maybe say you want to park for different amounts of time (15 minutes, 20, 30, 1 hour, etc.) and explain the best payment method.

I can really see this in an Alg 2 course where we are discussing solving multiple linear equations.

What happens when you walk up to the meter with 7 minutes still on it?

Does your thinking change if there are 28 minutes on the meter?

Now you get many different situations out of one picture, and some are cheaper with different coin combinations. This picture with a couple of simple questions could be an entire lesson on solving three equations in three variables.

How about, you’re meeting a potential date for coffee. There is 15 minutes on the meter, how much do you put in, and why?

1. Some may want an maximum amount of time, they will go for the best bargain within the 1 hour time frame;

2. Some may want a minimal amount to have an excuse to cut out (gotta leave, my meter’s running out), they will go for the minimal outlay.

3. Door number three, anything one can imagine.

You could do a unit on the economics of dating, and include your “How I met your mother” video showing the hot vs. crazy girlfriend linear relationship, and choosing a text plan based on your annual report graph.

I tried taking this picture over the summer (in a moment that was very Dan-inspired). Only my photos didn’t turn out well and I didn’t have a compelling hook for students who rarely see parking meters, unless I turned it into a city vs country thing. In which case, Dina’s story and selling the point of becoming a savvy consumer just might work. Combined with other situations that would encourage shorter time choices. (Two minutes for a date, not so much.) Shopping? Library visit? I’m not sure where meters are located, especially in relation to what your students do.

I like the dating-themed unit, though I’m not convinced I could pull it off. (Confession, I was searching the archives a few weeks ago for the bubble graphs of where single people live. Include it too.)

I like what A. Mercer did with it, I was thinking along the same lines.

You park downtown, and need to do some shopping for your Mom/ Boyfriend/ Girlfriend/ Significant Other. The meter has 13 minutes on it, and you know you will be in the store for at least an hour (after that you will have to take your chances!)

2N + 5D + 12Q = 47

I am missing something, because I can’t think of how to create the other 2 equations. Also, this equation would not give you the smallest dollar amount.

.05N + .10D + .25Q = ?

Hmm, perhaps my initial thought was off the mark. Anyone have any assistance on this?

1) how far can you walk away from the meter (given current city layout, street light timing, car and pedestrian traffic, your pace) before you have to turn around and put more money into the meter [assume the meter officer will promptly give you a ticket when the time exactly runs out]? Supposed the above variables were simplified? How does that change things?

How long can you linger in a store and how far away can you be from the meter to linger in a store a set amount of time?

Which meter do you want to park at given three or four stores that you absolutely want to shop at but minimize your return to the meter?

2) Reconstruct the background that is reflected in glass.

First thing that popped into my head when I saw this pic was Dan’s long ago post about the theives who unwisely stole bags of nickels. How much time would a pound of nickels buy compared to dimes and quarters?

I’m teaching slope in my 8th grade math class right now.

This would be a great example of a system of equations…or assuming that all of the rates represent one function…the idea of linear vs. non-linear functions.

Domain and Range could come into play here as well

Ask yourself: what is the point of pulling this image into your classroom? Is it simply a diversion? Is it simply a full-color re-imagining of a textbook problem?

I say, if we use this image correctly our students will begin taking a deeper look at the world. They’ll begin noticing these mathematical connections without our prompting and by the end of the term, they’ll start e-mailing the images to you.

But that’ll only happen if the teacher removes herself from the process as much as is humanly possible (after all, the teacher won’t be on hand when the student encounters the parking meter on her own), only asking questions that the medium supports, supplying information only as the student requests it, putting herself on the bench, essentially.

For example, I can ask the following question:

—————–

4 nickels, 3 dimes, and 2 quarters buys you 47 minutes
1 nickels, 5 dimes, and 1 quarter buys you 39 minutes.
2 nickels, 2 dimes, and 1 quarter buys you 26 minutes.

How much time is each coin worth?

—————–

Is it as obvious to my readership as it is to me that the teacher’s presence in this problem is categorically “strong”? That the teacher has to impose herself all over this problem for it function at all? That I may as well have asked, “I have three numbers in my head. If you multiply the first by 4, the second by 3, and the third by 2, you get 47,” etc.

That sort of questioning serves the same purpose as the problems in my textbook, to develop procedural fluency, but it doesn’t do anything to alter how my students see the world.

Does this make sense? Depending on our motives for bringing media into the classroom (and my motive is strictly “so that they will have a richer understanding of the world they live i”n) certain questions are counterproductive.

All of that said, I can see myself asking two questions here:

1) Which coin is the most valuable?
2) Which gets me more time on the meter? One dollar in quarters, dimes, or nickels?

These questions don’t rock the procedural fluency like the system of three equations, but they do build conceptual fluency like a system of equations doesn’t. I have the textbook for procedural fluency. I don’t have anything else but this for conceptual fluency.

1. The joy of your blog is that I can talk about my local without fear of alienating those I’m meant to coach. wait, I’ve said too much. Sorry for not sharing your blog. I’m a greedy dog.

2. Okay, but what can you do with this:

You have students entering your class three years behind grade level. They are in a ‘ramp up’ course, meant to prepare for next year’s entrees: Algebra 100 or Geometry. They are asked to solve for x, because they need to do this. It’s all part of the ramp up process.

But ask them to talk about this, a sign at a clothing retailer:

‘2 for 25’ or ‘3 for 30’,

and they are crippled.

You see, this at-heart English teacher, having spent years increasing vocabulary with words like ‘misanthrope’ and ‘invective’ (we can talk word choice at a later date), never realized that for a meaty portion of the student body, the word ‘for’ signed complex, unsolvable mathematical procedure.

So I’m all for images. I’m all for signs. But I’m shocked at the emphasis on solving for x when prepositions render students mathematically impotent.

Not a trick question. Just wondering why math spend time on ‘x’ when, forgive me, it can’t help them navigate their way through the retail world (and other variations of the world they stand to inhabit).

Good for you, Dan. I think we should consider the value of pulling back as much as possible (sounds fairly constructivist to me) and help students develop their own sense of their world as they can/should see it (mathematically/critically). If an image like this is cheaper/quicker/easier to to help students learn a prescribed skill that the textbook can do more efficiently, then it is simply a fancy distraction, but if the goal is something more, then let the instruction honor the complexities the students can/will draw out of it (with guidance from the teacher). Certainly, you need to continue to guide students in relating a view of the world around them (the image) to the relevant mathematical possibilities, but if the image does the same thing the text can do, then what’s the point? Also, cripple them, as Ken says will happen, and then provide the soft- or heavy-handed support to help them make the connections to other “real world” contexts.

This reminds me of the duct tape picture and the nickel stealing vs. dime stealing.

You could take it in another direction and compare the parking meter to a monthly rate in a covered garage by converting to the same units. There’s some interesting trade-offs there between not having to feed the meter, convenience of having a reserved space, inconvenience of winding through a narrow garage, etc. The units could also be converted to $/hour and compared to minimum wage.

I suppose since the coins are discrete, you can do some combinatorial problems as suggested above.

Yeah, that’s good stuff. The cents/minute angle is what interests me most, a fact which is only relevant inasmuch as I know I can sell that angle best.

The parking meter v. covered garage is useful in that it gets kids arguing feverishly but I can’t ignore the possibility that a class of 14-yo’s will have no opinion whatsoever on the matter.

Allowing the teacher to remove herself from the scene presupposes that we ( the teachers) have the autonomy to truly employ the Socratic method in mathematics courses.

In a world where teachers, particularly those working in high needs areas, are constantly plagued by standardized test requirements time is not on our side and as such…we have to somehow drive the instruction according to some arbitrarily created calendar or pacing guide.

I’m very interested in your position on textbooks because, quite frankly, it mirrors my own. Textbooks are dull and they tell kids what, how, and when to think about a problem. I want to encourage my students to become thinkers but the materials…and subsequent assessments pull students back into the real of passive recipients of my instruction.

However, assuming that franchising education and allowing students to create their own context for a problem situation are not incompatible…how do we go about delivering this type of instruction while still meeting specific standards in a limited time frame ?

Or is the real answer to the problem…to change the standards to fit real learning ? Instead of arbitrary “grade level appropriate” content is the answer to place students, regardless of age, where they can best function in the curriculum ?

Allowing the teacher to remove herself from the scene presupposes that we ( the teachers) have the autonomy to truly employ the Socratic method in mathematics courses

The teacher doesn’t really remove themselves from the scene. You are critically important for setting up the story, and for setting the beats that move it forward.

But, if it’s a good story, the kids tell it themselves, and you get to hide in the shadows while they do it. If the teacher weren’t crucially important for that, textbooks wouldn’t suck.

Which reminds me: I’ve got about 40 lbs of nickels lying around here. I’m quite sure that there will be an opportune time to drop a big canvas bag of them on some poor kid’s desk at some point.

Find a street nearby and count up the number of parking spaces. Assume that everyone is only using dimes (least efficient method). How many hours of continuous parked cars will it take to pay for the bailout of Wall Street?

Or to come at the the same issue from a less inflammatory hook: “Who cares about small change?” Why does the government pay such close attention to such small sums of money? Illustrates federal vs state vs local and leads into a discussion about the theory of laws like this one and why they exist.

Another humanities link, different direction. Put up alongside this one an image of the RFID-controlled parking meters that are the norm in many Asian countries (google image search – octopus parking meter – to see what we use in hong kong). “Why the difference?”

Could also do this one with any number of countries. Probably much harder to use this effectively in class…but it’s cool.

Playing off Steven P’s idea, why cents/min and not mins/cent? Although they are each ratios of the same concept, the philosophy behind the choice of measurement is interesting.

Someone choosing cents/min implies he is looking for the cheapest way to get a fixed task completed. “I need 30 mins in this store, how can I park cheapest?”

Someone choosing mins/cent implies he is looking for the maximum time allowed for the change in his pocket. “I’ve only got 50 cents, how long can I stay at the park?”

I think it’s an interesting question (especially when applied to the American mpg vs the European liters/100km). It could also bring up the idea of multiplicative inverses mathematically.

## 25 Comments

## Chris

February 16, 2009 - 12:01 pm -This meter is different with how you actually get a deal depending on how you pay. I would have students tell me the best deals. Maybe say you want to park for different amounts of time (15 minutes, 20, 30, 1 hour, etc.) and explain the best payment method.

## Glenn

February 16, 2009 - 12:05 pm -I can really see this in an Alg 2 course where we are discussing solving multiple linear equations.

What happens when you walk up to the meter with 7 minutes still on it?

Does your thinking change if there are 28 minutes on the meter?

Now you get many different situations out of one picture, and some are cheaper with different coin combinations. This picture with a couple of simple questions could be an entire lesson on solving three equations in three variables.

## Dan Meyer

February 16, 2009 - 12:24 pm -Can you give me an example of a compelling hook that leads into a system of three equations? That’d be great.

## A. Mercer

February 16, 2009 - 3:43 pm -How about, you’re meeting a potential date for coffee. There is 15 minutes on the meter, how much do you put in, and why?

1. Some may want an maximum amount of time, they will go for the best bargain within the 1 hour time frame;

2. Some may want a minimal amount to have an excuse to cut out (gotta leave, my meter’s running out), they will go for the minimal outlay.

3. Door number three, anything one can imagine.

You could do a unit on the economics of dating, and include your “How I met your mother” video showing the hot vs. crazy girlfriend linear relationship, and choosing a text plan based on your annual report graph.

## Sarah

February 16, 2009 - 5:43 pm -I tried taking this picture over the summer (in a moment that was very Dan-inspired). Only my photos didn’t turn out well and I didn’t have a compelling hook for students who rarely see parking meters, unless I turned it into a city vs country thing. In which case, Dina’s story and selling the point of becoming a savvy consumer just might work. Combined with other situations that would encourage shorter time choices. (Two minutes for a date, not so much.) Shopping? Library visit? I’m not sure where meters are located, especially in relation to what your students do.

I like the dating-themed unit, though I’m not convinced I could pull it off. (Confession, I was searching the archives a few weeks ago for the bubble graphs of where single people live. Include it too.)

## Glenn

February 16, 2009 - 5:56 pm -I like what A. Mercer did with it, I was thinking along the same lines.

You park downtown, and need to do some shopping for your Mom/ Boyfriend/ Girlfriend/ Significant Other. The meter has 13 minutes on it, and you know you will be in the store for at least an hour (after that you will have to take your chances!)

2N + 5D + 12Q = 47

I am missing something, because I can’t think of how to create the other 2 equations. Also, this equation would not give you the smallest dollar amount.

.05N + .10D + .25Q = ?

Hmm, perhaps my initial thought was off the mark. Anyone have any assistance on this?

## Michael

February 17, 2009 - 1:00 am -1) how far can you walk away from the meter (given current city layout, street light timing, car and pedestrian traffic, your pace) before you have to turn around and put more money into the meter [assume the meter officer will promptly give you a ticket when the time exactly runs out]? Supposed the above variables were simplified? How does that change things?

How long can you linger in a store and how far away can you be from the meter to linger in a store a set amount of time?

Which meter do you want to park at given three or four stores that you absolutely want to shop at but minimize your return to the meter?

2) Reconstruct the background that is reflected in glass.

## Ben

February 17, 2009 - 3:42 am -First thing that popped into my head when I saw this pic was

Dan’slong ago post about the theives who unwisely stole bags of nickels. How much time would a pound of nickels buy compared to dimes and quarters?## Jovan

February 17, 2009 - 6:13 am -I’m teaching slope in my 8th grade math class right now.

This would be a great example of a system of equations…or assuming that all of the rates represent one function…the idea of linear vs. non-linear functions.

Domain and Range could come into play here as well

## Dan Meyer

February 17, 2009 - 9:53 am -This conversation fascinates me.

Ask yourself: what is the point of pulling this image into your classroom? Is it simply a diversion? Is it simply a full-color re-imagining of a textbook problem?

I say, if we use this image

correctlyour students will begin taking a deeper look at the world. They’ll begin noticing these mathematical connections without our prompting and by the end of the term,they’llstart e-mailing the images to you.But that’ll only happen if the teacher removes herself from the process as much as is humanly possible (after all, the teacher won’t be on hand when the student encounters the parking meter on her own), only asking questions that the medium supports, supplying information only as the student requests it, putting herself on the bench, essentially.

For example, I can ask the following question:

—————–

4 nickels, 3 dimes, and 2 quarters buys you 47 minutes

1 nickels, 5 dimes, and 1 quarter buys you 39 minutes.

2 nickels, 2 dimes, and 1 quarter buys you 26 minutes.

How much time is each coin worth?

—————–

Is it as obvious to my readership as it is to me that the teacher’s presence in this problem is categorically “strong”? That the teacher has to impose herself all over this problem for it function at all? That I may as well have asked, “I have three numbers in my head. If you multiply the first by 4, the second by 3, and the third by 2, you get 47,” etc.

That sort of questioning serves the same purpose as the problems in my textbook, to develop procedural fluency, but it doesn’t do anything to alter how my students see the world.

Does this make sense? Depending on our motives for bringing media into the classroom (and my motive is strictly “so that they will have a richer understanding of the world they live i”n) certain questions are counterproductive.

All of that said, I can see myself asking two questions here:

1) Which coin is the most valuable?

2) Which gets me more time on the meter? One dollar in quarters, dimes, or nickels?

These questions don’t rock the procedural fluency like the system of three equations, but they do build

conceptualfluency like a system of equations doesn’t. I have the textbook for procedural fluency. I don’t have anything else but this for conceptual fluency.## ken

February 17, 2009 - 11:06 am -Two things, in ascending importance:

1. The joy of your blog is that I can talk about my local without fear of alienating those I’m meant to coach.

wait, I’ve said too much. Sorry for not sharing your blog. I’m a greedy dog.2. Okay, but what can you do with this:

You have students entering your class three years behind grade level. They are in a ‘ramp up’ course, meant to prepare for next year’s entrees: Algebra 100 or Geometry. They are asked to solve for

x, because theyneedto do this. It’s all part of the ramp up process.But ask them to talk about this, a sign at a clothing retailer:

‘2 for 25’ or ‘3 for 30’,

and they are crippled.

You see, this at-heart English teacher, having spent years increasing vocabulary with words like ‘misanthrope’ and ‘invective’ (we can talk word choice at a later date), never realized that for a meaty portion of the student body, the word ‘for’ signed complex, unsolvable mathematical procedure.

So I’m all for images. I’m all for signs. But I’m shocked at the emphasis on solving for

xwhen prepositions render students mathematically impotent.## Dan Meyer

February 17, 2009 - 4:08 pm -Uncripple them? Is this a trick question?

## Dina

February 17, 2009 - 4:30 pm -My God, Dan. Are you becoming… a *constructivist*??!

## Dan Meyer

February 17, 2009 - 4:50 pm -Nah.

## Ken

February 17, 2009 - 4:51 pm -Not a trick question. Just wondering why math spend time on ‘x’ when, forgive me, it can’t help them navigate their way through the retail world (and other variations of the world they stand to inhabit).

## apharbau

February 17, 2009 - 7:54 pm -Good for you, Dan. I think we should consider the value of pulling back as much as possible (sounds fairly constructivist to me) and help students develop their own sense of their world as they can/should see it (mathematically/critically). If an image like this is cheaper/quicker/easier to to help students learn a prescribed skill that the textbook can do more efficiently, then it is simply a fancy distraction, but if the goal is something more, then let the instruction honor the complexities the students can/will draw out of it (with guidance from the teacher). Certainly, you need to continue to guide students in relating a view of the world around them (the image) to the relevant mathematical possibilities, but if the image does the same thing the text can do, then what’s the point? Also, cripple them, as Ken says will happen, and then provide the soft- or heavy-handed support to help them make the connections to other “real world” contexts.

## Steven Peters

February 18, 2009 - 11:54 am -With my current obsession with units of measurement, I convert these to an appropriate unit, say cents/minute.

Nickel: 2 1/2 cents/minute

Dime: 2 cents/minute

Quarter: 2 1/12 cents/minute

This reminds me of the duct tape picture and the nickel stealing vs. dime stealing.

You could take it in another direction and compare the parking meter to a monthly rate in a covered garage by converting to the same units. There’s some interesting trade-offs there between not having to feed the meter, convenience of having a reserved space, inconvenience of winding through a narrow garage, etc. The units could also be converted to $/hour and compared to minimum wage.

I suppose since the coins are discrete, you can do some combinatorial problems as suggested above.

## Dan Meyer

February 18, 2009 - 4:27 pm -Yeah, that’s good stuff. The cents/minute angle is what interests me most, a fact which is only relevant inasmuch as I know I can sell that angle best.

The parking meter v. covered garage is useful in that it gets kids arguing feverishly but I can’t ignore the possibility that a class of 14-yo’s will have no opinion whatsoever on the matter.

## Dan Meyer

February 18, 2009 - 4:29 pm -Incidentally, Becky Blessing suggested these questions:

## Jovan

February 19, 2009 - 10:46 am -Allowing the teacher to remove herself from the scene presupposes that we ( the teachers) have the autonomy to truly employ the Socratic method in mathematics courses.

In a world where teachers, particularly those working in high needs areas, are constantly plagued by standardized test requirements time is not on our side and as such…we have to somehow drive the instruction according to some arbitrarily created calendar or pacing guide.

I’m very interested in your position on textbooks because, quite frankly, it mirrors my own. Textbooks are dull and they tell kids what, how, and when to think about a problem. I want to encourage my students to become thinkers but the materials…and subsequent assessments pull students back into the real of passive recipients of my instruction.

However, assuming that franchising education and allowing students to create their own context for a problem situation are not incompatible…how do we go about delivering this type of instruction while still meeting specific standards in a limited time frame ?

Or is the real answer to the problem…to change the standards to fit real learning ? Instead of arbitrary “grade level appropriate” content is the answer to place students, regardless of age, where they can best function in the curriculum ?

## Mr. K

February 19, 2009 - 6:38 pm -Allowing the teacher to remove herself from the scene presupposes that we ( the teachers) have the autonomy to truly employ the Socratic method in mathematics coursesThe teacher doesn’t really remove themselves from the scene. You are critically important for setting up the story, and for setting the beats that move it forward.

But, if it’s a good story, the kids tell it themselves, and you get to hide in the shadows while they do it. If the teacher weren’t crucially important for that, textbooks wouldn’t suck.

Which reminds me: I’ve got about 40 lbs of nickels lying around here. I’m quite sure that there will be an opportune time to drop a big canvas bag of them on some poor kid’s desk at some point.

## mrfollett

February 20, 2009 - 3:42 pm -How about just this question:

why?

## jeffreygene

February 24, 2009 - 1:17 am -Find a street nearby and count up the number of parking spaces. Assume that everyone is only using dimes (least efficient method). How many hours of continuous parked cars will it take to pay for the bailout of Wall Street?

Or to come at the the same issue from a less inflammatory hook: “Who cares about small change?” Why does the government pay such close attention to such small sums of money? Illustrates federal vs state vs local and leads into a discussion about the theory of laws like this one and why they exist.

Another humanities link, different direction. Put up alongside this one an image of the RFID-controlled parking meters that are the norm in many Asian countries (google image search – octopus parking meter – to see what we use in hong kong). “Why the difference?”

Could also do this one with any number of countries. Probably much harder to use this effectively in class…but it’s cool.

## David Petersen

April 21, 2009 - 11:57 pm -Playing off Steven P’s idea, why cents/min and not mins/cent? Although they are each ratios of the same concept, the philosophy behind the choice of measurement is interesting.

Someone choosing cents/min implies he is looking for the cheapest way to get a fixed task completed. “I need 30 mins in this store, how can I park cheapest?”

Someone choosing mins/cent implies he is looking for the maximum time allowed for the change in his pocket. “I’ve only got 50 cents, how long can I stay at the park?”

I think it’s an interesting question (especially when applied to the American mpg vs the European liters/100km). It could also bring up the idea of multiplicative inverses mathematically.