I think I just tweeted this as you posted the article:

https://twitter.com/farrarscott/status/522517332661633024
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Appeal to coins already an attempt at “real”/accsble . But strange constraints to know. Scrap&Stepback: 2eq2var: Whats intrstng?hmm
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I think you and Ryan Adams’ idea (coinstar) are good enough and rich enough to justify keeping the Coin contrivance. Not all contrived situations are bad, its how we leverage them.

I think though I might come from a different angle. And this might work well as a followup for someone doing yours or Ryan’s first:

Consider its a little strange to know how many coins you have but not how many of each denomination. Students can hang with you a little bit, but likely some loudmouth in your class will point this out along the way. #embracetheloudmouth

So on day 2, “you’re totally right, Lou D’mouth, a strange counting technique that counts without categorizing”

Now we can acknowledge the contrivance because its going to lead us to something special about this mathematical structure. We acknowledge and loosen our contrivance: parameterize the # of coins.

(Lets make the numbers smaller) Say we have $2.00 If we only use quarters and pennies… how many coins do we need? N=8 coins (8q0p) is one way… N=104 is another (ask how!).

1. spend some time finding Ns. mathematical practice standard: make use of structure! (How do we know when we’re done?)

2. To add up to $2.00 does every N (coin total) break down into only one quarter and penny partition? why?

3. Why do some Ns (105 e.g.) not have any solutions?

4. ***What is it about the quarters and pennies, Ns, and $ totals that forces us to have unique solutions or not?***

and 5. Can we contrive a coin-situation so that there are $s,Ns with multiple quarter-penny partitions? Is this possible?

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I remember a lot of handwaving and “worship” in my own education towards the fact that 2 equations and 2 unknowns has a solution. (and 3 eq, 3 unknowns, etc…)

But statements like “ok so this is 2 equations and 2 unknowns so we know how to do it” are exactly what contribute to the magic spellbook idea of math. Kids feel that those who are good at math know the right spells to incant, and “2 equations 2 unknowns” must be a pretty good spell if it makes answers appear out of nowhere!

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I think this would be a rare case in which I would NOT go for a graph quickly. Yeah, we know about intersections of lines. Yeah we can show these coin relationships are linear.

But can we justify to ourselves that this idea is generalizable? We start with quarters pennies sure. In the middle maybe we prove something about # of solutions of systems and we talk about other ideas like mixtures and such. But by the end? We need students to grok (http://en.wikipedia.org/wiki/Grok) this idea of systems and solutions.

Algebra is about understanding the constraints of the problem and finding a way to describe that relation. In this case, that means we have 42 coins. All of them are worth at least $0.10. Some are worth $0.15 more.

Therefore, the solution presents itself:

42 coins, dimes and quarters. Total Value = $6

– How much would you have if they were all dimes? ($4.20)
– How much short of $6.00 is that? ($1.80)

Since Quarters are worth 15 cents more than dimes,

– How many $0.15’s are needed to make up $1.80? (12)

Therefore, 12 quarters, 30 dimes (42 coins)

Check:

12 x 0.25 = $3.00
30 x 0.10 = $3.00
Total = $6.00

Before you ask, a final test is ‘will the method presented always work’?

The response is, this isn’t a method; it’s a logic application. And yes, it always works, on coin problems, 2 legs vs 4 legs, tricycles and bicycles, etc … all 2-var convergent problems.

Of course, it doesn’t require manipulation of identified variables and writing equations or drawing graphs, which may defeat the purpose of doing coin problems in the first place, but…

Like always, just Something to Think About. (STTA)

“When students are bored with these coin problems, the answer isn’t to change the story from coins to mobile phones.”

Why should it have come to this in the first place?

“The only math students who like these problems are the ones who grow up to be math teachers.”

I would give this problem a more generous read. It seems like a perfectly serviceable problem as written. It’s approachable by a range of students and through several approaches. There’s plenty of good work and discussion that can be prompted by it as-is. It isn’t “just solving a system of equations”, unless it’s embedded in a set of basically-identical problems in a “solving a systems of equations” section. All told, it doesn’t seem like a particularly tough sell, in the spectrum of textbook math tasks.

My sense is that it would be hard to make thoughtful estimates from watching the video–anything much more than shots in the dark.

Revealing information slowly and as-needed is a useful approach, here and often.

The numbers in the situation you pose are significantly larger than those in the original problem. To what effect?

Both versions involve both setting up and solving a system (if an equation-y route to a solution is taken). It doesn’t seem to me that “the work” in your version is so very different from in the original. (Excepting the estimating and slow reveal).

As I mentioned on Twitter, I think it’s much more important what happens *after* this problem, and before, rather than perhaps-in-its-stead. How do you see your recasting of the task as fitting into a sequence of lessons? After this problem is solved, what will students do next?

Because which seems like a better long-term strategy in making sure our students “aren’t bored”? By recasting standard problems as jazzier one-offs, perhaps involving some work that is less formal and rote? Or by showing them and involving them in how math grows, how there’s an arc, how problems beget new problems and new playgrounds? How with our tending math brings forth more ideas and more good work to do?

I, too, share similar thoughts with Justin. The original problem is accessible because it involves only $6 and two different types of coins. There were too many coins in the Act 1 video that I don’t even know where to begin to even throw out just one estimate.

I rarely set up a system of equations to solve this type of problem. I draw pictures instead.

https://docs.google.com/document/d/1zHwOTpTF6DPw0oeug8kh_pVtG6OR-qgErvM13xe_R_k/edit

Thanks, Dan.

as a math teacher I hated these contrived problems. That’s why I love IMP math. The systems problems in this program had realistic problems and my students could see practical applications for this math. The end problem involved a city planning project using matrices to make recommendations to the city for land use.

Building on the 20 questions idea:
“@ddmeyer Turn it into a 20-questions game: Stu reach into coin jar & grab handful, others ask Qs to find out what coins they have.”

I might get them to talk about what information they’d like to know about the handful of coins and how they could use that to help them figure out the answer. Once they had developed those word statements, they would be a lot closer to putting equations into a framework of meaning. I often have students do problems with ‘words first’ (no numbers allowed!!!). This gets them thinking about the concepts, meaning, and patterns in the problem rather than about a specific solution to a specific question. They find it quite frustrating at first to have to describe how to solve a problem without using any numbers in their explanations.

This is very similar to James Cleveland’s Archimedes Lab, which I’ve always wanted to try:

http://rootsoftheequation.wordpress.com/2013/03/27/the-archimedes-lab/

Or, show them the more creative solution:

Mark has 42 coins consisting of dimes and quarters. The total value of his coins is $6. How many of each type of coin does he have? Show all your work and explain what method you used to solve the problem

Assume they’re all dimes…42×0.10 = 4.20.

The extra $1.80 has to be distributed among the quarters.
1.80 / 0.15 = 12.(since 10 cents is already allocated previously, you divide by .15)

So 12 quarters is $3.00, and the 30 dimes make another $3.00.

Here is a video of a 6th grade Japanese lesson on this topic. Their approach is to focus the thinking on what happens as you trade a quarter for a penny — as Gerry did.

This approach also makes it clear, hopefully, why there is only one solution. The total $ will decrease each time you trade another quarter for a penny.

I have used a pseudo 3-Act format with a coin problem like this. I started by giving paired kids a stapled paper bag full of coins, and asking them to figure out what’s in the bag. It’s pretty obvious that the bags are full of coins, and I confirm that piece of info when they ask.

I gave them digital scales to use, and asked them to figure out how they might use the scale to figure out what’s in the bag. If/when they ask for the weight of different coins, I provided that as well. I had the bags separated into categories of easy/medium/challenging, which made for easy differentiation, and was deliberate about giving certain bags to certain students.

Although this is a pretty contrived problem, I have seen that kids are motivated to see what’s in the bag, and highly gratified when they get to the correct solutions.

Visual way to show one solution:

Imagine putting my coins in a long row with quarters first then pennies. Line up your coins below mine.

Q Q Q Q Q Q Q P P P P…..
Q Q Q Q P P P P P P P …

The only way we have the same $ is if the rows match. Otherwise, the person with more quarters clearly has more $.

I like the idea of building up questions so the students have a better understanding of what exactly is being asked. If you throw the numbers 42 and 6 at them, they might not know what is being asked. Start with smaller amounts of money and work your way up. THEN start to ask in-depth and critical thinking questions which will force them to understand the material at an even deeper level. Ask them why it works or how to make sure that they have the correct answer.
To be honest too, watching the clip did not really help me form any type of connection to the problem. If we want to form a connection, then we should have the students actually deal with coins and work it out with their hands. Then develop different equations to play with.

To me, you can always be more creative with problems.

Although I see a lot of great ideas in this thread for math tasks, very few of them (IMHO) achieve the same result that the classic coin problems are intended to achieve (whether they do or not may be a function of context and teaching rather than the problems themselves).

As I see it the classic coin problems are supposed to communicate two ideas:

1. You can construct systems of equations to describe situations where the variables are related to each other in some way. This might be because we want to solve for an unknown quantity, but there might also be other reasons (for example we may be trying to generalize or abstract an observed pattern).
This is an incredibly useful skill in a wide range of fields, and it is also one that we will continue to build on throughout algebra/geometry/calculus.

2. More specifically, when there are units that are distinct but related to each other with known ratios (in this case number of coins related to monetary value) we can decrease the number of variables by including the ratios as coefficients.
In other words, while we could describe the relationship in this particular problem as Nd + Nq = 42 and Tv = 6, this would mean that we have three variables and no particular way to combine these into a single equation or solve for any particular unknown. By expressing the value as 0.1Nd + 0.25Nq = 6, we have used our knowledge of conversion ratios to decrease the number of variables.
This again is an incredibly useful skill. Just think of how many equations people use every day (especially in financial contexts) relate value to quantity.

The challenge as I see it is that this is a “fake-world” problem in both the best and worst ways.
It is obvious to both students and teachers that using algebra to determine the numbers of coins of each denomination is a completely useless skill in its own right (even as a party trick it’s pretty lame).
On the other hand it is much simpler and more accessible than a “real-world” problem involving manufacturing constraints, economies of scale, cost of goods sold, or break-even points. None of the latter, while genuinely important and valuable for adults, are likely to matter to students any more than the numbers of coins in my pocket do, and they involve more complex contexts.

The worst of both worlds would be, as you say, to change the coins to mobile phones or something else that seems more relevant but is equally implausible and also removes the key point that we are relating quantity to monetary value using known value ratios.

It is less clear what the best of both worlds would be. For myself I would explicitly tell students that we are using this contrived (even silly) problem with coins to develop skills that we can apply to much more important problems later on. I have found many students (and myself, which may make me blind to some of the problems) who can easily enjoy classic coin problems (and even consecutive integer problems) as a game, and accept the implausible constraints as part of the fictional context of the game.

Nobody worries about whether chess is a plausible simulation of warfare as long as it is an engaging task which develops strategic skills.

I wonder if the video question students really want to answer is “How much money is it?” If it were my coins that’s what I want to know.

I suppose you could give them different “Act 2” information. It seems clear that there are more pennies than quarters and they could be given that ratio (after some guess/discussion) and the total number of coins. From there they could try to figure out the value of the coins. Unfortunately this set up reduces the need to reason between an expression about the value of the coins vs. an expression about the total coins

Perhaps we need a video starting with two containers. One filled with dimes and the other filled with quarters. Would they wonder which held more money? If so, they could be given the information of total coins and total value of the coins in Act 2. That leaves the necessity to model in order to answer their question about which container held more money.

The “who wins” type of question that Paul suggests has more appeal than “how many”. But, that question allows for a simpler guess & check approach:

Suppose half the $ were quarters: $3 is 12 quarters. That leaves 30 coins or $3 as dimes. It is a tie. The original question is an oddball example. But, the strategy of seeing what happens if half the money were quarters works for non-oddball cases as well.

You could do something like thispicture presentation of the problem for a minimalist approach.

I am wondering if there is a productive way to focus on the average value of the coins involved when approaching these problems: 42 coins worth $6 means the average value of a coin us $0.14 …