These Horrible Coin Problems (And What We Can Do About Them)

From Pearson’s Common Core Algebra 2 text (and everyone else’s Algebra 2 text for that matter):

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.

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

One fix here is to locate a context that is more relevant to students than this contrivance about coins, which is a flimsy hangar for the skill of “solving systems of equations” if I ever saw one. The other fix recognizes that the work is fake also, that “solving a system of equations” is dull, formal, and procedural where “setting up a system of equations” is more interesting, informal, and relational.

Here is that fix. Show this brief clip:

Ask students to write down their best estimates of a) what kinds of coins there are, b) how many total coins there are, c) what the coins are worth.

The work in the original problem is pitched at such a formal level you’ll have students raising their hands around the room asking you how to start. In our revision, which of your students will struggle to participate?

Now tell them the coins are worth $62.00. Find out who guessed closest. Now ask them to find out what could be the answer – a number of quarters and pennies that adds up to $62.00. Write all the possibilities on the board. Do we all have the same pair? No? Then we need to know more information.

Now tell them there are 1,400 coins. Find out who guessed closest. Ask them if they think there are more quarters or pennies and how they know. Ask them now to find out what could be the answer – the coins still have to add up to $62.00 and now we know there are 1,400 of them.

This will be more challenging, but the challenge will motivate your instruction. As students guess and check and guess and check, they may experience the “need for computation“. So step in then and help them develop their ability to compute the solution of a system of equations. And once students locate an answer (200 quarters and 1200 pennies) don’t be quick to confirm it’s the only possible answer. Play coy. Sow doubt. Start a fight. “Find another possibility,” you can free to tell your fast finishers, knowing full well they’ve found the only possibility. “Okay, fine,” you can say when they call you on your ruse. “Prove that’s the only possible solution. How do you know?”

Again, I’m asking us to look at the work and not just the world. When students are bored with these coin problems, the answer isn’t to change the story from coins to mobile phones. The answer isn’t just that, anyway. The answer is to look first at what students are doing with the coins – just solving a system of equations – and add more interesting work – estimating, arguing about, and formulating a system of equations first, and then solving it.

This is a series about “developing the question” in math class.

Featured Tweets

I asked for help making the original problem better on Twitter. Here is a selection of helpful responses:

2014 Oct 20. Michael Gier used this approach in class.

Featured Comments

Isaac D:

One of the challenges for the teacher is to guide the discussion back to the more interesting and important questions. Why does this technique (constructing systems of equations) work? Where else could we use similar strategies? Are there other ways to construct these equations that might be more useful in certain contexts?

About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

32 Comments

  1. Reply

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

    Can I get a hallelujah?

    Also, as someone who grew up to be a math teacher, I want to be on record as saying that not all of us liked these problems when we were math students. :)

  2. Reply

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

    https://twitter.com/farrarscott/status/522517332661633024
    ——-
    Appeal to coins already an attempt at “real”/accsble . But strange constraints to know. Scrap&Stepback: 2eq2var: Whats intrstng?hmm
    ——-

    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?

    ======================

    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!

    ======================

    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.

  3. Reply

    I just saw Denise Gaskins’s @letsplaymath tweet too… super good.

    @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.?

    ===============

    This is so good. **How many questions do you need?**

    Bam, we’ve just flipped this whole problem on its … head. ( http://instantrimshot.com/index.php?sound=rimshot&play=true )

    Because the idea is not finding the coins… its about how many questions we need in order to find the coins.

    Algebra is not about finding the answer. Its about when there is one.

  4. Reply

    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)

  5. Reply

    Someone had to program that CoinStar machine you’ve shown in the photo. That program has to calculate the exact amount of money it has taken in. A discussion of how the machine works would get at the math while also making the problem practical.

  6. Reply

    “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?

  7. Reply

    Justin Lanier: I share many of your reactions. I wish to state one idea that is in my mind. HS math suffers from needing kids to do certain things in certain ways. And we seem stuck in praising the glories of the SMPs (this decade, a different set of math practices or habits of mind another), yet only so long as students put those mathematical practices to use to get the result I want them to, on a rather prescriptive or restrictive task, created with an end in mind, applying specific mathematics I’ve just taught.

    Do we want kids to do math? I don’t really think so. We want them to do our math.

    If we wanted kids to not “be bored,” we would ask them to invent, create, find the numbers that would make 5 squares inside one square possible (see http://www.doingmathematics.com/blog/a-puzzling-situation)…

    Most simply, we would ask kids mathematical questions before they had tools at their disposal to solve them. And we would invite them to invent, what afterwards we might call mathematics.

  8. Reply

    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.

  9. Reply

    Last year with remedial kids with behavior problems, I needed a way to motivate them to want to do these coin problems. I displayed a bag full of coins and told them they could have the bag of coins IF they could me exactly what was in the bag. I have them nothing more until they asked… How many coins in total? What kinds of coins?

    I had the students describe how they figured it out. It was awesome.

  10. Reply

    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.

  11. Reply

    I have to agree with Breedeen. I couldn’t stand this type of problem. As a self-proclaimed “math memorizer” growing up, I would just try to find some sort of pattern to set this question up and follow a set of procedures. Wasn’t fun, but I managed to get the job done. I want more for my students than that.

    I like the video Dan shared because it can serve as a great conversation starter, but I see where Fawn is coming from with it possibly being too much, too soon.

    I wonder if rather than jumping into a question with tons of coins, we show a video of people putting their change into a city bus meter. Here, you must have exact change. Could serve as a good way to scaffold into a more challenging problem with a ton of coins.

    Regardless of the approach, I find that the question “as is” doesn’t have much to hook the kids into wanting to do some excellent work.

  12. Reply

    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.

  13. Reply

    Fawn,

    Nice diagrams!

    Interesting the choices you made for representation: finding a common divisor between the two items (quarters and dimes).

    Visually you went from 25:10 to 5:2. I like that this picture also has the 6 by 20 box (20 times 5 being $1).

    Your box provides the constraint of the $6, and then you explore how to fill it using the quarter and dime tiles.

    I wonder: how did you know how to hit 42 coins exactly? Was it trial and error? If a student solved like this, I might want to push them to say what if you trade out quarters for dimes or vice versa ?

    I might expect a student to see that the dimes make a dollar very easily, and the quarters make dollars very easily. So I can see a student finding a quick way to $6 and then nudging around to find 42.

    Note, if the question stated 45 coins, there might be some more thinking to do.

    I also like that the picture model nudges us away from the non-integer values for coins. A student that puts down 1 single yellow square can have their attention called to what that would mean… and then they could probably explain themselves why that wont work.

  14. Reply

    You could also include some work on averages and computation/answer checking. Pretend they have to program a CoinStar to work. They get the total amount, and the weight. Why count the coins? If you already know how many of each then it’s a waste of a problem.
    Average weight of a penny? (weigh 10 pennies, find average).
    Average weight of nickel?

    Then tell them the machine compares the weight of the coins against the final calculated value, to see how many of each there were.

    There is then a natural extension to 3 equations in 3 variables, etc as you add more coins. Then add in Cramer’s rule?

  15. Reply

    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.

  16. Reply

    Scott:

    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.

    Agreed. I actually have a number of regrets about how I prosecuted my case against “pseudocontext” a few summers ago. Your comment summarizes the regrets pretty well. The work students do matters more to me now than the context in which they do that work.

    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?

    I actually don’t know how I’d help students justify the fact that there’s only one solution outside of a graph. Help me out here.

    @Gerry, love the solution approach. And many many points awarded for its generalizability.

    Brian:

    Do we want kids to do math? I don’t really think so. We want them to do our math.

    Just as a fun thought exercise, let’s assume public school teachers are accountable to a certain definition of mathematics, a body of knowledge. Call it “our math.” Do you see any way to respect both a student’s development of mathematical knowledge as well as our math.

    Peter Monnerjahn, echoing on Twitter some comments here:

    Why would you want to make a question interesting that no one ever has asked? Why not work towards real student questions that need some of that math?

    My response there:

    Good question. Because I enjoy and learn from the exercise. Turning bad to good requires a lot of the same skills as creating great.

  17. Reply

    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.

  18. Reply

    > I actually don’t know how I’d help students justify the fact that there’s only one solution outside of a graph. Help me out here.

    Maybe this is something that the teacher holds in his or her head while only hinting at: this is where we honor the genius of the Cartesian plane a little bit.

    The graph informs us of:

    – the infinite nature of the linear models
    – the monotonic nature of the linear models
    – the difference in ratios between coinA:coinB and valueA:valueB (thus different slopes)
    – all of the ordered pairs that fit the conditions of the coin count
    – all of the ordered pairs that fit the conditions of the value total

    And combining our geometric knowledge about lines at different angles, mapping that onto the linear models, mapping THAT onto the coin situations… that’s how we are justifying our singular solution.

    How this all might play out in a classroom would be just that extra 15 seconds of wait time, that extra question for group discussion, that incremental food for thought…

    Not so much that we want a full answer, but that we want the students to take up the role of justification.

    T, to S: “Why DO we know there is one solution here…?”

    And if S reasons via graph, great!

    But S might say something like: “well if we take off a quarter, and add a time, we keep the coin total at 42, but lose value…. we’ll always lose value at this way so there can be no solutions with more quarters!”

    That’s (a) proof by contradiction and (b) using concepts that apply for theory of functions: decreasing, monotonic. (in this case its a sequence) But also the S has synthesized a hypothetical that generalizes to a larger case: take off a quarter, add a dime. Then they’ve upped their abstraction by arguing that any case like that will not provide a solution. (And similar arguments can work for adding a quarter, subtracing a dime)

    Then T has the opportunity to engage the class in a discussion of bridging the graphical and the deductive reasonings.

    It may very well end up becoming a discussion on why Cartesian graphs are so great, and all the better! Lots of students struggle with graphing because they are stuck on procedure without realizing what a graph represents specifically in an instance, or generally.

  19. Reply

    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.

  20. Reply

    This type of problem was done when I took Algebra I. The objective of these “contrived” problems–involving coins, ages, etc.–is to focus on:
    1. reading comprehension.
    2. identifying the underlying unknowns.
    3. defining variables to denote these unknowns.
    4. deriving the equations that the variables must satisfy.

    Many American students, who have taken so-called honors courses in high school, are so thoroughly pseudo-educated that they are unable to do any of the above.

    This was exposed so clearly in Andrei Toom’s landmark article, “A Russian Teacher in America,” originally published in the Journal of Mathematical Behavior and reproduced at:

    http://toomandre.com/my-articles/engeduc/ARUSSIAN.PDF

  21. Reply

    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 $.

  22. Reply

    I think the puzzliness of this problem will engage students enough. My students love these types of problems, although they often use a guess and check method (Math 8). I think the only thing I would do differently is make the numbers smaller to give them an entry point. I would either ask, “I have $1 in coins, what coins do I have.” or “I have 10 coins, how much money do I have?” I’d then proceed to get super excited over all the different answers. Maybe then ask more specific questions like “I have $1 in just dimes and quarters, what coins do I have?” I think all the students need to get interested in this is an open ended entry point, and then they can see that the more specific we get, the less open ended it becomes.

  23. Reply

    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.

  24. Reply

    I think that just giving students coins for the class and trying to guide them in some way, but just leave them to make conclusions for themselves throughout the class period would be a beneficial way to handle these types of problems. There would definitely need to be guiding questions to get that on the right track, but letting the students configure different situations and groups, I believe, would benefit their learning.

  25. Reply

    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.

  26. Reply

    I should also add (although this is probably obvious to readers of this blog), that traditional ways of using these problems are dangerous because they focus students on finding the unknown and grading them based on there mechanical skill in producing desired results (correct solutions).

    One of the challenges for the teacher is to guide the discussion back to the more interesting and important questions. Why does this technique (constructing systems of equations) work? Where else could we use similar strategies? Are there other ways to construct these equations that might be more useful in certain contexts?

  27. Reply

    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.

  28. Reply

    Paul:

    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.

    I like this a lot.

    I tossed this quote from Isaac D up to the featured comments section:

    One of the challenges for the teacher is to guide the discussion back to the more interesting and important questions. Why does this technique (constructing systems of equations) work? Where else could we use similar strategies? Are there other ways to construct these equations that might be more useful in certain contexts?

  29. Reply

    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 …

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