http://www.101qs.com/3232

The receipt is doctored (but correct) until I can get to coinstar.

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 …

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?

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.

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?

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.

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

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 $.