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.
I asked for help making the original problem better on Twitter. Here is a selection of helpful responses:
@ddmeyer I hide coins in a 35mm container & asked kids to guess the exact contents. Then I answer Qs about the total value, types of coins
— Jennifer Abel (@abel_jennifer) October 15, 2014
@ddmeyer Remove 42 coin restriction. How about.. Least # of coins? Max # coins? What pattern is there to coins needed to make $6?
— Jeff Harding (@GradesHarding) October 15, 2014
@ddmeyer If it’s only more probs like this one with numbers changed? Boo. But what about: 42 coins and $6. What denominations could this be?
— Justin Lanier (@j_lanier) October 15, 2014
@ddmeyer just spitballing: start w/ "Try to make $6 with 42 coins" (or whatever). "Can you make it w/ 43? 41?" "Can you make $3 w/ 24?" etc
— Geoff Krall (@geoffkrall) October 15, 2014
@ddmeyer (Probs more like "I have 37 cents, what coins could I have?") Idea to get kids thinking re constraints, multiple possibilities.
— Katherine Bryant (@MathSciEditor) October 15, 2014
@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.
— Denise Gaskins (@letsplaymath) October 15, 2014
@ddmeyer Make a video of me taking my coin jar from home to CoinStar. Make it take a long time…until kids ask how much money did you get?
— Ryan Adams (@MrRadams) October 15, 2014
2014 Oct 20. Michael Gier used this approach in class.
"I'm gonna solve this one, Mr. Gier!"
— Michael Gier (@mgier) October 20, 2014
— Michael Gier (@mgier) October 20, 2014
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?