Math problems aren’t written, they’re rewritten.
Here at Fishtank we write a lot of math problems. Recently, our Managing Director of Math Jami Therrien Wells estimated that in our recent push to develop Fishtank Plus materials for 3rd through 8th grade, we write about 400 new problems each month. As the person who writes most of the answer keys for these problems, I can tell you that that’s a lot of math problems.
If you’ve ever written math problems for your class, you know that your first draft usually isn’t perfect. There’s nothing like having 20 to 30 vocal copy editors critique your work in real time, pointing out spelling mistakes, laughing at an unreasonable answer (Really Ms. Gasaway? He bought 47 apples?), or glowering through a problem that requires the product of three fractions with prime number denominators.
Thankfully, our team collaborates, checking each other’s work and providing thoughtful feedback to help us help you avoid these problems and get the best possible information from your students’ work.
This is a real example of how a word problem I wrote evolved from its fun, but rough beginning into the rich, polished problem that ended up on one of our 7th grade Fishtank Plus assessments.
1st Draft
|
Anne-Marie buys a couch in the United States for $225. The sales tax on her purchase is 6%.
Part A: What is the total amount of money she pays for her couch, including the tax?
Part B: In France sales tax is 20%. How much would a couch in France have to cost before tax to have the same total price as Anne-Marie’s couch in the US?
|
I liked this original concept – I’d been thinking recently about how easy it is to buy unusual or rare things on the internet. I can put a book from England in my cart and let the magic of the internet take care of converting the currency, adding up the sales tax and shipping, and decide at the end whether it’s worth the total price. Yep, it’s lazy and I could add it or estimate it myself, but I do a lot of math for my day job, so…
Anyway, this standards-aligned two-part question set up an engaging real-world scenario with realistic numbers for the price and tax percentages that would be an excellent evaluation of students’ knowledge.
I passed this question along to Jami who had a very good point:
I like the idea of this problem, but I think it’s complicated by the different currencies/exchange rates. While most 7th graders might not be in tune to that, you may have some students who are, and it may muddle the problem for them. What if she finds a coupon for a percent discount, and students have to calculate the total price after discount and tax?
Fair enough. I like thinking about buying fancy French furniture, but it is unnecessarily complicated. So I rewrote the question.
2nd Draft
|
Anne-Marie buys a couch at BuyKea for $225. The sales tax on her purchase is 6%.
Part A: What is the total amount of money she pays for her couch, including the tax?
Part B: Heyfair, another furniture store, is having a sale where customers can take 20% off their total purchase price. How much would a couch at Heyfair have to cost to have the same total price as Anne-Marie’s couch at BuyKea?
|
I also really liked this problem. We lose the international element but gain two puns and test an additional skill: discounts. On balance a win for everyone. I sent the rewritten problem back to Jami and she quickly had another note for me: “When is the tax calculated? Do you use the coupon and then calculate it or find the total and then take off 20%?”
We sat down together and talked about what we wanted this assessment item to do and how we could communicate it more clearly without losing the richness of the double part application problem. Here’s what we came up with:
3rd Draft
|
Anne-Marie is looking for a couch at Buykea, a furniture company. She has a $225 budget and a 20% off coupon.
Part A: What is the highest price couch she could consider without going over her budget?
Part B: There is a 6% sales tax on furniture. How much more money would Anne-Marie need to cover the sales tax?
|
We thought this pared down question more clearly communicated what was going on in the scenario. We passed it along to two other members of the math team, Sarah and Zach, who solved it… and ran into problems.
“Wait, if she needs more money to cover the sales tax, does that mean she’s going over budget? Do I need to rework Part A to cover the sales tax too?” they both said. “Do we calculate the sales tax on the total before or after the coupon?”
One of the most important things a math problem should do is communicate the desired action/calculation clearly. A problem needs to be unambiguous so students have the chance to communicate their knowledge clearly back to their teacher. In the real world, you’d just ask the salesperson or look at your cart before checkout. On a math quiz, you might not be able to ask that question.
Also, depending on how you interpreted the problem, you might not be able to get Part B right if you got Part A wrong. We could do better.
Final Version
|
Anne-Marie is looking for a couch at Buykea, a furniture company. She has a $225 budget and a 20% off coupon.
Part A: What is the highest price couch she could consider without going over her budget?
Part B: Anne-Marie found the perfect couch at the top of her budget! It came to $225 after she used her coupon. Unfortunately, she forgot that there is a 6% tax on furniture. How much more would Anne-Marie need to cover the sales tax?
|
I was really happy with this final draft. The answer to resolving the ambiguity was adding more context and drama. Poor Anne-Marie! You can relate to her surprise and that moment of panic makes it clear that she’ll be over budget. Students clearly understand what they need to do and, as an added bonus, Part B is now computationally independent from Part A making grading easier for teachers and easing some anxiety for students who aren’t sure of the answer to Part A.
Most problems don’t need this much back and forth, but this example highlights some of the things we think about when reviewing and editing our work: Is it clear? Is it aligned? Is it accessible? Getting it right isn’t always easy but it is really satisfying when we do.
Find a problem in one of our courses that you think could use more clarity? Send us an email!
Kate Gasaway is the Curriculum Associate for Mathematics. She holds a Bachelor’s of Science degree in Psychology with research and business certificates from the Georgia Institute of Technology, and a Master’s in Effective Teaching from the Sposato Graduate School of Education. She started her teaching career at Neighborhood House Charter School, spending five years teaching 8th grade math and one year teaching 6th grade math.