Ask your students to write down which one they’d use. Some students will assume you should always use $20 off. Others will assume you should always use 20% off. Still others will (rightly) understand that it depends on the cost of the item you’re buying.

Our goal here is to get all of those responses on paper, emptied out of the students’ head. If one student in the class blurts out “It depends!” we’ll lose a lot of the interesting and productive preconceptions lurking about.

Take a show of hands. Ideally you’ll find some disagreement. At this point, students should try to convince each other of their position.

Offer the material from act two here: a bunch of items that will test out their hypotheses.

Once we reach the understanding that it’s better to take *a percentage* off the ~~large~~ expensive items and better to use *the fixed value* with the ~~small~~ cheap items, it might seem natural to ask:

Where’s the break-even point? Where do cheap items become expensive items? For what dollar cost should you use one coupon versus the other?

Then generalize some more:

If the coupons read “x% off” and “$x off”, where is the break-even point? Does your answer work for every x?

**BTW**. There’s a perplexing little pile of coupons assembling at 101questions right now. Great work, everybody.

**Featured Comment**

â€œIf you are allowed to apply one coupon, and then the other on a purchase, does it matter in which order you apply them?â€ is also a really nice question.

You need to be careful in your use of â€œsmallâ€ and â€œlarge.â€ An iPod is small (yet expensive) compared to a large bouncy ball (inexpensive).

**2014 Dec 9**. Shaun Errichiello has created a series of printable cards for students to sort:

We asked students to physically sort the cards into groups. One group contains all the cards where the 20% coupon is the better choice, the other group contains all the cards where the $20 coupon is the better choice. We changed one of the prices (the desk) to be exactly $100.

## 14 Comments

## Kate Nowak

March 12, 2013 - 8:36 am -“If you are allowed to apply one coupon, and then the other on a purchase, does it matter in which order you apply them?” is also a really nice question.

## Chris Hunter

March 12, 2013 - 8:46 am -Agreed, Kate. Scott Keltner has a great post on this question and composition of functions. Worked very well with a group of teachers at a pro-d workshop.

http://scottkeltner.weebly.com/1/post/2012/12/coupon-composition-just-in-time-for-the-holidays.html

## David Wees

March 12, 2013 - 10:03 am -Here another idea on how to extend this. Create another picture and replace 20 with x, and see if students can work out the same comparisons with variables (ie. if x = 100, you are always better off choosing the percent discount, assuming that no stores will ever give you money back on a discount).

## Norma Gordon

March 12, 2013 - 12:17 pm -Other possible follow up questions:

Which coupon coupon might a store let you use multiple times on the same item? Which one would they not?

If you could use multiple coupons, what is the minimum number/combination of coupon for a free item?

## Dan Meyer

March 12, 2013 - 12:40 pm -You guys should add these awesome questions to the question page.

## Norma Gordon

March 12, 2013 - 1:03 pm -Done – but a challenge to shorten the # of characters!

## Mary Hillmann

March 13, 2013 - 2:27 am -“Once we reach the understanding that it’s better to take a percentage off the large items and better to use the fixed value with the small items…”

You need to be careful in your use of “small” and “large.” An iPod is small (yet expensive) compared to a large bouncy ball (inexpensive).

## Julie Reulbach

March 13, 2013 - 2:32 am -This will be great to do if we have extra time after the double stuffed Oreos question in math club today. Thanks Dan! Keep them coming. You all are making my job super easy (and fun) these days! :)

## Dan Meyer

March 13, 2013 - 3:41 am -@

Mary, good tip. I’ve amended the post.## Mike

March 13, 2013 - 4:56 am -Another interesting component that could be integrated is taxes. Is it cheaper to discount before or after taxes? Does it matter? Showing this algebraically is a good exercise.

## Sean

March 13, 2013 - 10:04 am -Fantastic work here.

So I’m thinking of a kid who gets it, and says something like:

“At $100, they’re the same. Anything less that that, go with $20 off, anything more go with 20% off.”

And I think someone like Bret Victor is pretty satisfied with this student’s response. Are you?

What are the actual advantages to making the student represent this symbolically as x – 20 = x – .2x?

You’ve written before (and quoted Einstein I think?) that the formulating of a problem can more important than the solution to it. I wonder, though, about the kid who first solves this problem, next generalizes (“$100 is always the break-even point”), but gets stuck at modeling it algebraically and even more stuck trying to manipulate whatever it is he’s modeled.

## Dan Meyer

March 13, 2013 - 2:06 pm -Sean:For students comfortable with that level of abstraction, the equation is faster than guessing and checking for the break-even point.

I’m not sure about that.

As I read Victor’s treatise on abstraction, my sense is he would create a 2D visualization of the break-even point for every value n (where in this case n = 20). Then he’d use that visual as a club for whacking teachers. “Why are you teaching abstraction?” Imaginary Bret Victor would say. “Look at the power of the visual!”

Obscured by his technical artistry, though, is the fact that the visualization wouldn’t have been possible if he hadn’t already known how to use the abstraction.

## Crystal

March 13, 2013 - 4:47 pm -Questions can also be asked in terms of what is best for the store. “A customer is purchasing a pair of running shoes for $120 from your family store. The customer has a $20 off coupon and a 20% off coupon. Your family store does not have a policy against using more than one coupon per purchase. Does the order in which you apply the discounts matter?”