A coupleÂ weeks ago, I was discussing ratio tasks, including *Sharing Costs: Travelling to School*Â from MARS, withÂ a colleagueÂ whoÂ reminded me of a numeracy task from Peter Liljedahl. Here’s my take on Peter’s Payless problem:

Three friends, Chris, Jeff, and Marc, go shopping for shoes. The store is having a

buy two pairs, get one pair freesale.Â Chris opts for a pair of high tops for $75, Jeff picks out a pair of low tops for $60, and Marc settles on a pair of slip-ons for $45.

The cashier rings them up; the bill is $135.

How much should each friend pay? Try to find the fairest way possible. Justify your reasoning.

I had a chance to test drive this task in a Math 9 class. IÂ asked students toÂ solve the problem in small groupsÂ andÂ record their possible solutions on large whiteboards. Later, each student recorded his or herÂ fairest share of them all on a piece of paper. If you’re more interested in sample student responses than my reflections, scroll down.

The most common initial approach was to divide the bill by three; each person pays $45. What’s more fair than *same*? I poked holes in their reasoning: “Is it fair for Marc to pay the same as Chris? Why? Why not?” Students notice thatÂ Chris is getting more shoe for his buck. Also, Marc is being cheated of any discount, as described by Student A. (This wasn’t a happy accident; it’s theÂ reason why I chose the ratio 5:4:3.)

Next, most groups landed onÂ $60-$45-$30. Some, like StudentÂ A, shifted from equal shares of the cost to equal shares of the discount; fromÂ ($180Â âˆ’ $45)/3 to $45/3. Others, like Students B, C, and D, arrived there via a common difference; in both $75, $60, $45 and $60, $45, $30, the amountsÂ differ byÂ $15. This approach surprised me. Additive, rather than multiplicative, thinking.

Student C noticed that this discount of $15 represented different fractions of the original prices; $15/$75 = 1/5, $15/$60 = 1/4, $15/$45 = 1/3. He applied a discount of 1/4 to all three because “it’s the middle fraction.” Likely, this is a misconception that didn’t get in the way of a reasonable solution.

Student D presented similar amounts. Note the interplay of additive and multiplicative thinking. She wants to keep a commonÂ difference, but changes it to $10 to better match the friends’ discounts *as percents*.

Student E applies each friend’s percent of the original price to the sale price. This approach came closest to my intended learning outcome: “Solve problems that involve rates, ratios and proportional reasoning.”

In spite of not reaching my learning goal, I think that this lesson was a success.Â The task was accessible yet challenging, allowed students to make and justify decisions, and promoted mathematical discourse.

Still, to increase the future likelihood that studentsÂ solve this problem using ratios, I’m wondering aboutÂ changes I could make. Multiples of 20 ($100-$80-$60) rather than 15 ($75-$60-$45)? Different ratios, like 4:3:2 or 5:3:2, mightÂ help; the doubles/halves could kickstart multiplicative thinking. (Also, 5:3:2 breaks that arithmetic sequence.)

Or, I could make changes to my questioning.

When I asked “What do you notice?” students said:

- the prices of the shoes are different
- Chris’ shoes are the most expensive
- Marc’s shoes are the cheapest
- Chris’ shoes are $15 more than Jeff’s, whichÂ are $15 more than Marc’s
- Jeff’s shoes areÂ the fugliest

Maybe I could ask “What else could you say about the prices of Chris’ shoes compared to Marc’s?” etc. to promptÂ comparisons involving ratios. If that fails, I’m more comfortable connecting ratiosÂ to the approaches taken by students themselvesÂ than I am forcing it.

BTW, “buy one, get one 50% off” vs. “buy two, get one free” would make a decent “Would you rather?” math task.

h/tÂ Cam Joyce, Carley Brockway