## Fair Share Pair

A coupleÂ weeks ago, I was discussing ratio tasks, including Sharing Costs: Travelling to SchoolÂ 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 free sale.

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.

Sharing Pairs.pdf

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

## Always, Sometimes, Never

This week should have been my first official â€“ third unofficial â€“ week back. Instead, Iâ€™m starting this school year as I ended the last â€“ walking the picket line. I haven’t beenÂ upÂ to blogging since this started. Â Below is a draft from June. I never got around to finishing it. The ending has a “pack up your personal belongings” feel. I left it as-is; seems fitting that this post should come up shortâ€¦Â I mean, 10% of my pay â€“ and my colleague’s â€“ was being deducted at the time.Â

Recently, I invited myself to a colleague’s Math 8 class to try out Always, Sometimes, Never. In this formative assessment lesson â€“ originally by Swan & Ridgway, I think â€“ students classify statements as always, sometimes, or never true and explain their reasoning.

Because it’s June, weÂ createdÂ a set of statements thatÂ spanned topics students encountered throughout the course. Mostly, this involvedÂ rephrasing questions from a textbook,Â Math Makes Sense 8,Â as well as from Marian Small’s More Good Questions, as statements. That, andÂ stealing fromÂ Fawn Nguyen.

To introduce this activity, I displayed the following statement:Â When you add three consecutive numbers, your answer is a multiple of three.

Pairs of students began crunching numbers. “It works!”

“You’ve shown me it’s true for a few values. IsÂ there a counterexample?Â What about negative numbers? Does it always work? How do you know? Convince me.”

Some studentsÂ noticed thatÂ their calculators kept spitting out theÂ middle number, e.g, (17 + 18 + 19)/3 = 18. This observation lead to a proof: take one away from the largest number, which is one more than the middle number, and give it to the smallest number, which is one less than the middle number; each number is now the same asÂ the middle number; there are three of them.Â For example, 17 + 18 + 19 = (17 + 1) + 18 + (19 â€“ 1) = 18 + 18 + 18Â = 3(18).

I avoided explaining my proof:Â x + (x + 1) + (x + 2) = 3x + 3 = 3(x + 1). This may have been a missed opportunity to connect the two methods, but I didn’t want to send the message thatÂ my algebraic reasoning trumped their approach. “Convince me,” I said. And they did.

To encourage students to consider different types of examples, I displayed a ‘sometimes’ statement: When you divide a whole number by a fraction, the quotient is greater than the whole number.Â Students were quick to pick up on proper vs. improper fractions.

Next, students were given eight mathematical statements. We discussed some of the statements as a whole-class. Some highlights:

A whole number has an odd number of factors. It is a perfect square.

I called on a student who categorized the statement as always true because “not all of the factors are doubled.” WeÂ challenged doubled before she landed on square rootsÂ being their own factor pair.Â For example, 1 & 36, 2 & 18, 3 & 12, 4 & 9 are each counted as factors of 36, but 6 in 6 Ã—Â 6 is counted only once.

The price of an item is decreased by 25%. After a couple of weeks, it is increased by 25%. The final price is the same as the original price.

Like the three consecutive numbers statement above, students began playing with numbers â€“ an original price of \$100 being the most popular choice. I anticipated this as well as the conceptual explanation that followed: “The percent of the increase is the same, but it’s of a smaller amount.” I love having students futz around with numbers; it’s so much more empowering than having them “complete the table.”

The number 25 was chosen carefully in hopes that some students might think fractions: (1 + 1/4)(1 âˆ’Â 1/4) = (5/4)(3/4) = 15/16. None did. Thereâ€™s a connection to algebra here, too: (1 âˆ’Â x)(1 + x) = 1 âˆ’Â xÂ². Again, I didnâ€™t bring these up. Same reason as above.

One side of a right triangle is 5 cm and another side is 12 cm. The third side is 13 cm.

All but one pair of students classified this as always true. That somewhat surprised us. More surprising was how this one pair of students came to realize

BTW, the blog-lessÂ Tracy ZagerÂ has a crowd-sourced aÂ set of elementary Always, Sometimes, Never statements.

Update: I stand corrected.