A week and a half ago, I was at The Fair and noticed this:

Two different mini donut vendors, two different sets of prices. I wondered “What’s the best deal?” As much as I love asking students “What do you notice? What do you wonder?” when introducing problems–see this introduction to “I Notice, I Wonder” from The Math Forum—Ā I’m thinking about skipping this routine here and just presenting the context and problem (using these photos). Let me explain that decision later in this post.

I love best buy problems because they lend themselves to multiple strategies. *From students*, not “let *me* show you six different ways to solve these.”Ā For example, I anticipate that many–most?–students will determine and compare unit rates. It’s an intuitive thing to do. (Or not. See Robert Kaplinsky’s discussion of his Carnival Tickets task.) At *FunDunkers*, it’s $12 for 3 bags, or $4 per bag; at *those little DONUTS*, it’s $10 for 2 bags, or $5 per bag. Winner: *FunDunkers*. Students may also think common multiples (or scale up). At *Fundunkers*, it’s $12 for 3 bags, so $24 for 6 bags; at *those little DONUTS*, it’s $10 for 2 bags, or $20 for 4 bags, or $30 for 6 bags. We can easily compare ratios or rates when one term is the same, be it one bag or six.

After having some students present their solutions, I’d display these photosā¦

ā¦ and ask students if they’d like to revise their solutions. Now, students will likely determine and compare new unit rates. “One” has changed: dollars per one *donutĀ *instead of dollars per one *bag*Ā (#unitchat). Here some students may also consider one *dollar* to be the unit (and avoid fractions or decimals in doing so). At *FunDunkers*, it’s 36 donuts for $12, so 3 donuts per dollar; at *those little DONUTS*, it’s 45 donuts for $15, so 3 donuts per dollar. A tie. Less interesting than a reversal but, hey, “real world” numbers.

I like the teacher move of gradually providing students with new information and asking them if they’d like to revise their thinking. (It’s a strategy I used with Sinusoidal Sort and “Selfiest” Cities.) Not all the time. But in this task, if students wonder how many donuts are in each bag, then you kinda have to provide this up front. This means that we might not get the dollars per bag idea on the table at all–a missed opportunity to compare and connect strategies.

(Anyone else notice my donut hole-like tunnel vision in that last *FunDunkers* photo? One step back and maybe there’s a math task involving souvenir cups and pop refills.)

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