*Dan Meyer has struck again:*

“I spent a year working on Dandy Candies with around 1,000 educators… In my year with Dandy Candies,

, even in a crowd that included mathematics professors and Presidential teaching awardees. So now I’ll put that question to you.” Dan Meyer’s full postthere was one question that none of us solved

*What is it about a challenge like that?!? Of course I had to follow the thread! *

I read the over 100 comments as writers posed solutions, wrecked solutions posed by others, and even wrecked their own solutions! I watched as they systematically used the faults in their solutions as a springboard to better – but apparently still breakable – solutions.

I also heard a ghost of an admission that there may not be a single solution,

as timteachesmath writes, *“Which broken algorithm is best so far? An algorithm that fails for ‘720’ but works for 95% of really composite numbers less than 720 might be better than one that works for ‘720’ but only works for 80% of really composite numbers less than 720.”*

**There is a Lesson here!!!!**

I teach algenra 1 and geometry; that means 9th and 10th graders. I want to challenge them with Dan’s problem.

It’s simple, right? We are just talking about a box of CANDY!

I can see you now, shaking your head in disbelief: 9th and 10th graders able to frame an answer to a problem that *even 1000 math teachers couldn’t solve.*

**Not only that, I can give this lesson to both algebra AND geometry!**

Here is my explanation of the sequence of activities that would make the most sense to their budding understandings of math:

Essential Understanding: The best packaging involves the least surface area.

- The least surface area results from the tightest (closest) configuration of a cube’s side lengths.
- The surface area is a result of the combined areas of the six sides of the candy box.
- To find the minimum surface area for any number of candies,
a) if the number is prime: 1, 1, the prime; b) a perfect cube: root squared times six; c) numbers with three primes: use the three primes; d) numbers with four or more primes: Multiply groups of the prime factors back together to find three products. These three products will be the three factors that will be the measurements of the box.*check for the following conditions:* - Calculate the surface area from the measurements of the box.
- The box with the least surface area will have the factors that are closest to each other. It is possible for two of the factors to be the same number.

720 is a great example for (d):Prime factors of 720 are 2, 2, 2, 2, 3, 3, 5

While they can be multiplied back together to create numerous factors, not all sets of three factors will give us minimal surface area.

Some of the sets of three that can be created are:

4, 4, and 45;

8, 6, and 15;

10, 6, 12;

And so on, until we get the multiples 8, 9, 10;

Checking for the optimal area involves a handshake (multiplying) among each of the three numbers – 8 times 9, 8 times 10, and 9 times 10, adding the products together and multiplying by 2.

Does anybody else see the individual lessons embedded in this process? This one problem is incredibly rich!

*It’s not the solution, it’s the building of understanding*!

The interactive process of doing this by hand is a wonderful opportunity to teach finding primes (6n-1), (6n+1). Students might also feel the need to learn how to find prime factors (and learning that all numbers are products of primes!). The question would arise about the geometry of area vs surface area. (Think of the manipulatives! I wonder of my kids would feel silly stacking cubes of jello!!!)

We also wouldn’t be able to ignore the eminently practical side of saving the planet through minimal packaging – not to mention the extension of how many candies we should pre-package for the best shipping (i.e, how many boxes can fit into a bigger box? Can we afford to package odd sizes and still keep our costs low enough to generate profit and sales?) (ooh! I can teach my kids to design boxes – quadratics, anyone?) Here we could also lead the class into the sales curve (parabolas – more quadratics! I’m in Heaven!)

By Jove! I think I figured out why Algebra and Geometry finally got together! They complete each other!!!

And I love the fact that once my students come to this understanding of the problem, they could begin to write a viable solution, either in algorithm or in code. Or maybe their understanding leads them to the conclusion that a single algorithm isn’t possible – did somebody just whisper the word “proof”? *(You did just think that – you know you did!)*

Just think of the STEM project ideas this activity could generate…

As many of Dan’s commenters pointed out, this is tedious by hand. But the truth of the matter is – __they knew how to begin solving the problem by seeking to understand the problem to be solved!__ These are the skills our children need to learn. These are the lessons we need to teach. Let’s quit calling them 21st Century Skills; these skills really are useful for any age, anywhere. I’m living proof!* (I’ve made it this far on those skills, haven’t I? LOL!)*

As I see it Geometry and Algebra are alternative ways of seeing the world. Sometimes one is more successful than the other, sometimes each works, but gives different insights, and sometimes failure to get anywhere spurs the development of a new branch of math.

I have just been following Mean Green Math, with many ways, generated by students, of solving a very simple limit problem. Start with no. 1, and no. 4 will give you that “dammit, why didn’t I see that!” kick. It did me.

http://meangreenmath.com/2015/09/23/different-ways-of-computing-a-limit-part-1/