Let’s quit calling them 21st Century Skills; These babies are useful in any century!!!

Dan Meyer has struck again:

“I spent a year working on Dandy Candies with around 1,000 educators… In my year with Dandy Candies, there was one question that none of us solved, 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 post

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.

1. The least surface area results from the tightest (closest) configuration of a cube’s side lengths.
2. The surface area is a result of the combined areas of the six sides of the candy box.
3. To find the minimum surface area for any number of candies, check for the following conditions: 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.
4. Calculate the surface area from the measurements of the box.
5. 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!)

Best 1st Day of School??

Dateline: Monday, August 10, 2015 Gwinnett County, GA

Today is the day! The streets are filled with yellow buses; sidewalks have filled with backpack-covered children, as if a clutch of turtles are migrating. Cars fill Continue reading “Best 1st Day of School?? “

My students take their EOCT test tomorrow

I feel like a parent sending their child out into a storm. I’ve made sure their coat is buttoned, they have dry boots on their feet, and a good umbrella. They are prepared…. Or are they?

The state says this is an important assessment. I tell my students to answer the best they can, and leave no question unanswered. I think, I could have done more; why didn’t I go over this or that; we should have done more with probability; will they remember what capital B is in the area formula?! Oh well, it is their turn now. I want them to taste success, to let their grade reflect what they truly know, and, at the same time, afraid that their grade will reflect what they truly know!

EOCT : end of course test or end of career terrors…. Either way, we will know soon! No matter what, I’ve seen my kids grow. I am proud of what I’ve seen them learn. They will be okay, No Matter What! That test is just a snapshot of tomorrow, and after that it will be a blip in history. In the meantime, I hope they remember to use their umbrellas!!

#MTBoS30

Paper Cup Probabilities…

This is my first entry in #MTBoS30 Today I asked my 10th grade geometry class three questions (review time!)
1. What is the probability of rolling a one (using a regular six-sided cube)?
2. If you flipped a coin, would you expect it to come up heads?
3. What is the probability that a paper cup tossed in the air will land on its side?

The ensuing discussion involved certainty. First question:
1Ss: one out of six
Several other students chimed in in agreement.
Me: who can tell me how you can be so sure?
2 Ss: because the cube has six sides, and the sides are numbered one, two, three, four, (she is ticking off on her fingers; other students were nodding in agreement and telling her what to say) five, six, and there is only one side with one!
This class is usually not this involved. I think it had to do with the fact that they really KNEW this! (Confidence is a wonderful thing!)

Second question (key word here”” “expect”)
1 Ss: yes, well, no, (???) it could be heads or tails. I mean, you could expect a heads or a tails.
Me: why can’t you expect just heads?
1 Ss: because it’s 50%. (At this point, other students begin chiming in:
“Yeah, it’s 1/2!” “A coin has two sides” and similar statements.) The question wasn’t a straightforward question about a probability fraction, so I think that caused them to not feel as confident with the answer, until one student decoded it. Think: lemmings!

It was the third question that really threw them. I held up one of those small cups, like you find in a bathroom cup dispenser. I asked them to tell me what they thought the probability would be of the cup landing on its side when tossed. The guesses ranged from 1/2 to 340/500. As we looked at the cup, the guesses got more specific. Several students noticed that the cup had a top, a bottom, and a side. The reasoning followed that there should be a 1/3 chance of landing on its side. At this point there was quite a bit of agreement. This seemed very logical (and if the strongest kids in the class said so, it must be right! Lemmings, I’m telling ya!) Multiple students jumped on the bandwagon and agreed. (No one talked about surface ratios – I figured we could tackle that later!) Then I gave each student a paper cup and asked them to create 20 trials each. I deliberately refrained from telling them instructions for tossing the cup. I just walked around and watched. Some kids tossed (across the room!), some kids tossed on their desks. Some dropped the cups on the floor. I heard disappointment as Ss complained, “it’s landing on its side every time,” how do you make it land on its top?” “There is something wrong with this cup!”

The trials were listed on the board and tallied. The students really seemed puzzled as to why the results weren’t anywhere near what they expected. They were already arguing why this was so, so I put them in groups with the instruction to:
1. Compare the actual probability from the trials to the expected.
2. Come up with some reasons for the difference.
3. Pick a spokesperson to share their ideas with the class.

Then the bell rang! Okay. The debriefing happens Monday…