# Duck, Duck, Money Duck

When I moved to Minnesota, I learned a new game called Duck, Duck, Gray Duck.  This is similar to the game that the rest of the country cleverly calls “Duck, Duck, Goose.”  Evidently, in Minnesota, as you are tapping heads, you can call out absurdities such as purple duck or yellow duck.  Listening skills at work here; gray duck is the magic color.

[The preceding paragraph has nothing to do with this post, but if you’ve always wondered why Minnesota boasts Duck, Duck, Gray Duck rather than conforming to the rest of the country, now you know.]

Speaking of ducks, newestwas coincidentally timely with my probability and statistics progression.  Today’s learning target included expected value, so I thought we’d give it a go.

Act 1, Initial Questions:

• Can you actually buy one of those?
• Is that like the diamond ring candles?
• Do any of them have \$50, for real?
• Would it be worth it to buy a bunch to get the \$50?
• How much do those things cost?

I had them speculate a fair price for one of these duck soaps.  We had a discussion about what was meant by “fair” which was productive.  Most students settled on a price between \$3 and \$20.  The students also wanted to consider if shipping was included in our pricing.  Since we were looking at the price from the Seller’s point of view, it made us wonder if the shipping for Amazon Prime products is passed along to the seller or absorbed by Amazon.  We’ll have to address that another day.

Notables in Act 2:

1.  When deciding which probability distributions were impossible, students were quick to point fingers at E and F.

After making the connection that the total of all bars must equal one, most students were able to identify B and C as impossible.  Arguments ensued over D about whether the two bars would total 1.  The ruler confirmed that indeed the bars did not add up to 1.

2.  When looking at these distributions and determining how a \$5 duck would be bad for business, my students noticed something interesting.

We had some great conversation about which would be worse:  losing customers from a faulty product or losing money with too many rich ducks.

3.  When determining fair prices for these distributions, I was impressed with my class’s use of an area model.  I sometimes supplement the probability unit with activities from IMP’s The Game of Pig and liked their application of a ruggish diagram here.  This allowed for a more fluid connection between the value of the duck bill and the probability of that payout.

These are 9th graders, so only a few requested the sequel.    Overall, I was pleased with the outcome of this lesson.  I feel like the the money duck grabbed their attention more than previous attempts at real-world expected values such as pull-tabs or roulette.  I think the kids felt like soapy money is something they can access, and I think their attention to the task reflected that.

# A Desmosian Gem

I finally had a chance to do the Function Carnival with my classes.  Thank you to Desmos, Christopher Danielson, and Dan Meyer for their work on this project.

As David Cox captured in his blog previously, the real power of this activity is the immediate feedback.

When the graph looks like the one below and 8+ rocket men burst out of the cannon, the students see that right away and adjust for it.

Dan had mentioned in a blog post a while back that “this stuff is really difficult to do well.”  After seeing students work through this activity today, I can appreciate the difficulty in creating an online math activity that gives both students and teachers detailed feedback in real time.

Some observations:

• Students don’t realize at first that you can see their work live.  I allowed them to “play” for a minute, but some may need more encouragement.
• A tool to allow you to communicate digitally with the class would be nice.  Google chat, for example?
• Some students don’t realize that the bumper car SHOULD crash and make their graph to avoid it.
• A student or two misunderstood the graph misconception questions and went back and changed their graphs to look like the misconception graphs.
• It was interesting to see which students wanted their graphs to be perfect versus which ones said there’s was “good enough.”  It would be interesting to have a discussion about which is appropriate in the particular situation.

Bravo, Dan, Christopher and the Desmosians.  Thank you for creating an online math activity that gives me some faith in online math activities for the future.

# Pattern Power

If you have little kids and you’ve been privy to an episode of Team Umizoomi, then perhaps the title of this post evoked a little jingle in your head. You’re welcome; I’m here all day.

My daughter, although she doesn’t choose Umizoomi over Mickey Mouse as often as I’d like, picked up on patterns relatively quickly after watching this show a couple of times.  She’s 3 years old, and she finds patterns all over the place.  Mostly color and shape patterns, but a string of alternating letters can usually get her attention as well.  These observations of hers made me realize that pattern seeking is something that is innate and our built-in desire for order seeks it out.

High school students search patterns out as well.  For example, I put the numbers 4, 4, 5, 5, 5, 6, 4 so that the custodian knew how many desks should be in each row after it was swept.  It drove students absolutely CRAZY trying to figure out what these numbers meant.  I almost didn’t want to tell them what it really was as I knew they’d be disappointed that it lacked any real mathematical structure.

I’m not as familiar with the elementary and middle school math standards as perhaps I should be, but I’m confident that patterns are almost completely absent from most high school curriculum.  Why are most high school math classes completely devoid of something that is so natural for us?

Dan Meyer tossed out some quotes from David Pimm’s Speaking Mathematically for us to ponder.  This one in particular sheds light on this absence of pattern working in high school mathematics:

Premature symbolization is a common feature of mathematics in schools, and has as much to do with questions of status as with those of need or advantage. (pg. 128)

In other words, we jump to an abstract version of mathematical ideas and see patterns as lacking the “sophistication” that higher-level math is known for.  To be completely honest, this mathematical snobbery is one of the reasons I discounted Visual Patterns at first.  Maybe it was Fawn Nguyen’s charisma that drew me back there, but those patterns have allowed for some pretty powerful interactions in my classroom.   I’ve used them in every class I teach, from remedial mathematics up to college algebra because they are so easy to  differentiate.

I think high school kids can gain a more conceptual understanding of algebraic functions with the use of patterns.  For example, this Nrich task asks students to maximize the area of a pen with a given perimeter.   The students were able to use their pattern-seeking skills to generalize the area of the pen much  more easily than if they had jumped right from the problem context to the abstract formula.

I also notice that the great high school math textbooks include patterns as a foundation for their algebra curriculum.  For example, Discovering Advanced Algebra begins with recursively defined sequences.  IMP also starts with a unit titled Patterns.   I think these programs highlight what a lot of traditional math curriculums too quickly dismiss:  patterns need to be not only elementary noticings of young math learners but  also valued as an integral part of a rich high school classroom.

# Puzzling Perseverance

School mathematics has a bad reputation for being intellectually unattainable and mind-numbingly boring for many students.  Proclaiming the falsity of these beliefs is usually not enough to convince kids (or people in general) of their untruth.  Students need to experience their own success in mathematics and be given the opportunity to engage in curiosity-sparking mathematics.  For me, one of the very best moments in a classroom is when a self-proclaimed math hater fully engages in a challenge and is motivated to work hard to arrive at a solution.

Enter January 2nd and 3rd.  Students are back for a two-day week which they view as punishment and a rude-awakening from a restful winter break.  To boot, the Governor Dayton announced today at about 11 am that all Minnesota schools will close Monday, January 6th due to impending dangerously cold weather.  You can imagine where the motivation level was in school today.

As the CEO of room 114, I decided to make an executive decision and do a puzzle from Nrich (shocking, I know) in my probability and statistics class.  Technically, the students could use the mean or median to help solve the problem, so I wasn’t veering too far off of what I had previously planned.

The Consecutive Seven puzzle starts like this:

Initially, one student began by explaining to me that she took one number from the beginning of the set, one from the middle and one from the end.  Then she figured the other consecutive sums needed to be above and below that number.  (Spoiler alert:  These numbers actually end up being the seven consecutive sums, so I was very interested in her explanation of how she arrived at those particular answers.  )

It’s worth noting that this student’s first words to me at the beginning of the trimester term were, “I hate math and I hate sitting in the front.”  So you can imagine my excitement when she dove in head first into this particular task, happily and correctly.

Adding to my excitement about the class’s progress, another girl (who was equally enthusiastic about math at the beginning of the term) was the first one to arrive at a correct solution.  And although she probably wouldn’t admit it, she was thrilled when I took a picture of her work.  And I am more than thrilled to display it here:

If you were wondering about how math-love girl #1 fared in completing the task, she persevered and impressed her skeptical cohorts:

This phenomenon fascinates and excites me that students, when confronted with a puzzle, highly engaged and motivated throughout the lesson.  Dan Meyer summarized this idea nicely on his blog recently:

“The “real world” isn’t a guarantee of student engagement. Place your bet, instead, on cultivating a student’s capacity to puzzle and unpuzzle herself. Whether she ends up a poet or a software engineer (and who knows, really) she’ll be well-served by that capacity as an adult and engaged in its pursuit as a child.”

And who knows.  Maybe one of the girls featured above will become a puzzling poet.