Authentic, Value-Added Algebra

About a month ago, my algebra class was working on the Math Forum’s Free Scenario called Val’s Values.  

There was a lot to question here (which they did) and a lot to wonder (which they did as well).  Something that was unsettling, however, was that they did not know the age of Val or Amir which they felt was pertinent to answering a major question:  Who has spent more on jackets in his/her lifetime?

We made some age estimates and answered our own question as best we could, but it felt less authentic than it could have been.  So, we submitted a comment on Valerie’s blog and today our attention to precision was answered with a response.

I gave my class another go at figuring out who spent more on jackets.  Here are a couple of their responses:

IMG_5293 IMG_5291

What impressed me overall with their approach to this problem was not necessarily the mathematics itself.  The magic was in their careful identification of important variables and analysis of what mattered and what didn’t.  Additionally, they were able to look past the “right” answer and truly own THEIR answer from THEIR assumptions.  I had very few students ask Is this right?  Instead, they were communicating their methods with one another and challenging the reasonableness of their results.  The spark:  the flexibility of the scenario and the real response from a Math Forum Team Member willing to help add some authenticity to a classroom task.  Thank you, Valerie Klein.  We appreciate it.

 

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, Dan Meyer’s newest three-act lesson was 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.

moneyduck2.002

 

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.

IMG_5283IMG_5284IMG_5285

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.

moneyduck2.004IMG_5282

 

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.

 

 

Buried Bias

Another provocative post coming at you.  You’ve been warned.

Probability and Statistics classes always rejoice on the days that I teach them how to play dice games.  Today it was Pirate’s Dice (or Liar’s Dice).  I love using this game because it’s simple to learn, fun to play, and actually requires the use of probability rather than just luck.  I found this article in NCTM from December 2012 and have modeled my activity after theirs.  That article is behind a paywall, so this link will tell you how to play the game if you are interested.

Overall, the student’s enjoyed the game and most of them got into it.  I had to move a few groups away from the shared wall because they got a little too excited, but overall, I felt like my goal for them was reached.  Most of them used probability to create a strategy to help them win.  I had lots of students tell me that the game was fun, so you’d think I would just close the week and move on.

Here was the problem:  in one of my classes, I had a group of students who barely participated.  They were in groups of their choosing, alleviating the idea that they can’t work together.  They positioned themselves in the back corner and once I was off helping another group, all members promptly dug their faces into their phones, texting, tweeting, and snapchatting away as if they were sitting in the cafeteria rather than math class.

I definitely could have handled the situation better because I got MAD at these kids.  Not yelling (because I don’t yell ever), but angry, defensive, and accusatory.  They got back to work.  Sort of.

I stepped back from what just happened to assess why their actions set me off in such a way.  A few other kids were on and off their phones when it wasn’t their turn to bid, and I wasn’t angry at them.  Was it perhaps because it was an entire group of 6 people that were disengaged?  An easier target?

I’ve had enough psychotherapy to know that  this had little to do with the fact that those kids weren’t playing as directed.  This had to do with the fact that this group of kids were the “cool kids.”  These were the popular, tons-of-friends, high-status 9th graders who always have a place to sit at lunch, who have a locker in the center of the hallway, and who would have never given a kid like me the time of day in high school.

Nailed it.  My frustration and resentment toward this group of kids had more to do with how I was treated by their “type” when I was in high school than their inappropriate behavior at that moment. In fact, had I nicely told them to get back to work, who knows, they might have happily complied.  Maybe not, but that’s not the point.  The point is that I didn’t give them that opportunity because my reaction was out of emotions from my high school experience.

I don’t do this often, and I’m glad I recognized it right away.  Right or wrong, these kids deserve a teacher that fairly and consistently applies her classroom management philosophies.  And students shouldn’t have to bear the brunt of their teacher’s lasting scars from a high school experience. I’m glad I’m aware enough to recognize this and change my actions.

Etcetera, etc….

I love it when students figure stuff out.

I love it even more when:

A.  Students figure out things that, as a teacher, I didn’t  notice myself.

B.  Students who are labeled as “not good at figuring stuff out” figure stuff out.

Here’s what we did today in Algebra 2:

number pyramids

This is a SMILE resource from the National STEM Centre.  The problem I thought I would encounter is the word “etc.”  Kids don’t do well with “etc.” Etcetera is vague, non-committal, and easily dismissed.  To a student, etcetera usually means “I’ll ignore this and see if no one notices.”

It is helpful for me to be more specific with my expectations of students, especially when their mathematical well being is at stake.  But today, I was feeling a little vague and non-committal myself, so I handed out the sheet, explained what was going on and let them go…etc.

There are no words I love to hear more in my classroom than “Mrs. Schmidt, look what I figured out.”  And today was chock FULL of those statements.  Here are a few:

  • The triangles are always as wide as they are tall.
  • The sum of the base of triangles 3-wide is 3/4 of the top number.
  • As the triangles get larger, the percentage of the peak number gets smaller.
  • The percentage decrease is related to the size of the triangle
  • If the triangle has an odd numbered base, then the center number in the base is always related to the peak number.

There were lots more.  I was very proud of this class’s resolve in addressing the Etcetera.

 

Dice Wars

Nrich has an interesting activity called “Non-transitive Dice” that I’ve always wanted to use in my probability and statistics class.  I’m intrigued by the relationship between the strategy in choosing a dice and the probability of winning with that dice.

We don’t have blank dice, so I had my students make their own with cardstock.

IMG_5239

 

Initially, I had them choose which dice would win overall.  Then we let the rolling begin:  A vs. B, B vs. C, and C vs. A.  As they collected their data, they started predicting which dice would end up on top after battle.

Tomorrow, I’d like to sum up the probability representations of some of the dice match-ups.  I found this nice post by James Grime (yep, the Numberphile chap) with a few varieties of non-transitive dice.  Next year, I might start with his Grime set and have students collect data on different matchups.

If we are successful, hopefully we can workout the probability of these outcomes.

And finally, I know that my students will want to compare this dice game to Rock – Paper – Scissors – Lizard – Spock.

I kept digging into James Grime’s rabbit hole and realized, you can purchase this set of non-transitive dice.  Skippy.  I might do that!

#TMWYK – The Return of the Sand Pool

It’s a tough time of year for teachers, and I don’t say that to garner any sympathy.  But I’m going to take a moment to deviate from the regular musings of my classroom and write about my favorite topic:  my daughter.  The discussion won’t be completely unrelated as I have learned a great deal about my students’ development of mathematical literacy while watching my daughter make sense of numbers, quantities and shape.  And of course, Christopher Danielson’s development and facilitation of Talking Math with Your Kids has encouraged me to continue the conversation with my own child.  Specifically, I appreciate that his daughter is a few years older that Maria so that I know what I’m looking for and what to look forward to.

Maria (3.5 years old) loves to be outside.  As soon as the snow melted, she insisted that it was now summer and hence every activity from that moment forward must be done in the great outdoors.  A personal favorite is the sandbox, with water.  I’m not opposed to the sandbox overall, but mixed with water, it becomes more like a swamp.  Plus, let’s face it.  It’s Minnesota. It’s Spring, not Summer, and taking out the hose just isn’t in the cards just yet.

So we made a deal that when the temperature on my weather app reached 70 or above, we could take out the hose.  In the mind of my three year-old, this meant that the first of the two digits needed to be a seven.  On Saturday, this lucky girl got to take out the hose.

IMG_5229

Results as expected.

IMG_5230

 

Sunday, I decided to test Maria’s understanding of these numbers.  She again asked “is it seven on the phone?”  I instead showed her Chicago’s temperature which was a balmy 82 degrees.    As expected, her response was “Aww, it’s not seven so we can’t do water.”   I know she knows 8 is bigger than 7, but hasn’t yet connected that a temperature that begins with an 8 represents something warmer than a temperature that begins with a 7.

 

 

Give Me Sum Dice.

Prompt:  You are rolling the dice 99 times and finding the sum on the dice each time.  Make a graph showing your prediction of the results.

I’m not sure why every year I think that most of the students will know ahead of time that 6, 7, and 8 come up most often.  Of course, the activity is more fun given that they don’t have a clue what’s going to happen.

Uniform

Uniform

Uniformly Random

Uniformly Random

 

Just Random

Just Random

Peak in Center

Peak in Center

I ask the kids if there are any other strategies for their predictions other than the ones we have discussed.  One student added that maybe 2 would show up least often and 12 the most, increasing in between.  Discussion ensued.

It’s fascinating to me to actually see the students discover what happens and why.  I used to have them use the probability simulator on the TI-83/84 but I just don’t trust TI to do anything random anymore.  Plus, there’s something more “real” about actually rolling the dice.

They then can compare the graphs of their predictions to their results and discuss differences and reasons for them.  IMG_5215

A question that blew me over today:  If we roll two dice over and over, which will happen first  –  a.  rolling a sum of 7, sixteen times,  or   b.  rolling 100 times altogether?  

I’m wondering , in a class of 30, what the aggregate results will be on this mystery question.

 

#TBT Math Style – SMILE Cards

While perusing UK’s National Stem Centre website recently, I came across something called SMILE.  Here’s what the website has to say about them:

SMILE (Secondary Mathematics Individualised Learning Experiment) was initially developed as a series of practical activities for secondary school students by practising teachers in the 1970’s. These mathematics books are intended to be not only a source of ideas but to be a flexible resource that can be adapted to different circumstances and ability groups. 

Not that it takes much to ignite my mathematical excitement, but the 1970’s got my blood moving.  I was sold.

Here’s a sample:

photo

 

It sort of shocks me when I use these kinds of resources and kids ask, “why is color spelled wrong?” I wonder what they’d say if they knew the rest of the world says “maths” instead of “math.”

Anyway, I could spend about a day looking through the National Stem Centre.  If you’re going to check it out, make sure you have Evernote ready!

 

If you’re decently competent in the area of probability, you might know that your chances of winning fall below things like “death from a vending machine” and “having identical quadruplets.”  This doesn’t stop many people from playing.  I think playing the lottery is more about the chance to dream of what our lives would be like with that much money rather than actually believing we could win.

In the UK, the lottery consists of picking 6 numbers between 1 and 49.  Any player to match all 6 numbers is the grand prize winner.  The chances of this are certainly astronomically low.  A fun question to ask a class of students:  If we bought a lottery ticket for every different combination of 6 numbers to ensure we’d win, how high would that stack of tickets reach?

In the task Do You Feel Lucky, Nrich tackles the idea of evaluating advice given on raising your chances of winning this seemingly impossible lottery. Students are asked to comment on the validity of the advice given and one in particular caught my eye:

When picking lottery numbers, choose numbers that sum between 100 and 200 because the total is rarely outside this range.  

Whoa.  There are so many ways we could evaluate the validity of that claim.  So I sent my students off to the races. Most of them wanted to use a random integer selector and then gather the data from the class’s trials.

IMG_5135

GeoGebra Results:

Lottery

Lots for them to talk about here.  Lots of questions for them to ask as well.  Does the range seem too wide?  Do we have enough trials?  What do we make of the dip in the middle?  Should we change the bar graph to have different class sizes?  Would a box plot have been more appropriate?  What about the descriptive statistics?  Would those help us out?

I’m hoping next year to extend this into more of a class activity rather than an impromptu discussion.

 

Moments from MCTM

My brother wisely told me when he saw who I followed on twitter to stop following dumb celebrities and start following some real people.  The problem was that back then, I didn’t know which real people to follow.  Luckily, I soon discovered that there were math teachers on twitter.  Lots of them.

I’ve been to MCTM a couple of times and NCTM once or twice. I felt energized, and motivated after those conferences definitely, but this year was different than any conference I’d previously attended. The difference was my willingness to make a face-to-face connection with people I knew from twitter.   I’ve loved twitter for a long time for a variety of reasons, but meeting some tweeps in person and getting to talk math and more math was a real thrill.  It mattered less which conference sessions I attended, although they were great,  and mattered more who I took the time to interact with in between.   Although Christopher Danielson says that he doesn’t remember me as a snarky student in one of his math ed courses, I was grateful to get to spend some quality time talking with the man behind the hierarchy of hexagons. I met many others, and truly got to appreciate the wide range of awesomeness that make up Minnesota’s mathematics teachers.  

Next time, though:  book a hotel room right away.  Lesson learned.