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:

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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.

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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.

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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.

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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.

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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!

#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:

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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!

 

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.

 

Chipotle for Everyone

I’m hard pressed anymore to find a classroom of high school kids who don’t absolutely adore Chipotle’s menu options.  They all have a favorite, and they own it as THEIR burrito.  (I like Chipotle in particular because as a vegan, I can get a delicious meal, as can any non-vegan meal companion.)

I came across this article from Vox claiming Chipotle’s menu calorie disclosures were inaccurate.  I’m going to give Chipotle the benefit of the doubt here because their website contains a very detailed nutrition calculator which allows you to determine the number of calories for your  customized burrito.

The article references a study from the Journal of Public Health Nutrition which reviews a study in which customers are asked to estimate the calorie content of their meal. Some groups were given no information at all.  Some groups were given a range of calories in which burritos in general fell.  Last, additional groups were given example burritos containing the low and high values in the calorie spread.

I had a randomly selected student create a burrito.  Each class was obviously something different which made it kind of fun.

First, I had them estimate the number of calories in the chosen student’s burrito.

Second, I gave them the calorie range of 410-1185 claimed in which Chipotle’s burritos are claimed to land.  I had them adjust their estimate and give reasoning for their adjustment based on the additional information.

I then showed them the calorie range with an example from the Journal article’s study:

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Third, I wanted them to use the examples above to adjust their estimate once more.

We then talked about how the range of our estimates changed and why.  We also had a discussion about ‘averaging bias’ and how healthy ingredients make us assume that certain food are lower in calories than they actually are.

We were able to discuss the surveying methods done for the study and the demographics of participants, which led to a nice discussion about sampling.  (Evidently high school 9th graders find it odd and quite a bit creepy that participants in the survey were given a “flavored ice pop” in exchange for 5 minutes of their time.)

As long as I had their attention with food, I asked them to estimate whether the student’s burrito had more or less calories than my vegan burrito.  I’ll let you decide:

Student’s Burrito:  chicken, white rice, pinto beans, tomato salsa, cheese, and lettuce

My Burrito:  brown rice, fajita vegetables, black beans, tomato salsa, corn salsa, guacamole, and lettuce.

 

 

 

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.

Rocketman

 

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.

Class: 9th grade prob and stats. Topic:  Linear regression.  Enter: the Laundry Data.

The data sheet seemed to spark a LOT of curiosity.  In retrospect, I wish I would have given them some time to Notice and Wonder about the detergents.  Probably I’d also add some estimation first about these bottles of detergent rather than just handing them the data.  I should have known better.

Still, an interesting discussion ensued about ounces of detergent and loads of laundry.  We plotted the points on Desmos and wanted to choose two of them to create our linear model.  I teach three sections of this class and all three classes picked different points to make their equation.

One class picked (50, 33) and (200, 140), and after determining that they needed to find the slope in order to write the equation of the line, I posed that question to them.  How would we find the slope between these two points.  Crickets.

I want to note that a good minute of silence and eye-contact avoidance went by before one brave student spoke up.

S:  You FOIL them.

Me:  Can you explain what you mean by that?

S:  (coming to the board) You multiply them like this.

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Me:  What do we think of what S just wrote up here?  (at least 8 hands shot up in the air)

Me:  Please put your hands down and let’s discuss this.  What I like about what S just did here is he got us started somewhere.  He was willing to take a guess and risk being wrong.  Before S showed us his idea, no one was willing to volunteer their method.  Now that S has broken the silence, lots of you seem ready to discuss.  Thank you S for starting us somewhere.

After this student broke the ice, we came up with about 4 ways to determine the slope of this line and about 8 ways overall to figure out the equation of the line between these two points.  In the past, I would have said to this student, “No, we don’t FOIL, who has another idea?”  Now I know that allowing this student to explain his method does multiple things.  First, it helps the other students practice patience and courtesy when listening and responding to this student whose solution they know is incorrect.  Second, it is a great opportunity for students to engage in SMP #3: Construct viable arguments and critique the reasoning of others.  Third, it provides an opportunity to praise the value in providing the wrong answer.  So much of math class for these students has been about getting the right answers.  I’m glad this teachable moment came about for students to learn from the wrong one.