It’s probability time in my 9th grade prob and stats class.  Call me crazy for giving 9th graders dice and pennies with a month left of school, but it’s how I roll.  (Ha! I’m cracking up over here!)

I like to start with the Game of Pig, similar to the game used in the IMP curriculum.  I adapted it a little to have kids compare strategies for when playing with their own dice (or separate from their partner) to playing with the same dice as their partner.

It’s interesting to see their strategies develop here.  Some use very solid ideas like “I stopped when my round score reached 20.”  But I also get to see misconceptions like believing that a “one” will be rolled relatively soon after a “two” is rolled.  Having them share their strategies helps me to see where these misconceptions lie and deal with them before we start calculating any concrete probability.

Tomorrow, we’ll start by discussing which of these are legitimate strategies and which of them are not.

photo 1 (3) photo 1photophoto 4 (1)photo 2

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:


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.



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.


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.

The Satanic Sheep in Class

Here’s the backstory:    And I was overstressed by situations not involving my students.   My goal for the day was to not take my undue stress out on students.  A student in one of my classes finds what most would call odd (like satanic sheep) to be particularly amusing. I’m sure you’ve had many of these cherubs in your own classes. This day, it was the sheep, horns, fire, a devil’s tail, the works.  I was mildly intrigued at this point.  This kid then made a rather punny joke about his creature having steel wool.  This exchange of oddities had me forgetting about my tough day and laughing at the cleverness of a 9th grade boy.

Satanic Sheep

Since this event, the student has stepped up his math game tremendously.  He has demonstrated over and over his intuitive numerical abilities as well as his persistence in solving difficult problems.  He adds his own creative edge to my class as well as strives to thoughtfully engage in the activity at hand.

I kept the picture and put it in a prominent place on my desk as a reminder that what first comes across as outlandish and tough has soft bleating heart underneath.   I think a lot of people don’t see past the satanic sheep and miss the creative, caring, hardworking problem solver underneath.   My challenge to you: find a “satanic sheep” in your class.  Show them you appreciate the creativity they add to your classroom mix.  You may be pleasantly surprised at what comes next.


Earth Day Trash or Cafeteria Treasure?

I had intentions of scouring the internet for the perfect Earth Day activity. Luckily, I came across this:
I saw Beyond Traditional Math’s lesson on the Pacific Garbage Patch, and more than anything, I was impressed with his work and inspired to learn more about this floating mass of plastic-y mess twice the size of Texas floating in the Pacific Ocean. His focus on this Plastics issue made me examine where this permanent substance permeates our lives.  In short, every single piece of plastic that has ever been manufactured is still on earth today. Every. Single. Bit.

I showed this video from the blog post in my classes today:

I supervise the lunch room each day for about 30 minutes and I was in awe as I watched student after student empty his or her tray into the garbage.  Each time, I watched at least one plastic utensil fall into the trash.  One, after another, after another.  I inquired with the cafeteria staff how many of this flatware was disposed of each school year at our school.

Stop for a second:  1500 high school students.  How many plastic tableware do you think we toss out every year?

According to the super-helpful cafeteria staff, we started the practice of using disposable forks, spoons, sporks, and knives after it was discovered that much of our metal silverware was ending up in the trash.  (The pig farmer that collected the compost for feed was not too keen on metal spoons ending up in his pig feed.)  I was informed that we go through as a high school, about 3 cases of forks, 2 cases of spoons, 2 cases of sporks, and 1 case of knives each week.  That’s 8 cases, 1000 per case, every week, for 40 weeks.  320,000 pieces of plastic, every school year, in the trash.  And that plastic lasts forever.  In the trash.

Now, I didn’t want to go about suggesting to a hardworking group of school cafeteria employees that they should change their utensil type because I’m on an Earth Day mission.  But, I thought it was an interesting question to pose to my 5th hour Algebra class:  which would be better to use and why?

I was very impressed with the detail and consideration they applied to the question.  How much metal silverware would be needed to supply the entire school for a day?  Is there time to wash them between lunches?  Between breakfast and lunch even?  Should we consider that extra labor and water cost in the price of the metal tableware?  How would we estimate that?  How can we condition the entire school to stop tossing the silverware and throw it into a separate bin when returning trays? Wouldn’t the metal utensils end up in the trash, costing more money?

We found out that there was definitely a financial savings when purchasing metal utensils of approximately $1500 per year.  I asked the students to weigh in on whether this was worth the hassle of the switch.  I love the interesting answers I get from questions like this.  I didn’t necessarily know what to expect, but I got some very thoughtful answers which considered many of the variables in play in this situation.  The discussion seemed to indicate that the students recognized that even a small switch like plastic or metal utensils requires careful consideration and precise implementation and wouldn’t operate like an on/off switch.  A good day.

Catwalk Mystery

I wanted to quickly share an awesome activity that is usable in a variety of classes.  A website I use occasionally to browse teacher resources is TES.  This UK based site is a treasure trove of shared lessons and activities.  This gem, Amelie’s Fashion Mystery starts with a simple question:  Will Amelie make it to the catwalk in time?   Students work through mathematical clues in order to determine if this supermodel makes it on time to the fashion show.  The task requires students to utilize a huge range of math skills and is differentiated with two versions.   Thank you Jake Mansell for a great activity.  Here are the files:   AmelieFashionMysteryintroslideandvisibilitygraph AmelieFashionMystery(visibilityfromgraph) AmelieFashionMystery(visibilityfromformula) (1)

Stripping Down the Stock Photo

Smack dab in the middle of all of the awesomeness coming out of Rafranz Davis’s blog was a gem that stuck in my brain:  Addressing the Edu Stock Photo.  In short, Rafranz challenged the twitter/blogging teaching community to take a reflective opportunity to address a difficult issue in your school or classroom.  Taking on this challenge made me feel a sense of freedom from what’s frustrating in my classroom by taking off the shiny bow and acknowledging what I could do more effectively in my classroom.   Today’s Algebra class ended up being a great opportunity to reflect on what hasn’t been working in my classroom.  It started very typically by doing some estimation.  I walked around the room and noticed who was jotting down an estimate milliseconds before I wandered past their desk.  I saw who was more interested in their snap chat than participating in sharing estimates and reasoning.  I let the frustration build and boil over a little with my raised voice.  The breaking point came when a student literally talked over me in a regular conversation-volume voice as if I weren’t leading a class in an objective.  I sat down at my desk and felt in that brief moment like I was never going to get these kids to care about math.  Didn’t they know how much time and effort I put into figuring out how to help them learn?  Why didn’t they appreciate how much I cared about their learning. I kept putting the estimations on the board, not really saying anything.  It would have been very easy to shut down at that point.  There were about 25 minutes left in the week, and learning meaningful mathematics seemed out of the question at this point. Then I had a profound realization that changed my whole view of my class in an instant.  I was angry and frustrated at the wrong thing.  The kids in this class are stuck in the same cycle of schooling that they have been in for years.  They know that they are tracked in the “low level” math class, and they have come to accept that math is not something they’ll be expected to be good at. And they also have come to expect the same cycle of student/teacher frustration:  kids will talk and goof off, the teacher will get angry, yell, punish, and send kids out.  Things will be calm for a few days and then they can begin the cycle again.  It’s not the student’s fault, they don’t know any better. And it’s always worked before for them because they got themselves this far. I know this cycle is playing out in my classroom because these are nice, likeable kids.  They’re creative and interesting.  They’re emotional and sometimes dramatic.  And I love them.  I have loved the opportunity to get to teach them.  But I could do a better job than I am.  I could complain about the size of the class.  And I do.  Or I can change what I actually have control of, which is helping these students learn mathematics.  I do have control over giving them opportunities to interact positively with a discipline that they have been fearful of going all the way back to timed-arithmetic.

Much of what impacts our memory of particular events as positive or negative is rooted in how the story ends.  I believe the same can be true for education.  My incurable optimism tells me that something else will work for these kids and I believe in them and in myself.  To end that class period on Friday, I called on a good friend, Nrich.

One hundred percent of them participated, 100% of them engaged and wanted to be the one closest to 1000.  It was a small victory, but was absolutely essential in ending the week pointed in the right direction again.  Mathematical curiosity ensued for a brief moment (Why did one person get 1008 and another 992?  Who is the winner?)  Now Monday won’t feel like more of the same for my students and me.  It’s a new chance to bring them together with mathematics and hopefully have some fun in the process.

Alright, Mr. Stadel. We’ve Got Some Bacon Questions

Greetings, Mr. Stadel.  We know that you are very busy.  We appreciate your brief attention.  Rather than bombard you with tweets, we decided to bloggly address our questions and comments about your Bacon Estimates.

First of all, bravo.  You dedicated an entire section of your estimation180 blog to a culinary wonder some refer to as “meat candy.”  Even our vegan teacher felt compelled to engage us with these estimates.  (She says it is for the sake of the learning.)

Second, the time lapse videos of the cooking are pretty sweet.  Too bad the school internet wouldn’t stop buffering.  But nice touch, Mr. Stadel.  Nice touch.

A question:  Did you know that the percent decrease in length of bacon is 38% after cooking, but the percent decrease in width is only 23%?  We figured that out adapting your “percent error” formula to the uncooked/cooked bacon.  Do you have any initial thoughts about that discrepancy?  Is it bacon’s “fibrous” fat/meat striped makeup that allows it to shrink more in length than width, inch for inch?

Also, did you know that the percent decrease in time from the cold skillet to the pre-heated skillet is 29%?  That one was a little harder for us to calculate, because we figured out that we needed to convert the cooking times to seconds rather than minutes and seconds.

To summarize, we wanted to thank you, Mr. Stadel.  Our teacher tells us that you dedicate your time and energy to the estimation180 site so that WE don’t have to learn math out of a textbook.  We wanted to tell you that we appreciate it.  And the bacon.  We appreciate the homage paid to bacon.


Mrs. Schmidt’s Math Class

St. Francis, MN

When the Answer is E: He Falls Off the Roof and Breaks His Neck

Our annual state testing season is almost here. The juniors will partake in the Minnesota Comprehensive Assessments in Mathematics a week from Tuesday. Our department decided issuing a practice test to all of our juniors would help re-familiarize them with long lost skills. After distributing copies during our monthly staff meeting, I’m always curious if any teachers in other disciplines look at the practice materials. Much to my delight, the choir director approached me at lunch on Friday, test in hand.

Mr. Warren: Is this test just like the MCAs?
Me: Most likely similar. Why?
Mr. Warren: Ok, well look at this one.


Mr. Warren: I think the answer is E, Xai s going to fall and break his neck.

The conversation went on for another few minutes, with me agreeing  that what’s been called “math education” includes ignoring the context of situations and focusing on a procedure.  In fact, I was curious how many juniors who completed this practice test even noticed that the situation was outrageous.

Since we were running on a 2-hr delay schedule Friday, I thought it would be the perfect opportunity to present the problem to my algebra class. They are mostly juniors who have been continually frustrated with a mathematics curriculum that doesn’t make any sense in the real world.

Me: Read through this problem. Does it make sense?

Student: ok, it looks like 32.

I didn’t expect any of them to apply any trigonometry, so I thought we needed to approach the problem differently.  In fact, I wasn’t even concerned about the angle measure.  I wanted them to look at the scenario itself.

Me: Imagine this scenario. We’ve done a lot of estimating in here. We need to envision a 20-foot ladder, three feet away from a house. Does this seem reasonable?

Unfortunately, it did seem reasonable to most of them. I needed another approach.

Me: ok, how could we simulate this in classroom-scaled size?

Student: Get a ruler.

Me: Perfect. How close does it need to be to the wall?

Students: (a chorus of answers)

After exploring multiple methods of calculating exactly how far, we arrived at 1.8 inches.  With as much drama as possible, I set the ruler against the wall, exactly 1.8 inches away.

Me:  Does this look like a ladder that any of you would want to stand on? (of course, a few did).  Keep in mind, this is a TWENTY foot ladder, not a 12 inch ruler.

Student:  Yea, I don’t think anyone is climbing up that ladder and coming down in one piece.

Another Student:  What if they had a spotter?

A spotter!  Now we’re talking.  To be honest, I have no idea if a spotter could hold a 20-foot ladder so that it could be placed three feet from the wall.  But now I’m interested to find out!

I know Mathalicious investigated a similar scenario using a claim from Governor Janet Napolitano.

In my mind, these are the questions that should be circulating Facebook and aggravating parents.  This is the kind of math that should rile up Glenn Beck and company.  Our state of Minnesota opted not to adopt the Common Core State Standards in Mathematics, but requiring this kind of math instead is what is actually dumbing down the curriculum.  It assumes that the real world doesn’t apply, only rote procedure does.  “Just figure out the answer, don’t question the situation,” is what kids read and do over and over when problems like this are solved without real context.  A richer classroom experience for both teachers and students comes when we ask students to assess the reasonableness of situations, create new scenarios that are more appropriate, and solve the new problems they develop.  The CCSS Standards for Mathematical Practice tell students that it’s vital that they “construct viable arguments and critique the reasoning of others.”  I don’t think “critique the reasoning of others” should be reserved for only reasoning created in the classroom.  I’d like my students to critique the reasoning of the creator of these types of problems and others like it that have been deemed a necessary component of high school math success.

Thank you, Mr. Warren for igniting the exciting conversation in my classroom.