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.



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.


Results as expected.



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.



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.


Seeking Shelter

WARNING:  This post might challenge some of your views on the responsibility we have for failing students

From Simplifying Response to Intervention:  Four Essential Guiding Principles (Baffum, Mattos, and Weber)

It is disingenuous for a school to claim that its mission is to ensure that all students learn at high levels, yet allow its students to choose failure.  Unfortunately, at the secondary level, it is all too common for students to be “offered the opportunity” for help.   But if a schools gives students the option to fail, is the school teaching responsibility or merely punishing students for not already possessing the skill?  By “offering” help, the school expects students to either have an intrinsic love of learning or to fully grasp the lifelong benefits or life-damaging consequences of not succeeding at school.

I hear occasionally from teachers that we need to teach kids “responsibility” and we can’t force them to learn if they don’t want to.  This line of thinking bothers me a great deal as places the burden of being eager to learn on the student.  Some kids place “learning at school” very low on their priority list.  We must acknowledge that rather than disregard it with “he/she never came and asked a question.”  If we are being honest with ourselves, we know exactly which kids need the help but won’t outwardly seek it.  We know which kids won’t ask questions when they have them, and which ones won’t make an effort to turn in assignments that they’ve missed.   It’s not that they are incapable of seeking help, asking questions, and turning in assignments.  But by stating that “help was offered but not taken” we do not absolve ourselves from the responsibility to reach these students.

I am putting this in blog format to hold myself accountable as the end of the year approaches, but anyone who would like to join me is welcome.  I want to make a commitment to those students that struggle but don’t know how to seek help:  my job is not to teach students, but to make sure that they learned.  I want to do better at addressing those kids in my classroom.

#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:



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!


Puzzles I Pretend To Like

I realized after searching through my puzzle collection that I possess many of them, but have completed very few.  This is true of physical puzzles as well as puzzles on paper.  My reasoning: actually doing a puzzle scares the heck out of me because I might not be able to do it.  Ironically, I tell people I like puzzles because what crazy person would have a closet full of puzzles in her classroom if she didn’t love them?  Fortunately, I make no claims to sanity on this blog, but I’d like to get better at actually attempting puzzles.

I came across an app called Puzzlium.  This app highlights a little bit of all of their great puzzle apps, but the one that caught my eye was Puzzle Quizzes.  I was drawn to the short format of these puzzle. It seemed like a good place to start anyway.  Here’s an example:



Now, if you’re looking for a place to start in having students “make viable arguments,” this isn’t necessarily an obvious choice.  But the simplicity of the question coupled with the inviting color scheme lay down a framework that allows all of them to enter the conversation.  I had students who haven’t raised their hand all trimester actively engage in sharing their answers, summarize others, and thoughtfully comment on the arguments of their peers.

This task evolved as the day went on, and my question to my 5th hour class was:

Come up with as many reasons as you can for each of the patterns being the one that does not belong.

Tons of great responses using everything from which colors touched one another to number of blocks in each row/column to perimeter of the total figure.  I’ll bet each student in my 5th hour came up with a different argument.

The question that, of course, I wasn’t able to avoid was “So what’s the right answer?”  I turned that one right back on them.
“what do you THINK the right answer is? Was there one or more arguments that stood out as “better” than the rest?”  The overwhelming response was No.  The arguments given were solid and the instructions were ambiguous.

Still, they must know the “right” answer.  I made sure to praise their efforts on constructing and presenting their arguments.  However, here is the “back of the book” answer:



Showing them this seemed to invite the usual, “ha, I was right and you were wrong” mentality.  Maybe next time I’ll resist their pleas and stick with all of them believing they were right.