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.

 

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:

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!

 

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:

photo

 

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:

photo

 

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.

 

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.

 

My #MCTM Sub Stuff

Today my students will have a sub since I am attending our state’s math teacher conference (#mctm). Given the overall success of our Desmos Carnival activity from Monday, I decided that a computer lab activity might be fitting. Since we are starting a unit on probability, I took the opportunity to use some Nrich probability simulations.
I’m also attempting something new with Google Forms. I’ve observed my colleague, Dianna Hazelton, incorporate Google Forms, Sheets, and Docs quite seamlessly into her trigonometry and prob/stat classes. Her success with these apps made me eager to try them out as well. I like that I’m able to “see” what they did via the google form responses right away rather than have a pile of papers waiting for me on Monday.

Nrich’s Digit Doozy

If you are a math teacher who hasn’t taken some time to get lost in the problems on Nrich, stop reading this and go there  right now.  You’ll need to finish reading this post tomorrow because that’s how long you will be immersed in its seemingly endless array of engaging problems.

Today, my intention was to do a little starter activity with my 9th graders to help support their number sense.

Here’s the basis of the problem:

american billions C

For two out of three of my classes, it turned into a whole-class period problem-solving extravaganza.  Seriously.  30 minutes later, the brain sweat is still palpable in the room.  There were so many calculators in use, I think the smartphones were starting to get jelous.

Some chose to use whiteboards, some choose numbered cards 0 – 9 while some wanted to use paper.  It was so interesting to me to see them figure things out that must be true about the different number places.  A few remembered the divisibility rules for 3 and shared them.  Then they were able to put the divisibility rules for 2 and 3 together to get divisibility for 6.  I didn’t even know that there was a divisibility rule for 4 and 8!

Some student observations:

  • The 2nd, 4th, 6th, and 8th numbers need to be even.
  • The last number must be 0.
  • The 5th number must be five, since the last number must be 0.
  • The first three numbers have to add up to a multiple of 3.
  • The first 9 numbers need to add up to a multiple of 9.

I even had a student say, “How much longer do we get to play this game?”  Music to my ears.

It’s difficult to give students a task that you know most of them won’t solve which is why I’ve shied away from this one in the past.  I made sure to praise the efforts of those that were able to get their numbers to work for all except one of the digits.   (For example, their 2, 3, 4, 5, 6, 8, 9, and 10 digit numbers worked, but their 7 digit number didn’t).

Nrich gives another variation on this task by making it a game.  Basically, students take turns creating 1, 2, 3…digit numbers by choosing from the 0 – 9 digit cards until someone can’t use any more of the cards.  I think having them play this activity as a game would help alleviate some of the discontent of feeling like this problem was too difficult to solve.