1,714.285714what??

   
One of the most important habits a math teacher can develop is to do the problem first, always.

Tracy Johnston Zager

Becoming the Math Teacher You Wish You'd Had

pg. 46

     During one of our weekly planning sessions last month, Rich and I decided to roll out a new 3-Act task.  The idea came to me in the faculty lounge, a place where I find inspiration on a regular basis.  The Act One video showed four teachers in succession going up to the sink and washing their hands, each one receiving an automatic squirt of liquid soap.  The question I had was: How many squirts would be dispensed before it emptied?
     I had our custodian open up the dispenser.

She's gotten used to these kinds of requests.

She showed me a refill, and I found out that each bag held 1,200 mL of liquid soap.

But how much soap came out in each squirt?  A call to Georgia Pacific and I had my answer.

The dispenser has two settings: one for 0.4 mL and another for 0.7 mL.

We couldn't determine where the dispenser was set, and I worked out the answer for the 0.4 setting.
After a false start...

113 is wrong!

...I figured it out.  3,000 squirts!

   I was curious to see how Rich and Megan (our grade 5 resource room teacher) would attack the problem, so I asked them to have a go:

Rich's solution.  While I thought of four tenths as a decimal, he used a fraction.

Megan also thought in fractional terms. 

    I thought it was interesting that the three of us took three different solution paths, and we imagined we'd see a variation in the class responses.
    For Act 4, we thought we'd ask the kids to find out how many squirts it would take to empty the bag if the dispenser was set at 0.7 mL per squirt.  Megan had to leave, so that left Rich and I to work it out.

Rich got an answer, but he wasn't happy.  He was in a rush because he had to leave to pick up his class, and I told him I would save his work.  He came back the next day and found where his thinking had gone off the rails.

I went back to my table.  It had worked before.  Why not again?

My problem was that 0.7 didn't fit nicely with 1,200.  I narrowed things down until I got to 1,714 pumps, which would empty 1,199.8 mL from the bag, leaving just .2 of a mL left.  Now what?

     As I started to puzzle it out, the phone rang.  It was Rich.  He had gone back to his room, divided 1200 by .7 and gotten a wacky decimal.  "It's 1,714.285714," he said, "And it goes on."  I got out my phone calculator, punched in the numbers, and got 1,714.28571429.  I knew that the answer lay somewhere between 1,714 and 1,715, but where did that ugly decimal come from?  What did it mean?  If there was 0.2 mL left in the dispenser, wouldn't the answer be 1,714.2?  Or maybe it was 0.2 of 0.7?  But then that would leave an answer of 1,714.14 squirts.
     No kidding, I stared at the whiteboard for the better part of a week, thinking about decimals, automatic soap dispensers, and Robert Kaplinsky on productive struggle.

The decimal was killing me!

   I finally broke down and called my supervisor.  He was fascinated by the way I went about solving the problem, and he picked up a marker and started working in the bottom right corner.

For one thing, he wasn't crazy about my equal signs.  Then he explained that if 0.7 mL was a whole squirt, and only 0.2 mL came out, then the last person to wash their hands would get 2/7 of a squirt, assuming that an amount of soap that minuscule would even come out.  And what's 2/7 as a decimal?  Light bulb time!



  

     It was one of those knock yourself on the head kind of moments, but I didn't feel bad.  I had enjoyed the process.  I always tell the teachers I work with that if they have questions about the math they're teaching to please come to me and we'll figure things out together, but I know that some feel embarrassed.  They think that asking for help is a sign of weakness, but really it's a sign of courage. So it's important for me to put my money where my mouth is, and I reach out to our math supervisors for help when I have questions.  It's how I learn and grow, and it's a good habit to develop.
    So how did the 3-Act play out with the kids?  Stay tuned for a report.  But because we had worked out the problem beforehand, Rich and I felt ready for anything.

A postscript:
  Curious about the decimal, I did a little research.  Turns out it's a repeating decimal, and the part that repeats is called the period or repetend, and there are cool patterns that occur with the digits.  I tried dividing 2 by 7, first by hand...

...and then on different calculators:

 They were all different, because some of them rounded and some of them didn't!  I started to like the decimal, and had to revise my opinion.  What I first thought of as something ugly turned out to be rather cool!  

Tracy gets the final word:

     Playing with the problem ourselves activates our identities as mathematicians.  Just like reading teachers need to read and writing teachers need to write, math teachers need to do math.  Noticing which mathematical concepts and techniques come into play while we work focuses our thinking on content.  Rather than jumping right to planning activities students will do, we spend time thinking about the mathematics students may learn. 

     

       

Ball Don't Lie

Adapted from Urban Dictionary:

ball don't lie

     A phrase commonly used by former professional basketball player Rasheed Wallace; once famously yelled by the late coach Flip Saunders. 
    "Ball don't lie" is said when a player misses one, two or all three of his free throws after a questionable (read as: unwarranted) foul call is made by an official. The ball is, essentially, the unbiased judge who will not reward the player by going in if the apparent foul was indeed unwarranted.

     Recently I visited a fourth grade classroom where the teacher was conducting a lesson on comparing fractions.  She explained that the task would be made much easier if the fractions in question had common denominators, and she was reviewing the method they were to use:

       After several worked examples, the students were divided into two groups.  One group was directed to work on Chromebooks.  Their task was to look at two fractions written side by side and choose a comparison symbol from a drop down menu to make the expression true.  The second group had a similar task.  They were sent to a table with a stack of laminated cards.  Each card had two fractions with a blank box between them.  They were asked to copy the fractions onto a worksheet and select the correct comparison symbol.
     After giving the kids a few minutes to settle in, I started to circulate.  I happened upon a student working on a Chromebook.  The screen displayed two fractions, 7/8 and 3/4.  She had selected the correct sign and was ready to click to the next screen when I asked her to take a minute to explain how she arrived at her answer.  That's when she pulled a whiteboard from her lap.

She restated the teacher's explanation almost word for word.

  I decided to leave the multiplication error aside and press her understanding a little.  I wanted to see of she could compare the fractions by using 1 as a benchmark.

Me: Do you know how many eighths make a whole?
Student: Eight.  Eight eighths.
Me: Good!  And so do you know how far away 7/8 is from 1 whole?
Student: One.  
Me: One?
Student: One eighth?
Me: What about fourths?  Do you know how many fourths you need to make a whole?
Student: Four.
Me: And so how far away is 3/4 from a whole?
Student: One.
Me: One?  
Student: One fourth?
Me: And so which fraction is closer to 1 whole?
Student: They both are.  They're both one away.

I decided to try another line of reasoning.

Me: Let's look back at 7/8.  You said that eight eighths makes one whole.  So is 7/8 less
       than, more than, or equal to a whole?
Student: Less.
Me: What about twenty-eighths?  How many of those would make one whole?
Student: Twenty-eight.
Me: (pointing to the 7/8 and 28/28) So then these two fractions are equal?
Student: Yes.  See?  I did the butterfly multiplication.

     Leaving the student to continue her work in peace, I crouched down next to a student looking at a card with the fractions 4/6 and 8/18.  He had implemented the butterfly method, and written the expression 72/108 > 48/108 on his worksheet, and was ready to move on to the next card.  Not so fast.  I wanted to see if he could compare them using 1/2 as a benchmark.

Me: Nice!  You did some fancy multiplication there!
Student: (no response)
Me: I want to talk about these fractions for a minute. Let's look at 4/6.  Is that more or less 
       than 1/2?
Student: (silence)
Me: Well do you know how many sixths you'd need to make 1/2?
Student: (more silence) 
Me: (picking up a nearby pencil) Well, say I drew a rectangle and divided it into sixths...

Me: ...Could you color in half the rectangle?

Me: Great!  Now how much of the rectangle is colored in?
Student: Three thirds?
Me: Carry on.

     A teacher, looking at the Chromebook data and the turned in worksheet, might conclude that both students have a firm grasp of the relative size of fractions.  Like the basketball referee making the questionable call, he sees what he sees.  But the truth about what these students understand about fractions won't be found in the colorful charts and impressive graphs generated by the computer program.  And it's not on the worksheet.  So where is it?  Sometimes the truth is on a whiteboard. Sometimes it's scribbled in the margin.  Sometimes it's written on a piece of scrap paper, and if you look hard, sometimes you can even see it in the faint trace left after it's been erased.  The truth is in the minds of our students, sometimes out in the open, sometimes hiding in a dusty corner. That's where we need to look.  So let's keep our eyes on the ball, because ball don't lie.