Guess what day it was today!

Lots of math fun in the primary grades.  Kudos to all the kindergarten, first, and second grade teachers for all their hard work!

Grade 1: Making a 100 day crown with 100 tally marks.

Kindergarten: Making a necklace out of 100 Fruit Loops.

Grade 2: Roll to 100.

Kindergarten: Counting from 1 to 100.

Kindergarten: Stacking 100 cups.  This was crazy!

Grade 2: Working with a 100s chart.

80 more days to go!!

"Fraction of" problems

Nicora Placa's latest post has motivated me to blog about a method we've used for helping kids develop some conceptual understanding for what happens when they find a fractional part of a set.  She calls them "tape diagrams".

We start with fraction tiles.

We have lots of these in my school.  I think the previous math specialist got a special deal on them.  

We have the kids make booklets.  They trace the bars.  Each page has a whole bar, and then the whole made with unit fractions.  I like them to write both the fraction and the words, and a number line as well.  

   Most kids will make it through the eighths, and some will go all the way to the twelfths.  This takes about a period.  But it's worth it.

We have the kids make up their own "fraction of" problems.  We check them over and then have the kids write them on index cards.  In order to solve this problem, you need 18 counters and your thirds page.  I really like how this looks, because the kids can now see that the set of 18 is the "one whole".  We've transformed the whole from a candy bar, or a brownie, to a set of individual objects.

Now it's just a matter of dividing the 18 counters into 3 equal sets.

We've collected lots of cards like these.  I've used these with third, fourth, and even fifth graders.  After a while they can transition into using drawings, which look like Nicora's "tape diagrams".  And now when we talk about the relationship between the numerator, denominator, and the whole number, we have a concrete model to look at.

And they're self-checking, so you don't have a pile of papers to correct!

Old School

It felt really retro on Monday when Rich and I spent some time at the chalkboard trying to give the kids some options for multiplying mixed numbers.  I thought we might play around with the partial products algorithm.

Rich started out by reminding them how we organized multi-digit multiplication by breaking the numbers into tens and ones, finding the partial products, and then adding them up.

We tried it with mixed numbers.

It worked!

Then something really interesting happened.  Flashback: about a week ago, Rich and I had spent some time at his back table playing around with the fraction tiles, seeing if they could be put to practical use in a fraction multiplication "area model".

We turned the tiles on their sides.  We started with 1/2 X 1/3.

We defined the whole.

We built it out.  I liked the way it divided the whole into six 1/2 by 1/3 rectangles.  But it wasn't so practical.  The tiles kept toppling over.

 So Rich copied frames that measured 1 square unit bar and he put them  in plastic protectors.

The tiles could be used to measure off the halves and thirds...

...and then the whole could be divided into 1/3 by 1/2 rectangles.

1/3 x 1/2 = 1/6

We tried it with mixed numbers.

2  1/2 x 2  1/3

5  5/6.  See it?  I wasn't sold on the idea.  The bell was about to ring.  We decided to revisit it later.

Flash forward: So, after we had used the partial-products box for multiplying mixed numbers, a boy raised his hand and asked if he could come up to the board and draw something.  "I overheard Mr.Schwartz and Mr. Whalen talking about this last week," he said.  "I watched them work with the tiles.  Those boxes reminded me of that."

Rich and I looked at each other.  Who knew he had been paying attention?

After that, a student piped up:  "What about lattice?  Would that work?"

We tried it out.  Instead of "tens and ones" we used "whole numbers and fractions".  The kids went wild.

Another student volunteered to come up to the board.  At this point the chalk dust was really flying.

The old "turn the mixed numbers into improper fractions, multiply, and then convert back into a mixed number" method.

.

We asked the kids to work on some problems and try out some of the different strategies.

Still a work in progress.  But I hadn't been covered in so much chalk dust since...well, since Eddie Shore laced up his skates.

"Old time hockey! Like Eddie Shore!"

It's the real thing...

Yesterday was day 1 of our estimation180-style "grams of sugar" extravaganza.  I was very excited to find out how the kids would respond.

How many grams of sugar in the bottle of Coke?

I even passed around a little baggie with a gram of sugar (that's 1/4 teaspoon, according to my sister the nutritionist).

   The estimates were interesting.  The lowest "too low" was 5 grams, the highest "too high" was 60 grams.  Most "just rights" were in the 15g-30g range.

They were really surprised! 

  Many of the number lines had to be extended onto the next page of their notebooks.

Thanks to Jeff for letting me use his class to test this one out.  And there's lots more sugar to come!

Building Towers

The fourth graders have been getting a work-out with fractions.  They spent some time using fraction tiles to build highways and number lines, a project I describe here.  In order to help the kids understand how to decompose fractions and mixed numbers, and express them as sums of unit fractions and as products of whole numbers and unit fractions (if you don't believe this is really something a fourth grader needs to know,  it's 4.NF.B.3a and b and 4.NF.B.4a) we decided to let them explore with fraction towers.

First we let them play around with the tower pieces.  We compared them to the tiles, which were used during the highway sign project.  This was a perfect opportunity to hammer home the idea that the fractions were meaningful only in relation to their respective wholes.  

Our idea was to let them build towers using the same color pieces (unit fractions)...

...then swap out and make a mixed number equivalent.  This student showed how 7 sixths is equivalent to 1 whole and 1 sixth.

Both towers needed to be drawn and labeled.

Rather then using a horizontal number line, like we did in the highway sign project, we thought a vertical number line provided a better representation.

Describing towers using number models.

This activity will not revolutionize the profession.  But there was something about seeing the improper fraction tower next to the mixed number tower next to the number line that made me happy.  It seemed that snapping the pieces together screamed addition more than putting tiles together. And I had a few other ideas, but not enough time (what else is new), including:

• separating the pictures from the number models and have the kids work on matching them up
• taping several pieces of paper together to allow for really big towers

  And towers made me think about castles and forts, and Fawn and Andrew's  Hotel Snap project.  I wish that the pieces could snap on more than just top and bottom so the kids could use them to build something other than towers and then calculate the "value" of their structures.  Anyone from Lakeshore listening?

Nix the Tricks

Several weeks ago  a colleague asked if I had ever heard of the "butterfly method" for adding fractions with unlike denominators.  I said I hadn't, and she drew this:

She was pretty excited.  She explained how this made adding  fractions with unlike denominators much easier for her kids.   I mumbled something about trying to make sure that the kids had a good conceptual understanding of what was happening, and later looked it up on Nix the Tricks.  Sure enough, it was in there.
  If you haven't seen Nix the Tricks, you should take a look.  It's is a downloadable book that explains how these shortcuts circumvent conceptual understanding, and offers alternative, more meaningful ways to approach the concept.
Yes I am sympathetic, but I'm also sympathetic towards teachers using these shortcuts.  When teachers feel pressured to make sure their kids have "mastered" a skill (adding fractions with unlike denominators, for example) and have a limited amount of time in which to accomplish the task, this is what happens. Unfortunately there is not always time to let some of these skills, and the concepts that underpin them, unfold in meaningful, organic ways.
     Here's something my daughter (grade 8, pre-algebra) drew for me one night several months ago while she was doing a homework assignment that had to do with multiplying integers.  She explained that her teacher had shown her this in class.

It helped her complete the assignment, but she pretty much had no idea what was really going on.

   Nix the Tricks is a work in progress.  You are encouraged to comment on submissions and add your own.  Take a look and let them know what you think.