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.

Let's Go To the Movies!

     If you haven't checked out the lessons and activities at Robert Kaplinsky's site, you should.  That's where we ripped off the highway sign activity, which I described here.  And don't miss his blog,  which is very thought-provoking and deserves a read.  Earlier this year Rich and I adapted his movie theater project and turned it into a week long problem solving exploration for our fifth graders.
     We started by giving the kids a folders with the following information from the local multiplex:

The first page included pricing information, movie times, ratings, and summaries, and the capacity of each theater, which Rich found out from the manager.

The second page had the rest of the movie listings.

The last page had the food prices, which Rich copied down from the theater.

We asked the kids to take a look.  They started reading each synopsis, talking excitedly about which movies they had seen and which ones they wanted to see.  They laughed about "Jackass Presents: Bad Grandpa".  They whispered about the "R" rated movies.  They debated the merits of seeing "Thor: The Dark World" in 3D.  They drooled over the snacks.  Rich and I exchanged a look.  The hook was baited, now all we had to do was reel them in.
   I decided to use the strategy I had tried last year with Shannon's fourth graders.  They would create the problems themselves, and then solve them.  We borrowed a phrase from their ILA teacher: "thin questions vs. thick questions".  We encouraged them to write some questions that were straightforward and could be solved in a relatively easy manner (thin questions) as well as more challenging, complex problems (thick questions).  They started on their own, then met in small groups to discuss and refine their questions.

Here's an example of two questions from a student notebook.

I had to include this one, which provoked a very interesting "discussion" between the question's creator and another student that I was fortunate to capture on video.  The student insisted that the problem made no sense.  No one would pay to see a movie, leave halfway through, and then go to see the second half of another movie because that would be a "waste of money".  The student who created the question tried to explain that the scenario was not supposed to be taken as anything that might realistically be expected to happen; he just thought it would be a challenging problem to solve.  The other student would have none of it, and this back and forth went on for a good 5-7 minutes.

Groups got together and put their most interesting questions on chart paper.  Rich and I looked them over and selected about 15 for the class to work on.  They varied by topic and skill level.  I encouraged Rich to post them around the room.  The kids had several days to look them over before they were asked to dig in. This time, we assigned each question a letter and asked the kids to list their top 3 choices in their notebooks.  We gave them time during class to tour the room and take a careful look at each one before they made their choices.

The following day they entered the room and saw large, blank pieces of construction paper and markers along with the questions arrayed on desks and in corners all over the room.  They were told to get working on their first choice, but that if more than three people were already working on a problem, they were to try an alternate choice and come back later.  We also provided post-its in case kids wanted to provide comments.  Some worked alone, but most solutions were the result of collaborative efforts.

This was problem C.

These kids tried to find the difference by subtracting and got the wrong answer.  This led to an interesting discussion about why adding and subtracting time is different then adding and subtracting whole numbers.

This group used a number line.  Their work was correct but the answer is wrong.  In the middle post-it the group explains that their answer was a "typo".

This group also used a number line, which they termed "useless".  One of the students who answered the question plotted it out and revised their comment on the number line to "not useless".

This was question F.  It involved calculating with decimals. 

This group was close.  When they went to find the total, they copied $87.75 as $87.00 and were off by $0.75.

This group added incorrectly and was off by $10.00.

A correct answer!

My favorite: question A.  The reason: great example of a problem with a big vertical scale.

This student added up the costs of each individual item in each meal or combo package, found the total cost, and then calculated the difference between that and the price of each package.  It took him two days.  Talk about perseverance!  He claimed that for one of the choices you'd be better off buying each item individually.

A very basic response.

Also basic, but this student has a notion that perhaps better value may be found in another option due to the relative sizes of the drinks and foods that are offered.

The student who wrote the question also provided an answer.  I'll let his work speak for itself. 

  As the experience progressed from beginning to end, Rich and I engaged in a continuing conversation about what was happening.  Our reflections:

• We felt that this was an effective model for setting up conditions where students can be engaged in both traditional and non-traditional problem-solving scenarios. 
• Motivation and engagement came from: student-designed questions that originated from a high-interest topic, and student choice regarding which problem(s) to solve.
• Mathematics arose from a need, not the other way around.
• Low barrier to entry and plenty of room to scale up.
• Tremendous amount of opportunity for communication; both verbal (in the discussions students had during the question formation phase and also during the problem-solving phase) and written (in the way students express their solutions on paper.)

We liked this "movie theater" project so much that we started contemplating another, similar experience for later in the year.  Great Adventure anyone?