math

Multiplication Models

I wish I could remember how I was taught to multiply.  I have absolutely no memory of who first taught me about multiplying…which is kind of weird because I remember all of my elementary school teachers. I do remember doing “mad minutes” in Mr. Goodreau’s 6th grade class. I would bounce around doing all of the 1s, 2s, 5s, and a few other random problems that I actually had memorized. And I remember being in high school algebra class and writing down my skip counting for problems with 6s and 7s. Heaven forbid Mr. Creager should ask a question that involved 8s or 9s.

Excuse me while I go deal with some residual anxiety related to multiplying.

As a teacher, I’ve often wondered what would have been different for me had someone along the way pulled out some manipulatives, or showed me an array, or, you know, explained that there’s more than one way to answer a multiplication problem.

I introduce multiplication to my grade 3 students using a “groups of” model. Using lessons from Marilyn Burns “Lessons for Introducing Multiplication” book, I let them investigate (though I’m not now 100% sure I have been allowing them to investigate enough…topic for another night!) ways to figure out how many legs there are in their group, or how many pencils we’d have if I bought 4 boxes, or any number of other questions designed to push them to use skip counting or repeated addition and then connect that to multiplication.

I also teach my students about arrays. I get out the colour tiles and ask them how we can arrange them to show 4×3, for example, and we move on from there.

Why arrays?

Well, because Marilyn Burns told me to, at first.  Then I stuck with it because it made so much sense to me.  And years later I am still doing it because I see it as a model that makes sense to kids.  There are arrays everywhere, so it is easy to find ways for them to practice using them, as well as a way to connect them to why multiplication is meaningful.

Today I wanted my grade 3s to work on arrays.  Usually, I would show them an array and then we’d practice them together and we’d all walk away pretty happy.  However, today I decided to let them discover the array on their own.

I pulled out the colour tiles and put them in the middle of the group.  “We’ve been playing “Circles and Stars” and I am wondering if that always has to be played with chalk and a chalkboard, or if we could play it with these colour tiles.”  The colour tiles were our manipulative, and the array was the tool I wanted them to work toward understanding.  I am going to admit to being a bit nervous about this, and thought it might be a disaster.  But who am I to question Cathy Fosnot, right?  So I kept going.

A few of them suggested that they could use tiles to make circles and then fill the circles with other tiles.  That would be like circles and stars.  I asked them to show me, and they started arranging the tiles into circles.  They soon decided  that it is hard to make a circle out of squares.  Then one boy said, “You could think of the sides.  It has 4 so you could do 5 x 4.”  This is what he made.

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“That’s a Minecraft circle!”  someone said.

I asked, “You think we could do it like that?  For every problem?”

A students said,”That would be hard.  That would be a lot of work!”

Another student had been watching the whole thing.  She finally said, “I have 2 plans now.  You could use the tiles to make dice….wait…you could have imaginary circles.  You don’t actually need the circles, just the groups of tiles.”  HOORAY!!

We took turns modelling that:

I asked them to count what they had, and some interesting counting things happened.  I know Melissa will be sorry if I don’t explain that, but it is kind of a tangent. πŸ™‚  Let’s just say that even though they had groups of 4, they still counted by 2s and 1s.  (sigh)  But then another child said, “Well if we put them in lines, we can count by 2s.”  and he made this:

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Someone counted by colours in groups and then they started, on their own, talking about other ways to arrange the 16 tiles we had randomly started with.  Like, seriously, they started making the division connection ALL ON THEIR OWN!

(By now I had switched out all the tiles so they were the same colour.  I wanted them to think about organizing the arrays without worrying about making designs with the colours.  This is a real problem with colour tiles and can often get in the way of the math learning.  I mean, patterns are important too, but making checkerboard patterns for everything doesn’t seem to help with the counting or understanding in anyway. It just slows kids down and causes them to hate one another because someone is always hoarding the good colour that everyone else wants.)

We talked about how each of the arrays we had made matched with the dice we use in “Circles and Stars”. Then I asked, “Now do you see how to play ‘Circles and Stars’ with colour tiles?”  And they did!

So I sent them off with a partner and a pair of dice.  We are now using a 10-sided die to show the number of groups and a regular 6-sided die to show how many in the group.

Here are some highlights:

  • On their own, one group found out about ones and zeros and what happens with them in multiplication.  πŸ™‚
  • On their own, each group figured out that organizing the groups into straight lines made them easier to count, and easier to count without having to go back to ones.
  • One girl started counting by ones.  I stopped her and encouraged her to try a bigger number. (Array pictured below) “I can’t count by 2s because I get confused!” I said, “Then try something besides 2s.”  I thought she’d do 5s, but she counted by 4s.  πŸ™‚
  • Slowly 2 of the 3 groups moved to this sort of group arranging:

I feel like my goal today was accomplished, and then some! The students discovered the array.  I felt like I was scaffolding, not rescuing them.  (A personal pursuit of mine.) And I think there was a healthy bit of discovering things that can lead to other things for us.  (Division, for example.)

I have an idea for my lesson tomorrow.  Two of my grade 3s were away today, and should be back tomorrow.  They will need to be in on this discovery.  And I want to really solidify what we did today.  BUT…instead of sharing my idea, I’d love to hear ideas from others.

And by the way, the grade 2s were all doing their Guided Math rotations while the 3s and I did this.  I blogged about my experiment with GM earlier in the week. I am still on the fence about it, but will finish out the week before I talk about it again.

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