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.


“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. ๐Ÿ™‚  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:


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.  ๐Ÿ™‚
  • 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.  ๐Ÿ™‚
  • 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.

Finally, because I know that many people visiting this post are new to blogging, I want to mention that it’s great to leave a “drive by” comment.  But the best conversations happen when people return to read comments left after their own, and then comment again!  After you write a comment, WordPress blogs allow you to click a box that will “subscribe” you to a post.  Every time someone comments on that post you’ll get a notification that will keep you in the conversation.  I recommend it!

You’ll have to log in to comment on my blog, leaving your name and e-mail address.  Rest assured that only I will be able to see your e-mail address.  It won’t be visible to anyone else.  I have this setting on because it helps to eliminate many spam comments.

11 thoughts on “Multiplication Models”

    1. It’s a fine line for me sometimes. I think it’s easier this time of year because we are a community and they know they can do hard things. That’s not always so earlier in the year.

  1. Really interesting post – good to get inside your thinking to see how discovery can work in the classroom – thanks for this. Good instructions on posting comments too – thanks!

  2. I have also been thinking about how I learned to multiply and divide over the past few weeks. I teach in the junior division and a big stumbling block is always the long division algorithm. I actually only use the flexible algorithm now and I can now do way more in my head than even ten years ago. I suppose that it goes without saying that the improvement in our teaching means an improvement in our learning. I am also looking forward to hearing about how the guided math went this week.

  3. I still remember my stern grade four teacher standing at the blackboard with her pointer, she’d call out a timestable (we didn’t call them facts) and we would repeat it, over and over. I think I understood multiplication as repeated addition but we certainly didn’t get the opportunity to model it, nor any other math concept. Much of my number sense came from playing card and dice games with my family. I love how you used the tiles here and how you allowed the students to discover what you wanted them to learn. When I did a similar activity with my grade 4/5 class, some of my students resorted to using open arrays…”It just saves time Mrs. V.” I applauded their efficiency! By the way, this is my first comment on a blog! Now to create my own…

    1. Hooray for your first comment! ๐Ÿ™‚ It’s great that your students used an open array. They must be great at finding area and perimeter. So many times students are slow to make that jump from counting boxes to using the length X width formula. And that now makes we wonder how we feel about teaching kids formulas.

  4. Hi Lisa, I appreciated the opportunity to read your blog. Doing a lot of thinking about the array these days as we learn alongside Dr. Fosnot. I have been working with a couple of Gr. 3 boys on Cathy Fosnot’s minilessons for multiplication. So far, we have been using the visuals and lessons that are in her book – both boys have been mostly skip counting to find the totals #s of objects i.e. the muffin tin questions. Last week, they literally tore through my house looking for arrays they could use. A highlight was watching them work through a problem I set up where they figured out how many tiles I would need to replace the tiles in my kitchen – 3 are cracked.
    What fun they had and so proud that they helped me solve a really impt problem. While they are still skip counting I have noticed they are starting to expand to using groups of ten and take one away and they are both starting to notice the number of rows and groups they are working with. They will actually say and gesture that 5 groups of 10 are 50. 10, 20, 30, 40 50. I am thinking about the unitizing…..So – your blog is timely – I have acquired some tiles (I am not in a school but have a couple of children of friends who work with me at night – for my own learning); next week work we are going to construct the arrays – or should I say they will construct their own arrays. Your blog has really helped me think through the approach I will use as we enter this next step in their and my learning! It is fantastic to see and understand the progressions they are making on the continuum – I love that they see this mathematics – working on multiplicative thinking – as both fun and challenging – and that we get to celebrate together the learning moments as they progress in their strategies and thinking. Looking forward to sharing my adventure a little more next week when I contribute my own blog! This work is addictive – Thanks again for sharing.

    1. Thanks for your thoughtful comment! I love that we can all share our experiences this way. The tile question is a good one because, as you said, it is real! It’s a lot more interesting for kids, I think, to do “real” work.

  5. Hi Lisa!
    Thank you SO MUCH for sharing this journey with us. Like you, I can’t remember for the life of me how I was taught to multiply (or divide, for that matter). It must have just been by learning the algorithm. As a secondary teacher, I’m sad to say, I never questioned how students would have learned to multiply, or what their landscape of learning might have looked like as they approach multiplication problems. But now that I’m participating in the Not A Book Study, and reading posts like yours, I’m seeing how so many of these learning conversations can also happen in grade 9 math. I’m looking forward to when I can be back in the classroom so I can rediscover the learning with my students!

  6. Hi Lisa,
    I loved reading your blog! Getting to hear the “why’s” behind your educator decisions is very insightful! I really like how you were so intentional with the use of the square tiles, allowing students to connect back to a game they were familiar with, using your goal for them to discover and make sense of the array. You were also very in tune with the role of the tiles to provide an opportunity to make rows/columns, but noticed when the manipulative became a distraction. When students began to make patterns, rather than paying attention to the numbers of rows and columns, you changed it so that all the tiles were the same colour. All of these subtle decisions are so important, and we don’t often talk about them enough. It’s true how much these small decisions can make or break an inquiry-based lesson.
    I also think your point about scaffolding vs rescuing is really important. We need to know the difference between these two (again subtle) ways of interacting with kids.
    Thanks so much for sharing this commentary on your kids and your lesson! You’re right too – I would be interested in hearing how they counted :), but just am glad to know that this too was one of your professional noticings!
    Overall, as I’m reflecting on your blog, I’m struck again by all of the subtle, but very important components that are involved in teaching. It’s amazing how when we learn more, these subtleties become more apparent.

    1. Yes! I like to think I’ve been a responsive teacher all along, but I feel like my responsiveness is becoming more specific.

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