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Welcome fellow Recovering Traditionalists to Episode 154: Virtual Math Summit Preview: Helping Students Understand Multiplication

Before we get into the episode, this week’s positivity comes from Julie and it’s another throwback from last year’s Virtual Math Summit.  Julie typed this in the chat during the summit: “If all I learn today is what I learned here, it is sooo worth it.”  I wanted to share this one because I want to remind you that even though this year’s Virtual Math Summit has 24 sessions, you don’t have to attend them all.  Go look through the speakers and their sessions at and pick a few that you want to watch.  If you take just one thing that helps your students, then all the time we put into getting the Virtual Math Summit ready for you is worth it.  One thing you learn, helps a classroom full of kids…and we get to do that for the tens of thousands of educators who sign up for the Virtual Math Summit every year.  I’m so thankful to be able to do this for all of you.

The free registration gives you 10 days to watch this year’s sessions. We do have a VIP access for the summit which gives you access to this year’s sessions through the end of March.  Or if you’d like to access the past 7 years of summits, you can become a member of the Build Math Minds PD site.  Information about all 3 options is available by going to

This week’s episode is our second preview of some of the sessions from the upcoming 2024 Virtual Math Summit.  This week, I’ve curated snippets from 4 sessions that each talked about helping students build their understanding of multiplication (and some address division as well).  All of these sessions show some kind of visual to help you better see what they are talking about.  So I highly suggest you actually go watch this podcast instead of listening to it.  You can watch it over at my YouTube channel,

Shannon Olson has done sessions for the Virtual Math Summit in the past and I always love learning from her.  This year she is sharing about Using Progressions and Learning Trajectories to Guide Intervention in Multiplication and Division.  Even though her session is focused on Intervention, these ideas apply to all students who are building their understanding of multiplication & division and I also loved learning her take on the difference between a Learning Progression and Learning Trajectory as those phrases are often used as one & the same.

“…So notice that when we’re working with one-digit numbers, the research does not recommend starting with 1s, 2s, 3s, 4s, 5s in that specific order. But starting with what is called Foundational Facts first.  So 2s, 5s, and 10s first. Um, things are used to doubling, skip counting, and then working with zeros and ones and then nines as they derive those from tens, and then 3s, 4s, 6s, 7s, and 8s.  Then notice the place value progression as we multiply within 100, then multiples of 10, and then those multi-digit whole numbers.  With the standard algorithm being the very last thing that students are expected to learn and that they’re expected to learn it for division after multiplication.  Strategies and Representations; you’ll see a lot of physical objects along with visual drawings. Um, beginning with equal groups, then arrays, area models, and number lines. Lots of other different strategies.  We’ll move into repeated addition, repeated subtraction, and then also looking at partial products and partial quotients before moving into algorithms as well.  So again, this is linked in your handouts. I highly recommend using it as a reference as you support student learning of where to go next as we’re building student thinking. Um, just a couple ways to summarize how I’m defining Progressions and Trajectories. So I view Learning Progressions as describing how big Ideas advance through the grade levels and the way standards connect over time they tell us WHEN students learn concepts.  I’m defining Learning Trajectories as the pathways children naturally take to develop mathematical understanding and they’re telling us HOW students learn concepts. Um, a couple examples…”

In this clip from Juli Dixon’s session, Multiplication Fact Fluency: A School-Wide Solution, she shares a different approach for which facts to start with and build upon.  It’s slightly different from what Shannon talks about in her session but one thing is the same: it’s about laying a foundation of facts that kids know and then helping them see connections between those and the other facts. Here is Juli:

“…So the grounding tactic begins with the assumption that students understand multiplication as groups of objects. Where the groups are the first factor, the objects are the number of objects… I’m sorry where the first factor’s the number of groups, the second factor is the number of objects in each group. Beyond that, after we’ve made sense of the meaning of multiplication, we want students to already know their 2s, 5s, and 3×3.  Let me help you to see why.  We build multiplicative reasoning by using facts we already know to get to other facts.  And so, if we don’t know any facts there’s nothing to base the multiplicative reasoning upon. Students learn 2s and 5s pretty quickly through skip counting or they can think of doubles through their work with addition in Grade 2. So we can think of 2×7 as 7+7 to give us 14. 7 * 2 would be the 2,4,6, and so forth. So 2s, 5s, and 3 * 3.  Why is 3 * 3 in there? Well I needed this starting point. I needed a student to know some facts so they can build on those facts to get the other facts and 3 * 3 fit in that pretty well. Once we’ve grounded multiplication now we want to link multiplication facts from one fact to the next. This is The Linking Tactic.  The order of the facts really matters if we want to develop reasoning, multiplicative reasoning and algebra readiness. By that I mean, we don’t want to just do all the 1s, the 2s, the 3s, the 4s, and so forth because then students are encouraged to learn by adding another multiple, skip counting only, rather than multiplicative reasoning. We don’t even include the one or zero facts in this program because those students should just know if they understand multiplication.  If they understand that they have no groups with anything in those no groups they still have nothing.  If they have one group times anything in that group they have the number in that group. So those aren’t facts that students should need to practice, they need to understand them and then use them. So how many facts? Well when we think about multiplication beyond the twos, the fives, and 3 * 3, there’s really only 20 facts to focus on.  So what are those facts? Well if you look at the green shaded region those are the facts that are left. The yellow are the commutative property of those facts, right? The reverse order of the factors, I’ll call those the partner facts. So we have these facts, in this order. The columns that you see here.  We begin with 3 * 4, then 3 * 6, and 4 * 4 and so on. What’s important is this order and connecting one fact to the next. I’m going to make sense of that by focusing on these two facts….”

I’m a huge fan of using visuals and ten frames to help kids build their understanding of multiplication but in Thuc-Khanh Park’s session, Developing Multiplicative Reasoning to Learn Basic Facts Beyond Memorization, she shared something that I don’t talk much about.  In a part of her session she talks about the importance of Positionality.  Which is basically how you position the visuals.  How you position them inside of a ten frame can impact whether or not your students see the relationships you are hoping they see.  She talks about using N-groups and N-tiles it sounds like ‘in’ but it’s actually the letter N to represent NUMBER:

“…The first two type of materials that we’re going to talk about are: 

  • the N-groups which is based off of dice patterns, dice dot patterns
  •  and N-tiles which are simply a row of dots 

N-groups and N-tiles encourages students to see group of groups. Let me introduce you real quick to these settings or these materials.  First we have our N-groups and as you can see they are based off of dice dot patterns. I simply have a stack of all these cards, they’re about a 2×2 card square.  And then in addition to those, I also use what we call N-tiles and N-tiles is just a row of dots; 5, 4, 3, and 2s. They come in blue and they also come in red.  Let’s take these manipulatives, or materials, of the N-groups and N-tiles and be intentional and purposeful and think about how we’re going to lay them out.  The settings; we’re going to talk about three specific settings. The first one is Positionality. Positionality uses the structure of the 10 frame, which kids are very familiar with, and what the 10 frame does is it encourages the kids to subitize, or instantly see, the number of groups.  Decide what multiple that the kids will be practicing. Will they be multiplying by 2s, 3s, 4s, or 5s?  Once you have decided that, think about how you’re going to lay these cards out in the structure of the 10 frame.  You can do the exact same thing with the N-tiles. So for instance, let’s see what Positionality could look like using N-groups. What if I lay out three groups of fives? Once the child knows instantaneously that it’s 15, we’re going to build on that and leverage that understanding and knowledge so that when we ask them ‘there are six groups of fives,’ they can figure out ‘oh well three groups is 15 another three group is 15, and 15 and 15 makes 30.’  What does it look like with N-tiles? How can Postionality of the structure of the 10 frames help kids quickly derive multiplication facts using N-tiles.  Let’s say you tell the students, “Hey, I have seven tiles, two dots on each tile. How many total dots is that?’  If they can see that the top row of the 10 frame has five groups and 5 twos equals 10 and 2 twos on the bottom has four dots, 10 + 4 is 14.  Therefore 7 twos, or 7 x 2, must also be 14. We’re going to watch a clip of Jackson again and he was given the task of 10 groups of threes…”

Brittany Hege runs Mix and Math and is well known for showing mathematics visually in the upper grades.  Her session is full of visuals to help us make sense of the mathematics and in this clip she is talking about extending students’ understanding of the basic multiplication facts and helping them make connections to other multiplication problems, but what you don’t see in this audio is the visuals she is showing.  So once again to get the full effect of this clip check out the video at

“Now we may or may not use an open area model with students when they are working on developing this fact, but if your standards call for you to work with problems like this it absolutely makes sense to start with what students know and maybe connect that to a representation that is going to be more long-lasting for students. One that is going to work with situations that have much larger numbers, that are no longer model-friendly. So we could start with ‘Hey remember when you worked with multiplication in this situation? We could also represent it with an open area model’ and we would show our five groups of four here and our one group of four here.  And so now we’re making connections between these two different types of representations. We’ve got our concrete or physical, and then we’ve got our visual. But now they’ve got this other model in their tool boxes that they can use for a problem like this (16×4).  It also shows them that they bring knowledge to the table that allows them to be successful with the problem like this. Some students may still decide to pull out 16 groups of four and that’s perfectly okay, but more connections we can make showing students that they bring knowledge to the table that this builds on something they already know and we connect back to what they already know the more successful they are going to be and the more we bridge that gap so it doesn’t feel like what students are learning about multiplication in third grade is completely different than what they’re learning about multiplication in fourth grade. It’s really the same exact type of thinking applied to situations with larger numbers. So here we’ve connected between two different types of representations but we’ve also connected the same representation to two different problems to show them how this continues or progresses.  Now speaking of making connections between different representations and problems we want students to see that the strategies that they use with whole numbers are the same strategies that they can approach fractions with as well. Now in order for students to successfully multiply fractions they’ve got to have a deep understanding of multiplication so taking that extra time to really develop students’ understanding of multiplication and working with whole numbers really pays off when you move on to working with fractions…”

Again, those are just 4 of the upcoming sessions that will be at the 2024 Virtual Math Summit, starting on February 24th.  I hope you got some inspiration and ideas to help your students build their understanding of multiplication throughout the upper grades, but to get the full impact of these sessions make sure you register for the summit so you get access to the entire session.  Go to to get your spot at the summit. 

Until next week my Fellow Recovering Traditionalists, keep Building Math Minds.

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As you start off the school year, I want you to keep in mind what is really important as we're trying to teach mathematics to our students.