Slow to Go Fast: Multiplication Progression

Oct 22, 2020

It’s about that time of the year…

If you are a 3rd grade teacher, you’ve probably been reviewing addition and subtraction from 2nd grade and working on concepts of place value, and now it’s time to help kids understand their multiplication facts and patterns!

If you’re a 4th or 5th grade teacher, it might be that time of year for you too, as you review what your students learned in 3rd grade and determine where they might have weaknesses in their understanding of multiplication.

I can remember when teaching multiplication usually started off with a little bit about arrays, knowing how many groups are in a column versus a row, and then it just seemed like my book said, Okay, now the kids start doing those timed tests because they should magically get those facts!  

As the title of the blog indicates, that’s not the pattern we want to continue as we teach multiplication to our students today. We want to go slow to be able to go fast. We want kids to really be able to develop a depth of understanding around multiplication, but in a way that’s developmentally appropriate.

The sequence we’ll talk about today, the same one we always talk about when we look at multiplication, is great to follow when you’re helping students in 3rd grade who are just learning, but could certainly benefit 4th, 5th, and even 6th grade students that might need to solidify their skills.

times up

Of course, I can’t talk about this without talking about my favorite book – Time’s Up on Timed Tests, from my friends Kristin Hilty and Eliza Thomas Sorte, which we have available in our store as a digital download! These two fabulous ladies created a book that is truly amazing and filled with all kinds of activities for addition and subtraction, as well as multiplication and division. But what makes this book so incredible, in my opinion, is that it really looks at the foundation of fluency, which (spoiler alert) is not knowing facts quickly and being able to regurgitate answers under pressure.

Additionally, Kristin Hilty also designed one of my most favorite products – the Patterns of Multiplication journal. The sequence she outlines in this journal really helps 3rd graders to understand their facts in a different way. 

When I taught multiplication in the classroom, we would do the x2, x3, x4, x5 and start to be really proud of how well our class was doing! But then we rounded the corner of x6, x7, x8 and x9, and things started to fall apart.

Instead of going in numerical order to learn the multiplication facts, the Patterns of Multiplication Journal looks at the patterns, or the families, that we find within multiplication to help students understand it more fundamentally. 

We start with x10 and x5 – of course those go together! They have lots of things in common and you could easily and rhythmically count and hit those numbers based on the patterns. 

The next set of family patterns would be our x2, x4 and x8. If kids learn their x2, the x4 will make more sense, and then the x8 will come along with it. 

Next, we look at x3, x6, and x9, which also hit that same pattern.

The x7 is kind of the odd ball, which we usually teach separately. We decompose it into x5 and x2 because students can then anchor to a fact that they know. Sometimes kids have a fear of the x7s, but if we were to think of seven decomposed into 5 and 2, it’s a lot easier to figure out 7 x 7. I know that 7 x 5 is 35. And 7 x 2 is 14. When I add 35 and 14 together, it gives me that answer of 49. 

In this tutorial video, we’ll walk through the progression to help you get the idea of how patterns of multiplication are really developed.

Some teachers like to do this as a station, taking a fact per week. Some teachers prefer to do a fact all in one day. In some of the classrooms in our project schools, we actually plan a break from our math book to take a good 15 days and use the Patterns of Multiplication journal to create a solid foundation that students can use going forward.

Five Sections of Multiplication Journal

The journal has five different sections that we go through with each fact. Want to see the part of the video for a specific section?

Patterns and Equations

We have the students use an abacus or unifix cubes to talk about the patterns they see within each number. Teachers that use this sequence every year tell me, “Shannon, I really thought the kids would have gotten the idea of these patterns” and every year, they’re blown away at the things you and I might think is completely obvious but that students just don’t see.

In combination studying patterns, students write out the equations. They combine the concrete tool (maybe an abacus) with a grid that they can color in to reveal patterns, and then put it into words with equations and practice using appropriate language.

Let’s say we’re working on our fours. I have no groups of four, what does that look like? 0 groups of 4 = 0. So students could be writing 0 x 4 = 0, but their verbiage when they say “times” really should be “groups of.” If students understood the concept of “groups of” early on, man would it help them as they get older! Especially as we start to look at the multiplication of fractions.

As they start concretely, they might push over four beads on an abacus, and say, “Now I have one group of 4. 1 x 4 = 4.” So they color that in, Look at it on the abacus, and then create that equation. 

Now I’m going to push over another 4 beads. Now I have 2 groups of 4. 2 groups of 4 is 8 or 2 x 4 = 8. Push over another group of 4 beads. I now have 3 groups of 4. I know that total is 12. 

Really spend time here! This is not busy work to give your 3rd graders to complete at their seat or at home (if they’re virtual). WIthout the purposeful process, students will notice enough patterns to complete the chart (“fill in this one, skip the next two, fill it in, etc.), but they won’t really “get” multiplication.

Creating area models

It was brilliant of Kristin Hilty to put area models into her journal, because this really helps students to really understand the idea of the commutative property. As I draw 1 group of 4, and I’m kind of going across, I ask myself – how many groups do I have? I have 1. How many are in each group? 4. So 1 x 4. When you turn that paper around, or turn it 90 degrees, you would look at that differently. How many groups do I have now? I have 1 group, 2 groups, 3 groups, 4 groups. How many are in each group? 1. Well, that’s now going to be written differently – 4 x 1 or four groups of one. The same thing can be done for all the different parts is they’re creating the area models up to x10.

If I were to make six groups of four, I could go ahead and color in six groups of four. How many groups do I see? I see 6. How many are in each group? 4. Turn it 90 degrees – same idea. I now see 4 groups, but there’s 6 in each. So it’s 4 x 6. 

This page is a great reference for students and their understanding of that area model. It will help them connect things like square numbers, and to find things like missing parts of an area. 

Real World Concepts

Day four is another one of my favorites where students create their own real-world problems. We want them to use the patterns of multiplication they’re learning, and this helps us see if they are really understanding multiplication as they’re trying to apply it.

There’s also an area for a visual representation, where we’d like students to be able to make a visual model, an area for the answer, and one for the answer sentence. 

You have kids work in pairs where one student might do write the story problem and the other creates the visual model to go with it. Maybe one student creates a visual model and the other student has to write the story problem. Either of those options are really good ways to engage students’ understanding of visual models.

This step is easily differentiated with three levels. The first level is a very basic problem, multiplying your number by anything up to 10. Level 2 usually takes it from 11 groups to 99 groups of the number. And then level 3, the most advanced level, has kids coming up with two or three digit numbers times the number you’re working on.

Extensions

This is a fun day where students can apply the pattern that they’re working on. Here are some ideas for extensions:

  • Fact Flap Cards. Check out the video on creating a multiplication and division fact flap card, which is actually 20 flashcards in one! 
  • Multiplication Bump. Watch the video and download the corresponding game boards.
  • Multiplication Speed! This game helps students practice their speed if they’re working on the patterns. Check out the Blog Post!
  • Number Bond Cards. Practice number bond concepts with multiplication  and division. The video explains different ways that you can do multiplication and division number bonds. We have a downloadable, printable version in our store, and if you’re an M³ member, you can download it for free!
  • Skip Count. Help your students get comfortable skip counting forwards and backwards by the number they’re working on. If they’re working on x4:  4-8-12-16-20-24-28-32-36 and then 40. Then ask your students if they can skip count backwards? They may want to utilize the abacus to do this. Showing 40 beads. Push 4. Now I see 36. Push 4. Now I see 32. Push 4. Now I see 28. Push 4. Now I see 24, and so on all the way back to zero. 
time's up blog

The Time’s Up on Timed Tests book has a ton of resources where you can really look at how to collect data on fluency as you’ve gone through the different sequences. After you’ve completed your x5 and x10, are kids starting to understand this pattern? Are they able to understand the multiplication or the division within those patterns? 

So, if you’re in 3rd grade or (as I sometimes say) if you’re doing a “clean up on aisle 5” and having to start by determining which facts your students might still have difficulty with in 4th or 5th grade, using the Patterns of Multiplication or any of the extension resources we talked about, will really help students to “get” the patterns of multiplication.

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