As we begin to introduce multi-digit multiplication, the tendency is to dive right into the standard algorithm. While the standard algorithm is an efficient strategy, it is very procedural and many kiddos mimic the steps without understanding the process. A great approach for building conceptual understanding is to move through the concrete, representational, and abstract stages of learning. Let’s see what that looks like.
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Concrete (hands-on) learning
The first concrete step is making equal groups with manipulatives. Since we’re using larger numbers, and multi-digit multiplication is all about place value, it makes sense to use base-ten blocks. To solve 6 x 24, students simply make 6 groups of 24. To find the product, it’s likely they will begin by counting the tens. They might count them by tens (10, 20, 30…120), or they might recognize that each group has 20 and count by twenties (20, 40, 60…120). Encourage multiple strategies! Next, they will count the ones. Again, some students might count all the ones one-by-one, others might count by 2s, while still others might make groups of ten out of the ones. Be sure to have students share their strategies and ask students to reflect on which strategies were more efficient.
After students have practiced using the equal-groups method, let them know that you have a more efficient way of organizing the materials to make it more efficient to count the tens and ones. Introduce the area model using the base-ten blocks. Notice that the 24 along the top and the 6 down the side are showing our length and width, and we are counting the blocks inside the area. Make the connection that we still have 6 groups of 24, but now the groups are equal rows. In the second picture notice that we have shown 24 in its expanded form, 20 + 4, and we are grouping both tens and ones into groups of ten to make counting more efficient.
Transitioning from concrete to representational (pictorial)
Our next step is to transition students from the concrete area model to a pictorial pencil-and-paper version. I would probably throw a little problem-solving into the process. After students build the area model using base-ten blocks for 6 x 24, I would show them the divided rectangle with no numbers. Have a group discussion asking what students notice and wonder about the new pencil-and-paper model. Probe students about how they think the two divided sections are related to our concrete area model. When they make the connection that it’s like the tens and ones in the area model, ask Hmmm, how do you think we should label the top and the left side then? From there, you can have them offer suggestions for how to fill in the numbers in the two sections of the area model.
When I’m ready to introduce using this model for 2-digit by 2-digit multiplication, I’d again use problem-solving. Presenting them with the problem 24 x 16, I’d say, Hmmm, I wonder what our area model would look like for this problem? It’s a little different. Then I’d let them work in partners to try to figure it out. It’s really not that much of a stretch! If both factors are 2-digit, I need another row.
Using the area model requires students to work with multiples of 10 and 100. You might check out this blog post for an activity that builds fluency with multiples of 10 and 100. Another great activity is to have kids skip-count by multiples, so instead of counting 3, 6, 9, etc., they count 30, 60, 90, etc. That works great with this whole group Sparkle game.
Moving from pictorial to abstract
The area model is a pictorial representation closely connected to an algorithm called partial products. Students who have mastered the area model will make this transition smoothly. Here you see the two strategies side-by-side, and it’s easy to see the similarities. Help students to organize their thinking by modeling how to show the multiplication for each of the partial products, writing 6 x 4 beside the product of 24 and 6 x 20 beside the product of 120. Let students practice showing the area model and partial products algorithm side-by-side to solve problems until you feel that they no longer need the pictorial support.
Finally, the standard algorithm!
The strategies leading up to the standard algorithm are not meant to replace the standard algorithm. They are designed to build an understanding of the standard algorithm. Once students are confident using partial products, we can put it alongside the standard algorithm and show students that the standard algorithm is just a shortcut. While this example shows 2-digit by 1-digit multiplication, it works just as well for 2-digit by 2-digit or 3-digit by 1-digit.
There you have it—the road to the standard algorithm for multi-digit multiplication with understanding! Remember that this process takes time and practice with each new model/strategy. Sometimes it’s misleading because I’ve covered all the models and strategies in one short post. But you want to make sure students are secure with each model or strategy before moving to the next. This will require differentiation since we know students don’t progress at the same rate.
Practice makes perfect!
Of course, your students will need meaningful practice, and I have a simple game for that. There are versions for 2-digit by 1-digit, 3-digit by 1-digit, and 2-digit bt 2-digit. They can play the game anywhere along the process I laid out in this post. For example, they might play the 2- by 1-digit game using base-ten blocks, then using partial products, and finally with the standard algorithm.
Download your free game using this link.