The transition from additive thinking to multiplicative thinking is very difficult and abstract for some kiddos. Think about it, for as long as they’ve been working with numbers, the number 2 has represented two things. Two ducks, two cookies, two linking cubes, etc. Now, with multiplication, 2 x 8 means 2 groups of 8. That’s huge. So, of course, it’s important that students get LOTS of concrete practice with the concept of multiplication—building 2 groups of 8. And then they should represent that same idea by drawing it (The representational stage of CRA).
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Once students have a strong understanding of the meaning of multiplication, we can extend that understanding by bringing in place value concepts and using multiples of 10. This is an essential understanding that must be in place prior to moving to multi-digit multiplication.
A great way to start this journey is with a version of Count Around the Circle from the book Number Sense Routines, by Jessica Shumway. Students sit or stand in a large circle. Start out skip-counting by a single-digit number, for example, 2. Go around the circle, and each student says the next multiple of 2—2, 4, 6, 8, 10, etc. Next, do the same routine with students counting by 20s instead. You want them to see that it’s the same pattern. Feel like a challenge? Try 200s. Make this a regular routine in your classroom. It’s a great way to practice multiplication facts and to stretch students to think about larger numbers.
Next, it’s time to bring all of your CRA stages together. I created a little work mat with number cards that you can use for small group instruction.
I printed the first 2 pages of numbers, which are the factors, on pink card stock, and the last two pages, the products, on yellow card stock. There are a LOT of numbers, so you’ll want to pick out the set of numbers you want to work with. In the example above, I’m using the variations of 8 x 2. Have students build a model of 8 x 20 with base-10 blocks and skip count by 20s to find the product. You also want them to draw a representation (see the picture below). The work mat provides the abstract number sentence, so you have all three stages of CRA—concrete (manipulatives), representational (drawing), and abstract (symbols).
Here I turned the factors around and had the student build (and draw) 20 x 8. Once the student has built the model with units, encourage them to group them into tens for easier counting.
This photo shows the representations of 8 x 20 and 20 x 8.
Here’s a shortcut way to draw representations of larger numbers.
Now let’s bring in some problem-solving. Notice here that I’ve given the student one factor and the product, and the student has to determine the missing factor. Encourage the student to draw a model to solve it!
This one will be a little trickier. If students have realized the pattern of zeros from their previous experiences, they will know that 200 goes in the blank—three extra zeros in the factors means three extra zeros in the product. Be sure to have students justify their solutions and explain their thinking!
Ooh, what a great class challenge. Let students work in pairs to come up with as many solutions as possible!
Ready to try it? Download the work mat and number cards using this link.