Multi-digit division (or long division as we called it back in the day) is one of the trickiest concepts students learn in elementary school. In fact, most adults I know (including myself) have terrible memories of learning how to divide. Unfortunately, this is because most of us were only taught to divide using the standard algorithm. The long division standard algorithm involves memorizing multiple steps and doesn’t seem to connect to the place value of a number. Students often forget steps or don’t realize when they get an unreasonable answer.
That’s why I recommend using the CRA (Concrete-Representational-Abstract) approach when teaching students to divide multi-digit numbers. By starting with hands-on models and visual representations, students build a solid understanding of what it actually means to divide larger numbers. This makes the transition to abstract computation smoother and more meaningful—setting them up for success when they move on to dividing using the standard algorithm.
What is the CRA Method?
CRA stands for Concrete-Representational-Abstract. This is the process students should move through in order to build conceptual knowledge, which can be applied across multiple mathematical concepts. Ultimately, this is what we want! We don’t want students who know a formula but can’t figure out how or when to apply it. We want them to use it in real life across multiple situations and scenarios!
Here’s an overview of CRA:
- Concrete – In this stage, students use objects to model problems. For example, using base ten blocks or other objects.
- Representational – Students move from manipulatives to drawings in this stage. Students can use grid paper (graph paper) to represent the concrete models.
- Abstract – This is the stage people typically think of with math: algorithms and equations! Here the actual mathematical symbols, notation, and algorithm will be used.
Introducing Multi-Digit Division with Conceptual Understanding
When introducing multi-digit division, it’s helpful to remind students that dividing means to create equal groups. To get students thinking about equal groups, begin with the “dealing out” method. This is where students build the number they are dividing (dividend), decompose it in a helpful way, and split it into groups (similar to “dealing out” cards).
Let’s use the example 138 ÷ 6. Students should build 138 using base 10 blocks. Then they need to figure out how to split it into groups (the divisor). They might notice that they need to regroup almost immediately–they can’t split a hundred block into 6 groups. But if they regroup their hundred into 10 tens, they are able to start “dealing out” their tens.
Since division can be represented 2 ways, you may want to give students time to explore whether it’s more efficient to make 6 groups or to make groups of 6. They should quickly see that making 6 groups is more efficient when the dividend is so large. It will also help them to transition to arrays, where it will help them to think of creating 6 rows.
Once students have a good understanding of creating equal groups with their base 10 blocks, you will want to encourage them to start forming arrays. This time when they build, they should think about splitting their dividend into 6 rows. The number of columns will be their answer. You should discuss what they should do if they need to regroup again.
It’s important for students to get comfortable with arrays before moving on to the representational phase. The “dealing out” strategy will quickly become inefficient when students start drawing, and when they eventually begin dividing by two-digit divisors.
Dividing Using Drawn Representations
When students are ready, start modeling problems using drawn representations. Some students may need to continue to build arrays with base-10 blocks before they draw them on paper–allow them to do so! This allows them to “bridge” their thinking from the concrete to the representational phase.
When students begin to draw their arrays, it’s very helpful to allow them to do so on graph paper. Using the same example, 138 ÷ 6, students should think about splitting 138 into 6 rows. I like to encourage students to begin with a “friendly” number, such as 10. How much of the 138 is shown if you make 6 rows of 10?
As you can see, creating 6 rows of 10 took a large chunk out of 138, but there is still some left to split. Ask students if they are able to take out another chunk of 60. They should see that they can!
When you are no longer able to use friendly numbers, use students’ knowledge of math facts to help them. In this case, students can think about the fact 18 ÷ 6 = 3. They should see that they can split the remaining 18 by adding 3 to each of the six rows!
When students are pretty comfortable with arrays on grid paper, it is time to transition them to the open area model for division. The major difference between the array and the open area model is that students can’t see the individual blocks.
When I first transition students to the open area model, I like to have them include the “dimension pieces” on the outside of the boxes. This helps them keep track of the different partial dimensions they are using.
As students get more fluent with using arrays and area models, the partial dimensions that they use should change. Instead of using 10 each time, they may get more comfortable with using numbers such as 20, 30, 40, etc. This is a flexible model!
Division Using Abstract Thinking
After students have built a solid understanding of multi-digit division with concrete and representational models, they can start solving using abstract thinking! Some students may still need to draw models to go with their abstract work as a “bridge.”
The strategy I prefer to use with students entering this phase is the partial quotients strategy. This is the strategy I wish my teachers had taught me growing up–and it’s my preferred method when I personally solve division problems. This strategy is great because it directly relates to the open area model!
To use this strategy on the same example, 138 ÷ 6, students set up the problem similarly to the standard algorithm, but they add a line down the side to help organize their work. To use this strategy, students should think about the number of groups of 6 that are in 138, starting with friendly numbers. For example, if they know they can fit at least 10 groups of 6 in 138, they can show that on the side and subtract it from the dividend. Then they continue to see how many groups they can fit!
The great part about the partial quotients strategy is that it is another flexible strategy! For example, students could use 20 instead of 2 tens.
Depending on your grade level and state standards, you may stop here with your division strategies. The Common Core standards don’t have students work on the standard algorithm until 6th grade! If the standard algorithm isn’t in your standards or if your students don’t seem ready, I suggest encouraging them to continue using the partial quotients strategy.
Helpful Resources
If you’re feeling overwhelmed with teaching division, try out some of these resources. They are filled with examples, guided practice, and more to help students master multi-digit division.
#1 Guided Notes: These division guided notes for 4th grade and 5th grade will help you easily scaffold students’ learning. The notes go over multiple strategies for dividing multi-digit numbers. The notes really come in handy when reinforcing the representational and abstract portion of the CRA method, and you can use the problems in the notes to introduce the concrete stage.
If you love these guided notes, you can grab the entire 4th Grade Math Guided Notes Bundle or 5th Grade Math Guided Notes Bundle.
#2 Interactive Worksheets: If you have students that are struggling with division or need extra help, the interactive worksheets for 4th grade and 5th grade make it a breeze. Each worksheet includes a QR code that links to a mini-lesson video.
If you love these guided notes, you can grab the entire 4th Grade Math Interactive Worksheet Bundle or 5th Grade Math Interactive Worksheet Bundle.
#3 Math Riddles: Add a bit of fun to multi-digit division with sports-themed Division Math Riddles. As students solve each division problem, they get closer to figuring out the answer to a fun riddle. The riddle part of the worksheet not only makes it fun for students, it makes the activity self-checking! This makes for great morning work, early-finisher practice, and independent work.
Love these ideas and want more? Check out these posts:
