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The Standard Algorithm for Multidigit Addition and Subtraction in 3 Easy Steps

Are you a 2nd Grade teacher who loses sleep over teaching the standard algorithm for multidigit addition and subtraction? Or a 3rd or 4th grade teacher trying to remediate students who can’t successfully add and subtract multidigit numbers? What if I told you it takes just 3 easy steps? Well, I’ve got good news! It’s true. Following these 3 easy steps is the key to mastering the standard algorithm for multidigit addition and subtraction. It’s not magic! Used correctly, we overlap methods and build a strong understanding of the place value concepts required to understand regrouping when adding and subtracting.

standard algorithm multidigit addition and subtraction

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Step 1: Direct Modeling

Direct modeling involves using manipulatives and a counting strategy to find the solution to a problem.

ADDITION

Begin with smaller, 2-digit numbers. Ask students to use base-ten blocks to add 23 + 45. Having students work with a partner cuts the manipulatives needed in half and fosters mathematical discourse.

To solve 23 + 45, students will likely build 23 and 45 and then they will add the tens, followed by the ones. They might push the tens together and the ones together, but they might just count them as they are. And they might use different counting strategies—some counting the tens by tens (10, 20, 30, etc.) and others counting the tens (1, 2, 3, etc.) and adjusting for place value (6 tens is 60). Give students the freedom to count the materials in whatever way they like initially. If you see students continue to use inefficient counting strategies, work with them to move to a more efficient counting strategy.

You can see that using direct modeling students can easily add numbers we would traditionally think would need regrouping. They’ll just have more than ten ones. It’s not necessary to have them trade ten ones for a ten. Let them discover that it’s easier to count if they do trade a ten for ten ones.

Do not rush this! Let students use base-blocks when solving word problems. The standard algorithm is all about place value, and the more practice they get with direct modeling, the better their place value understanding.

SUBTRACTION

Subtraction with base-ten blocks works much the same way. The big difference is that instead of building both numbers, we only build the number we are subtracting from. From that number, we take away the number we are subtracting. So, if we are solving 45 – 23, we build 45 and then remove 23, leaving 22.

If the problem requires regrouping, we must trade a ten for ten ones before we can remove the blocks. So, if we are solving 45 – 17, we need to trade a ten for ten ones. After trading it’s important to ask students, Do we still have 45? It’s a common misconception that after regrouping the number has changed. Now we remove 17, leaving 28.

standard algorithm multidigit addition and subtraction

Remember, provide LOTS of practice doing this! It builds understanding for the regrouping in the standard algorithm.

SUBTRACTION OVER ZEROS

You know how much you dread teaching subtraction over zeros? If students learn the process in the direct modeling stage, it’s a breeze by the time you get to the standard algorithm.

Here, we’re trying to solve 105 – 62. Having had plenty of practice trading a ten for ten ones with smaller numbers, many students will realize that we have to trade that hundred for something. I say that, because they might not immediately realize that they’ll trade it for ten tens, and then trade one of the tens for ten ones. Let them struggle with it a bit. That’s problem solving!

                                                standard algorithm multidigit addition and subtraction

Remember that you are not trying to tie this process to an algorithm yet! Just continue to let students use the blocks and a counting strategy to solve problems using direct modeling until you’re sure they have a very secure understanding of the place value behind the process.

STEP 2: EXPANDED FORM

Adding and subtracting using an algorithm based on expanded form is also called partial sums or partial differences. It is a written form of the direct modeling students have been doing.

To introduce the expanded form method, we’re going to put it right alongside the direct modeling, so students make the connection between the two processes. I’ve created a template you can use in a plastic sleeve with dry erase markers to help students line up the place value parts. There’s a link to download it at the bottom of this post.

Do the two methods side-by-side until students don’t feel they need the base-ten blocks anymore.

Subtraction is a little trickier, because it can be a little confusing initially when students see the plus signs in a subtraction problem. To clear up any misconceptions, be sure to use the base-ten blocks alongside the algorithm and specifically address why there are plus signs: We are subtracting 60 + 7, another way to write 67, minus 20 + 5, another way to write 25.

standard algorithm multidigit addition and subtraction

Students will already know how to handle regrouping using the base-ten blocks, but you’ll need to show them what that looks like using the expanded form algorithm. Once students trade a ten for ten ones using the blocks, show them how we record that with the algorithm, renaming 60 + 5 as 50 + 15, which is just another way to decompose 65.

Remember that you’ll phase out the direct modeling once you feel that students have a strong understanding of the expanded form algorithm. And it’s likely that milestone won’t occur at the same time for all students.

Before moving on to the standard algorithm, allow students plenty of practice with the expanded form algorithm.

STEP 3: THE STANDARD ALGORITHM

Finally, you are ready to introduce the standard algorithm as a shortcut to the expanded form algorithm.

Once again, we’ll put the new method (the standard algorithm) alongside the method students are secure with. Because there are conventions associated with the standard algorithm, direct teaching is requires.

Explain to students that with the standard algorithm we start by adding the ones. Add the ones in the expanded form problem, getting 12. Show students that when we add the ones in the standard algorithm, we record the 2 ones in the ones place and regroup the ten over with the other tens.

Next, add the tens in the expanded form problem, getting 80, and add 80 to 12 for the sum, 92. Now switch back to the standard algorithm, showing that we add 6 tens with 2 tens and then add the extra ten we regrouped, getting 9 tens, or 90.

standard algorithm multidigit addition and subtractionSee what a great connection there is?

Let’s skip right to subtraction with regrouping, since subtraction without regrouping is pretty straightforward.

Show the regrouping first using the expanded form algorithm, reminding students that if we can’t subtract the ones, we rename the number we’re subtracting from. In this case, we’ve renamed 65 as 50 + 15 because we can’t subtract 7 ones from 5 ones. I can’t stress enough that if students are not yet secure with this process, they are NOT ready for the standard algorithm. Next, model how to show the regrouping using the standard algorithm, making the connection between 50 + 15 and 5 tens and 15 ones.

Finish the problem by subtracting the tens.

standard algorithm multidigit addition and subtraction

Continue having students solve problems with the two methods side-by-side until you feel that they are secure enough with the standard algorithm to drop the expanded form algorithm.

And there you have it—three easy steps! Download a PDF version of this post with materials you can use in your classroom using this link.

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One Comment

  1. I love the idea of letting them explore with a partner using base ten blocks. They generate their own understanding and insights.

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