Thursday June 1, 2023
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There are many different ways to model two-digit subtraction when solving for an unknown. In this post, we’ll explore five different methods and discuss the benefits of each for different types of two-digit subtraction problems. We’ll also look at when it’s appropriate to use each method. Finally, we’ll talk about some of the misconceptions that students often have about subtraction.

## What is the best way to explain how to solve for the unknown in these two-digit subtraction problems?

23 + ____ = 39

55 – ____ = 13

Before I get started on ways to model two-digit subtraction problems, I have a great blog post all about models and strategies for solving two-digit numbers. It walks through the differences between a model and a strategies and gives several examples of use cases for each of them.

## Two-Digit Subtraction Solve for an Unknown

Below are some idea for helping students solve subtraction problems with an unknown or missing part.

### Part-Part-Whole Strategy with a Bar Model

If you’re working with two-digit subtraction problems, the part-part-whole strategy can be a helpful way to visualize the problem and find the unknown. In this strategy, the whole is the total amount, the part is the amount you know, and the unknown is the amount that’s missing.

A bar model, or a tape diagram, is a great visual for part-part-whole. Can you see the relationship between models and strategies here? Part-part-whole is the strategy. The Bar model or tape diagram is the model.

For example, let’s say you’re trying to find out how many candy bars are in a box if you know there are 39 total and 23 have already been bought. In this case, the part-part-whole model would look like this:

#### Part-Part-Whole with Base-10 Blocks

You can also model two-digit subtraction using base-10 blocks. For a missing unknown problem, build both numbers with base-10 blocks. Ask students to find how many more blocks they need to get to the larger number. This is one way to use base-10 blocks with a count-up strategy. In this case, part-part-whole with the base-10 blocks becomes the model and count up becomes the strategy.

Models and strategies are nuanced. I would not teach students the differences between the two, but, as a teacher, it’s good to be able to differentiate between them so that you can move students toward efficiency in both areas.

### Teach that the equal sign means “the same as”

The equal sign does not mean the answer. it’s a balance and both sides have to be the same for it to balance!

As you are demonstrating math problems with students, write the equations with the answer on the left side. See if any students say that it’s wrong. Ask them why it’s wrong. See if they can explain their thought process. Ask other students if they agree or disagree. Then draw it out or use manipulatives to prove that the answer can go on either side of the equation.

### Use Inverse Operations

Inverse operations are a great way to help students understand how subtraction works. When you use inverse operations, you are using the opposite operation to solve a problem. For example, if you are subtracting two-digit numbers, you can use inverse operations to help you get the answer.

Use a number bond with smaller numbers to establish the concept of inverse operations. Help students see the pattern with larger numbers. Start with 13-5 and write all of the corresponding equations.

Inverse operations can also be used to check your work. If the answer to a subtraction problem is incorrect, reversing the operations will usually give you the correct answer.

### Use an Open Number Line to Subtract

Either count down the number line or find the missing parts. Both strategies work because the problem is not contextualized; it’s not a word problem.

Show students to count up or down the number line by friendly numbers, like tens, rather than counting by ones. Always move toward efficiency when solving math problems.

### Introduce Algebraic Concepts

This is a great opportunity to introduce algebra to your students. Put a shape or a letter in the blank. Tell students that the symbol needs to be by itself. Whatever you do on one side you must do on the other side.

This goes along with the suggestion above about making sure students understand that the equal sign means the same as, not the answer. It also goes along well with the inverse operations suggestions above. Are you starting to see how all of these strategies are related to each other?

It’s all about balancing the equation.

### Don’t use a Number Bond to Model Two-Digit Subtraction

Number bonds are great for single digit addition and subtraction or to show how to break apart a number. They show the relationship of the numbers in an equation. That’s all they do. They show a relationship of the numbers.

Always move students toward efficiency in math. A number bond is a great tool in grades K-1 and the beginning of second grade. As students move through multi-digit addition and subtraction, they should move toward more efficient models.

Don’t use a number bond to show all of the digits in a two-digit subtraction equation. Use it to show single digit relationships and use it to break apart numbers to group them easier.

Number bonds are also a great way to break apart larger problems. For instance, you could break apart 23 into 20 + 3 and show it using a number bond, or two lines below the number. This is a way to use number bonds to break apart larger numbers.

## Misconceptions that Students often have about Subtraction

Many students have misconceptions about subtraction. For instance, they often think that subtraction is just counting down.

These misconceptions can cause students to struggle with solving math problems. In order to help students overcome these misconceptions, it’s important for teachers to be aware of them and address them in class.

Here are some of the most common misconceptions about subtraction:

• Subtraction is just counting down: Counting down is one strategy when subtracting, but students can also count up or find the difference between two numbers.
• You can only subtract from larger numbers: As students will move through their math education, they will learn that, you can subtract from smaller numbers and work with negative numbers.

As a teacher, it’s important to be aware of these misconceptions when you’re working with a variety of two-digit subtraction strategies.

## Resources that Model Two-Digit Subtraction

Here are a few resources that you might like for your classroom. These resources use the models and strategies you’ve read about above.

## More Subtraction Ideas

Are you teaching two-digit and three-digit subtraction? Here are more ideas that expand on the concepts in this blog post.

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