Subtracting by Adding Up with the Think Addition Strategy

Throughout this series on computational fluency, we've explored how students develop increasingly sophisticated strategies by building on what they already know. Students first develop foundational fact relationships, including +1 and +2 facts, doubles, and combinations that make 10. They then use those known relationships to solve problems more efficiently through strategies such as Near Doubles and Make a Ten. Subtracting by adding up continues this progression by helping students recognize that addition and subtraction are closely related operations.

Like the strategies featured in my previous posts, subtracting by adding up is not about learning another trick. It is about helping students recognize important mathematical relationships so they can solve problems with increasing flexibility, efficiency, and understanding.

A Second Grade Lesson I'll Never Forget

One of my favorite examples of subtracting by adding up happened during a second-grade lesson built around a Compare, Smaller Unknown story problem. Instead of beginning with specific numbers, we first focused on making sense of the situation.

Students first made sense of the relationship between the quantities before assigning values to the story.

Before choosing numbers, we discussed a simple question:

Who completed more jumping jacks?

Students discussed which quantity represented the greater amount and how the two quantities were related. Once that relationship was clear, they chose numbers to complete the story.

After discussing the relationship, students selected values for the story before solving the problem.

One student chose 100 for Sam and 49 for "how many more."

As soon as he looked at the completed problem, Sam said,

"I don't know how to subtract across zeros."

Instead of explaining a procedure, I simply asked,

"What plus 49 equals 100?"

Without hesitation, he smiled and answered,

"51!"

In that moment, he had solved the subtraction problem without ever subtracting.

He didn't suddenly learn how to subtract across zeros. He already understood the relationship between 49, 51, and 100. The obstacle wasn't subtraction—it was believing subtraction had to be solved using a particular procedure. By reframing the problem as an unknown-addend situation, he immediately accessed mathematics he already understood.

That's the power of subtracting by adding up. Rather than relying on a procedure, students use what they already know about addition to solve subtraction problems efficiently.

But this way of thinking isn't something students invent in second grade. Its foundation is intentionally developed much earlier.

What Is Subtracting by Adding Up?

Subtracting by adding up is a subtraction strategy that treats subtraction as an unknown-addend problem.

Instead of solving:

• 13 − 9 = ?

  
  

students think:

• 9 + ___ = 13

  
  

Rather than thinking, "Take 9 away from 13," students think, "What do I add to 9 to make 13?" They might reason, 9 plus 1 is 10, and 10 plus 3 is 13, so the answer is 4. In doing so, they build on the same benchmark-number reasoning developed through Make a 10 while strengthening the relationship between addition and subtraction.

Rather than counting backward or relying on a subtraction procedure, students determine the amount needed to reach the larger number. In doing so, they strengthen one of the most important relationships in elementary mathematics—that addition and subtraction are inverse operations.

The Standard Behind Subtracting by Adding Up

The idea behind subtracting by adding up isn't simply a mental math strategy. It is embedded in Grade 1 mathematics.

Grade 1 Standard 1.OA.B.4 states:

Understand subtraction as an unknown-addend problem. For example, subtract 10 − 8 by finding the number that makes 10 when added to 8.

This standard broadens students' understanding of subtraction beyond simply "taking away." It helps them recognize that subtraction can also be viewed as finding an unknown addend.

This understanding is foundational to students' mathematical development. It helps them connect addition and subtraction as related operations, build fluency with related facts, solve a wider variety of story problems, and develop increasingly flexible computational strategies—including subtracting by adding up.

Story Problems Bring the Standard to Life

Students develop this understanding through the variety of addition and subtraction situations they encounter in Grade 1. Both the Common Core State Standards and the New Jersey Student Learning Standards for Mathematics expect students to solve Add To, Take From, Put Together/Take Apart, and Compare situations with all possible unknowns.

Addition and subtraction situations by grade level. This table, published by Achieve the Core, highlights the variety of addition and subtraction situations described in the Common Core State Standards and the New Jersey Student Learning Standards for Mathematics. By solving problems with different unknowns—not just Result Unknown situations—students develop the unknown-addend reasoning described in Standard 1.OA.B.4, which supports the development of subtracting by adding up.

Consider this Add To, Change Unknown problem:

Two bunnies were sitting on the grass. Some more bunnies hopped over. Then there were five bunnies. How many bunnies hopped over to the first two?

Although this is an Add To situation, students can represent it with either:

• 2 + ___ = 5

  
  

or

• 5 − 2 = ___

  
  

Both equations describe the same mathematical relationship.

These experiences reinforce the understanding described in Standard 1.OA.B.4 by helping students see that subtraction can be used to find an unknown addend. As students solve Change Unknown and later Compare situations, they continue strengthening this way of thinking.

This understanding also sets the stage for viewing subtraction as finding the difference—or distance—between two numbers. Grade 2 is the first time the standards explicitly introduce the number line as a representation for reasoning about addition and subtraction. On a number line, students often discover that moving from 48 to 52 is more efficient than thinking about taking 48 away from 52. This shift from take-away to distance is one reason subtracting by adding up becomes such a powerful and flexible strategy.

 For 52 − 48, thinking about the distance from 48 to 52 requires only one jump of 4, while a take-away interpretation may involve removing 48. Viewing subtraction as finding the difference often leads to more efficient reasoning.

Extending Subtracting by Adding Up Beyond Basic Facts

As students become more confident in reasoning about number relationships, they extend the subtracting by adding up strategy beyond basic facts to solve problems with larger numbers. 

For 342 − 156, they may reason:

• 156 → 160 (+4)

• 160 → 200 (+40)

• 200 → 342 (+142)

• 4 + 40 + 142 = 186

  
  

Notice that students are no longer counting every number. They are using benchmark numbers, place value, and known relationships to efficiently determine the difference.

Another Step Toward Computational Fluency

If you've been following this computational fluency series, you've probably noticed a common theme. Although each strategy looks different on the surface, they all help students reason from what they already know. Each new strategy builds on understandings students have already developed rather than replacing previous ones.

Whether students are using Near Doubles, Making 10, subtracting by adding up, or later reasoning with place value through Partial Sums, computational fluency develops through connected mathematical ideas—not isolated procedures.

Computational fluency isn't about using one strategy for every problem. It is about selecting an efficient strategy based on the numbers involved. Subtracting by adding up is another example of how students develop flexibility by making sense of mathematics rather than memorizing procedures.

Building Connected Mathematical Thinking

Subtracting by adding up is much more than another subtraction strategy. It reflects students’ growing understanding that mathematics is a connected system of ideas. By recognizing the relationship between addition and subtraction, students build fact fluency, develop flexible reasoning, and learn to choose efficient strategies based on the numbers in front of them.

If you'd like to use the same Grade 2 story problem that sparked the classroom conversation in this post, I've shared it as a free download below. Simply complete this form to download the activity.  I'll also send occasional classroom activities, teaching tips, and standards-based mathematics resources to support meaningful mathematics instruction.