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Mental Models for Adding and Subtracting
Addition and subtraction are not just procedures to memorize. Students can solve problems in many flexible ways by thinking about how numbers are related. In this topic, we look at mental models and strategies that help students make sense of addition and subtraction, rather than relying only on standard algorithms.
For addition, strategies such as doubling, making ten, compensation, and incrementing help students use number relationships to make problems easier. For example, a student might solve 8 + 7 by thinking “8 + 8 is 16, so 8 + 7 is one less, or 15.” Another student might solve 9 + 6 by making ten: “9 needs 1 more to make 10, so I can take 1 from the 6 and think 10 + 5.”
For subtraction, strategies such as number lines, adding up, and constant difference can help students think about the distance between numbers instead of only “taking away.” For example, 52 − 48 can be thought of as “How far is 48 from 52?” instead of requiring a standard subtraction algorithm. These strategies help students build number sense, choose efficient methods, and explain their thinking clearly.
As future teachers, it is important to recognize that different strategies make sense for different problems. The goal is not to force every student to use the same method, but to help students develop a toolbox of strategies and learn when each one is useful.
Student Learning Goals
By the end of this topic, students should be able to:
Describe several mental strategies for addition and subtraction.
Explain how doubling, making ten, compensation, and incrementing can make addition easier.
Use number lines, adding up, and constant difference to solve subtraction problems.
Choose an efficient strategy based on the numbers in a problem.
Compare different solution methods for the same problem.
Explain how mental strategies support number sense.
Discuss how teachers can help children move from counting strategies toward more flexible reasoning.
Key Vocabulary
Mental Math - Solving problems by reasoning about numbers, often without writing a standard algorithm.
Doubling - Using a known double, such as 6 + 6, to solve a nearby fact.
Making Ten - Rearranging numbers to create a group of 10.
Compensation - Adjusting one number to make the problem easier, then correcting the answer if needed.
Incrementing - Adding or subtracting in smaller, convenient steps.
Transformation - Changing a problem into an equivalent or easier problem.
Number Line - A visual model that shows numbers and distances between them.
Adding Up - Solving subtraction by counting or adding from the smaller number to the larger number.
Constant Difference - Changing both numbers in a subtraction problem by the same amount so the difference stays the same.
Common Student Misunderstandings
If a student thinks mental math means “doing it in your head with no strategy,” then the student may not realize mental math is based on reasoning and structure.
If a student believes there is only one correct way to add or subtract, then the student may need more experience comparing strategies.
If a student Uses compensation but forgets to adjust the answer, then the student may understand part of the strategy but not how to keep the value balanced.
If a student thinks subtraction always means taking away, then the student may need experience with comparison, distance, and missing-addend situations.
If a student misuses constant difference by changing only one number, then the student may not yet understand why both numbers must change by the same amount.
Example Strategies
Doubling - 7 + 8 - “7 + 7 is 14, so one more is 15.”
Making Ten - 9 + 6 - “Move 1 from 6 to 9. That gives 10 + 5 = 15.”
Compensation - 39 + 24 - “Think 40 + 24 = 64, then subtract 1. The answer is 63.”
Incrementing - 46 + 28 - “Add 20 to get 66, then add 8 to get 74.”
Transformation - 25 + 19 - “Change it to 24 + 20, which is 44.”
Number Line - 63 − 27 - “Start at 63, jump back 20 to 43, then back 7 to 36.”
Adding Up - 52 − 48 - “48 to 50 is 2, and 50 to 52 is 2, so the difference is 4.”
Constant Difference - 83 − 49 - “Add 1 to both numbers: 84 − 50 = 34.”
Teacher Connection
Mental strategies help students see numbers as flexible and connected. A student who solves 49 + 36 by thinking “50 + 35” is showing strong number sense. A student who solves 72 − 68 by adding up from 68 to 72 is recognizing subtraction as a difference, not just a take-away action.
As a future teacher, you will want to ask students to explain their thinking, not just give an answer. These strategies can also help you notice whether students understand place value, properties of operations, and number relationships.
Helpful teacher questions include:
What made you choose that strategy?
Could another strategy work here?
How did you keep the value the same?
Where do you see the numbers changing?
Which method feels most efficient for this problem?
Quick Reflection Question
Why might 52 − 48 be easier to solve by adding up instead of using the standard subtraction algorithm? What does this reveal about the meaning of subtraction?