Wednesday, January 23, 2019

Mental Addition and Subtraction Strategies

Have you ever met someone who could do amazing computations in their head, faster than you could even pull out a calculator?

Conversely, have you seen older students or adults who still have to add and subtract by counting up or down from the starting number?

One way Singapore Math differs from other curricula is that it takes what people who intuitively figure out how numbers work and can calculate mentally, and makes it explicit so every student can use these same strategies.

The approaches begin in early grades, such as Kindergarten and first grade, where children go beyond knowing the counting sequence and learn the different ways any number within 10 can be decomposed. For example, they practice with:
5 is 4 and 1
5 is 1 and 4
5 is 3 and 2
5 is 2 and 3
These are first introduced with decomposing with objects, and then those are related to number bonds with objects, both in concrete and picture form, then to number bonds with pictures, and finally to equations, first in word form as above, then with symbols.
When number bonds are introduced, part-whole thinking can develop, leading to ease with both composing (adding) and decomposing (subtracting). Both the part-whole models and bar models build upon that.

Fluency with these compositions within ten provides the foundation for all the subsequent computations. For example, if each subsequent decade is understood as 10 or a multiple of 10 plus one of the known compositions, it’s a simple matter to apply the same patterns. No longer is each number a discrete entity; each is made up of the same type of puzzle piece as the ones they have already learned.

So students learn to apply the rules they learned to compose and decompose with larger place values, using different manipulatives to become acquainted with the values of the different digits and how they work together. This leads to more complex strategies starting in second grade, such as addition and subtraction compensation.

Addition Compensation

One form of addition compensation is the “make a ten” strategy. This is where one addend is decomposed to make a ten, or a multiple of ten, to make adding easier. Some examples of strategies are the number bond decomposition and the “arrow way,” or step-by-step approach. For example:

Subtraction Compensation

This depends on the understanding of subtraction as finding the missing part of a whole and as a difference. The strategies are a bit different here. Number bonds cannot be used in the same way; instead, the students learn that to find the difference, the same number can be added to both minuend and subtrahend, and the difference will stay the same. Two examples here are making a ten out of the subtrahend, and step-by-step subtraction.

There are so many other strategies and so many ways to use these strategies. Sharing solution strategies can be a great teaching tool. Both my students and I have gained flexibility and ease with numbers the more we practice with these. We can also mentally check our answers if we use an algorithm or a calculator, always checking, “Does that make sense?”

Do you use a similar or alternate strategy? Try these out or share your own!
Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at

The Matholia Team

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