Suggestons to create sequences to teach strategies to solve multi digit addition and subtraction

This page includes instructional suggestions and sequences to help develop people's ablity to solve addition and subtraction problems mentally. Sequences to introduce different strategies, which everyone is capable of learning and using.

Prior to teaching or learning these strategies it is helpful for learners to have had experiences to develope their spacial abilities. Abilities to visualize the relationships of numbers. For example, relationships of different values, as represented on a number line or hundred chart as well as addition and subtraction with arrow math on a hundred chart. Also beneficial, is numerous experiences with single digit addition and subtraction. Instruction which has included deconstruction and reconstruction of values to add and subtract. For example:

Strategies like these can be taught with a general instructional sequence and a list of problems as will be described:

General instructional method


Periodically review:

To help learners remember and use the different strategies, a spiral approach that periodicaly returns to each presented strategy unexpectedly is helpful. Periodically randomly presenting a problem from each strategy to review.

The problems in the following strategies are provided as samples, which portions can be used across all grades. The order of the problems are not the only order in which strategies can be presented. Depending on the student's experiences with mathematics and their use of a strategy, the list may vary from the orders presented here.


Decide where to start, what problem to follow each problem with, and when to stop if the learners are not ready to move through a complete list.

Jumps of ten

Background: counting by tens.



Randomly select a number and have students call out ten more:



Using tens & adjusting

Background: counting by tens, making tens & left overs




Making jumps of ten backward

Background: counting by tens backwards, reducing or adding to tens with left overs. If no one uses the strategy of ten. Give problems from the jumps of ten list. When they are back on track, return to problems with nine.




Using known facts

Background: some basic facts


Initally help students learn the addition facts they struggle to remember, which an include most of the facts with an addend above five.


Counting on, counting back, adding on or removing

Background: Because addition and subtraction are related so are adding on and removing.



When numbers are closer together it is better to add.

When they are far apart it's better to work backward.

Structure your sequences with that in mind to help learn when to add or remove.


Subtraction with addition

Background: It is important to learn subtraction problems can be solved with addition strategies.





double dieStrategy to introduce doubles to young children:

Randomly say numbers and ask for their double.


Doubles, near doubles with addition, doubles with plus and minus

Double dominoBackground: doubles.




Mix them up:


Doubles and near doubles with subtraction

Background: See doubles and near doubles above.




Sliding differences

Adding 51 + 49 by adjusting each to 50 + 50, doesn't work the same way with subtraction.

Subtraction is the difference. 51 - 49. Visualize a number line. The difference between the numbers has to stay the same. Therefore, to keep them the same, a value of adjustment can be added or subtracted to both. For example: to make it easier for some to see, add one to both, or sliding both. 51 and 49; slide one to 52 - 50; which is easier to see a difference of 2.

Another: 52 - 34. Could add 6 to each; 58 - 40, which is easier to see a difference of 18. You can use a number line to model how the differeence between 52 and 34 stays the same (18) if the numbers are slid from 52 t0 58 and 34 to 40. can slide in either direction.

This strategy is Not for below third grade. This is a difficult strategy for children to understand. However for problems such as: 1436 - 188, adding 12, makes a much easier problem 1448 - 200,


Canceling out common amounts

Is the same as sliding differences. 120 - 109, cancel 100 from both makes 20 - 9; 11.

This can be modeled with a double number line to show that the difference between the numbers stay the same as the numbers are slid along the number line. Which, is the same as canceling the 100's from each to leave 20 - 9; 11 or the difference between 20 and 9 is 11. The same difference between 120 and 109.

20342 - 10012; 10330


Decompose, compose and create mental algorithms

Children usually decompose numbers left to right by place values.

28 + 44 is decomposed into

Learners will solve problems by decomposing by place values. It takes a lot of experience before the are able to understand and operationalize addition and subtraction with two or more digits, each with different place values. Eventually, they will understand to add 23 and 34, theymust operate on all numbers within their place values:

To help learners move to a more efficient strategy, you may need to encourage them not to decompose both numbers, but to select one to decompose and one to add on from.

In today's world with calculators and spread sheets the strategies presented here along with mental algorithms created with the use of decomposition, compositioni, and working from left to right can proved probably all the a person needs for everyday calculations. The beauty of the traditional algorithm isn't in using it when other strategies are more efficient it is in its elegancy of working in all situations, particularly in advanced mathematics.

Removing to friendly numbers by counting back

Background: friendly numbers are tens and fives


Subtract parts by removing amounts to work with friendly numbers.



Adjust to the next friendly number

Background: friendly numbers are tens and fives




Swapping digits

Background: This is one that probably will need modeling.




Commutative property of addition

Background: Commutative propery of addition. Commutative property of addition is the order in which a pair of addends is added does not affect the sum. For all real numbers a and b, a + b = b + a




Dr. Robert Sweetland's notes
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