Addition, subtraction, multiplication and division are foundational skills that are applied to many mathematical concepts. Often, when we are hoping for student automaticity and fluency in numbers, number operations are what we are talking about.

Mathematical Models

Models are the way we are representing numbers so that we can do number operations. There are a number of different models that are helpful to students understanding number operations.

Models that Emphasize Partitioning Number

The Importance of Partitioning Numbers

Regardless of what number operation we are talking about, it is important that children are able to break numbers into parts.

Friendly Numbers – children are often able to understand number operations with ‘friendly’ numbers like 2, 5, and 10. Breaking a 7 into a and a 2 allows us to use number facts that are more familiar.

Place Value Partitioning – when we are working with multi-digit numbers, it is helpful for us to break numbers up into the values of their digits – for example, 327 is 300 + 20 + 7.

Number Operation Strategies

There are many different strategies that children use to perform number operations. A misconception is that all children need to know and use all strategies. It is important for us to expose children to different strategies through classroom discussion and routines such as number talks and number strings. When combined with Margaret Smith’s ideas around Orchestrating Classroom Discussion, we can set a task for students and

Predict what strategies they might use. Order these from least to most complex.
Observe students doing mathematical tasks – using white boards allows us to see their thinking. We can then identify different strategies being used.
Have students share their thinking in an order from least to most complex. This should not include every child sharing for every task. A small handful of children sharing in a logical order can help students understand the next more complex solution. In this way, children are being exposed to other strategies, will be able to understand those that are close to their own, and increase the sophistication of their thinking.

Strategies	Connection to Addition	Connection to Multiplication
Counting: This is a common strategy when one of the numbers is small.	Addition by counting or counting on from one number. Ex: 25 + 7 = 25, 26, 27, 28, 29, 30, 31, 32.	Skip counting by one of the numbers being multiplied. 9 x 5 = 9, 18, 27, 36, 45
Decomposing Numbers: breaking numbers apart.	Adding friendly numbers. Ex: when you need to add 12, breaking it into +10 and then +2 more. Making 10. Ex: when adding 5 + 7, recognizing that 5 + 5 = 10, and so it is 10 + 2 more = 12. Breaking one or both numbers into place value. Ex: 23 + 47 is 20 + 40; 3 + 7	Multiplying friendly numbers. Ex: when you need to multiply by 6, break it into x 5 and 1 more. Partial Products: Breaking one or both numbers into place value. Ex: 23 x 47 is (20 + 3) x (40 + 7)
Compensation: this is very common when a number is close to 10.	Rounding one of the numbers to a friendly number, then compensating the answer at the end for the difference. Ex: 36 + 9 is close to 36 + 10, subtract 1. Ex: 36 + 11 is close to 36 + 10, add 1.	Rounding one of the numbers to a friendly number, then compensating the answer at the end for the difference. Ex: 99 x 5 is close to 100 x 5, subtract 5 Ex: 101 x 5 is close to 100 x , add 5
Double/Half	Recognizing that 4 + 4 is double 4, or 8. Recognizing that 4 + 3 is almost double 4, subtract 1.	Recognizing that 5 x a number is the same as ½ of 10 x a number. Ex: 9 x 5 is half of 9 x 10 = 45
Standard Algorithm	Traditional algorithm, symbolic regrouping.	Traditional algorithm, symbolic regrouping.

A Bridge between Addition and Multiplication: Doubles

Doubles are one way to think about adding a number to itself, as well as the start to multiplicative thinking.
Doubles are an important bridge between adding and multiplication.
You can read more about teaching doubles here.

Addition and Subtraction

Addition is the bringing together of two or more numbers, or quantities to make a new total.

Sometimes, when we add numbers, the total in a given place value is more than 10. This means that we need to regroup, or carry, a digit to the next place. There is a great explanation of regrouping for addition and subtraction on Study.com.

Subtraction is the opposite operation to addition. For each set of three numbers, there are two subtraction and one addition number facts. These are called fact families. For example:

For the numbers 7, 3, 10:

7 + 3 = 10

10 – 3 = 7

10 – 7 = 3

Fact families can be practiced using Number bonds or Missing Part cards.

As we move from single digit to multi-digit addition and subtraction, it is important that we maintain place value, and continue to move through the concrete to abstract continuum.

A helpful progression for teaching addition and subtraction can be found on the Math Smarts site.

Multiplication and Division

Conceptual Structures for Multiplication

Repeated Addition

This is the first structure that we introduce children to.
It builds on the understanding of addition but in the context of equal sized groups.

Rectangular Array/Area Model

This is often the second representation of multiplication introduced It is useful to show the commutative property that 3 x 4 = 4 x 3 = 12

Number Line

A number line can represent skip counting visually.

Scaling

Scaling is the most abstract structure, as it cannot be understood through counting.
Scaling is frequently used in everyday life when comparing quantities or measuring.

Single Digit Multiplication Facts

Multiplication facts should be introduced and mastered by relating to existing knowledge. If students are stuck in a ‘counting’ stage – either by ones or skip-counting to know their single-digit multiplication facts, it is important that they understand strategies beyond counting before they practice. Counting is a dangerous stage for students, as they can get stuck in this inefficient and often inaccurate stage. Students should not move to multi-digit multiplication before they understand multiplication strategies for single-digit multiplication.

It is important that students understand the commutative property 2 x 4 = 8 and 4 x 2 = 8.
2 x 4 should be related to the addition fact 4 + 4 = 8, or double 4.
Using a multiplication table as a visual structure is helpful to see patterns in multiplication facts.

Mental Strategies Continuum

Same as (1 facts)
Doubles. (2 facts)
Doubles and 1 more (3 facts)
Double Doubles (4 facts)
Tens and fives (10, 5 facts)
Relating to tens (9 facts)
Remaining facts (6, 7, 8 facts)

Conceptual Structures for Division

Equal Grouping

In an equal grouping (quotition) question, the total number are known, and the size of each group is known.

The unknown is how many groups there are.

Equal Sharing

In an equal sharing (partition) question, the total number are known, and the number of groups is known.

The unknown is how many are in each group.

Number Line

Ratio

This is a comparison of the scale of two quantities and is often referred to as scale factor. This is a difficult concept as you can’t subtract to find the ratio.

Division Facts

Relate division facts back to multiplication facts families:

Ex) 6 x 8 = 48

8 x 6 = 48

48 ÷ 6 = 8

48 ÷ 8 = 6

Once students have understanding and fluency with single digit multiplication and division fact families they are ready to move on to multi-digit fact families.

So What do Students DO with Number Operations?

Simple computation is not enough for children to experience. They need to have opportunities to explore and wonder about numbers and how they work together. Regardless of the routine or task, children should be encouraged to use different concrete and pictorial models to show their thinking.

Some examples of rich interactions include:

Number Talks

Number talks promote classroom discussion. Combining number talks with visual or concrete models can help us see what students are thinking.

Number Strings

Number strings can help children see the pattern in number operations. They are helpful for children to see the pattern in number operations, which is the foundation for algebraic thinking.
You can see the structure for building number strings here.

SPLAT

SPLAT encourages both additive thinking and subitizing. More complex SPLAT lessons are also great for encouraging algebraic thinking with unknowns.

Problem Stories

Building problem stories are powerful for children to understand contexts of mathematics in their every day life.
Using real objects or pictures encourages children to see math in their environments.

Invitations

Invitations that are created to inspire mathematical thinking encourage exploration, vocabulary building and wonder.

Games and Puzzles

There are so many games and puzzles that can have children play with number operations.

Open Middle Problems

Open middle problems allow for flexible thinking and exploration. You can see a sample here.

Teaching Number Operations