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.

# 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

1. Predict what strategies they might use. Order these from least to most complex.
2. Observe students doing mathematical tasks – using white boards allows us to see their thinking. We can then identify different strategies being used.
3. 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.

## 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.

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

• 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.

### 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

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.

Patterns are everywhere. Exploring and identifying patterns can help children understand our number system, operations, spatial understanding and the foundations of algebra. Mathematics is the study of patterns and exploring them through play can begin mathematical and algebraic thinking in early years. Click here for a downloadable version of this post.

There are several big ideas related to patterns:

1. Patterns exist and occur regularly in the natural and man-made world.
2. Patterns can be recognized, extended and generalized using words and symbols.
3. The same pattern can be found in many different forms – physical objects, sounds, movements and symbols.

The progression of patterns through Saskatchewan Curricula:

When viewing patterns, it is useful to know the following terms:

• Element – an action, object, sound or symbol that is part of a sequence.
• Core – the shortest string of elements that repeats.
• Pattern – a sequence of elements that has a repeating core.

Children will develop their ability to recognize and manipulate patterns differently. Some children will move through the following progression:

Exploring patterns also gives children practice and exposure to other mathematical ideas, including:

• Counting and cardinality – counting the number of items in the unit of a repeating pattern, or how many items are added in an increasing pattern.
• Adding and subtracting – generalizing about an increasing or decreasing pattern – how many more or less.
• Position and spatial properties – which element comes next, which element is between two others, reversing order of elements.

## How might you teach patterns?

As with many mathematical ideas in early years, it is important to connect ideas. Learning is not linear! It is important that children use physical materials from their environment to build and explore patterns rather than relying on drawing and colouring patterns. Buttons, toys, linking cubes and natural materials can all be used to create patterns.

The Measured Mom has a list of fun ways to engage young children in exploring patterns. It is fun to take children outside. Megan Zeni describes how you might have children explore Patterns Outside and in Nature.

## Repeating Patterns

Repeating patterns can be introduced using concrete objects, sounds, body movements or symbols. Exploring with a variety of materials can help children identify what is creating a pattern.

Pattern Strips can be made using any shape or object. Students can work independently or in groups to copy the pattern on a strip using real objects. These patterns can then be extended. Watch whether they are copying each element separately or if they have identified the core of the pattern and are able to place all of the elements of the core at the same time. This might look like:

• If the pattern is red/blue/red/blue – children will place the red and blue at the same time.

A significant step in understanding patterns is when children are able to identify that the same pattern exists even when the materials are different. Using some type of symbol, children are able to code a pattern and compare it to other patterns. If they choose to code the pattern using the alphabet, they might describe it as A-B-A-B or A-A-B-A-A-B. An extension with pattern strips is to create the same pattern with different materials.

Pattern Match can happen in many forms.

• You can give each group a set of different pattern strips, and they find which strips are showing the same pattern.
• Children can work in groups, one child is the pattern caller. They choose 3-4 pattern strips and lay them face up on their table. They then ‘secretly’ choose one of the strips and calls out the pattern code. Their group members try to identify which strip is being read.

## Growing Patterns

In Saskatchewan, children begin to explore increasing patterns in grade 2, and decreasing patterns starting in grade 3.  The beginning of understanding growing patterns is for children to experience building them with concrete objects.

It is important for children to record their observations. A table can help students record the number for each step in the pattern. Using a table, students can predict how many items are needed to create a certain step in the pattern.

## Patterns with Numbers

Number patterns are woven throughout our number system, how we perform operations and the ways we represent numbers. John Van de Walle and LouAnn H. Lovin (Teaching Student-Centered Mathematics K-3, 2006) have created a number pattern activity that has students identify how a number string continues by identifying the pattern present.

### Skip Counting

Skip counting is an excellent source of patterns. We often limit skip counting to small numbers like 2, 3 or 5. We also often start skip counting at 0. Children can explore the patterns that are created when we skip count by larger numbers, changing the start number. It is a great idea to use a calculator, so students don’t get bogged down in computation!

• If we start at 7 and skip count by 5’s, what pattern do we see?
• 7, 12, 17, 22, 27, 32…
• If we start at 7 and skip count by 55’s, what pattern do we see?
• 7, 62, 117, 172, 227, 282, 337, …
• What do we notice about these two patterns?

### Patterns in the Hundreds Chart

#### Start and Jump on the 100s Chart

Using a hundreds chart, have students colour in the pattern created by one of the Start and Jump Numbers sequences given. If different students represent different patterns, what do they notice?

• How do patterns change when
• the start number changes?
• The jump number changes?
• Which skip count numbers create columns?

#### Let’s Build the 100’s Chart

• Using a pocket 100s chart or interactive 100’s chart. Place the following number cards in the pockets:
• 4, 10, 17, 32, 48
• Gather students so they can see the 100’s chart easily.
• Hold up a number related to one of those in the chart, such as 18. Ask “who would like to place this number?”. Explain how you know where to put it.
• Choose numbers to place based on the number concepts you are working on:
• If you are working on adding 10, choose numbers that emphasize that concept.
• If you are working on skip counting, choose numbers that emphasize what you are counting by.

#### Game: Arrow Clues

• Clues can be created on cards or written large enough for all players to play the same clues.
• Arrow clues can look like:
• Differentiation:
• Students can play with or without a 100s chart to refer to.
• Have students describe the impact of each of the types of arrows on the VALUE of the number.

#### Missing Number Puzzles

Using the patterns in the 100’s chart, children can figure out the missing numbers when only a part of the 100s chart is provided.