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.

Place value and number sense are foundational concepts on which others build over the years in mathematics. Some of the big ideas within place value include:

## Concept Progression Over Time

In Saskatchewan, our curriculum identifies the following ideas:

• In Kindergarten, children learn that counting tells us how many. The whole numbers are in a particular order and there are patterns in the way we say them that help us remember their order.
• In Grade 1, children understand place value in individual numbers – they look at 17 as a quantity. We can compare and order numbers.
• In Grade 2, children understand that the value of the digit depends on its location or place.
• In Grade 3, children consolidate their understanding that the place determines a number’s value.

## Ideas for Teaching Place Value

### Rekenreks, 5 and 10 Frames

Number sense is a foundation of place value. Relating numbers to ‘friendly’ 5 and 10 are key ideas that can move children past counting.

Try This – Use a rekenrek to show the following:

1. Representing numbers – how might children use these tools to represent 7? 3? How do they know?
2. Quick flash – flash a number of beads on a rekenrek and have children tell you what the number is. How do they know this is the number? Are they counting? Or comparing to the ‘friendly’ 5 or 10?
3. Model numbers in a number string – showing 4, then 5, then 6. Some children will see the pattern of 1 less than 5, 5, and 1 more than 5. You can then repeat with 3, 5, and 7.

Now try to think through these activities using 5 and 10 Frames and linking cubes to show numbers. How is this the same and different than using a rekenrek? There are a number of games and activities involving dot cards and 10 frames that can emphasize 10.

You can find out more on the Building Math Minds Rekenrek activities site.

### Subitizing

Subitizing is a foundational skill and occurs when children know that a number of objects is present without counting. Subitizing can occur with random displays of objects or dots, or patterned dots like you would see on a dice, dominoes or ten frames.

Try ThisBuilding Math Minds has a great site for subitizing games. You can find some ideas in this Evergreen Games Overview.

### 100s Chart

The hundreds chart is an important tool for children to see patterns in our number system. There are a number of games and activities that you can try to emphasize different math ideas.

Try This – There are a number of blogs and vlogs that teachers have created to highlight the 100’s chart. Buggy and Buddy does a good job curating ideas from a number of sources. You can also have children try to find the missing numbers on a 100’s chart to emphasize the patterns in our number system.

### Base 10 Blocks

Based 10 blocks are a foundational manipulative to help children understand our number system.

Try This – Go to Hand2Mind website and scroll down to view the lessons provided. These are organized by grade band so that you can find what might fit your students best. Use the base 10 blocks provided to try to work through some of these lessons

## Place Value Misconceptions

Misconceptions can be created by a mis-applied pattern, or incomplete understanding of number concepts. The following are some place value misconceptions that occur in Early Years, and some possible instructional strategies to address them.

Misconception: A number is a number, and does not represent a bundle of 10, 100, 1000 etc. objects regardless of its position in a number.

Example: 1 means one, so when it is placed in a number 17, it still represents one rather than 10.

What to do about it? Use the concrete to abstract continuum to represent 17:

1. Place value blocks or other counters, such as coffee stir sticks.
2. Arrow Cards
3. Find the digit on the 100’s chart

Misconception: Students represent numbers after 100 as they sound.

Example: Students think that the number after 100 is 1001, then 1002, 1003, etc.

What to do about it: Use a chart that goes beyond 100, have children fill in the next numbers after 100.

Misconception: The student orders numbers based on the value of the digits, instead of place value.

Example: 67>103 because 6 and 7 are bigger than 1 and 2.

What to do about it: Have students represent numbers using base 10 blocks and then write out expressions using > and < when comparing.

What to do about it: Have students show numbers on a number line to see which numbers are further from zero to the right.

Misconception: The student struggles with the teen numbers, as they are different from the pattern in other decades.

Example: Students may say “eleventeen” or may not understand that 16 is ten and six. They may also think that sixteen is 61 because we say the number six first.

What to do about it: Christina, The Recovering Traditionalist, has curated a number of games and ideas for addressing how to teach the teens.

## Having Fun with Math

Mathematics should be playful, and there are a number of games that can build fluency in mathematics.

### Combo-10

This game allows students to see how numbers fit together to make 10 using domino-like game pieces. It is for groups of 2 – 4 players.

Try This – Play with at least two people or groups. Each group needs 1 set of dominoes. Lay them face down. Each person/group draws 7. The rest are the draw pile.

• The player with the highest double (or most dots if there are no doubles drawn) plays first. A piece can be played if the number of dots on one side of the domino adds to 10 with a domino on the table. Doubles can be laid sideways, allowing more arms to grow.
• A wild card is a domino whose dots add to 10. If you play a wild card, you can play twice.

### Snap

Try This – One player has a stack of 10 cubes behind their back. ‘Snap off’ part of the stack and show the part that is remaining to your partner.

The partner tries to guess how many were snapped off and hidden from view. The unknown part is revealed.

Variations:

• Using more or fewer blocks in the stack.
• Breaking the 10 cubes apart and hiding some of them underneath an opaque glass or container.

## Race to 100

The goal of this game is to get to 100 first without going over.

Try This – Play the Game

Each player starts at 1. The first player uses a spinner or dice to generate a number. They can move up the 100s chart by their number of tens or ones until one player gets to 100 without going first.

Variations:

• Each player gets 6 turns. The closest to 100 without going over wins.
• Continue playing until a player lands exactly on 100. If the roll takes them over 100, they lose that turn.

### Math Swat

Flyswatter math combines the fun of moving and slapping with the chance to learn number recognition and solving math problems.

Creating the game board: The game board can be as small or as large as you would like and include the number range and type of numbers that you are working with in your classroom.

Try This – Play a Game with two lines of players. Each line has their own swatter.

• Counting: swat the numbers in order – in either direction.
• Number recognition: say the number and have learners swat the correct symbol.
• Counting and 1:1 correspondence: give a number of counters, blocks, etc – they count and then swat the number.
• Addition or subtraction facts: give the fact, swat the correct sum.
• Addition facts: give the sum and one addend, swat the missing part.
• Skip counting: swat the numbers as they count by 2s, 5s, etc.

## Using Technology in Mathematics

Technology can be used to enhance mathematics in a number of different ways:

### Place Value Online Games

As you know, not all online games are created equally! Sometimes, they are just online worksheet with little engagement. Sheppard Software is a site that encourage practice through play, including flexible thinking about place value.

Try This – Try playing one of the place value games, Underline Digit Value, on Sheppard Software.

### Interactive Whiteboards

These whiteboards all allow you and students to share thinking. They can include audio, pictures, and mark ups. Some apps are free, while others require a subscription.

Try This – Log into one of the interactive whiteboards below that you have not used before. Use the username and password provided on the sticky note!

### Interactive Manipulatives  – ICT Math

These interactive manipulatives can be used to explore math ideas. These tools are web-based and do not require a log in or download.

Try This – Go to the Arrow Cards tool in the “Teaching Tools” at ICT Math. You can show the value of numbers using arrow cards along with either rek-n-reks or base 10 blocks simultaneously. Show the value for 3299. What happens when you add one more ones digit?

### QR Code Scavenger Hunt

This teaching idea comes from Kristin Kennedy and is available free on Teachers Pay Teachers. It would be relatively easy to create your own based on this idea.