Primary Resources: Maths: Calculations: Multiplication & Division

Multiplication & Division: A Comprehensive Guide

Multiplication & Division:

Explore the fundamental concepts of multiplication and division, their properties, methods, and real-world applications. This guide provides detailed explanations, examples, and mathematical expressions to enhance your understanding.

General Multiplication & Division

Multiplication and division are two of the four basic arithmetic operations, alongside addition and subtraction. Understanding these operations is crucial for solving a wide range of mathematical problems, from simple calculations to complex equations.

Multiplication is Commutative

The commutative property of multiplication states that changing the order of the numbers being multiplied does not change the product. Mathematically, this is expressed as:

\[ a \times b = b \times a \]

Example 1: Commutative Property

6 × 7 = 7 × 6 = 42

This property simplifies calculations and allows flexibility in solving multiplication problems.

Multiplication (Arrays)

Arrays are a visual representation of multiplication, showing how groups of objects can be arranged in rows and columns. This method helps in understanding the concept of multiplication as repeated addition.

Example 2: Multiplication Using Arrays

To calculate 3 × 4 using arrays, arrange 3 rows of 4 objects:

\[ 3 \times 4 = 12 \]

This shows that 3 groups of 4 equal 12.

Multiplication and Division Rules

Understanding the rules governing multiplication and division is essential for accurate calculations. These rules include the properties of operations, order of operations, and the relationship between multiplication and division as inverse operations.

\[ \text{If } a \times b = c, \text{ then } c \div b = a \text{ and } c \div a = b \]

Example 3: Multiplication and Division Inverses

If 5 × 8 = 40, then:

\[ 40 \div 8 = 5 \quad \text{and} \quad 40 \div 5 = 8 \]

Multiplication Facts (Derivation)

Multiplication facts, often referred to as multiplication tables, are foundational for efficient arithmetic. Mastery of these facts allows for quicker calculations and forms the basis for more advanced mathematical concepts.

Example 4: Multiplication Fact

7 × 9 = 63

Inverse and Division

Division is the inverse operation of multiplication. Understanding this relationship helps in solving equations and performing complex calculations.

\[ a \div b = c \quad \text{if and only if} \quad a = b \times c \]

Example 5: Inverse Relationship

If 56 ÷ 7 = 8, then 7 × 8 = 56

Multiplication and Division Investigation

Investigating multiplication and division involves exploring patterns, properties, and relationships between numbers. This deepens understanding and enhances problem-solving skills.

Example 6: Pattern Exploration

Consider the pattern of multiplying by 5:

\[ 5 \times 1 = 5 \\ 5 \times 2 = 10 \\ 5 \times 3 = 15 \\ \dots \]

Notice the consistent increase by 5 each time.

Multiplication Grids Starter

Multiplication grids are tools that help visualize multiplication tables and understand the relationships between different factors.

Example 7: Multiplication Grid

Here is a partial multiplication grid:

1 2 3 4 5
1 1 2 3 4 5
2 2 4 6 8 10
3 3 6 9 12 15

Repeated Addition / Repeated Subtraction

Multiplication can be viewed as repeated addition, and division as repeated subtraction. This perspective helps in understanding the operations more intuitively.

Repeated Addition as Multiplication

When you add the same number multiple times, it is equivalent to multiplying. For instance:

Example 8: Repeated Addition

4 + 4 + 4 = 12 is the same as 4 × 3 = 12

Multiplication as Repeated Addition

Multiplication simplifies the process of adding the same number repeatedly by providing a concise notation.

Example 9: Simplifying Addition

Instead of adding 5 five times: 5 + 5 + 5 + 5 + 5, you can write 5 × 5 = 25

Division Using a Number Line

Visualizing division on a number line helps in understanding how a number can be split into equal parts.

Example 10: Division on a Number Line

To divide 20 by 4, start at 0 and make 4 equal jumps to reach 20. Each jump represents 5:

\[ 20 \div 4 = 5 \]

Division by Repeated Subtraction

Division can be understood as subtracting the divisor repeatedly from the dividend until what remains is less than the divisor.

Example 11: Repeated Subtraction

Divide 15 by 3:

\[ 15 - 3 = 12 \\ 12 - 3 = 9 \\ 9 - 3 = 6 \\ 6 - 3 = 3 \\ 3 - 3 = 0 \]

Subtracted 3 five times, so 15 ÷ 3 = 5

Multiplication Build Up

Building up multiplication involves gradually increasing the multiplicands to understand the scaling effect of multiplication.

Example 12: Building Up

Start with 2 × 2 = 4, then 2 × 3 = 6, and so on, observing the pattern of increasing by 2 each time.

Changing Adding to Multiplication

Transforming addition into multiplication simplifies calculations, especially when dealing with large numbers.

Example 13: Transformation

Instead of adding 7 + 7 + 7 + 7, write 7 × 4 = 28

Division as Sharing

Division can be conceptualized as sharing a quantity equally among a given number of groups. This interpretation is particularly useful in real-world scenarios like distributing resources.

Division by Sharing Equally

When you divide a number by another, you are essentially sharing it equally among the specified number of groups.

Example 14: Sharing Equally

If you have 24 candies and want to share them equally among 6 friends, each friend gets:

\[ 24 \div 6 = 4 \text{ candies} \]

Division by 3

Dividing by 3 involves splitting a quantity into three equal parts.

Example 15: Division by 3

18 ÷ 3 = 6

Each group receives 6 items.

Sharing Counters

Using counters as a visual aid can help in understanding how division distributes items into equal groups.

Example 16: Using Counters

Distribute 20 counters into 5 equal groups:

\[ 20 \div 5 = 4 \]

Each group has 4 counters.

Simple Division Questions

Practicing simple division problems builds proficiency and confidence in handling more complex calculations.

Example 17: Simple Division

30 ÷ 5 = 6

Division 'Rounding Remainders'

Sometimes division results in a remainder. Understanding how to handle and round these remainders is essential in various applications.

Example 18: Rounding Remainders

25 ÷ 4 = 6 with a remainder of 1. Rounded up, it becomes 7.

Sharing Zoo

Using scenarios like sharing items in a zoo helps in contextualizing division problems, making them relatable and easier to grasp.

Example 19: Sharing in a Zoo

If there are 12 apples to be shared equally among 3 monkeys, each monkey gets:

\[ 12 \div 3 = 4 \text{ apples} \]

Making Groups

Creating groups is another way to visualize division, ensuring that each group has an equal number of items.

Example 20: Making Equal Groups

Divide 16 marbles into 4 equal groups:

\[ 16 \div 4 = 4 \]

Each group has 4 marbles.

Division as Sharing 1 & 2

Further exploration of division as sharing reinforces the concept through multiple examples and applications.

Example 21: Advanced Sharing

Share 45 stickers equally among 9 students:

\[ 45 \div 9 = 5 \text{ stickers per student} \]

Division as Grouping

Division can also be seen as grouping items into a specific number of groups, emphasizing the distribution aspect of the operation.

Dividing Using Groups

This method involves dividing a total number of items into a specified number of groups, each containing an equal number of items.

Example 22: Grouping

Divide 28 books into 4 groups:

\[ 28 \div 4 = 7 \text{ books per group} \]

Dividing by Grouping with Remainders

When the total number of items doesn't divide evenly into groups, a remainder exists.

Example 23: Grouping with Remainders

Divide 22 candies into 5 groups:

\[ 22 \div 5 = 4 \text{ R } 2 \]

Each group has 4 candies, with 2 candies remaining.

Division by Grouping

This approach focuses on dividing a set of items into a number of groups, ensuring each group has an equal number of items.

Example 24: Equal Grouping

Divide 36 pencils into 6 groups:

\[ 36 \div 6 = 6 \text{ pencils per group} \]

Division with Remainders

Sometimes, division does not result in an exact quotient, leaving a remainder. Understanding how to handle remainders is vital for accurate calculations.

Division Word Problems with Remainders

Word problems involving division with remainders often appear in real-life contexts, such as distributing items or scheduling tasks.

Example 25: Word Problem with Remainder

Jane has 50 apples and wants to distribute them equally among 6 baskets. How many apples will each basket contain, and how many will remain?

\[ 50 \div 6 = 8 \text{ R } 2 \]

Each basket will have 8 apples, with 2 apples remaining.

Remainders as Fractions and Decimals

Remainders can be expressed as fractions or decimals to provide a more precise division result.

Example 26: Remainder as Fraction

7 ÷ 3 = 2 R 1

\[ 2 \frac{1}{3} \text{ or } 2.333... \]

Rounding Up and Down After Division

Rounding the result of a division operation can simplify the answer, especially when precision is not critical.

Example 27: Rounding

19 ÷ 4 = 4.75

Rounded up, it becomes 5. Rounded down, it is 4.

Simple Divisions with Remainders

Practicing simple divisions that result in remainders enhances understanding and calculation skills.

Example 28: Simple Division with Remainder

23 ÷ 5 = 4 R 3

Weetabix Division with Remainders

Applying division with remainders in practical scenarios, such as distributing food items, makes learning more relatable.

Example 29: Distributing Weetabix

Distribute 17 Weetabix biscuits into 4 bowls:

\[ 17 \div 4 = 4 \text{ R } 1 \]

Each bowl gets 4 biscuits, with 1 remaining.

Division Problems with Rounding

Solving division problems that require rounding ensures answers are in a practical and usable form.

Example 30: Rounding Division

52 ÷ 7 = 7.428...

Rounded to the nearest whole number, it is 7.

Word Problems with Remainders

Engaging with word problems that include remainders enhances problem-solving skills and application of division concepts.

Example 31: Word Problem

Tom has 31 marbles and wants to divide them equally among 5 friends. How many marbles does each friend get, and how many are left?

\[ 31 \div 5 = 6 \text{ R } 1 \]

Each friend gets 6 marbles, with 1 marble remaining.

Word Problems Involving Multiplication & Division

Problem 1: Calculating Total Items

Sarah buys 8 packs of pencils. Each pack contains 12 pencils. How many pencils does Sarah have in total?

Solution:

To find the total number of pencils, multiply the number of packs by the number of pencils per pack:

\[ 8 \times 12 = 96 \]

Sarah has 96 pencils in total.

Problem 2: Sharing Equally

A teacher has 45 apples and wants to distribute them equally among 9 students. How many apples does each student receive?

Solution:

To determine how many apples each student gets, divide the total number of apples by the number of students:

\[ 45 \div 9 = 5 \]

Each student receives 5 apples.

Problem 3: Combined Operations

A factory produces 150 widgets each day. How many widgets are produced in a week (7 days)?

Solution:

Multiply the number of widgets produced each day by the number of days in a week:

\[ 150 \times 7 = 1050 \]

The factory produces 1,050 widgets in a week.

Problem 4: Fractional Division

If 64 candies are divided equally into 8 bags, how many candies are in each bag?

Solution:

Divide the total number of candies by the number of bags:

\[ 64 \div 8 = 8 \]

Each bag contains 8 candies.

Problem 5: Distributing Resources

There are 100 books to be placed on 5 shelves equally. How many books should be placed on each shelf?

Solution:

Divide the total number of books by the number of shelves:

\[ 100 \div 5 = 20 \]

Each shelf should have 20 books.

Problem 6: Grouping for Events

At a party, there are 48 balloons to be grouped into sets of 6. How many groups will there be?

Solution:

Divide the total number of balloons by the group size:

\[ 48 \div 6 = 8 \]

There will be 8 groups of balloons.

Problem 7: Calculating Perimeter

A rectangle has a length of 7 units and a width of 3 units. Calculate its perimeter using multiplication and division.

Solution:

The perimeter \( P \) of a rectangle is calculated as:

\[ P = 2 \times (length + width) \]

Substituting the given values:

\[ P = 2 \times (7 + 3) = 2 \times 10 = 20 \text{ units} \]

The perimeter is 20 units.

Problem 8: Scaling Recipes

A recipe requires 3 cups of flour for 4 servings. How much flour is needed for 10 servings?

Solution:

First, find the amount of flour per serving:

\[ \frac{3 \text{ cups}}{4 \text{ servings}} = 0.75 \text{ cups per serving} \]

Then, multiply by the number of servings:

\[ 0.75 \times 10 = 7.5 \text{ cups} \]

7.5 cups of flour are needed for 10 servings.

Problem 9: Dividing Money

Linda has \$120 and wants to divide it equally among 8 friends. How much does each friend receive?

Solution:

Divide the total amount by the number of friends:

\[ 120 \div 8 = 15 \]

Each friend receives \$15.

Problem 10: Distributing Seats

A theater has 360 seats divided equally into 12 rows. How many seats are in each row?

Solution:

Divide the total number of seats by the number of rows:

\[ 360 \div 12 = 30 \]

Each row has 30 seats.

 

General Multiplication & Division:

  • Multiplication is Commutative (Margaret Carr) MS Powerpoint
  • Multiplication (Arrays) (LT) MS Powerpoint
  • Multiplication and Division Rules (Nicola Edwards) MS Powerpoint
  •  T1 U9 Multiplication (David Arthur) ActivStudio
  • Division – Repeated Subtraction (Andrew Woodcock) MS Powerpoint
  • Division Snap (Nadine Turner) PDF
  • Multiplication Facts (Derivation of)
  • Gingerbread Multiplication (Sheena Florey) PDF
  • Toes Multiplication (Sheena Florey) PDF
  • Divisor Spiders Sheet 1 (Larissa Hughes) PDF Sheet 2 PDF
  • Arrays (Hamish Hobkinson) PDF
  • Function Machines (Mark Stallwood) PDF
  • Division (Amy Hedges) Bottom PDF – Middle PDF – Top PDF
  •  Division Problems (Elaine Smith) DOC
  • Dividing Money by 4 & 5 (Joanne Gordon) DOC
  • Division Dominoes Game (Ashley Staniforth) Rules DOC – Dominoes DOC – Small DOC
  •  Mental Methods Revision (Louise Pickering) DOC
  • Sums Square Multiplication Bingo (Campbell Airlie) 
  •  Inverse and Division (C. Williams) DOC
  • Division Dominoes (Joanne Nalton) DOC
  •  Division Machines (Vicky Dowding) PDF
  • Choose the Product Games (Vicki Foy) PDF
  •  Commutative Multiplication (Peter Smith) DOC
  •  Division (by 2, 3, 5, 10) (Mark Wilson) DOC
  •  Straight Division (Michelle Culliford) PDF
  •  Straight Division 2 (Michelle Culliford) PDF
  •  Using Known Facts ( BlkA U2) (Brenda Vaughan) DOC
  • Multiplication and Division Investigation (Kate Lowndes) DOC
  • Exploring Patterns in Linked Division Calculations (Louise Forster) MS Powerpoint
  • Exploring Patterns in Linked Division Calculations (Louise Forster) DOC
  •  Multiplication Grids Starter (Paula Alty) MS Powerpoint
  •  Division Grids Starter (Paula Alty) MS Powerpoint
  •  Division Grids with Remainders Starter (Paula Alty) MS Powerpoint
  •  Cooking Multiplication (Robert Bentall) DOC
  •  Division Picture (Victoria Adams) DOC
  •  Division Triangle Jigsaw (Peter Barnett) PDF
  •  Division Hexagon Jigsaw (Peter Barnett) PDF
  •  Missing Numbers: Multiplication & Division (Dhipa Begum) DOC
  •  Multiplication Word problems (x2, x5, x10) (Cindy Shanks) DOC
  • Division Duck (4x) (Sarah Dickens) MS Powerpoint
  •  Mental Multiplication Using Factors (Gemma Finbow) MS Powerpoint
  • Quick Multiplication Question Generator (R. Lovelock) MS Powerpoint
  • Multiplication “Jeopardy” Game (Helen Newton) MS Powerpoint
  • Multiplication Problems (Sheila Black) PDF
  • Multiplying two digits (Sheila Black) PDF
  • Multiplication (Ian Mason) Sheet 1 PDF – Sheet 2 PDF
  • Multiplication (Ian Mason) Sheet 3 PDF – Sheet 4 PDF
  • Factor Spiders (Larissa Hughes) Sheet 1 PDF Sheet 2 PDF
  • Division (Kevin Kerr) PDF
  • Division Spiders (Gareth Rein) DOC
  • Bear Multiplication (Judith Brayshaw) DOC
  • Multiplying and Dividing by 4 (Jon Fordham) DOC
  •  Multiplication & Division Term 1 Unit 2 (Fred Daynes) Day 1 MS Powerpoint Day 2 MS Powerpoint Day 4 MS Powerpoint Day 5 MS Powerpoint
  • Multiplication Worksheet Generator (Campbell Airlie) 
  •  Simple Multiplication Problems (C. Williams) DOC
  • Multiplication (Lots of) (Liz Hazelden) DOC
  •  Multiplication & Division Relationship (Rich Robinson)  
  •  Multiplying by 4, 5 and 20 (Richard Queripel) DOC
  • Multiplication Games (Vicki Foy) PDF
  •  Multiplication Questions (Anne Richard) DOC
  •  Multiplication & Division Methods Poster (Ali McNamara) DOC
  •  Y3 Division Test (Sharon Richard) DOC
  • Division Splat (Jim Usher) DOC
  • Associative Multiplication (Andrew Woodcock) XLS
  • Division Sheets (Linda Cook) DOC
  •  Multiplying Measures (Michelle Culliford) PDF
  •  Trio Triangles (Caroline Stares) DOC
  • Dividing 2 digit by 1 digit Numbers Mentally (Louise Forster) DOC
  • Pick and Match Calculations (Known Facts) (Louise Forster) DOC
  • Using Known Facts (Louise Forster) DOC
  •  Animal Multiplication (Mez Miles) DOC
  •  Octopus Multiples (Mez Miles) DOC
  •  Using Multiplication Facts to Solve Division Questions (Paula Alty) DOC
  •  Missing Numbers Division (Joanne Pooley) 
  •  Multiplication & Division Arrays (Emma Bagley) DOC
  •  Linked Division (Louise Forster)  
  • Multiple Monsters (Joanna Cayley) MS Powerpoint 

 

Repeated Addition / Repeated Subtraction:

  •  Repeated Addition as Multiplication (Katherine Gronert) MS Powerpoint
  •  Repeated Addition & Multiplication (Raj Barard) MS Powerpoint
  •  Multiplication as Repeated Addition (Amy Sheppard) 
  •  Multiplication Arrays (Julie Stead) 
  • Division Using a Numberline (Toni Boucher) 
  •  Repeated Addition (Valerie Ryan) 
  • Multiplication Build Up (Gill O’Neil) PDF
  • Changing Adding to Multiplication (Carol Wright) DOC
  •  Division by Repeated Subtraction (Richard Queripel) DOC
  • Counter Array (Andy Cork) DOC
  • Groups of (Early Multiplication)  (Shirley Lehmann) Easiteach
  •  Repeated Addition (Lesley Bratton) MS Powerpoint
  • Number Line for Repeated Addition (Morag Watson) 
  •  Division as Repeated Subtraction using a Number Line (Paula Alty) DOC
  •  Multiplication & Division Arrays (Robert Jinkerson) MS Powerpoint
  •  Division Using a Numberline (Kate Major) MS Powerpoint
  •  Division by Repeated Subtraction (Chris Kirwan) MS Powerpoint
  •  Multiplication (Repeated Addition and Arrays) (Andrea Harrison) MS Powerpoint
  •  Division by Sharing (Avani Chotski) 
  •  Introduction to Multiplication (Jude Kuscher) MS Powerpoint MS Powerpoint
  •  Introduction to Multiplication (Dawn Atkin) DOC
  • Division using repeated subtraction (Chez Owen) DOC
  • Patterns of Three (Tamsin Hall) PDF
  • Multiplying Sets (Carol Wright) DOC
  •  Multiplication & Division on a Numberline (Ali McNamara) DOC
  •  Dividing on a Numberline (Robert Jinkerson) MS Powerpoint
  •  Division by Repeated Subtraction (with a numberline) (Richard Queripel) DOC
  •  Multiplication Arrays (M A Crook) DOC
  • Counting in Groups (Lucy Hall) DOC

 

Division as Sharing:

  •  Division by Sharing (Claire Robinson) Smart Notebook (zipped)
  •  Dividing and Sharing (3 levels) (Naomi Hass) DOC
  • Division by 3 (Rachael Durneen) Smart Notebook (zipped)
  • Sharing Counters (Liz Hazelden) DOC
  • Sharing Equally  (Shirley Lehmann) Easiteach
  •  Smarties Share (Andy Cork) DOC
  • Simple Division Questions (Helen Bell) DOC
  • Division ‘Rounding Remainders’ (Michelle Rundle) Smart Notebook (zipped)
  • Sharing Sweets (1) – (2) – (3)
  •  Sharing Zoo (Lesley Bratton) MS Powerpoint
  •  Sharing Zoo (Lesley Bratton) DOC
  •  Making Groups (Paul Smith) Smart Notebook (zipped)
  •  Division as Sharing 1 (Carly Pitman) Smart Notebook (zipped)
  •  Division as Sharing 2 (Carly Pitman) Smart Notebook (zipped)

Division as Grouping:

  •  Dividing Using Groups (Tim Pool) MS Powerpoint
  • Dividing by Grouping (Zoe Mayston) DOC
  •  Division by Grouping with Remainders (Matt Lovegrove) Smart Notebook (zipped)
  •  Division by Grouping (Vicky Frampton) DOC

Division with Remainders:

  •  Division Word Problems with Remainders (Shazia Hussain) DOC
  • Division Picture PDF
  •  Remainders as Fractions and Decimals (Richard Queripel) DOC
  •  Rounding Up and Down After Division (David Peynado) DOC
  • Simple Divisions with Remainders (R. Lovelock) Smart Notebook (zipped)
  •  Weetabix Division with Remainders (Heidi Morris-Duffin) DOC
  •  Division with Remainders (Jenny Synnott)  
  •  Multiplication & Division (T3 Unit 2) (Joanne Robson) Day 1 MS Powerpoint Day 2 MS Powerpoint
  •  Simple Division with Remainders (C. Williams) DOC
  •  Rounding Up After Division (Fiona Bell) DOC
  • Remainders Game (J. Balmer) DOC
  •  Division Problems with Rounding (Helen Langford) DOC
  •  Division with Remainders (Victoria Adams) DOC
  •  Word Problems with Remainders (Steve Abey)