# Multiplication

## Introduction to Multiplication

You are at the grocery store, and you buy 5 bags of chips each costing Rs 15/-. But before you get to the checkout line, you want to figure out the total and hand in the right amount of change to the cashier. How do you calculate? 15 + 15 + 15 + 15 + 15 = 75

Multiplication might seem as a tough task at first, but when you break it down and practice the basics, you will have it figured out in no time. All multiplication is, is repeated addition

## The big idea: What is multiplication?

To simplify multiplication, it is nothing but a process in which a number is added to itself for a particular number of times.

To break this down further, the **multiplicand** is the number being multiplied, and the **multiplier** is the number performing the multiplication. The outcome achieved is known as the **product**. Multiplication has a **commutative property**, where numbers can be multiplied in any form and way.

In the above example of 2 x 3 = 6, 2 is the multiplicand, 3 is the multiplier, and 6 is the product. Because of the commutative property, the above numbers can also be multiplied as so

Multiplication on the two numbers can be performed in the following ways since addition is the core of any multiplication;

- 2 + 2 + 2 = 6
- 3 + 3 = 6

Instead of repeatedly using a number to add up to find the sum, we directly multiply two numbers to find the product.

### The Multiplication Table:

Multiplication is furthermore simplified when it is taught from the basics. A simple addition is all it takes to understand what happens when 2 single digit numbers are multiplied. A complete understanding of the times table can make any form of multiplication easier. To easily understand how addition is used in multiplication, we introduce the multiplications table or the times table, which represents groups of numbers of the same size.

Using the above table as an example, we use two numbers, 5 and 2. 5 is taken from the column, and 2 is taken from the row above;

5 x 2 = 10

This means that there are 5 groups of 2. If we were to add them, it would be;

2 + 2 + 2 + 2 + 2 = 10 or

5 x 2 = 10

### Adding groups of numbers

A graphical representation of how numbers are multiplied is shown here.

Using the above table, we multiply **7 x 6 = 42** which means there are **6 groups of 7** stacked besides each other to give us the graph we see;

**7 + 7 + 7 + 7 + 7 + 7 = 42 or
7 x 6 = 42**

### Long multiplication:

Two, single digit numbers can easily be multiplied when you know the concept of addition and groups. But what happens when you have to add larger numbers with each other? The concept of addition of groups could make it a little harder to add up so many numbers. For example, 46 x 25

According to the addition of groups it would be 25 + 25 + 25 + 25 + 25 ……… + 25 = 1150

Adding 25, 46 number of times would become very long and tiring. To quickly do this form of multiplication, we introduce the carry forward. One way to master any long multiplication is to memorise the multiplication table

### The carry forward:

Taking the example of 46 x 25

**Step 1:**

**Step 2:**

Add a zero to the tens place on the next line.

**Step 3:**

**Step 4:**

Add both rows of numbers after they have been multiplied.

Numbers increase from left to right of the Multiplications Table. When larger numbers are multiplied, their product increases each time. Below is a chart representing the multiples of each number from 1 to 10.

If we look at an example of 3 x 8, you can see 24 squares inside the block. Which represents 3 groups of 8.

8 + 8 + 8 = 24 = 3 x 8

## Importance of multiplication

Multiplication is widely used not just in the classroom but all throughout life in general. As one of the 4 mathematical operations, multiplication is used almost everywhere every day.

Understanding the basics of multiplication as a child has a significant impact as adults. Failing to understand the basics or not being able to memorise the multiplications table could be problematic when they are faced with other more robust mathematical equations taught in higher grades. This leads to a negative build up and lack of motivation towards mathematics in general.

### Common Errors

The most important factor is to master the method of performing long multiplications. Common errors occur when a child forgets to add zeroes to each product at every row as they multiply multi-digit numbers. Another error is when a child forgets to carry forward and add numbers to the next product.

Memorising the entire multiplications table is the first step at acing multiplications on the whole. 2 x 8 and 8 x 2 are both 16, which means that memorising half of the table is equal to memorising the remaining half, making your task easier.

## Common mistakes or misconceptions

- When
**multiplying by 2 or more digits, children may miss adding a zero when multiplying by the digit in the tens place**. This happens if they have not truly understood the reason for placing that zero or if they are in a hurry. See if they repeat this mistake with other similar problems. If yes, then it’s probably a conceptual error. Else it’s a silly mistake.

- Children start by learning multiplication of whole number. This leads them to build a view that
**when two numbers are multiplied, the product will always be larger than the two numbers**. This is not a problem as long as they are dealing with whole numbers but may become a stumbling block when fractions and decimals are introduced. This view does not hold when multiplying by quantities less than 1.

## Tips and Tricks

- In multiplication,
**the order of numbers does not matter**(called the commutative property). So, choose the order that you are more comfortable with. When relying on memory of multiplication tables, compared to 9 x 4, students may remember 4 x 9 more easily. **When multiplying three numbers, choose the two that can be multiplied easily.**

E.g. 5 x 17 x 2 will be difficult if we try to multiply 5 x 17 first. Instead, multiplying 5 and 2 gives 10 which is easily multiplied by 17 to get 170.- When
**multiplying a 2-digit number with a one-digit number, it sometimes helps to break the two-digit number as per the place values. Then multiply each part and add**.

E.g 37 x 4 can be solved mentally by breaking 37 as 30 + 7.

Then 30 x 4 = 120 and 7 x 4 = 28. So the final answer is 120 + 28 = 148.

While this may seem more tedious when written down, it is much easier to do mentally. - Even
**if you don’t remember the multiplication fact, it can be easily mentally figured out**. E.g. 17 x 9 is difficult to remember. But restructure this mentally as 17 x (10 - 1). So the answer will be 170 - 17 = 153.