Lattice Multiplication: Definition, Methods, Examples

Lattice Multiplication: As we all know, multiplication is simply repeated addition. When a number is added to itself several times, it is a faster technique to find the sum. Lattice multiplication is a grid-based technique of multiplying two huge integers. Lattice multiplication was discovered in India in the tenth century and was subsequently used by Fibonacci in the fourteenth century. This approach is also known as the box method.

Multiplication by lattice is a simple, interesting, and enjoyable approach to learn. This article will go through how to multiply integers using lattice multiplication. Join along and learn more about lattice multiplication.

Lattice Multiplication

The lattice multiplication method is used to multiply two numbers in which at least one of them is a two-digit number or greater. In this method, we organize the number in boxes, and thus this method is also named the box method. This method breaks the multiplication process into smaller steps, which makes the multiplication easier.

Lattice multiplication is a convenient method, which helps break down large numbers into simple maths facts. It can help to envision the different steps and support a better understanding of multiplying numbers. The lattice method of multiplication is an alternative to the method of long multiplication. Although both ways consist of breaking up the numbers, multiplying them, and then adding them together, there are still some key differences in how it is written down.

Multiplying by Lattice Method

Below given are the steps which will help us to multiply the numbers with the help of the lattice multiplication method, and they are:

1. Drawing the lattice
2. Labelling the lattice
3. Multiplying the numbers
4. Adding the numbers

Let us understand the application of the upper given steps by taking an example. Let us multiply \(45\) by \(36\).

1. Drawing a Lattice

The first thing required is to draw a grid that matches the number of digits. In this example, \(45\) and \(36\) are both \(2\)-digit numbers, so we need a \(2×2\) grid.

Then draw the diagonal lines in each box. These lines need to go from the lower-left corner to the upper right corner of the boxes.

2. Labelling the Lattice

Now that we are done and ready with the lattice, the next step is to label it. The first number goes on top of the lattice. For example, if we take \(45\), then we need to put each digit over each box.

Then, the second number on the right of the lattice. Again, one digit should go next to each box, as displayed below.

3. Multiplying the Numbers

The next step is to multiply the numbers. This multiplication can be done on the numbers that meet on the grid. First, multiply \(4\) and \(3\) to get \(12\) and then write in the first box above and below the diagonal as shown below:

Similarly, multiply other digits and fill the box.

4. Adding the Numbers

Since we are done with the multiplication, the next and final step is to add the numbers in each diagonal. Start from the bottommost diagonal. Add and write the sum below each diagonal row, and if there is any carry, move that to the next diagonal row.

As \(0\) is alone in the bottommost diagonal row, write \(0\) just below that, as shown in the figure.

Similarly, add each diagonal row.

And once we are done with the addition of digit present in diagonal, we need to read and write the numbers starting from the top one on the left and moving till right.

i.e., read from \(1\) on the top left to \(0\) on the bottom right to get \(1620\).

That is, \(45×36=1620\).

Lattice Multiplication Practice Problems

Before we start to multiply bigger numbers by lattice multiplication, let us be thorough with the multiplication of the smaller number through the lattice.

If the number after multiplication is a single-digit number, then put zero at the tens place in the upper part of the grid.

Example 1: Multiply \(14\) by \(56\).

Arrange the multiplicand and multiplier as shown in the figure.

Multiply the numbers and place them according to their ten’s and one’s position.

After multiplication, add the diagonal rows.

And finally, after multiplication, read the answer from left-top to right bottom. Ignore the \(0\) at the beginning, as it does not have value at the beginning of a number.

Hence, \(14×56=784\).

Example: Multiply \(2314\) by \(157\).

Let us understand the lattice multiplication method by multiplying slightly bigger numbers step by step in an elaborated manner.

1. Draw a grid with \(4\) columns and \(3\) rows, depicting the multiplicand and multiplier as shown in the below figure.

2. Draw a diagonal through each box from the upper right corner to the lower-left corner. (Continue the lines as short way past the grid).
3. Multiply each digit of the multiplicand with one digit of the multiplier at a time. 
4. Write the product in each square such that the tens are in the upper half of the square and the ones are in the lower half. 

5. If the product does not have a tens digit, mark it with \(0\) in the upper half.

6. Now, add the numbers in the grid along the diagonals, starting from the lower right corner.
7. Add and carry over to the sum of the next diagonal.

8. To find the answer, read the digits starting down the left of the grid and continuing across the bottom, ignoring the initial \(0\).

Here the product of \(2314×157=363298\).

Long Multiplication 

When we multiply two numbers, the first number is called the multiplicand, and the second number by which the multiplicand is multiplied called the multiplier. The result of the multiplication is called the product of the multiplication. Let us quickly recall the steps involved in multiplication.

1. Write the numbers in the place value chart one below the other. The multiplicand is placed above the multiplier.

2. Multiplication is done column-wise, from right to left.

3. If the multiplier has more than one digit, then multiply the multiplicand by each digit separately. Start with the multiplier digit at the smallest place. When multiplication by one digit is complete, then move to the next digit.

4. If there is a carry-over, add it to the product of the next column.

Let us see one example to understand the multiplication by the long method.

Example 1: Multiply \(31268\) by \(204\).

Solution: \(204=2\) hundreds \(+0\) tens \(+4\) ones

Hence, the required answer is \(6378672\).

Lattice Multiplication vs Long Multiplication

Compared to the long multiplication method, lattice multiplication is less popular, but still, lattice multiplication is used by educators to simplify multiplication by breaking it down into easy steps.

As it follows the standard multiplication algorithms, it shows what these algorithms mean in an accessible way for children. This is also an easy method for supporting visual learners, as some kids learn better by visualizing the process.

Even if it is not included in the maths curriculum, lattice multiplication is an excellent method for expanding children’s multiplication knowledge and number sense.

Solved Examples on Lattice Multiplication

Q.1. Multiply \(42\) by \(35\). 
Ans:

Hence, the required answer is \(1470\).

Q.2. Multiply \(32\) by \(27\) through the lattice multiplication method.
Ans: The steps are shown as below,

Hence, the required answer is 864.

Q.3. Multiply \(349\) by \(63\). 
Ans:

Hence, the required answer is \(21987\).

Q.4. Multiply \(165\) by \(223\).
Ans: 

Therefore, the required answer is \(36,795\).

Q.5. Find the value of \(119×23\).
Ans:

Hence, the required answer is \(2737\).

Summary

In this article, we learned about lattice multiplication, its origin, and then its adaption. We learned through the steps how to multiply numbers through lattice multiplication. In addition to this, we also learned about the difference between lattice multiplication and long division.

Frequently Asked Questions (FAQs) – Lattice Multiplication

Q.1. Explain lattice multiplication?
Ans: The lattice multiplication method is used to multiply two numbers in which at least one of them is a two-digit number or greater. In this method, we organize the number in boxes, and thus this method is also named the box method. This method breaks the multiplication process into smaller steps, which makes multiplication easier.

Q.2. How to solve lattice multiplication?
Ans: The steps are as follows to solve a multiplication problem through lattice multiplication.
1. Draw a grid with columns and rows depicting the multiplicand and multiplier.
2. Draw a diagonal through each box from the upper right corner to the lower-left corner.
3. Multiply each digit of the multiplicand with one digit of the multiplier at a time. 
4. Write the product in each square such that the tens are in the upper half of the square and the ones are in the lower half. 
5. Add the numbers in the grid along the diagonals, starting from the lower right corner.
6. Add the carry-over, if any, to the sum of the next diagonal.
7. To find the answer, read the digits starting down the left of the grid and continuing across the bottom.

Q.3. What is the benefit of lattice multiplication?
Ans: Lattice multiplication is an alternative method of multiplying the numbers. Lattice multiplication is a convenient method, which helps break down large numbers into simple maths facts. It can help to envision the different steps and support a better understanding of multiplying numbers.

Q.4. Where does lattice multiplication come from?
Ans: Lattice multiplication was first found in India in the \({10^{th}}\) century and later adopted by Fibonacci in the \({14^{th}}\) century.

Q.5. How do you do \(3\) digit by \(2\) digit lattice multiplication?
Ans: Draw a grid with \(3\) columns and \(2\) rows, depicting the multiplicand and multiplier. The first number goes on top of the lattice. Then, the second number on the right of the lattice. Again, one digit should go next to each box. The next step is to multiply the numbers. This multiplication can be done on the numbers that meet on the grid. Lastly, add the numbers in each diagonal, and write the numbers starting from the top one on the left and moving till right.

Now that you are provided with all the necessary information about lattice multiplication and we hope this article is helpful to you. If you have any queries on this page, post your comments in the comment box below and we will get back to you as soon as possible.