Multiplication Using Napier's Method - Year 7 PDF Download

Key points

• Napier's method of multiplication is a technique employed to multiply large numbers without a calculator. It relies solely on multiplication facts, which are instant recall pieces of information about numbers. For instance, a multiplication fact for 20 could be 4 x 5, and a division fact for 20 could be 20 ÷ 5 = 4, extending up to 9 x 9 and addition operations.
• An advantage of Napier's method is its exclusive use of multiplication facts for calculations. These facts facilitate quick mental arithmetic and encompass various operations, enhancing mathematical fluency.
• To reinforce times-table knowledge, an engaging game called "Guardians: Defenders of Mathematica" can be utilized as a fun and interactive tool for practice and learning.

How to use Napier's method of long multiplication

• Napier's method of long multiplication involves using a grid structure to multiply numbers efficiently.
• The layout of the grid is determined by the numbers being multiplied. For instance, multiplying 64 by 77 would require a 2x2 grid.
• When multiplying numbers like 321 by 36, either a 3x2 or a 2x3 grid can be utilized, as multiplication is commutative.
• Commutativity in multiplication means that the order of numbers being multiplied does not affect the result. This property is also present in addition.
• For example, 4 x 3 is equal to 3 x 4, and 4 + 3 is the same as 3 + 4.
• In the grid, a diagonal line is drawn to distinguish between the tens (T) and units (U) places in each box.
• Each cell in the grid represents a two-digit product of the multiplication, with a tens digit and a units digit.
• For instance, when multiplying 4 by 5, the product is 20, represented as 4 x 5 = 20.

Commutative Operations

• An operation is commutative if the order of the numbers involved does not affect the result. For example, in multiplication and addition, changing the order of numbers does not change the outcome. (e.g., 4 x 3 = 3 x 4 and 4 + 3 = 3 + 4)

Grid Multiplication

• A diagonal line is used in the grid to separate the tens (T) and units (U) places in each box, aiding in the multiplication process.
• Each cell in the grid corresponds to a two-digit answer resulting from multiplying two numbers. For instance, when multiplying 4 by 5 to get 20 (4 x 5 = 20), the answer is split into a tens digit and a units digit.

Understanding Multiplication Using Napier Grids

In multiplication, the product is the result of multiplying one number by another. For example, if we multiply 4 by 5, the product is 20 because 4 x 5 = 20. Multiplication is essentially repeated addition.

Example: Calculate 37 x 91

When multiplying two-digit numbers like 37 and 91, a Napier grid can be a helpful tool. This grid is structured to facilitate the multiplication process by breaking it down into smaller steps.

• Draw a 2 by 2 grid to multiply a 2-digit number by another 2-digit number. Label the grid with figures at the top and the right-hand side.
• Draw diagonal lines for each cell in the grid from top right to bottom left and extend them outside the grid. This step aids in organizing the multiplication process.
• Each cell in the grid represents a two-digit answer to a multiplication, consisting of a tens digit and a units digit. This breakdown helps in understanding the composition of the final product.

By using the Napier grid method, complex multiplications can be visualized and solved systematically, enhancing understanding and accuracy in mathematical calculations.

Image caption: Draw a 2 by 2 grid to multiply a 2-digit number by another 2-digit number.	Image caption: Draw the diagonals for each cell in the grid from top right to bottom left and extend outside the grid.	Image caption: Each cell in the grid represents a two-digit answer to a multiplication made of a tens digit and a units digit.

Multiplication Using Napier's Bones

Napier's Bones are an ancient calculating device used for multiplication.

How Napier's Bones Work

• Napier's Bones consist of a set of numbered rods or strips that can be used to perform multiplication through a series of simple operations.
• Each rod is inscribed with a multiplication table for a specific digit or set of digits.
• By aligning the rods in a particular way, one can easily find the product of two numbers.

Example Calculation

Let's consider the multiplication of 3 and 9 using Napier's Bones:

• When multiplying 3 by 9, we find that the product is 27, comprised of 2 tens and 7 units.
• The result can be visually represented on a Napier grid, as shown in the accompanying image.

Similarly, the process can be repeated for other multiplication combinations using Napier's Bones.

Summation of Diagonals

• Once all products are filled in the grid, the summation of each diagonal provides part of the final answer.
• The right-to-left diagonals represent different place values: units, tens, hundreds, and thousands.

Calculating Multiplication using Napier Grids

• Explanation on Adding Diagonals:
  • Instructions on Calculating Diagonals: Add up each diagonal, remembering to carry for two-digit answers.
  • Example with Calculation: Illustrated guidance on summing diagonals.
• Writing Calculations using Arrows:
  • Step-by-Step Guide: Follow arrows to write and solve 37 x 91 = 3367.
  • Visual Representation: Image depicting the multiplication process.
• Utilizing Napier Grids for Multiplication:
  • Grid Drawing Instructions: Draw and label a grid for multiplication.
  • Understanding Grid Cells: Explanation of cells representing two-digit multiplication answers.
  • Highlighting the Grid Answer: Identifying the answer within the grid for an example calculation.

Example: Calculate 653 x 72

Napier Grid Method in Multiplication

• Using Napier Grid for Multiplication
  • The Napier Grid method is a visual way to multiply numbers efficiently.
  • It involves breaking down multiplication into smaller steps using a grid.
• Grid Completion and Interpretation
  • Each cell in the grid represents a partial product of the multiplication.
  • The completed grid provides a structured way to calculate the final product.
• Summing Diagonals for the Final Answer
  • Adding the numbers along specific diagonals in the grid gives different place values of the final product.
  • The process involves summing up units, tens, hundreds, thousands, and ten-thousands separately.
  • Carrying is necessary when the sum exceeds a single digit in a particular place value.

Understanding Multiplication Using Grids

The Concept of Commutativity in Multiplication

Using Grids for Multiplication

Illustrative Example

Benefits of the Grid Method

• Practise Napier's Method

  • Question: Identify the Error

    Try to identify mistakes in your work.

  • Practice: Napier's Method Quiz

    Test your skills in long multiplication using Napier's method with this quiz. Grab a pen and paper to aid your calculations.

• Real-world Mathematics

  

  When you lack a calculator, Napier's method offers a manual approach to long multiplication.

  Multiplication plays a vital role in determining areas, batch production quantities, necessary construction materials, and more.

  Professionals in catering, engineering, construction, and various industries rely on multiplication skills for their daily operations.

  Mastering multiplication can significantly benefit individuals in the professional world.