Napier's Bones, Without the Bones.




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Napier's bones are an easy way of multiplying large numbers without losing track of all the columns, rows, carrying...

The original version (repeated in this instructable) consisted of sticks (bones) with numbers marked on them, but that's not so portable.

The process can, though, be repeated with pencil and paper.

In school, this method is suitable for classes of most ages who are getting to grips with multiplying larger numbers.

In the UK, KS2 and upwards.

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Step 1: The Grid.

You start with a grid, sized to match the digits of your numbers.

For instance, if you are multiplying 748x43, you need a grid of 3x2 squares.

Draw diagonal lines across the grid (top-right to bottom-left), extending them to below the grid (see the examples in the images).

Write your numbers outside the grid (in the templates, I have drawn dotted-squares to show you where).

If you are not used to using the grids, or are just too lazy to draw them yourself, you can use the templates I have added to this Instructable.

The large sheet, with every size of grid on it, is a resource I created for my maths class, some of whom have poor motor skills, so can't draw straight lines without help.

Step 2: The Easy Bit.

You don't get your final answer by magic, but it's not too hard as long as you know your basic tables up to 9.

Multiply each digit on the top row by each digit on the side column.  This can mean doing quite a few calculations, but they are all simple.

If the result of a calculation has two digits, the "tens" digit gets written above the diagonal line, and the "units" digit gets written below it.  For instance, 6x7=42, so write the 4 above and the 2 below (4/2).

If the result of a calculation is only one digit, write a zero above the diagonal line (2x3=6, so you write 0/6)

(You would not normally do this in coloured ink, but I have used it for clarity.)

Step 3: The Other Easy Bit.

Once you have done all the multiplications, it's time to add.

Starting at the right of the grid, add up all the single digits in a diagonal.  Write the answer in the row below the grid.

If the total is over 9, carry the extra digits over to the next diagonal, and add them on.

When you run out of diagonals to total, you have finished!

Read the numbers from left to right, as usual, and you have your answer.

Step 4: Free Mini-ible

No good at your nine-times table?

Do you have a full set of fingers and thumbs?

Good - you can count instead.

Hold your hands out in front of you, palms up, fingers spread.

If you want to find, say, 4x9, you count four fingers from the left, and fold the fourth finger down.

Now count the number of fingers and thumbs to the left of that finger (3) and to the right of that finger (6).  The answer is 36.  The same goes for any multiple of nine, up to ten.

Step 5: Why Use the Bones Method?

Other traditional methods of multiplication work, so why use the bones?

It's fairly simple; it lets you check your working more easily.

If you use the traditional column method, with lots of rows above and below the answer box, and having to keep track of place-value at the same time, it can be hard to check your workings, or to find out where you went wrong.

With the bones, you are checking a set of separate calculations, all relatively simple, with small answers.  Easy to check, easy to fix.

Go on.  Try it.

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    18 Discussions


    7 years ago on Introduction

    my 9 yr old daughter finds maths very difficult...they cant use calculators in school...only paper, pen, and their brains!...needless to say...she loved the finger method of the 9 time tables...wish i could find other methods for the rest!

    4 replies

    Reply 7 years ago on Introduction

    You can do your 11 times table by adding the number to itself, but with one number moved one place to the left.

    eg 11 x 42 (forgive the layout - you can easily do this in your head)

    _42 +



    Reply 7 years ago on Introduction

    Thank you! that is soooo there some more methods 8 times and 7 times and 6 times??? :-)


    Reply 7 years ago on Introduction

    Well, 8s are just your 4s, but doubled, but I'm afraid that 7s and 6s are best learned the old-fashioned way.


    7 years ago on Introduction

    Thank you Kiteman. That's brilliant.
    Too late to soften the hell that was my maths class at school, but I will certainly be using these techniques from now on, and teaching them to my children.


    9 years ago on Introduction

    Cool ! another perspective. I love finding different ways to do the same thing.


    9 years ago on Introduction

    I've been using this method for long multiplication for a long time - it's the best by far. Good 'ible on it, as always.

    1 reply

    9 years ago on Introduction

    Awesome ible. While the title made me laugh I don't think it'll help many people find it :( Hope your keywords are good!

    3 replies