Napier's Bones: Quickly Multiply, Wooden Style

I first encountered Napier's Bones back in Mrs. Hutnick's 4th grade EL math class. We had a 'build day' of sorts in which we made our own Napier's Bones, and at the time I thought they were really cool. Now, a bunch of years later, I still think they're really cool--they're very simple to make and to use, they are durable, and they can be personalized in any way you see fit.

My old pair is quite sloppy (though not bad for a 10-year-old, I'd say. Good job, former me), so I decided to make a new and improved set--documented here on Instructables.

Now for the important part: what Napier's Bones are. They are a multiplication tool/math toy that allows you to multiply 0-9 with large numbers extremely quickly with only slabs of wood. As long as you add small numbers and read correctly, you will get the correct solution every time.

Step 1: How They Work

Watch the video below while relaxing to the smooth tunes in the background:

Step 2: Materials

You'll need:

Wood. Unless you have a table saw to cut wood lengthwise, your best bet will be to get 1" x 1" or 3/4" x 3/4" poles. You will need at least 30 inches of wood, but more likely around 50 inches if you want the ability to mulitply more numbers. The wood needs to be square.

Hardboard. Also called Masonite, you need a piece 10 inches high by approximately 4 inches. I got a 10 x 14 sheet and cut it in one direction.

Wood Glue, and a clamp to keep pressure while it dries.

Sharpie Markers. If you want a different look, you can use some other writing/painting instrument.

A saw (I used circular) and sandpaper.

Step 3: Cut the Wood

Before you cut the wood, you want to decide how many pieces you want in your set. A set of two bones would be the minimum: you would be able to multiply some two digit numbers. I chose to make a set of four free bones, which covers every two digit number*, most three digit numbers, and about half of the possible combinations that entail four digit numbers. Also consider the size of your set: too many bones will no longer make your set portable and easy to use.

*NOTE: This numbering system does not include 1s and 0s in them--this is because 1s and 0s are not very interesting mathematically here, or not interesting enough to warrant more bones, in my opinion. If you want to include these numbers, you'll have to renumber some bones or make more.

For a set of four free free bones, you need to cut:
Five 9" pieces of your 1"x1" or 3/4" x 3/4" pole of wood
One 3.5" piece of the pole of wood

You need one more than the number of free bones because one bone needs to be fixed. The other piece is the base, so the length is dependent on how many bones you will make. (More bones, longer piece.)

I used a chop saw and a ruler to make the cuts, then sanded down anything not smooth.

You also need to cut the hardboard: measure out a size that is the width of one large piece and the length of the small piece added together, and then cut. I used the circular saw again, flipping it to cut the whole thing.

Step 4: Pencil in the Lines

I tend to make mistakes when I do stuff. To avoid this, I draw everything in pencil first. I recommend this method.

On the small 3.5" piece, do not draw anything.

On one of the 9" pieces, draw horizontal lines at 1" intervals along one side.

For the other four 9" pieces, draw horizontal lines every 1" down the rod on each side. To do this most effectively, measure and make a small mark at each 1" interval on all of the sides; then, using scrap wood or other pieces as right angles, draw lines at each mark. Make sure your measurements are careful--you want the lines on each side to line up with each other!

You should now have 9 boxes on each side of these four bones (formed by lines and edges). Draw a diagonal line from the upper right-hand corner of each box to the lower-left hand corner of that same box. Don't draw a diagonal in the uppermost box for all pieces and all sides!

Step 5: Sharpie Round 1

Go over all of the lines you just drew with a sharpie or equivalent artistic drawing device. Ask your inner artist how you want the bones to look, and color/decorate accordingly.

If you are using a sharpie, be careful: the marker lines will 'bleed' a little on the wood.

Step 6: Pencil in the Numbers

Now it's time to enter the numbers.

Don't draw any numbers on the 3.5" small piece. This piece will remain unmarked.

On the 9" piece with only horizontal lines on one side, write the numbers 2 through 0 (2,3,4.....,9,0), one in each 'box', in that order.

Take one of the remaining 9" pieces and draw the number 2 in the top box on one side. Turn the piece 90 degrees and draw the number 3 in the top box on that side. Repeat with 4 and 5 and the remaining two sides.

Repeat these exact steps with another 9" piece.

For the last two remaining 9" pieces, replicate the process with the numbers 6, 7, 8, and 9.

Take one of the [2,3,4,5] pieces and place it on the table with the 2 facing up. Place the first (fixed) piece (the one with only horizontal lines on one side) next to this piece, but offset so that the 2 on the fixed piece is aligned next to the highest empty box.

In that empty box, write in the two digit result of 2 x 2 (the upper number times the number on the left). Since the answer is 04, write the 0 on the left side of the diagonal and the 4 on the right side. In the next box, write the answer to 2 x 3 (the upper number times the number on the left) in the same fashion. Repeat this pattern for all of the numbers.

Step 7: Sharpie Round 2

Go over the numbers in sharpie or equivalent artistic drawing device. I consulted my inner artist and decided to use a different color than I used for the lines, and so the numbers are red and the lines black on my set.

For the smaller numbers, I found it easier to use a fine point sharpie as opposed to a regular tip.

Step 8: Glue Fixed Bones

The piece with just horizontal lines and one side of numbers is the fixed vertical piece. Using wood glue, glue it down on the left side of the hardboard. The bottom of the piece should be flush with the bottom of the hardboard, but there should be space at the top. Clamp and let sit according to directions on the glue.

Glue the small piece perpendicular to the vertical fixed piece and flush with the bottom of the hardboard. Follow directions again!

Step 9: Practice and Impress Your Friends!

Your set of Napier's Bones is now complete--all you need to do is practice! The free bones should align with the fixed numbers, and you read across left to right, adding inside the parallelograms (see step 2 for full use instructions).

Remember that not all numbers will work with the numbers of bones that you've made. With four bones, you cannot multiply the number 9999 because you only have two '9' digits. A number like 9471 will work perfectly, though.

Once you know how to use them, they are fun to try and to show people, especially if you've made them yourself. Everyone (all ages and backgrounds) I show the bones to thinks they are really cool and interesting. A "what's that sitting over there on your messy desk?" usually leads to an "oooh" or a "wow." Plus, you can challenge people to multiplication games: you, with the bones versus your arch-nemesis, armed with only pen and paper and his/her/it's measly little brain.

This can be a great project for kids, too, if you cut the wood first and let them insert the numbers themselves, decorate, and glue. They will have their own set of Napier's Bones, just like I did back in 4th grade that I have kept for nearly a decade.

Mathtastically,

joshf