Napier' Ligaments

Napier's Bones are a mathematical calculating device that increases the speed of multiplication, while reducing errors. They expand the steps of normal long multiplication, making them ideal for young students to better understanding the process of long multiplication. What would make them easier to use is if the students got to use them earlier. Napier's ligaments are bones for addition. That can be used at a far younger age.
Napier's ligaments are a mathematical addition device the represent answers in a lattice format. It also expands the steps of multi digit addition allowing the student to better understand the process. Included with this Instructable is a PDF file to print a set of Ligaments on two pieces of 11" x 17" paper. Also included are two SVG files to make a set with a laser using two pieces of 12" x 12" thin (1/8") plywood or MDF. The set contains the 100 two digit combination and some commonly used extras. So make a set and follow along.

A ligament, or "LIG" for short, is a 1 by 3 rectangular tile with two single digit values in the top and middle and their sum in the bottom written in lattice format. This lattice format allows them to be place together and easily do multi digit addition.

You can easily add as wide of multi digit values as you wish. The limitation is the number of LIGs you have. This basic set, without the extras, only has one each of the 100 combinations.

Now an easy solution is to make another set of LIGs, but with a couple tricks it is surprising how many digits you can do with only one set.

It is very common to be adding numbers of different length. To make them match, leading zeros have to be adding to the shorter number. This will lead to frequent use of the LIGs the have a lower digit of zero. The solution to this is to make spares of these LIGs. The patterns for these LIGs have space for 112 of them. With 100 standard LIGs, that leave 12 spares. Nine of these are used to make these lower zero LIGs.

One option for if you need a LIG you have already used, is to toggle its values. The answer remains the same but it now uses a different LIG.

One option for when you have already used a LIG you need, is to push/pull the value of the two digits in the LIG. That is add a one to one digit and subtract one from the other. The answer remains the same but it now uses a different LIG. If this new LIG has also been used, you can toggle or push/pull is again.

Of the 12 spare LIGs, Nine were used for extra spare lower zero LIGs. This leaves three to be defined.

Of all the 100 possible LIGs only two cannot either be toggled or push/pulled. They are the [9,9] LIG and the [0,0] LIG. So these became the last three spare LIGs.

If you want to multiply 456 and 123, you would either use long multiplication, Napier's bones, or Genaille rods to generate the three following values to be added:

1368

912 shifted left one digit

456 shifted left two digits

Use the LIGs to add the first two values.

The use the LIGs to add thi result to the last value.

This particular problem only required 7 LIGs, none that were duplicates. You can do a lot with a single set of LIGs.

What I like about these LIGs is the allows a kid to play around with the numbers and stop thinking about math as so rigid. A child can toggle or push/pull digits and become comfortable with manipulating them. (Numbers become play doo.) In the next few step I will explore some ideas to make learning addition to be fun and interesting. For the following steps please remove the 12 spare LIDs and only use the 100 basic ones.

With the answer easily covered with a thumb, these make excellent flash cards. Also viewing the answer gets the child more familiar with the lattice format. Have a child draw a random LIG and let them state the sum. Even but some reward system for correct answers. Something like:

"Dad, can I use the tablet?"

"You can use the tablet for 30 minutes if you can answer 10 LIGs"

Have a contest and see who can answer to most randomly selected ones in a row.

You get the idea. You will be surprised to see them practicing on their own.

Have the child or group of children assemble the multiplication table. They can make it 10 LIGs wide by 10 LIGs tall, or (as shown) 20 LIGs wide by 5 LIGS tall. You will be surprised the patterns they will discover. Ask them how many different sums the LIGs can have and how many LIGs there are for each sum.

Find or make a game board and use the LIGs to determine the movement. Place all 100 MIGs in a container and let a child, on his turn, draw one out. They must say what is on the LIG and move that amount. Something like, "Six plus five is eleven." and then they move there marker that many spaces.

A more advanced version is have some referee draws the LIG and holds their thumb over the answer. If the child answers correctly then let them move their marker. If not the child to the left (or right) gets they opportunity to answer it move instead.

How about dominos? They sort of look like dominos so figure out the rules to use them as such.

For game players; A random selected LIM's value is the same as a 2d10 dice roll.

Possible games boards are a go board, chess board, Candy land board, or the square tiles in the class.

Teachers, let the students make the rules! (A possible rule is that if you draw a double value LIG you must instead to draw another and move back the new LIG's sum.) All sorts of fun rules are possible. Let the students make it their own game. It will be interesting to see how their game evolves as the year goes on.

If any of you education students or researchers are interested in studying the use of Napier's Ligments to teach addition, you are welcome to contact me. I would by happy to discuss ideas with you and supply you with some LIGs.