What is Abacus Addition and Subtraction?

What is an abacus?

Ever heard of a counting board, totaliser or counting machine and wondered what they were or how to use them? Well, you're in the right place. They are simply different words for the good old-fashioned abacus, one of the oldest maths tools in existence.

An abacus is a Maths tool that uses lines of beads to represent numbers. By moving the beads, you can show different amounts and work out totals. It's both a visual and tactile method, where seeing and/or feeling the beads will tell you the answer. This means that abacuses are particularly useful for blind and partially sighted people, and they're great for teaching people with different learning styles.

An abacus is sometimes known as a counting frame or counting board, and can be thought of as the earliest form of calculator. People have been using abacuses for thousands of years, and there are lots of different kinds that have developed over this time. You can think of it as the great-great-great-great grandparent of today's humble calculator.

What are the different kinds of abacus?

There is a vast array of different abacus names and styles that covering them all could take too long to calculate. There is time however to delve into the depths of the three that are most common (and useful) for teaching addition and subtraction. They can be thought of more as broad categories, with lots of variation within them.

School abacus

This is the popular name for the most common type of abacus and indeed you might have first thought of this type when you came to this page. A school abacus has ten colourful beads on each of its horizontal rows (it usually has 10 of these as well - but not always).

It most closely represents the Russian style of counting board, because the beads go across rather than up and down. As well as being great for teaching counting and simple mathematics, they're also fun toys. Bear that in mind if you're introducing them to your class - there's bound to be some shakey shakey maths going on!

Modern abacus

We'll use this name to talk about the kind of abacus where the beads go vertically (that's up and down), and have a separation bar going across all the columns. These kinds of abacuses come from the original Chinese and Japanese designs. The beads above the separation bar are worth five, (skip counting anyone?) while the beads below the bar are worth one.

There are different numbers of beads, depending on which kind you use. They'll either have one or two beads above the bar (the fives), and either four or five beads below the bar (the ones). For this guide, we'll talk about how to use a modern (or Soroban) abacus with one bead at the top and four beads at the bottom.

No matter how many there are, the system is the same: the beads touching the separation bar are the ones you count.

Place value abacus

A place value abacus, or counting board, also has vertical rods for the beads, but this type is different from all the others because they're not kept inside a frame. This is an open abacus, where you can remove beads completely, instead of sliding them to a new position.

These abacuses are specifically designed for use when partitioning numbers. The beads you place on the rightmost pole are your ones, those to the left of them are your tens, hundreds are to the left of them and, if you had more poles, it would keep going like that. Usually, a place value abacus would have two or three poles to play with.

Place value is important for every kind of abacus, with each row, wire or pole representing a different set of units in your calculation. But, if your focus is specifically on exploring place value, then this kind might be the best for you and your learners.

What is abacus addition and subtraction?

As well as being great for simply counting (hence the term 'counting board'), abacuses are fantastic tools for calculations. Abacus addition and subtraction is where you use an abacus to work out addition and subtraction problems.

You need to start by setting your numbers. That means making the abacus show the starting number. Then you want to count out the beads for the number you're adding or taking away. You'll either be moving some of your starting beads back (for subtraction), or bringing more beads to join them (for addition). Exactly how you need to move them will depend on the kind of abacus, but it's always about moving them in or out of the section you're counting. This is a great way to use maths manipuatives to show your students the physical process of maths. The use of an abacus allows them to see how amounts get larger or smaller.

When you've done that, the abacus will already be showing you your answer! You just need to count up the beads you can see or feel in your counting area.

Maybe you are working on a multi step problem and haven't quite finished adding or subtracting. That's no problem. Your answer becomes your new set number, so you can keep going with more calculations. If you have a long list of numbers to add and take away, then an abacus is the perfect way to keep track.

If you feel ready to tackle some abacus addition and subtraction now, that's great! Feel free to skip forward to our resources section for some lovely abacus activities. To find out more, read on as we go through how to add and subtract on our cool counting boards in more detail.

School abacus addition and subtraction

On a school abacus, your counting area is one side of the frame, with all the beads you don't want to count kept neatly to the other side. Use whichever side works best for you and your children, but make sure to be consistent; you don't want to end up getting confused about which side you're counting.

So, move all your beads to your non-counting side to set the abacus to zero. Then, choose whether you want your ones to be the top row or the bottom row. You can either start at the top and increase the place value as you go down, or start at the bottom with your tens, hundreds, thousands and so on above you.

Now it's time to set your numbers. Move the correct value of beads over to your counting side. For example, to set six, count out six beads on your ones (or units) row as you slide them to your counting side.

Say you wanted to add two, you'd simply move two more beads on the ones row. You'd then have eight beads on the counting side of that row. That's your answer! 6 + 2 = 8.

Whenever you have a full row, you need to swap them out by moving them all back to the non-counting side and using a bead from the next row to show ten. Try counting up to twelve on your abacus. Once you got to ten, move these all back and put a ten bead on the counting side, then keep counting eleven and twelve as you move two more beads across on the ones row. This will help you when adding digits that equal more than nine.

How do you think you would you set three hundred and forty-two? You need to move two beads on your ones row, four beads on your tens row and three beads on your hundreds row. To add one hundred and twenty-five, you'd move across one more hundred bead, two tens beads and five ones beads. Your abacus would now be showing four beads on the hundreds, six beads on the tens and seven beads on the ones. What does that mean your answer is? 342 + 125 = 467.

But how do we subtract on a school abacus? It's the same, just the other way around! To take away numbers, move their beads away from the counting side and back to the start. So if you wanted to work out six minus two, you still set six, then move two of them back across. That leaves only four beads left on your counting side and that's the answer.

Again, it's the same with bigger numbers. Let's take the four hundred and sixty-seven from earlier. To subtract sixty, move six beads from the tens row back across. You now have four hundreds, no tens and seven ones. But don't forget the tens row! The answer isn't 467 - 60 = 47. Remember to count any empty rows in between your other place values. Your abacus is actually telling you that 467 - 60 = 407.

But what if a row goes below zero? We can work that out too. Remember back to when we swapped a full row of ten for a single bead on the row after it. Now we want to do the opposite, and swap a single bead for ten beads on the row before it.

Imagine you need to take five away from twelve. First, set twelve, with one tens bead and two unit beads. Now you want to put five ones back to your non-counting side. After moving beads one and two, you'll have run out of unit beads to move. So, slide a tens bead out of the counting area, and slide all ten ones over to fill its place. Now you can move beads three, four and five back out again to complete the problem. You'll have seven beads in the ones row, and none in the tens row. We've worked out that 12 - 5 = 7.

This is why an abacus is great to use if your children are struggling to understand how they carry over numbers in complicated sums. This gives them a multisensory way of learning why the concept works, so they can apply it to written working out (such as the column method) in the future.

How to use a modern abacus

Using a modern abacus follows the same main principles, so have a read of the section above to make sure you understand them. There's still different wires for place value and the same movement techniques. However, now we're moving beads up and down rather than side to side, and the counting area is the separation bar.

Once again, choose which wire you're using for units.We usually use the middle bar, to leave space for larger place values on the left, and decimals on the right. yNow you can count by moving the bottom beads up one by one towards the separation bar. This lets you count zero, one, two, three, four.

When you get to five, you need to move all these four down again, and move the top bead down to touch the bar. Then, you can start moving the bottom beads up again to count the numbers six to nine. To get to ten, you need to move all the beads back to their original positions and move a bottom bead up on your tens wire to the left of it.

The important difference to remember with this system, is that you aren't making a simple trade between one way of representing five or ten and another. Every time you make these swaps, it's because you're going up or down from what the abacus showed before. For the modern abacuses with more beads per wire, you will do ordinary swaps like on a school abacus.

Now you know how to use a modern abacus, you're ready to try addition and subtraction, by moving different numbers of beads closer or further away from the separation bar. If you get stuck, remind yourself what we did with the school abacus.

Addition and subtraction on a place value abacus

Time to take a breather. The place value abacus is much easier to use, so if you nailed the examples above, you'll be perfect at this. Set, add and subtract numbers by partitioning them, and either putting beads on the poles or taking them off.

One exciting thing you can do with a place value abacus, is keep hold of the beads you remove to find the difference. For subtractions, you can reset the abacus to the second number, and count up how many beads you needed to take off each pole to make that happen.

An abacus is a fantastic way to help your children understand partitioning and place value, and that's why we have lots of great resources for place value abacuses.

Resources for abacus addition and subtraction

If you don't have a place value abacus (or enough of them to share around the class), your children can still benefit by using a Blank Abacus Template for 2-digit, 3-digit or 4-digit numbers.

This Open Abacus Investigation doesn't need a real abacus either. You could use it as support for your abacus work, or as an alternative, paper-based way to explore place value.

We even have a complete lesson on solving problems with an abacus.

You can also use any of our resources with addition and subtraction questions to set your class some abacus challenges.

The history of the Abacus

Abacus counting has a long and rich history. To get the mathematical device’s origins, we have to look at ancient times. The ancient Greeks conceived of The Salamis Tablet, which was a slab of white marble, over one metre in length. It was a counting board with lines and symbols, used for strategising and calculating costs in markets.

The Romans later developed equivalents called the Roman Calculi and the Hand-Abacus. Sadly, many of them were destroyed because they were made out of stone. However, those made out of metal are able to tell us that Romans used a design like an abacus with vertical lines and dots.

In the Middle Ages, coin boards with horizontal lines started to appear in Western Europe and East Asia. The Suan-pan, the Chinese Abacus, was first written about around 1200CE. Unlike the Roman’s design, it was made out of wood with metal reinforcements. The design of the Suan-pan survived unchanged until 1850 when it changed to the format we know today.

Abacus counting continues to evolve, however, as it has since been turned into a portable computing device that, along with a traditional abaci, is used for education in Chinese schools and other Asian countries.