### Understanding Place Value

I have searched the internet, read numerous research articles and found plenty of tasks designed to help kids gain a deeper understanding of place value. This post contains a brief explanation of place value and why it is so important that kids really understand it. I have also included a list (with descriptions) of some of my favourite tasks. My next post explains how a good understanding of place value can be used to help kids learn to read and write numbers.

(To see a full list of available posts click here.)

What is place value?

Our number system is position based - the value of the number depends on its position within that number. We use the decimal, or base 10 system, as standard, meaning that each place has a value of 10 times the place to its right. Understanding place value is vital for understanding and being able to work with many areas of maths. It is more than knowing how to break numbers into 1s, 10s, and 100s; it is about understanding how a number is composed and knowing its relationship to other numbers.

Understanding place value precedes:
• an understanding of using zeros as place holders
• knowing what numbers after a decimal point represent
• being able to order numbers in terms of size
• an ability to estimate and round numbers
• a true understanding of how multiplying and dividing by factors of 10 work
• why we 'carry' in column addition and subtraction
• and much more...
Do kids really understand place value?

I can't remember ever being taught place value, I just developed an understanding and feel for it through many years of using our decimal number system. I don't think I had ever really properly considered what it would mean to not understand place value until I had a lesson at university that was taught in base 8 (instead of our base 10 decimal system). We were not told that it was base 8; we had to work this out for ourselves. For the first time, I really understood how students must feel when we ask them to carry out calculations when they do not really understand place value.

I think that most people get a natural feel for place value, as they use numbers every day. However, time and time again I have seen examples of students' lack of understanding of place value. For example:
• If you can simply add a zero when multiplying 3 by 10 to get 30, why can't you just add a zero when multiplying 3.3 by 10 to get 3.30? Here, there is no understanding that the zero on the end makes no difference to the value of the number, as the rest of the numbers are still in the same position so represent the same value. There is also no sense of the value that a number after the decimal point represents.
• 564 rounded to one significant would be 6. I have seen answers like this numerous times, even after lengthy discussions of how rounding is giving a rough idea of the size of a number. The students do not understand that the 6 needs to remain in the same position of the number to keep its value and that you need two zeros after the 6 as place holders.
• 8 + 3 is 11 so 0.8 + 0.3 is 0.11. I have seen many students struggling with column addition of numbers (and not just decimals) as they do not understand why you carry numbers to the column to the left. A true understanding of place value would help them to understand this.
These are just a few of many examples of where students struggle with place value and they have made me passionate about finding ways to help kids really understand our number system and how place value works. I have known students about to sit their GCSEs who were unable to place numbers on a number line. Surely if they had grasped a better understanding of place value at a young age, their progress in maths would have been made easier throughout their school years.

Use an abacus
I believe that using an abacus is such a powerful tool in helping kids gain a deep, fundamental understanding of the number system that they use every day. If they start at the bottom, once they have counted to 10 they cannot count any more. So, they realise (with a bit of discussion) that they can use the balls in the next column up to keep a record of how many times they have counted to 10 on the bottom row. This will give them a visual representation and a natural feel for how each ball in the second row represents 10 times each ball in the first row.

This pattern can be followed on up the abacus and can be extended in so many ways. If you turn the abacus sideways, it links directly to how we write numbers with pen and paper. This also allows kids to see why we need zeros to hold place value.

The best thing about an abacus is you can add a decimal point in the middle of two rows. This really helps kids to visualise what a number after the decimal point means. They can see that 10 balls in the 'tenths' row are needed to make up 1 ball in the units row, and that 10 balls in the 'hundredths' row are required for each tenth. It is then easy for them to come to the conclusion themselves that 3.3 and 3.30 are the same number, as the zero just means that there are currently no balls counted in the hundredths row.

Of course, we need to talk about when a zero is required to hold place value and when it is unnecessary (we need the zero in 3.03 but not in 3.30), but this can all be learnt through child led discussions, just by asking them to count with an abacus.

A great extension is to get kids to perform calculations on the abacus. They could try multiplying and dividing by factors of 10, using the abacus to order or round numbers, column addition/subtraction... . By using an abacus rather than pen and paper, kids can really 'see' the patterns and often come up with the rules themselves that we are normally so quick to tell them (e.g. why we carry, why we can add zeros to whole numbers when multiplying by factors of 10 but not with decimals...).

I found a fantastic resource on primaryresources.co.uk that can be used if you do not have a real abacus to hand. What I really like about this pdf is that each ball has the value of the ball written on it (1, 10, 100, 0.1 etc.), helping kids to see the value that each ball represents just because of its position within the number. I would prefer the kids to come up with the value and write it on themselves (otherwise they are being led to the conclusion, rather than realising it for themselves) but I love the idea. The only trouble with this resource is the amount of cutting required!

Use Dienes Blocks
Dienes blocks are a fantastic resource for helping kids to visualise the value of each number. In the example picture, kids can easily see the relative value of 1 yellow block (one unit) and 1 blue square (one hundred), for example. If you do not have any Dienes blocks, it is easy to make yourself some paper sets from units to hundreds (larger numbers become more difficult).

Before giving children tasks to do with these, it is important to go through the process of counting how many little blocks are in each larger block. This way, they know that one green stick has ten little blocks and so represents 10 yellow blocks.

A nice starter is to ask the children to make up different numbers using their Dienes blocks and then discuss the different possibilities. For example, when asked to represent 23, some children may count out 23 yellow blocks, some may use 1 green stick and 13 yellow blocks and others will use 2 green sticks and 3 yellow blocks. The children can lead the discussion in how using 2 green sticks and 3 yellow blocks uses the least number of different blocks and is easier and quicker to count. You can then introduce them to the position based number system, using a table with column headers as below. You can talk about how you cannot have more than 9 in each column so will need to use the next bigger column when necessary.

If you find that the kids are struggling to get to grips with how many little blocks are in each big block, NCETM suggest starting with straws. Get the kids to count out 10 straws and tie them together. This way, they know that their pack of straws represents 10.

Although Dienes blocks are a great resource for introducing place value, I think that the learning potential is limited. It is difficult to extend it on to decimals or really large numbers. I also do not think that it really explains why each position is worth 10 times the position to the left.

There are plenty of worksheets and tasks available on the internet if you need inspiration or ideas for what to do with your Dienes blocks. TES resources is just one place where you can find some.

If you have never come across nrich, I would strongly recommend it. It is a website packed full of maths resources for all ages and abilities and across all topics. The resources are designed to enrich mathematics learning and are often investigative and problem solving tasks.

Some of my favourite place value tasks are listed below. I believe that these tasks allow children to investigate and discover the way our number system really works:
• 6 beads - this task asks students to investigate the numbers that they can make using 6 beads on a tens/units abacus. This task really requires students to understand that putting a bead on the 10s means that it has a different value to putting it on the units.
• Five steps to 50 - this task requires students to move in jumps of 10s and 1s to get as close to 50 as possible in just 5 steps. Although it doesn't link directly to the position based number system, it gets kids thinking about 10s and 1s and the fact that adding a 10 makes a much larger difference than adding a 1.
• Largest Even - This task asks students to find the largest possible even number given specific numbered cards. It gets kids thinking about the fact that having a large number in the 10s column is much more important than having a large number in the units column, again leading kids to think about the different value that a number has depending on its position within the number.

Poly Plug Patterns
I also got this idea from NRICH. A Poly Plug is essentially a 5 by 5 grid and counters of different colours. When asked to make numbers up to 25 using their grid and counters, kids are able to do so using a 1 to 1 value. However, ask them to make larger numbers and they will start using different coloured counters to represent 2, 5 or 10. This task allows kids to discover for themselves that something can be assigned a different value based on its properties - in this case colour. You can then take this understanding and extend it on to our number system by putting the different coloured counters in different columns (larger values to the left). You can then get rid of the different colours and get the kids to discover that they know the different values based on the position of the counters.

The Universal Abacus
I found the universal abacus on GeoGebra. It is suitable for older children who already have some understanding of place value but can be used to really deepen and embed their knowledge and understanding. The abacus allows you to represent numbers of any base and really enables you to see how a position based number system works.

By playing around and trying to represent different numbers using different bases, students can discover how each position is related to the position to its right. For example, in base 8, the most right hand position (if there are no decimal numbers) is worth a unit, whereas the position to the left is worth 8 units, and the position to the left of that is worth 64 units (8 times again). Each position is worth 8 times the position to its right.

I strongly believe that kids gaining a deep understanding of place value early on can help overcome so many difficulties in the future. We teach kids numbers and how to count, we teach them the patterns of our number system and how to write the numbers down. It is all too easy to miss the opportunity to teach them a true understanding of place value before they gain all their own ideas and misconceptions. Let’s get kids counting with an abacus, investigating our number system at a young age and leading their own learning towards a fundamental understanding of place value and our decimal system.