### 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.

__Tasks to help develop understanding__

Use an abacusUse 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.

__NRICH Tasks__
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 PatternsPoly 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 AbacusThe 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.**

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