Renaming is critical

My PhD research into place value identified the need to focus our teaching around six key aspects (you can read more about this here): Count, Calculate, Compare/Order, Make/Represent, Name/Record and Rename.

While each aspect is important, for me, the most critical is renaming.

When I work with teachers, renaming is the place value aspect they are the least familiar with, and thus the least confident to teach. For many of us (me included) renaming was not taught to us at school, so it is a skill we need to spend time learning before we can teach it to our students.

Whilst I, (and others such as my PhD supervisor Professor Di Siemon) have been advocating for the importance of quality instruction around renaming for many years, it is pleasing to note that in Version 9.0 of the Australian Curriculum, rename is specifically mentioned in Level 2,3,4 and 5. This highlights to teachers the importance of this skill in both whole number and decimal contexts.

Standard and Non-Standard renaming

Essentially renaming is rearranging a number into a different form without changing its value. We can rename numbers in standard and non-standard ways.

For example, the standard ways to rename 345 are:

3 hundreds, 4 tens , 5 ones

3 hundreds, 45 ones

34 tens, 5 ones

345 ones

There are also many non-standard ways, such as:

33 tens and 15 ones

30 tens and 45 ones

2 hundreds 11 tens and 35 ones

Both standard and non-standard renaming build upon a deep understanding of part-part whole. For example, to rename 45, students must have the flexibility to think of 4 tens as:

 1 ten and 3 tens

 2 tens and 2 tens

 3 tens and 1 ten

They then need to recognise that each ten is equivalent to 10 ones and apply this knowledge to rename the number. For example, 45 is 3 tens and 15 ones.

To distinguish between standard and non-standard renaming I think of a number expander.

Now, I would like to point out here, that I am not a big advocate for using number expanders to teach renaming. I know some people like to use them, but for me, as a learner, they never helped me to conceptually understand renaming- they just ‘magically’ gave me the answer (and there was a lots of cognitive load taken up correctly folding)!

For this reason, I don’t use them in my teaching. I consider this more a personal preference, than a guideline to follow, as I acknowledge that there may be students in your class for whom numbers expanders provide a useful visual cue. I just wasn't one of them!

It is important to remember that a number expander is an abstract representation of renaming, and it is important that you also support students to develop their understanding through concrete representations such as base 10 blocks.

Another constraint of a number expander is that they only show students standard forms of renaming. This is because the same digits remain on the expander and students simply open and close folds to rename the number. Thinking of this helps me to distinguish between standard and non-standard renaming. Standard renaming can be found using a number expander. Non-standard cannot, because the digits change. It is important to allow our students to explore both standard and non-standard renaming.

Whether we rename in a standard or non-standard manner, the value of the number remains the same. The form in which it is presented changes.

To help students to better understand the idea of renaming, I make links with renaming files on the computer. When we copy a file, we can choose to rename the file. In this process the contents of the file remains the same, it is simply the file label that changes.

Renaming is particularly difficult for students to master as it requires a deep understanding of our “10 of these is one of those” place value system. It also requires multiplicative thinking- something we know takes time to develop in students.

Renaming requires students to understand that there are smaller units “hiding within” larger units. For example, to rename 420 in term of tens, a student must understand there are 2 tens in 20 but there are also 40 tens “hiding inside” the 4 hundreds. When asked how many tens are in 420, children are accustomed to focusing their attention on the tens column, as they believe this is the only place the tens “live”. It is this narrow focus that leads students to become ‘Independent Column Thinkers’ (Rogers, 2018). Students who think in this way, fail to appreciate the multiplicative relationship between columns, making it near impossible to rename.

For this reason, I always encourage teachers to emphasise the word altogether to encourage students to think more broadly when renaming a number. For example, how many tens are there altogether in 420?

Without doubt renaming is the most difficult place value aspect to teach and learn. It is a very abstract concept, therefore visuals are critical. I like to introduce the idea of renaming using Russian Babushka dolls. These dolls allow students to see the smaller dolls hiding inside the larger, just like in place value where smaller units like tens ‘hide inside’ hundreds or thousands. I encourage students to look in each column in a number and consider which units are ‘hiding’. For example, in 420 I would ask:

 “Are there any tens in the ones column?”, No, they are too big to fit in there.

 “Are there tens in the tens column?”, Yes, there are two.

“What about the hundreds column?” Yes, there are 40 tens hiding.

Therefore, we can rename 420 as 42 tens. We want students to appreciate that the value of the number remains as 420, it is simply written in a different form. Analogies can be made here- with their nicknames- for example, I can be renamed as ‘Ange’, ‘Dr Ange’ and ‘mum’, but I am the same ‘value’.

Similarly, whether 420 is written as 42 tens or 39 tens and 30 ones, its value remains the same.

Make the links

Algorithm

We must also make students aware of the contexts where renaming is useful. One such context is when completing the formal algorithm. Teachers often use terms like “borrow”, “trade” or “carry” when modelling the algorithm (because that is what we learnt growing up).

I always ensure I use the term ‘renaming’ when modelling the algorithm. This helps students to see that the renaming they have been ‘doing’ in place value, is very useful and can be applied in the algorithm. In the subtraction algorithm example shown below, I would move slowly to unpack the act of renaming and model the following place value language:

Can I take 7 ones from 5 ones? No (not unless I move into negative numbers)

At the moment I have 5 ones in 135, how can I rename 135 so I have more than 5 ones? (You could rename one of the tens as 10 ones)

How many tens do we have altogether in 135? (13 tens)

OK, let’s rename the one of the tens as 10 ones (see image below)

What is the value of the top number now that we have renamed? (The value has not changed- it is still 135)

How many hundreds are in the hundreds column now? (one hundred)

How many tens are in the tens column? (two tens)

How many ones are in the ones column? (15 ones)

Great, can we take 7 ones from 15 ones? (Yes, it leaves 8 ones)

Record 8 ones

Can we take 1 ten from 2 tens? (Yes, it leaves 1 ten)

Record 1 ten

Can we take zero hundreds from 1 hundred (Yes, it leaves 1 hundred)

Record 1 hundred

The consistent and accurate use of the word renaming allows students see how the algorithm links directly to their place value knowledge. Students also come to appreciate that when we rename a number in an algorithm we are not changing its value, we are writing it in a form that is simpler to compute.

Converting Units

Renaming also has important links with the work we do when converting units of measurement.

For example, if we are going to convert 230cm to metres, we can use our knowledge of renaming. We are essentially asking how many hundreds (because there are 100cm in a metre) are there in 230? There are, of course, 2.3 hundreds.

If we were to consider how many centimetres are in 520mm, we are asking: How many tens are in 520 (because there are 10mm in each centimetre). The answer is 52. In both cases we have not changed the value of the original measurement, we are simply using a different unit as the reference point. This can be clearly seen when we rename time. We can rename 49 days as 7 weeks, or 1 year as 365 days (assuming it is not a leap year!). Again, we are not changing the total amount of time, we are simply changing the units we are using to describe the time.

Another example of this is at the petrol station.

When we see the price, it is presented in the unit of 'cents' (because once upon a time the prices were less than $1!). You can see in the image below that currently the price of unleaded petrol is 190.9 cents per litre. We could rename this in dollars as $1.909 because we would be thinking 'how many hundreds are there altogether in 190.9'?... this is because there are 100 cents in each dollar.

In summary, a deep understanding of renaming takes time to develop. It is a very complex skill within place value for both teachers and students. However, it is through renaming that students can develop a deep understanding of our multiplicative base 10 system. I encourage you to provide repeated opportunities for students to rename so as to develop their skills and confidence in this critical aspect of place value.

So this week as you drive past the petrol station, I encourage you to have a little think about renaming and some of the different contexts in which we can apply this skill!