Numbers can be represented using standard base-ten groupings and non-standard groupings.
Different representations of the same number are equivalent in value.
Whole class
Lolly Shop PowerPoint
A large quantity of Unifix or interlocking cubes (ideally 2cm).
Each student
How many? Student sheet
Task
Use Lolly Shop PowerPoint to continue the story of Ms Fizz:
Ms Fizz is writing down how many boxes, rolls and loose lollies she has for four different collections in her shop. This is what her record chart says:
| Total number of lollies | Boxes | Rolls | Loose lollies |
| 1 | 9 | 14 | |
| 2 | 0 | 4 | |
| 1 | 8 | 24 | |
| 1 | 5 | 54 |
Ms Fizz needs to record the total number of lollies in each row.
Pose the problem: What is the total number of lollies in each row on Ms Fizz's Record Chart?
Renaming
I guess the most important thing is that renaming is a really critical concept in teaching and learning place value. It's a multiplicative idea, renaming, and it's something that probably we weren't taught when we went to school and so it's hard for teachers to know the importance of it and how to teach it and what children need to understand.
But I would say that renaming is really critical in children coming to understand place value as a whole.
And when would you introduce that idea? How early on would you introduce that idea of renaming? Maybe how would you do that?
Yeah, pretty early, so I'd definitely be doing it with two digit numbers and looking at two digit numbers, asking children to say, you know, let's say we've got 42, how many different ways can you make 42?
So, getting them to explore that it could be 3 tens and 12 ones, or it could be 2 tens and 22 ones. Just getting them to understand that equivalence, that it's still the same value. But we're just writing it or representing it in different ways.
I always talk about…I call myself Ange. You know, some people call me Mum. There's not many people that call me that - there's four! But so, you know, I'm also called Mrs Rogers. I'm called Dr Ange. So I have different names, but I'm still the same value. I'm still the same person and those different representations, you know, they have applications in, in different places.
So, you know, Mrs. Rogers is when I'm a teacher, usually Mum’s when I'm at home. And so when we look at numbers, we can think about, you know, we can represent them in different ways, rename them in different ways. And that helps us in different scenarios. So, you know, if we're, for example, doing an algorithm, it might be easier to think of 42 as 3 tens and 12 ones. That might help us.
So it's still the same value, but that different representation helps us to understand and think about that, that number in a, in a more efficient way.
This task explores the idea that a number can be represented in different, yet equivalent ways. When we represent a number differently, we are renaming that number. In this video, Dr Ange Rogers talks more about the importance of renaming and how we can teach this important idea in the classroom.
Provide students with How many? Student sheet, and allow students time to explore the problem. Have interlocking cubes available for students to use if they would like to.
Students will notice that the total number of lollies in each row adds up to 204.
- How many rolls of 10 are there in 204? Why is there a zero in the “rolls” column on Ms Fizz’s Record Chart?
- There are 20 rolls of 10 in 204. The zero represents that there are no loose rolls. Zero holds value in a number.
- How can four different representations add up to the same number?
- Numbers can be represented in different yet equivalent ways. It is possible to show that each of these representations are the same by grouping and ungrouping lollies.
- Can you see connections between the different collections of boxes, rolls and loose lollies?
- Students may notice that in each row, it is possible to make 10 tens by adding the tens that appear in the “loose lollies” column with the tens that appear in the “rolls” column. 10 tens can then be grouped to form 1 hundred.
Observe how students work out the total number of lollies in each row. Do they:
- Represent with cubes then count - If students count to find the total, prompt them to think about how they might use the rolls and box structure to support counting.
- Represent with cubes then regroup - Students may group 10 ones to make 1 ten and 10 tens to make a hundred. Ask them to consider if they are changing the value of the number by regrouping. We want them to have confidence that the different representations are equivalent in value.
- Don’t represent with cubes - Students working fluently with the numbers presented in the chart are working abstractly. It is likely that they have generalised the idea that 10 of these is equal to 1 of those. Ask them if all numbers are equivalent in value and how they can show that they are.
Zero holds value
In the Lolly Shop sequence, students explore the idea that zero holds value in a number.
Take for example the number 204. It is not uncommon for a number like this to be described as 2 hundreds, no tens, and 4 ones. But is that really the case? Let’s take a closer look.
204 is made up of 2 hundreds, but those 2 hundreds are made up of 20 tens. There are just no tens remaining after they have all been grouped to make hundreds.
So, as we talk about zero, it is important to stress that zero does hold value in a number.
Invite some students to share the strategies they used to work out the total number of lollies in each row.
Discuss:
- How many rolls of 10 are in 204? How do you know?
- There are 20 rolls of 10 in 204. These 20 rolls can be packed into 2 boxes, however this does not change that there 20 rolls. Emphasise that zero holds value in a number.
- What connections can you see between the different representations? Can you show how they each equal 204 lollies?
- For each of the rows with only one box recorded, it is possible to make a collection of 10 rolls by adding together the number in the “rolls” column and then the tens digit in the loose lollies column. These 10 lollies can then be placed in a box to create two boxes.
- How can four different representations add up to the same number?
- Numbers can be represented using standard base-ten groupings and non-standard groupings. These different representations of the same number are equivalent in value.
Pose the question: Can you find another way to represent 204 lollies?
Allow students time to explore other possible representations of 204. Record as a class some of the different representations that are found. There are many possible answers to the problem.
Connect phase
Focus this Connect phase on the fact that numbers can be represented using standard base-ten groupings and non-standard groupings. These different representations of the same number are equivalent in value. Each of the representations look different in Ms Fizz’s Record Chart, but the number of lollies in each row are equivalent in value.
We also selected numbers that would introduce the role of zero in a number. In a number, zero holds value. For example, there are 20 tens in 204. These tens have been grouped together to form 2 hundreds—there are just no loose tens in 204.
Focus this Connect phase on the fact that numbers can be represented using standard base-ten groupings and non-standard groupings. These different representations of the same number are equivalent in value. Each of the representations look different in Ms Fizz’s Record Chart, but the number of lollies in each row are equivalent in value.
We also selected numbers that would introduce the role of zero in a number. In a number, zero holds value. For example, there are 20 tens in 204. These tens have been grouped together to form 2 hundreds—there are just no loose tens in 204.