Investigating teen numbers with 10 frames
A thinking mathematically targeted teaching opportunity focussed on using a ten-frame to notice and wonder about teen numbers
Syllabus outcomes and content descriptors from Mathematics K-10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2021
You will need:
- colour pencils or markers
- something to write on.
Watch Investigating teen numbers with ten-frames video (9:56).
[White text on a navy-blue background reads ‘Investigating teen numbers on 10-frames’. Further white text below reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team)’. In the bottom right corner, the NSW Government red ‘waratah’ logo.]
Hello there. little mathematicians. Today we thought we'd spend some time exploring the 10-frame structure, and think about how it could be really helpful in investigating and exploring the quantities and numbers from 11 to 20.
[A black rectangle on a white background is divided into 10 squares.]
Now, before we talk about those teen numbers, let's discuss the important things we know about 10-frames. What do you know about this structure? Right. They always show us 10 boxes.
[Below the 10-frame text is typed out. It reads ‘There are 10 boxes’.]
And we can trust that because it's what we call a mathematical pattern, because it always has that same structure. What else do we know about them? Take a moment to think about how it's organised. Great noticing. There's a line in the middle that separates 5 boxes on the top and 5 on the bottom.
[To the right of the 10-frame further text is typed out and reads ‘5 on the top, 5 on the bottom’.]
Now, that's a really helpful feature of this structure because it means we don't have to count every dot or every object inside a 10-frame, and we can use what we know about the number 5 and its relationships to other numbers.
[The 10-frame is rotated vertically and further text is typed on either side (as read by speaker).]
Now, even if we turn the 10-frame on its side, we can still see that there are 10, because there is still a line in the middle and there are 5 along one side and 5 along the other side.
[The 10-frame is returned to the horizontal and a second 10-frame is added below it (further steps outlined by speaker).]
OK, so since we want to explore numbers bigger than 10 today, we can use a second 10-frame to help us understand the numbers and the quantities. So, let's investigate how many boxes we have now. What clues can you see that help us to think about how many boxes there are in 2 10-frames? Yeah, I was thinking the same, it's 10, and 10 more. And yes, you're right, you can say that as 2 10s. So how many boxes are there altogether? I can hear some people thinking. Some of you are saying, that's 20. We could use counting to check that there are 20. Here we know that there are 10.
[The top 10-frame is outlined in blue and the bottom 10-frame is filled with red numerals (as explained).]
Then we can keep counting like many of you were doing, 11, 12, 13, 14, 15. 16, 17, 18, 19, 20. I also heard some counting that started from a different number. Oh, because you used another clue that the 10-frame structure gives us, that each row is 5 more. So, we know that one 10-frame is 10, and the top row of another 10-frame is 5 more, and we know that we can rename 10 and 5 more as 15, so we can just count the rest. 16, 17, 18, 19, 20. 20 empty boxes.
Now, did anyone know that there were 20 in another way? Oh right, yes of course. One 10, 2 10s, well that can be renamed as 20. Great. OK, so let's take a closer look at some of those teen numbers and explore how the 10-frame structure helps us to represent the quantity. I'm going to show you a number represented on the 2 10-frames and I want you to tell me how many there are and how you know. So, what clues are you using? Ready? OK.
[Red circles appear in the boxes of both 10-frames – 10 in the top 10-frame and 4 more in the bottom 10-frame (as explained by speaker).]
Yes. I can hear lots of you saying 14 and I can also hear lots of different ideas about how you know the total and what clues you used. Nearly everybody saw the 10, because the first 10-frame was 4 and some people said that they knew the second 10-frame was showing 4, because the row was almost full, which would have been 5, but one was missing. Then they renamed 10 and 4, as 14. Some other people said they saw the 10 again, but double 2 this time and knew that that was 4 and again, they renamed 10 and 4, as 14.
[Two vertical 10-frames side by side (further steps explained by speaker).]
I'm going to show you another number represented on 2 10-frames, but you might notice that the 10-frames are now in a different direction. But because it's a mathematical pattern that we know and trust, we can still think about numbers using this 10-frame structure in exactly the same way. So, let's have a go. I want you to tell me how many there are now and how you know. Remember, you also need to share what clues you used to find the total. Ready? OK.
[The left 10-frame is completely filled by red circles. The right 10-frame is filled except for one empty box.]
Yes, I can hear lots of you saying 19 and I can also hear a lot of different ideas again which is great. It means that we know that the best thing about numbers is that we can think about them in different ways, using different relationships. So, everyone saw the 10 in that first full 10-frame, because it was full again and then some people said that they knew that the second 10-frame was showing 9 because the 10-frame was almost full, which would be 2 10s or renamed as 20, but one was missing. So they renamed 10 and 9, as 19.
Now, some other people saw that there were 10 in the first 10-frame, and then 4, and 4 again and knew double 4 was 8 and one more is 9, and renamed 10 and 9, as 19. I also heard some other people reason that they saw the full 10-frame of 10, 5 because there was another full row and then the 4, because it was almost a full row, but one was missing, and again renamed 10 and 9, as 19.
[White text on a blue background reads ‘Let’s investigate!’]
Now, I hope your brains aren't too sweaty yet, because there is something else I want your help with investigating. When I ask some other little mathematicians about some other numbers from 11 to 20 shown on 2 10-frames, there was a little confusion around 2 different representations.
[On the left, a 10-frame is completely filled by red circles and another 10-frame alongside it is filled with 6 red circles. On the right, a 10-frame is completely filled by red circles and another 10-frame has only one red circle inside it (further explained by speaker).]
This representation of 16 and this representation of 11. So let's talk about what's the same and what's different about these representations. First of all, I want you to have a think about, what do you notice is the same? Yes. They both have a complete 10-frame and are made up of one 10. What else do you notice? Oh that's right, they both have a row or a column where there is one dot and the rest of the boxes are empty in that row or column.
[Red text below (read by speaker).]
Alright, well now let's think about what are some important differences about these representations. Yeah, that's some great noticing, that first representation of 16 has 3 5s. One 5, another 5, so 2 5s, and another 5 which is 3 5s, and then the one more. Whereas, the second representation has one 5, 2 5s and then one more. What else did you notice that was different?
Wow. What a great noticing. The first representation only has 4 more to make 2 complete 10-frames. And some people explain that as 16 only needing four more added to it to make 20. But when we look at 11, we can see that it needs 9 more to make 2 complete 10-frames. Yeah, that means 11 is 9 away from 20, or 11 and 20 have a difference of 9.
[Black text reads ‘Let’s compare! Below, two sets of 10-frames (as explained by speaker).]
OK mathematicians, now it's your turn to compare. Have a look at this representation of 12 on a 10-frame and this representation of 17 on a 10-frame, and I want you to think about what is the same and what is different. Over to you.
[White text on a blue background reads ‘What’s (some of) the mathematics?’]
[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker).]
So, what's some of the mathematics? We know that we can think about and describe teen numbers as one 10 and some more. We can also say 10 and some more, or be even more precise like with 14, and say 10 and 4 more, or one 10 and 4 more. We also know that numbers can be renamed in lots of different ways, just like us.
[Black text on a white background reads ‘What’s (some of) the mathematics?’ Further text below (read by speaker) On the right, a cartoon image of a brown-haired woman.]
For example, this is my mum. I call her mum. Dad calls her Linky Lettuce. My brother calls her mumsy, but my granny calls her Lyn. All of these names all still mean this person, my mum. Numbers are like that too. We can talk about them in lots of different ways and it still means the same amount. Maths is pretty cool like that, huh?
[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]
[End of transcript]
Use your student workbook or a piece of paper to write or draw your noticings.