Doubling and halving: number talk
Stage 3 – A thinking mathematically targeted teaching opportunity focussed on exploring doubling and halving to solve multiplication problems.
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:
pens or pencils
plain paper, grid paper or dot paper.
Watch Doubling and halving video (8:44).
[White text on a navy-blue background reads ‘Doubling and halving? A number talk’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right corner, the NSW Government red ‘waratah’ logo.]
You will need some resources to create a mathematical model. You will need something to draw or write with, some scissors, some sticky tape or glue, and some plain paper, grid paper or dot paper.
[A blue text on white header reads ‘You will need…’ Four bullet points below (as read by speaker). Below, in a still colour image, a sheet of grid paper and another with rows of blue dots, a pencil, a black pen and an orange-handled pair of scissors.]
Hello there, mathematicians. We were doing some problems the other day, and one of those was exploring some of the strategies we could use to solve 25 times 8.
[A sheet of white paper on a light blue background. A person places a small square of paper onto the larger sheet of paper. It has ’25 x 8’ written on it in black pen.]
Some of you may have been working on solving 25 times 18.
[A second square is placed beside the first and has ’25 x 18’ written on it.]
And in both of these problems, we saw the same strategy being used where you double one number and halve the other.
[a rectangle piece of paper is placed at the top of the sheet. It has ‘double or halve’ written on it in black pen.]
So, let's have a look at how Policeman used the strategy double and halving in 25 times 8 to make sense of how this strategy works.
[The person places a ‘LEGO’ policeman figure on the left of the ’25 x 8’ square of paper and removes the ’25 x 18’ piece. A blank square of paper replaces it and is used for the subsequent mathematics.]
Firstly, Policeman used the commutative property of multiplication to rethink the problem as 8 x 25, or 8 25s. Then he halved the 8, which he knows is 4, and then doubled the 25, which he knows is 50. He still wasn't sure what 4 50s was, so he doubled and halved, or halved and doubled, again. So, half of 4 is 2, and then double 50 is 100. Then he could use renaming to know that 2 hundreds is 200.
Now, we know that symbols don't really give us the understanding of what's happening. They're good for recording, but not necessarily good for climbing inside someone's brain. So, let's use an array now to climb inside Policeman's brain.
[On a new sheet of paper, a small rectangle in the centre has rows of purple dots on it. In the upper left, the previous mathematics to reach 200 on the small square of paper.]
Here we have the 8 x 25, which means I have 8 rows, and there's 25 in each row. 8 x 25.
[‘8 x 25’ is written below the rectangle in green marker pen.]
The first thing we saw Policeman do was halve the 8. So, I can cut this array here to show how he halved 8.
[The rectangle of dots is cut lengthways with the orange-handled scissors. The two halves are then placed next to each other.]
And now I've got 4 x 25, and another 4 x 25. And if I move this part of the array up here, may have to slide this down to see if we can fit this in. I think I might even sticky tape this one together. 8 x 25 has been reshaped or reformed into 4 x 50. So, I just accidentally doubled the 25. So cool. And now we can also see that 8 x 25 is equivalent to 4 x 50.
[‘8 x 25 = 4 x 50’ is written in green marker pen in a line below. ]
There's been no new dots added and none have been taken away. So, the array now looks different and we describe it differently, but it still has the exact same number of dots. So, because Policeman wasn't yet confident with knowing and being able to recall a number fact for 4 x 50, he said, “I'll keep doubling and halving”. He halved 4, which is 2 and we can show that here.
[The rectangle of dots is cut once again lengthways and placed side by side. ‘= 2 x 100 = 200’ is added to the line below.]
And then the array reforms into 2 x 100. Wow. Again, we see it doesn't matter how we formed and reformed the array. It didn't change the total number of dots.
[The cut rectangle is replaced by a smaller rectangle filled with rows of tiny dots. A black pen is used to write ‘8’ down the left-hand side and ‘25’ at the bottom.]
Let's see how that plays out with an array, a teeny tiny array, of 8 x 25 that will fit into our frame. So, Policeman started with 8 x 25. And then halved the 8, which reformed it into 4 x 50.
[A second longer and thinner rectangle of dots is placed below the first. A black pen is used to write ‘4’ down the left-hand side and ‘50’ at the bottom.]
And then halved the 4. And when he halved the 4 to 2, it suddenly doubled the number in each row to 100.
[A third even longer and thinner rectangle of dots is placed below the second rectangle. A black pen is used to write ‘2’ down the left-hand side and ‘100’ at the bottom.]
So, you could apply the same thinking if you didn't know, for example, 6 x 8.
[On a blank sheet of white paper a black marker pen is used to write ‘6 x 8’. Further steps explained by speaker as they are written down.]
You could apply it because you could double 6 to get 12 and then halve 8, which is 4, and then double 12, which is 24. And then halve 4, which is 2. And then double 24, which is 48, and then halve 2, which is one. And now you have 48 ones, which is 48.
So, now over to you, mathematicians, to create a model to show how you could use the doubling and halving strategy to solve 6 x 8. You might like to use grid paper or some dot paper, or you could even create a comic strip to show all of the steps of using the doubling and halving strategy to solve 6 x 8.
[A small sheet of grid paper and a small sheet that has rows of red dots on it. A larger sheet has a comic strip of 6 squares with simple black pen drawings in each square.]
Have fun being creative. Over to you, mathematicians.
[White text on a blue background reads ‘Over to you to create a mathematical model to show how you could use doubling and halving to solve 6 x 8.’]
What's some of the mathematics?
[White text on a blue background reads ‘What’s (some of) the mathematics?’]
[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ A bullet point below (as read by speaker) and two of the small squares of hand-written mathematics from earlier.]
When multiplying two numbers, if we double one number and halve the other, the product still remains the same.
[A second bullet point (as read by speaker) and the dissected rectangle of dots from earlier with hand-written mathematics below.]
Reforming or reshaping an array doesn't change the total. This reminds us that quantities like 200 can look different and still have the same value.
[A third bullet point (as read by speaker) and three images below of the two different rows of dots and the comic strip from earlier.]
We can use models and representations to help us make sense of and explain mathematical ideas.
[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]
Create a mathematical model to show how you could use doubling and halving to solve 6 x 8. You could use grid paper, dot paper, or even draw a comic strip!
Find 3 examples where doubling and halving isn't an efficient strategy.
Why is it more efficient for some problems and not for others?