Let's explore – number talk (23-19)
A thinking mathematically targeted teaching opportunity focussed on reflecting on and identifying the most effecient strategy to solve 23 - 19
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, 2023
You will need:
- pencils or markers
- something to write on.
Watch Let's explore – 1 video (5:12).
OK mathematicians, let's explore.
So, one of the strategies that we uh shared was counting back in 10s and ones to solve 23 - 19.
Now some of you might know this as the split strategy and the jump strategy. We're using a combination of them.
Some of you also know, that what we're actually doing here is partitioning the numbers into chunks of 10s and ones so using knowledge of place value too. And knowledge of the backward count.
So, let's have a look at how many steps at talk to solve this problem.
So, here's the 2 tens and 3 representing 23.
[Screen shows 23 boxes placed horizontally across the page. The first 10 boxes are red, the next 10 are blue and the last three are red. Underneath the boxes is an empty number line.]
And here's a number line underneath to help us keep track of the number of steps that we're taking.
So, the first thing that we did was to take away a big chunk of 10.
[10 boxes are removed leaving 10 red boxes and 3 blue boxes.]
And we can record that here and so that means we've taken one step.
[A curved line is drawn from the right side of the number line to signify the subtraction of the chunk of ten.]
Then what happened was to remove the other 9.
And so, this is what that looks like.
[The presenter draws 9 curved lines to represent the subtraction of 9 blocks. As each block is taken away, the presenter adds an extra curved line or ‘jump’.]
Yeah, and that's now the second step and we take away a block.
We do the jump.
That's the third step, another block, another jump.
That's the 4th step, and so we keep going on until.
Yes, we removed all 9 ones.
And that's what it looks like ah uh.
So that was one strategy, counting back in 10s and ones.
Let's have a look at another one that we used.
Let's explore it further.
[Screen displays 23 boxes placed horizontally across the page. the first 10 are red, the next 10 are blue and the last 3 are red.]
So, in this one we adjusted the numbers.
So, we used our knowledge of landmark numbers as well as using known facts and some other bits and pieces of mathematical knowledge.
So far this shows 23 and we haven't taken any steps yet.
[Screen shows an additional box being added to the right of the original boxes. Underneath the equation reads 23 minus 9 equals 24 minus 20.]
And then what happened was we could re imagine the problem because we know that 23 - 19 is equivalent in value to 24 - 20 and so now we're going to record this on a number line to show, actually there is a little jump there, to increase, uh huh, the quantity in our collection from 23 to 24.
[A blank number line is drawn between the horizontal row of boxes and the equation. The presenter creates a curved jump towards the right arrow of the empty number line to show the increase from 23 to 24.]
So, there was one step and then a giant step of removing 20 uh-huh.
[Screen shows 20 boxes being removed leaving only 4 red boxes. The presenter draws a curved line to signify the subtraction of 20 which spans the width of the 20 boxes.]
And that's what that looks like on a number line.
And that was the second step taken that got us to the conclusion of 4 things remaining.
Yes, so adjusting the numbers only took 2 steps.
So now if we, compare and contrast one strategy counting back in 10s and ones took us 9 steps to solve the problem and adjusting the numbers.
[Screen shows both strategies used to solve the equation. On the left it reads: Counting back in tens and ones (‘split’ and ‘jump’), steps taken: 9 and on the right in reads: Adjusting the numbers (using landmark numbers and known facts), steps taken: 2.]
Only took us 2 steps to solve the problem, so what's some of the mathematics here?
So, what we really wanted to draw attention to today is the idea of efficiency.
So, efficiency is defined by the number of steps you take to solve a problem.
It's not about speed.
It's about the number of steps you take.
In fact, sometimes a more efficient strategy takes a little bit longer to think through as you're becoming confident in using them and as you become more confident.
It does tend to make it a little bit faster for you, because it's a more efficient strategy with less steps.
It also means that you've got much less chance of making small computational errors.
If I do 9 steps, I've got nine chances of making a small error if I do 2 steps.
I've only got 2 chances of making a small error.
So, the other thing about efficiency though, is it's a really personal.
So how efficient a strategy is to you, is based on how much you know about numbers, operations and mathematics.
You can't use strategies and knowledge that you don't know about just yet.
And 3 other things to be careful of.
Some strategies are pretty similar in their efficiency, so then you get to choose which one you would like to use.
And sometimes there's a difference of one or two steps, and you might think, well, I'm really confident, using this strategy, so I'm OK to go ahead with it.
Sometimes a strategy is efficient in one context, but not in another, which is why it's really important that we learn lots of different strategies.
Yeah, and not just get locked into one way of thinking.
And when we are tired or stressed or not feeling our mathematical bests, we might choose to use less efficient strategies as it's the best that we can do as a mathematician that day.
And that's OK too.
Yeah, so over to you now, mathematicians, I'd like you to reflect on the strategies you used to solve 23 minus 19. (Or 230 minus 190, or 2.3 minus 1.9) and identify the most efficient strategy you used. How many steps did it take? Do you think you could use the same strategy with other problems? Note your thinking in your workbook.
Or you might have been working on 23 minus 19 or 1 tens and 3 ones minus one 10 and 9 ones and identify now the most efficient strategy you used.
Yeah, how many steps did you take, and do you think you could use the same strategy with other problems and note down, your thinking in your workbook?
OK mathematicians, I hope that's helped you clarify some ideas about efficiency and until next time we meet have a lovely day.
[End of transcript]
Reflect on the strategies you used to solve 23 – 19 and identify the most efficient strategy you used.
How many steps did it take?
Do you think you could use the same strategy (keeping a constant difference) with other problems? Write 1 or 2 problems you could use this strategy with.