What is the final answer if I start with 4?

What is the final answer if I start with 6?

Will the answers to the puzzle above always be odd, even, or does it depend on the starting number?

Start with 6. What is the final answer?

Pick a number

﻿Hello, I'm Mr. Langton. And today we're going to be investigating the algebraic representation of odd and even numbers. All you're going to need is something to write with and something to write on. Make sure you're in a quiet place with no distractions and when you're ready, we'll begin. We'll start with the try this activity. Choose different values for n, and substitute them into these expressions. Can you organise the expressions into different groups? There's always odd, always even, or it could be odd or even. And can you explain why they always give these results? Pause the video and have a go. When you're ready we'll go through it together. Pausing in three, two, one. How did you get on? This is how I grouped them. Compare these with your answers and in a moment we're going discuss why they gave the results that they do. So we'll start off by looking at 11. How do you know that 11 is an odd number? I'm going to show you two ways that you can show that 11 is an odd number. I've represented 11 here as 11 dots. And I can split 11 into groups of two. Now if it goes exactly into two, there won't be any left over, but the fact that one there is leftover tells me 11 must be odd. Another way I can do it is split 11 into two equal groups. I've got five in that one and I've got five in that one. And once again I've got this one left over. So that's two different ways to show that 11 is odd. I'm going to look at that same thing now, algebraically. So 2n + 1 is two lots of something. So I don't know what it is, but I've got two of these Ns. And I've got two of them, then they must be even. I then got that one that's added on afterwards, and that's what makes 2n + 1 odd. I can split it into two even parts, but there's still one leftover. Now 2n + 6, I'm going to represent this like this- I've got an n and I've got another n, that's my 2 Ns. Plus six, now I'm going to do my individual numbers as dots or circles like I did before. And can you see that I can make two equal groups, each one of them is n + 3, but because I can make exactly equal groups with nothing leftover, then that means the value of 2n + 6 is always even. Finally we've got 4n + 3. So this time I've got four Ns. Now I can come up with two pairs of Ns can't I? And then I can separate my three into a dot, the dot, and there's another dot leftover. So I can't make two equal groups, I almost can. I've got two equal groups, but I've got that one leftover again. So 4n + 3 must be odd. So what we're going to do now, is apply that knowledge to these number grids. If I start with n and if I add two, I get n + 2. Now multiply that by three, I've now got three lots of n + 2, which is equal to 3n + 6. If I subtract four from 3n + 6, I've still got my three Ns, but now I've only got two. And when I multiply all of that by four- Oh, sorry, I'm getting ahead of myself. That's four lots of 3n + 2, which is equal to 12n + 8. One half it. Now we need to half my 12n and I need to half my eight. I'm going to subtract one, which gives me 6n + 3. And then I'm going to divide by three. So 6n divided by 3 is 2n. 3 divided by 3 is 1. So this is my final answer. 2n + 1. So let's see if if try and represent that as a diagram, I can split the Ns into two equal groups, I've got one leftover, can't be split up, and as such, 2n + 1, must be odd. So now it's your turn to have a go. I want you to work through each puzzle starting with n. So you try and develop a formula to link the input with the output, with that final answer. And can you say if it will always be odd, always be even, or if it could be neither of them. Pause the video and have a go, access the worksheet, and when your ready un-pause it and we can go over through it together. Good luck. How did you get on? Let's go through them together. So the first one n multiplied by 4 is 4n. If I add an eight, I get 4n + 8. If I half that I get 2n + 4. And if I add 10, I'm going to get 2n + 14. Now I can split my 2n into n and n, can't I? And I can split 14 into two groups of seven, which means I've got two equal groups, which means that this one is always even. Looks at the next one. Start with n and multiply by 10 and we get 10n. Now subtract five and we've got 10n takeaway 5. Now I'm going to divide by five, so divide that first one by five and I'm going to get 2n, divide that by five and I get takeaway one. If I add two I get 2n + 1. Which we've show before is always odd. So whatever number I start with on this one, I'll always get an odd number. Right finally. This one's much longer. I'm going to start by adding four. I'm going to multiply by two, to get 2n + 8. Now subtract eight and I've got 2n. So at this stage, this number here will always be even. Let's keep going. Multiply by three, and I've got 6n. Add 12, 6n + 12. Half it, 3n + 6. Add nine, I'm going to get 3n + 15. Divide that by three and I'm getting n + 5. Subtract five and I get n. Add one and once again I've got n + 1. Now n + 1, that's a might tricky one. Because I can't draw that very easily. I can't split it into two equal groups, but it's totally going to depend on whether n is odd or even. If n is odd, and then I add one, an odd number plus one must be even. If instead, we started with an even number, and we add one, then the answer would be odd. So we can't say for this one whether it would be odd or even, it could be either. Totally depends on what number we start with. Now this is a rather tricky explore task. What you need to do is fill in the blanks to make each one work. Sometimes there's more than one possible answer that you could use. Good luck trying to work them all out. On that last one you've got to make sure the final answer is an even number, so that's going to be really tricky. Pause the video and have a go. When you're ready, we'll go through it together. You can pause in three, two, one. So how did you get on? Here are some possible answers. Certainly in the case of the first one, you might have come up with some slightly different ones, and in particular I'm thinking here to get from 10 to 20, you may have multiplied by two, you may have added 10 instead, you may have done something slightly different. So there's more than one possible answer. It's the same here as well. Instead of subtracting 18, you might have divided by two. So there's more than one possible answer but here is something you've got that you could use. Looking at this last one here, I've left that last one blank. If you follow it all the way through to n + 4, we've got all of these steps done on the left hand side, but this last bit here, if we need to make this even, then we're going to need to multiply it by an even number. So in this box as long as you've written an even number- oh sorry I'm in the wrong box, here. You've written an even number here, then that will make your answer an even number here. That was a really tricky task. I hope you got through it. That's the end for this lesson. See you later. Good bye.

Is the value of 2n always odd, always even, or either?

Is the value of 4n - 2 always odd, always even, or either?

Is the value of 3n + 1 always odd, always even, or either?

What operation could I perform to 4n + 1 to make it always even?

What operation could I perform to n + 4 to make it always even?

Odd or Even?

In this lesson, we will investigate the algebraic representations of odd and even numbers.

Lesson overview:Odd or even?

Core Content

Algebra and problem solving: