Fill in the gaps: Multiplication by a ___________________ can be visualised as a stretch in the ___________________  direction.

Calculate 2 x (-9)

Calculate (-2) x (-9) x (-1)

Given n=-3 what would the value of 4n be?

Given n=-3 what would the value of -4n be?

The product of two numbers is 20. What could the two numbers be?

Further Multiplication Quiz

﻿Hello, my name is Mrs Buckmire. Today I'll be teaching you about division. Now first, make sure you have a pen and paper. Remember to pause the video when I ask you to, so that's when I want you to consider or think of some ideas or have a go at question. But also pause whenever you like, if I'm going too fast or you need some more time to do something, copy something down, just pause it then. Also if you need to rewind, if you need to hear it again, that's really good thing to do because that can help you to understand it just by hearing it again. Okay, let's begin. So for your try this, I have a grid here with lots of numbers on it. Now they haven't been picked out randomly. There are some patterns and I want you to spot the pattern. So what I want you to do, is for each of the numbers in the blue row, and this is the blue up here. I want you to write each numbers in the form of three times the number. So for example, for negative nine, well, that's three lots of negative three. And then you do three lots of, for each one working them out. And then can you write using the numbers in the red row, this row, as negative two times something. So for example, for negative two, I could write negative two. Let's write in black. Negative two times one and do that for each one. And then have a go at writing each of the other rows as a multiple of an integer. I'm not going to give you the integer, but you're going to think of it yourself. Again, so do pause the video and spend some time on this task. So here you have all of the answers, so hopefully, you weren't working on each one, when you started to spot some patterns. So do pause the video and study this carefully and check your work. And if you didn't quite understand it, maybe copy out certain lines and just write about what you notice. So I wanted to link with you our stretching model with division as well. So here we have negative eight, so that's four, lots of negative two. So it's kind of like we have negative two that has been stretched by a scale factor of four. And we can also see that here. So if I put this as repeated addition, well there there's negative two, negative two, negative two, negative two. So that links with division because negative eight can be split into four to get negative two, good. We can see the arrow is still going left, so it's still go negative. And each one is of length two, which we can also see. So like if I changed my a to let's say three, so having three, lots of the negative two. That means we can see the link between negative six divided by three equals to negative two. So here I'm just using GeoGebra app. I think Mass Mastery made this one. So when it is positive, we can see it's in the positive direction. And when it's negative, we can see what's going on. Let's make b equals negative three. So here we have negative three lots, three lots of it by negative nine. So negative nine divided by three equals negative three. So that's what we're going to do more of today. It's kind of linking our multiplication box with our division as well. Okay, so from that, we can actually form calculations that connected to this image. So for example, we have negative 12 and you can see there, that negative four is scaled by a factor of three. So we have negative this, would be worth negative four, and then we have three lots of it let's even write at the end this being negative four. And so therefore if that was the case, we could have the sum as three times negative four equals negative 12. But we could also actually see negative 12 has been scaled. How has negative 12 being scaled? Excellent, by a third. So we can also have a third times negative 12 equals to negative four. So link into that, we could actually create the division. So actually negative 12 divided by three equals negative four. Cause we can see if we split negative 12 into three equal parts on each one's negative four. And we could also actually write it as negative 12 over three. So that's the same as negative 12 divided by three. So I want you to describe two parts and a division, linked to the following. So linked to this diagram and linked to this diagram. This sentence then could help. So I can see that something is scale by fact or something. So that might help you with multiplication. And then you can derive a division factor from that. Or looking at a diagram, thinking about how it matches. It might be helpful to actually write in what this could be worth. Pause the video and have a go. So for A, now I'm going to see that well, it's divided into three. So 30 divided by three is actually positive 10. So I can actually have 10 times three equals 30. Or I can think of it as it's being scaled, 30 has been scaled by a factor of a third as a third times 30 equals 10, as well. As a multiplication factor A shows with 30 divided by three equals 10. So what about B? Yes, and B is negative. So actually this one would be negative 31. So I got three times negative 31 equals negative 93, or you could think of it as being negative 93 as been scaled by a scale factor of a third. So we get negative 31. So negative 93 divided by three equals negative 31 is the division factor used. So for your independent tasks, I want you to fill out these related calculations for A, B and C. Then for question two. And I want you to evaluate each of the expression below with the value of, n at the centre. At the centre of the network is n equals negative 12. So substitute those into the different expressions and work them out. You might see some relationships between some of them that might help. If you like, you could draw a diagram that might help you as well. So make sure you're do page one and page two. So for the first row I got negative eight. I got 0.5 or half, and I got two. On our next row, what did you get? Excellent, three, negative nine, and negative three. And finally, yes, negative one. Positive one fifth. So make sure that's a positive one fifth for C, negative five times a fifth. And we could also write the last one out. As negative one divided by five equals negative one fifth. And you might have slightly different answers, depending on what do you interpreted bear. I want to go through this one more thoroughly. So, because we haven't done a lot of substitution recently. So if I'm substituting n in instead of n, I'm going to have negative 12. So negative 12 divided by three. So I have negative 12, three equal parts. So each one's going to be negative four. So a third times n, so third times negative 12. So if I have negative 12, and then a scale factor or a third while it's becoming smaller. So actually this is the same as this one. These two are equal to each other. So also get negative four. So that's why I'm saying about relationships, it can make it easier. So here, n over three times negative one, well, n over three actually is equivalent to n divided by three. So this is the same as this answer, except your timesing it by one. So negative four times, sorry, negative one times negative one equals to four. So actually it ends up as positive four. N over four. So we had negative 12 divided by four. So there's my negative 12, four parts. Four equal parts even. So it's going to be negative three. N over four plus 15. What I already worked on negative three is n over four. So this is equals to negative three plus 15, which equals to positive 12. And finally, n divided by two. So negative 12 divided by two. I know that it is negative six. Well-done if you got those answers correct as well. Do feel free to rewind and listen to it again if there's anything you're not sure of. So for exploring I want you to consider each of the following statements and decide they're always, sometimes or never true. Now, here b represented any number. So the first one is b divided by three equals a third times b. Is that always, sometimes, or never true? What about for b over 10, less than zero. And b over two great then b. So for each separate one, try out some numbers and then just try and think about it. Maybe using your mathematical sense to think, is it always, sometimes, or never true? Have a go now. If you weren't sure I thought I'd give you some tips. So for this first one, I would actually draw a diagram. So for example, if this is my b, and I'm divide it by three, so I've got kind of parts like this, and then thinking about, okay, if I had a third time b, what would that look like? To think about if a third was a scale factor of b, what would that look like? And try and reason through that. With this one, I would probably try out some different value. So I try out like, when b equals to maybe two, maybe four, maybe negative two, negative four, and just substitute it in. Find out what the left-hand side equals. Find out what the right-hand side equals. And is the right-hand side less than the left-hand side or not. Try and think, oh, when is it? And when it's not? Maybe you can even come up with an inequality for it. If you didn't, and you're hearing this now, maybe go back and actually do think of an inequality. And this once again, I would just try out some values maybe. So maybe try out 10, maybe try out zero, maybe try negative 10. So you see how I kind of pick out a number and then just play around. Zero is always a fun thing to try out. So even here you could try out zero. It's just because it's a bit of a different number. It's not, you know, positive or negative. And so it's quite a nice one to try. So you have a go on substituting those values in and make some decisions. So for b divided by three equals a third times b. Now, hopefully my diagram helped you. So I'm going to do it again because it helps me. So I'm dividing it by three. So here are my three equal parts. So here I have, if we have b divided by three, but it's the same as having a third b, with a scale factor of a third. Because then actually we know its shrinks it by, times one third. So actually these are always true. They're always equal to each other. So what about b over two, great than b? What did I say to try, minus four was one of them. So minus four over two equals, to half of negative four, sorry. Equals a negative two. Is negative two great than negative four? Yeah, true so definitely works sometimes. Let's try positive four. So if four divided by two equal to two, is two great than four. So doesn't work. I've found the time it does work. I found the time it doesn't work. So it means it must be sometimes true. Again, for this one, I think I tried 10, zero and negative 10, maybe. So let's try 10 first. 10 divided by 10 equals to one. Is one less than zero? Let's try zero. Well, actually I can seen zero divided by 10 is zero. Which equals zero. So no, it doesn't work. Does it ever work? And what was the other one I said we could try, negative 10? So negative 10 divided by 10. So I have negative 10 this way, and it's divided by 10. Let's say there were 10 of these. Yeah, it was going to be negative one, isn't it? Still in the negative direction. Each one's one is negative one less than zero? Yes, so again, sometimes true. Well-done if you got that. Here are some official ways of writing them. So if you used inequality, you can check your inequality carefully. Well-done everyone, if you had a go to try this, you listened in on the connect, maybe made some extra notes and you had to go at the independent tasks and explore. I think you've done a fantastic job today. So really try and remember the relationship between division and multiplication, how actually we can use diagrams to really help us out. And we can derive facts from multiplication about division. I hope to see you again in another lesson, have a lovely day, bye.

Have a go at the questions below to show off what you know.

Use the diagram to select the 2 correct calculations.

Calculate (-123) ÷ 3

Calculate (−24) × (1/8)

Given n=3 what would the value of (-15) ÷ n be?

Use	a selection of the integers: −9 ,8 and −4 (without repeats) what is the  largest absolute value possible from this expression?

Will this calculation be positive or negative?

Division Quiz

In this lesson, we will explore a model for division involving counting backwards and analysing a number line.

Maths

Y7 Maths

Key Stage 3

Year 7

Division

Positive and negative numbers

Number

THEME

UNIT

Negative numbers in context

Order and absolute value

Modelling Addition

Further Addition

Subtracting positive numbers

Subtracting negative Numbers

Multiplication as scaling

Negative scale factors

Further multiplication

Further division

Axioms and negative numbers

Calculate (-29) + 29

Which word best completes the gap: -14 and 14 are additive inverses as they sum to ____

Which calculation gives the smallest answer?

True or false: The absolute value of a number is always greater than the number itself.

Find the absolute value of the number 0.

Which word best fills in the gap in this sentence: We can think of addition as _____________ a point on the number line.

What is the additive inverse of -4.2?

Joanna thinks the following calculation pair are additive inverses: 10 + (-50)  + (-18) and (-10) + 50 + 18. Is she correct?

Calculate (-2) + (-5)

Fill in the gap: In the diagram below the top arrow can be visualised as stretching -10 by a scale factor of_____.

Calculate the following product: 0.5 x (-12)

What is the temperature 10 degrees warmer than -30 degrees?

Lesson overview:Division

Core Content

Positive and negative numbers: