1 bar of chocolate costs £2.10.  How much do 3 bars of chocolate cost?

Four tickets to the cinema cost £42. How much do 8 tickets cost?

Four tickets to the cinema cost £42. How much do 9 tickets cost?

Fill in the gaps: You can use .............. .............. to work out the best value for money.

x and y are directly proportional. What happens to y if x doubles?

Proportion in Context

﻿Hi there and welcome to another maths lesson with me Dr. Saada. In today's lesson, we'll be learning about inverse proportion. Don't worry if you don't know what that is, you will by the end of today's lesson. For this lesson, you need a pen and a paper. So if you don't have these handy, can you please pause the video and go and grab them, and when you're ready, we can make a start. We will start today's lesson by looking at this question. It takes 24 minutes for Anthoni to tidy Miss Jone's classroom on his own. It takes 12 minutes for Anthoni and Binh to tidy the classroom. It takes eight minutes for Anthoni, Binh and zaki to tidy up the classroom. How long do you think it will take for Antoni, Binh, zaki and Yasmin to tidy up the classroom? Please pause the video have a go and try this task. I would suggest that you look for patterns in the numbers that can be given in the question. The try this task should take you about five minutes to complete. Resume the video once you're finished. Welcome back. How did you get on with the try this? Did you manage to spot some patterns, and answer the question? Okay, let's have a look at the question together. It takes 24 minutes for one person for Anthoni to tidy up the classroom. It takes 12 minutes, this time for Anthoni and Binh so we have two people, so two students take 1/2 the time. 12 minutes is 1/2 of the 24. It takes 8 minutes for 3 students to tidy up the classroom. Eight minutes in relationship to 24 that's a 1/3 of it really good. So three students take a 1/3 of the time. How long do you think it would take four people to tidy up the classroom? But if 2 students took 1/2 the time, 3 students took a 1/3 of the time then four students will take a 1/4 of the time, really good. So 4 students will take a 1/4 of the time. A 1/4 of the 24 original time is 6 minutes. So they should take 6 minutes to tidy up the classroom. Did you get that Right? Good job. Let's look at this in a bit more depth. So I've created here a table that has the number of people, the time taken, and I'm going to calculate the number of people multiplied by the time. Now if one person tidies up, it takes them 24 minutes, two people, it takes them 1/2 the time, that's 12 minutes, three people, it takes them a 1/3 of the time, a 1/3 of that original times a 1/3 of 24, which is 8 minutes. For 4 people we said it takes them a 1/4, So a 1/4 of 24 is 6. If we have 5 people cleaning, then it will take them a 1/5 of the time, So that would be 4.8. Because if one person is cleaning, they take 24 minutes, they need to do everything on their own. But if you have 5 people everyone is chipping in and everyone is helping, then that time is divided among the five of them. And If you have six people cleaning, then they will finish in a 1/6 of the time. 1/6 of the 24 is four minutes, they'll finish in four minutes. Let's have a look at what the value of number of people multiplied by the time so the first one, 1 times 24 is 24. 2 multiplied by 12 is 24. 3 times 8 is 24. Are we starting to make any observations here? Really good, 4 times 6 is 24. 5 multiplied by 4.8 is also 24, and 6 times 4 is 24. So what do we notice here about this number? This number is always constant. What does that number represent, the number of people multiplied by the time? Well, it's a constant value, it's telling us the relationship between the number of people and the time taken. It's telling us that constant of proportionality. Now when we were doing direct proportion, remember we divided y divide by x and that was constant. with inverse proportion and this is inverse proportion y multiplied by x gives us that constant value. So if we have two quantities, two variables and we multiply them multiply the numbers together, the product is going to always be constant, if the relationship between the two variables is the inverse proportion. So, two quantities are said to be inversely proportional, If as one increases, one of the quantities increases, the other decreases at the same rate. This is different to direct proportion. With direct proportion one increases the other increases with the same rate to maintain that ratio. With inverse proportion one increases, the other decreases by that same rate. The number of people and time taken are inversely proportional in this example, their product, ie product means when we multiply the number of people by the time taken, the product is always the same constant value. And in this case, it's always 24. Now, thinking about real life examples, where else can you think of inverse proportion when something increases, something decreases at the same rate. So we looked at cleaning up a classroom, the more people you have the less time it will take, anything else that you can think of? Say it to the screen. Okay. Really good. I would like you to make a note of the definition of inverse proportion as it appears on the screen. So please pause the video and copy the two sentences down resume the video once you've finished. Okay, let's have a look at another real life example. It takes three maths teachers six hours to mark all of year eight assessment papers. How long would it take one teacher to mark all of the papers? How long would it take for 7 maths teachers to mark year eight assessment papers? Now the fastest step here is to identify is this direct proportion or inverse proportion? I have three maths teachers marking papers, if all the papers are left to one person to mark on his own, will, it takes them longer or will it be three hours Or will it be less? Of course it will take them all longer because there's no one else there to help them. So that one math teacher will take a lot more than the six hours. Now what about 7 maths teachers if you get instead of having three teachers to asking other people to come and help, and you have seven teachers sat in one room marking the papers. They will definitely finish quicker. So it will take them less time. So in one case, we have more teachers less time, in another case, we have less teachers, one teacher, we end up with more time. So this is an inverse proportion question. You need to be able to spot that when you read the question. Now I want to answer the questions I start by saying teachers, hours. Okay, what do I know from the question? I've been told that three teachers will take six hours to mark the papers. I want to know one teacher, how long will one teacher take? So what have I done here from three to one, I divide it by 3. Now, if I divide the other part by three, if I divide 6 by 3, that will give me 2 hours we know that that's not going to happen. One teacher needs a lot more time than six hours. So we do the inverse we multiply by three and that gives us 18 hours. So three teachers will take six hours to mark the papers but one teacher will take 3 times the time so end up taking 18 hours to mark the assessment papers. Now if I want to find for 7 maths teachers, it's really helpful that I have that one teacher and how long it takes that one person, because 7 is not a multiple of 3, and it will make it a bit a bit trickier for me to multiply. So how do I get to 7? I multiplied by 7. So what do I need to do to the number of hours now I'm having more teachers helping. If I multiply by 7, I'm going to 7 teachers will take a huge amount of time, and I know that's not the case. It will take less time so I'm going to divide by 7. That gives me roughly 2.6 hours. Now, there's a way for us to check if our answers are correct. Do you remember that constant of proportionality that we looked at earlier today? We said it's always the same, isn't it? So this is how you check when you do it this way. You say to yourself 3 times 6 is 18, 1 multiplied by 18 is 18. 7 multiplied by, and we had the 2.6 to two decimal places, so we've rounded it here, so it's not going to be exactly 18 but it will be close enough. So in order for me to check, I can multiply and check that the product is always the same, the product has to be the same If we have inverse proportion, so 3 multiplied by 6 is 18 1 multiplied by 18 is 18 and 7 multiplied by 2.6 is 18.2 but we know that this is the case because we round it. Okay. Now, can you think of another real life scenario where we may use inverse proportion? So one quantity increases the other decreases by the same rates, Can you think of any other scenarios? So we've had maths teachers marking exam papers we've had students tidying up classroom, Can you think of something maybe outside the classroom for me? Have a little think, say to the screen. Okay, really good, so these are couple of my examples, any building work, so if you've got builders do any work at home, whether it's painting, whether it's building, whether it's doing the grass in the garden, the more people you have, the less time it will take. The less people you have, the more time will take. The speed and time, Okay, if you're driving at a certain speed, if you increase that speed, you will take less time so if you go faster, you will take less time if you go slower, you will take more time, so that's inverse proportion there. It's time now for you to have a go at the independent task. The first question, I've given you some scenarios and I want you to think about them and decide whether these scenarios represent direct or inverse proportion. For Question two, I've given you a real life problem that I would like you to solve. The independent tasks should take you about 10 minutes to complete, so please pause the video and complete it to the best of your ability. Resume the video once you're finished. Welcome back. How did you get on with this task? let's mark and correct the work together. So the question one, yet the cost of apples and the quantity of apples and we know that this is direct proportion. The more apples you buy, the more you're going to pay. The speed is travelled and the time taken. we've just discussed that that's inverse, didn't we? The faster you're going, the less time it will take, the slower you're going somewhere, the more time it will take you to get there. Third one, the time it takes to build the house compared with the number of builders all working at the same rate. So if you have builders building us a work, the more builders we have, the less time they will take to build and complete the job. So that's inverse proportion. And the last one, the circumference of a circle and its diameter. Well, we know that the circumference of a circle and diameter are directly proportional, the bigger the diameter of the circle is the bigger the circumference. So that's direct proportion. Did you get all of these Correct? Really good, Well done. Let's have a look at question number two. The Beta construction company is building an extension. Three builders take 15 days to complete the project. How many days would it take one builder? And how many days would it take to complete the project if we had six builders? So let's have a look at how we write and answer this question. So I can start by saying builders, time. Block three builders, we need 15 days, if I have one builder, so having less builders now, obviously will take more time. How did I get from three to one? I divided by 3 so I need to do the inverse. So if one thing decreases by three times the other quantity increases by three times. So it's multiply this by 3 and that gives us 45. So we know that one builder now would take 45 days to complete the project. Now how about we have 6 builders. Now let's write 6 on increasing the number of builders, I have 6 at much I have 6 times as many builders working on the project. So the time will be, it will be a 1/6 of the time it will take for one person or one builder. So we divide by 6 here and that gives us 7.5 days. So we know that the project would be complete in 7.5 days if we add 6 builders, did you get this correct? Really good, job well done. Okay, this brings us to that explore task. Now remember, in that independent task, I gave you some scenarios and I asked you to look at them in a table and decide whether they showed direct proportion or inverse proportion. It required you to read the scenario and think about the relationship between the two quantities or variables that we have there, and what happens if one increases to the other what effect it has on it. We can find out if two quantities are directly proportional or inversely proportional using just the maths without even knowing what the scenario is. And this is what you're going to do into this explore task. I have given you here three tables with values for x and y. I want you to tell me for each table if x and y are directly proportional to one another, or inversely proportional to one another. if you feeling confident about making this document, please pause the video now and off you go. If not, I will be giving you a hint in three, in two, and in one. Okay, I'm going to remind you about something that we have done when we looked at direct proportion. When we looked at direct proportion, we had tables and we had for example, the number of monks and the cost of monks and then we divided the cost of monks by the number of monks to find the cost of one item, or one monk. And we did it throughout the table and we were looking to see if that number was always the same, that was y divided by x. And we said when it was constant, it meant that the two quantities that we're looking at were directly proportional. So here, if you look at these tables, if you calculate y divide by x, and you find that it's the same value through out the table, it means that x and y are directly proportional to one another. Now, what about inverse proportion? I asked you earlier on today to pause the video and make a comment to copy down the definition of inverse proportion. And it had something about the constant of proportionality there. What did it say? the product, really good. So the product means if we multiply them, if we multiply x times y and it's always the same number, it means that x and y are inversely proportional. So now you should be able to make a start, off you go. The explore task should take you about 10 minutes to complete, so please pause the video and have a go at it, resume the video once you're finished. Welcome back, How did you get on with the Explore task? Okay, let's have a look at the answers together. So for the first table, if you calculate x multiplied by y, so the product of x,y you will see that it's always different 27, 75, 108, 507 we've never had the same number. So we cannot make any conclusions about it. if you try y divided by x the answer is always 3. This shows that we have x and y are directly proportional to one another. This shows that x and y are directly proportional to one another because y divided by x is always a constant value. Table two we get x multiplied by y was always 60, 2 times 30 is 60, 3 times 20 is 60, 5 times 12 is 60, 6 times 10 is 60. Y divided by x was always a different number, the product of x, y is always constant So this tells me that x is inversely proportional to y as one increases, the other decreases at the same rate. Now the last table was interesting, had lots of decimal numbers. So product x, y was always the same. y divided by x was not the same, because the product was always the same this shows that the two quantities x and y are inversely proportional to one another. Did you get this correct? Really good, well done. This brings us to the end of today's lesson. I hope you enjoyed today's lesson. I want you to spend two minutes thinking about inverse proportion, I want you to write down the one most important thing that you've learned from today's lesson. It's entirely up to you what that is. Once you're finished, I'd like you to complete the exit quiz. This is it from me for today, enjoy the rest of your day and I'll see you next lesson. Bye.

Fill in the gaps: Two quantities a and b are said to be directly proportional if one quantity is always a constant ........ of the other.

Fill in the gaps: Two quantities x and y are said to be inversely proportional if x increases as y ............... at the same rate, so their product is constant.

Fill in the gaps: Two variable quantities are said to be ........................ proportional if the product of the two quantities is constant.

Three builders take 15 days to complete a project. How many days does would it take one builder?

It takes 4 friends 6 hours to clean the garden. How long will it take 2?

Inverse Proportion

In this lesson, we will explore when two quantities are inversely proportional to each other. We will learn about the term inverse proportion and how to identify an inversely proportional relationship.  


Maths

Y8 Maths

Key Stage 3

Year 8

Inverse proportion

Direct and indirect proportion

Ratio and proportion

THEME

UNIT

Looking at proportional relationships

Direct proportion (Part 1)

Direct proportion (Part 2)

Direct proportion (Part 3)

Graphs of proportional relationships

The constant of proportionality

Proportion in context

Inverse proportion in context

Graphs of inverse proportion problems

Direct or inverse proportion?

Pens are sold in boxes. Each box contains 6 pens. How many pens are there in 5 boxes.

A chain costs 8p per 2cm. The cost and length are directly proportional. What is the cost of 6 cm?

Look at the table below. Choose the correct statement that describes the relationship between the number of mugs and the cost of the mugs.

The number of rulers are directly proportional to the cost. What is the missing number B?

x and y are directly proportional. What is the missing number in the box?

Which graph shows direct proportion?

Complete the sentence: When two quantities are in direction proportion, when one variable is zero, the other is .............

Which table shows direct proportion?

What is the constant of proportionality between m and n?

Decide whether the scenarios below represent direct or inverse proportion: The cost of apples and the quantity of apples.

It takes 2 people 3 hours to clean the house. How long does it take 6 people to clean?

Lesson overview:Inverse proportion

Core Content

Direct and indirect proportion: