True or False? Inverse proportion  can be represented graphically with the line 𝒚 = 𝒂𝒙 passing through the origin.

True or False: Inverse proportion can be represented graphically using 𝑦 =𝑎/𝑥. This is a curved line.

Direct, inverse or neither?

Graphs of inverse proportion problems

﻿Hi there and welcome to another maths lesson with me Dr. Rim Saada. In today's lesson we will be looking at direct and inverse proportion. This is our last lesson in the unit. So it's just to wrap up everything that we have learned so far. For this lesson, you will need a pen and a paper. So if you do not have these handy, please pause the video go grab these and when you're ready, we can make a start. I would like you to have a look at the six graphs given here. Do these representations show direct proportion, inverse proportion or neither? Suggest a possible scenario for each graph. If you're feeling confident, please pause the video and have a go at this. If not, I will be giving you a hint in three, in two and in one. Okay, so my hint for you is to remember that direct proportion have this relationship between the x and y. y divided by x is going to be a constant number. Inverse proportion will have the relationship of xy=k, equals a constant. So if you multiply x by y will always get a constant number. Now with this hint, you should be able to have a go at this. Please pause the video and have a go at this task. This should take you about 10 minutes, in particular, because you will be coming up with scenarios to match each graph. Resume the video once you're finished. Welcome back. How did you go with this task? Really good. Did you manage to write the scenario for each of the graphs? Okay, let's have a look at this. The first one show direct proportion. The second one, what did you write down? Really good. It's neither it doesn't show direct proportion, it's not showing us inverse proportion. Next one. Excellent. It's neither. Next one. Really good job. It's it's showing us inverse proportion. As the X is increasing, the Y is increasing and they are increasing at the same rate. Next one. Excellent job. If you wrote that neither, remember even though it's a linear graph, and it's showing us as something as x increases, the Y is also increasing. However, it's not passing through the origin. So it is not showing direct proportion. And therefore the last one is showing us direct proportion. Really good. I wonder what scenarios you wrote down for each of them. For the direct ones, you could have suggested things about the cost of items being increased. If you buy more you like to spend more without obviously any offers. For the inverse proportion, you could have suggested something to do with speed and time. And for the neither you would have suggested something that doesn't take the direct or the inverse. For this one here in particular, you could have suggested something like a taxi where you get in a taxi and then they tell you that there is a minimum charge. So they start off with a minimum charge of let's say, two pounds and 50 and then they charge you per mile depending on the distance that they cover. Really good job. Let's move on to our connect task. And as I said today we're going to just wrap up everything that we have learned about direct and inverse proportion. A table of values for a and b is shown below, a is directly proportional to b. Complete the table. So you can see here the table, I've given you the first row values for a second one chose values for b, and the third one, b divided by a. Now, if a is directly proportional to b, we know that b divide by a is always going to be constant is going to have a constant value, and one of the values has been given to you as five. This tells me that the rest of the table here should have fives every time we divide b divided by a, it should always give us a constant number and that is five. Now with this information, you can find the values of b, the missing values b. So I know that the first one must be 10 because 10 divided by two is five. Next one, what do you think it's going to be? Good job. 35, 35 divided by 7 is going to be 5. And next one. Really good. You can think about it as 5 times 20, and that is 100, and the following one, have a little think. Good job 250. And the last one. Excellent 500. Now let's look at the second part of the question. A table of values for m and n is shown below. m is inversely proportional to n, completely the table. So what do we know about inverse proportion? We know that the two quantities so in this case, m and n would always multiply to give us the same product. So m multiplied by n is always going to be a constant value, and in this case, it is given to us as 500. So I look at the table and I need to think four multiplied by what number gives me 500. I need to find that out. I can start by just writing the 500 set because that is always going to be constant. Now four multiplied by what is 500? 125, you can do the inverse and just divide 500 divided by 4 to find out the 125. Next one, it's an easy one. Good job, 100. And the following 20 multiplied by something gives us 500. Really good 25. And the last one, really good 12.5. Now what do you notice about this what's happening to the value of n? As m is increasing, the value of n is decreasing, whereas in the first statement with direct proportion, as a was increasing b was also increasing. Now it's time for you to have a go at the independent task. I want you to read the questions carefully and answer them to the best of your ability. If you're feeling confident, please pause the video and have a go at the independent task. If not, I'll be giving you a hint. In three in two and in one. Okay, and my hint for you here is that you can start by drawing a bar model for the first question. Let's read it together first. This is a recipe for 12 pancakes and you've been given the recipe. It says that you need 100 g of plain flour, you need two large eggs, 300 ml of milk, three lemons and 30 grammes of caster sugar. Ashley has the following ingredients in his cupboard. So what does Ashley have? Ashley has 250 g of flour, six large eggs, one litre of milk, eight lemons, 100 g of caster sugar. So is he going to have enough for how many cakes? Does he have enough 12? We know he has enough for 12 cause he has more than what the recipe says. Now I can draw a bar model to help me answer this question. So I started with flour and I said, Okay, well 100 g are put it in one equal part. he needs 100 g to make that 12 pancakes and another 100 g next to it for another 12 pancakes. he still has more, he still has another 50 right? So 50 grammes will give him only six pancakes. So in total he's going to make 30 pancakes. 12+12+6 is 30 pancakes. So using the flour alone the flour is enough for 30 pancakes. But does he have enough of the other ingredients to make 30 or not? Then I looked at the eggs. He has two eggs that will give him the 12 and other two eggs that will give him the 12. And then he'll still have two more eggs, so in fact he has enough eggs for 36 pancakes. Obviously we know he cannot make 36 pancakes because he doesn't have enough flour for it. Now, what about the milk? What about lemons? What about sugar? Does he have enough for 30? Or for less? If you complete the bar model, you should be able to answer this question easily. So with this hint, you should be good to go. Please pause the video and complete the independent task. This should take you about 10 minutes to complete. Resume the video once you're finished. Welcome back. How did you go on with this task? Did you manage to complete the bar model? Really good job. Let's go through this together. So we already looked at the flour and we know that he can make 30 pancakes. We looked at the eggs that he has and he can make 36 pancakes. So let's look at the milk. For making he has one litre, which is equal to 1000 ml. Now for 12 pancakes he needs 300 ml. So if we look at the bottom of that and we put 300 here, we know that he can make 12 pancakes, another 300 will give him another 12 pancakes and then another 12 another 300 ml we'll give him another 12 how many mls does he left with? He's left with 100 mls of milk, which are actually enough to a third of the 12 pancakes. So they are enough for four pancakes. So 12 multiplied by 3, add the 4 that gives 40 pancakes So he has enough milk to make 40 pancakes. What about lemons, he has a 8 lemons he needs three to make 12 pancakes. So if we start with the first three, and then another three that's already 24 pancakes he is left with two lemons which will give him two thirds of that 12 pancakes. Therefore in total, he can make 32 pancakes. With the sugar. What's happening here? He has 30 and another 30 and another 30. So will give him 36 will have 10 grammes left. So you can make a third of the pancakes. So in total he can make 40 pancakes, just using the sugar. Now if we look at the quantities that he have, can he make 40 pancakes? No, because he doesn't simply have enough eggs and flour for that. Can he make 32? Again, he cannot because he doesn't have enough flour. So how we have to look at the minimum number here. He can make a maximum of 30 pancakes because he will have enough ingredient, enough of each ingredient for 30 pancakes and he will obviously have some ingredients leftover. Did you get that right? Well done. Let's go through a question number two together. The time in days T, it takes to build a house is inversely proportional to the number of builders ,N, working on the house. Eight builders take 25 days to build the house. How many days would it take 4 builders? How many days would it take 10 builders? So let's have a look at how I answered this question. I started by writing time to builders as a ratio. I know that 8 builders will take 25 days to complete the job. So I said, what about if I have four builders? What have I done to the number of builders, I have that? So what do I need to do to the time taken? That would be double? That would take 50 days to complete. Did you get that right? Well done. Now that are so many different ways you could come up with the same answer. You could have said like this, but I know 25 times eight is 200. I know that the relationship between the time and the number of builders is inversely proportional. So I will always have the product of the time, of the time taken multiplied by the number of builders I'll always have that as 200. So if I know that I have four, for builders, four multiplied by something gives me 200. And that is 50 days. You can see that it's a nice way of checking that whether answers are correct, because I use two methods, they both gave me the same answer. Now, the second part, how many days would ten build a take? So I could have started using exactly the same method as the first one saying 25 days, it will take eight builders in 25 days. Now, I don't want to go to 10 directly, because it might be a bit tricky for me to because I'm not multiplying or dividing by integer. So I'm going to find time for one builder, how long will it take? Divide by eight, so therefore the time only need to multiply it by eight. So one builder will take 200 days to complete the job. But I don't have one builder, I want to know for 10 builders. So for 10 builders, I'm multiplying by 10. So what am I going to do to the time? Am divide it by 10 it will take less time. If I divide it by 10 that gives me 20 days. Again, I could have done exactly the same as the second method 25 multiplied by eight is going to be 200. So I know that the product is always going to be a constant value of 200. So I could have said, 20 multiplied by something gives me 200. I know that the product has to be 200. Now, this will also give me 10 days. And you can see here that I'm getting the same answer with the two methods. And now I could have used the second method to find out the answer, just like some of you may have done it that way, I could have said 25 multiplied by eight is equal to 200. So I need now a number multiplied by 10, to give me 200. And that number must be 20. So again, it's a really nice way of quickly checking that our answers are correct. I wonder how many methods you have used to answer this question and whether you have used more than one method just to double check your answers. It's a really, really good strategy. And through our explore task guides. I would like you to draw a Flayer model for the direct proportion and one for inverse proportion. if you have not done a Flayer model before well Flayer model is it's telling us what's the same and what is different between direct and inverse proportion. To do that, you need to draw a diagram similar to the one shown here on the screen. with direct proportion right in the middle, then you need to have four sections definition, facts and characteristics, examples and non examples. You need to complete that for direct proportion, and then do another one for inverse proportion. You will find it really really helpful to look back at your notes from the past few lessons, because you have been copying lots of examples about direct proportion, lots of examples about inverse proportion. You've copied definitions of these, you've copied facts about them. So please use that information in your book in order to complete the Flayer model. This whole task should take you between 15 to 20 minutes to complete. Please pause the video and complete it to the best of your ability with some research and looking back at your notes. Resume the video once you're finished. Welcome back. How did you go on with this? Did you manage to look back at your notes and use some of the information that you have in your book? Really good. Okay, let me just share here my thoughts on the Flayer model for a direct proportion. So for a definition, I thought that I could write this two quantities are directly proportional if one quantity is always a constant multiple of the other. The facts and characteristics is that as one increases, the other increases by the same rate or at the same rate. Also, I know that the relationship is y divided by x is always a constant number. I know that the graph should look like a linear graph, it should pass through the origin and it should look like this. Some of the examples are money conversion graphs, Ingredients required for a number of servings just like we looked at the pancake question now, and we looked at baking questions earlier in our previous lessons, the cost of one item, and the cost for items and so on or more items provided, obviously, we do not have any offers. So you're not getting any discount because you're doing multi-buy. Non examples are taxi fares where you have to pay a minimum charge or an initial charge, and then a charge per how many miles you know your journey is. The price when it changes after a certain amount. Sometimes you're told that if you buy this much, this is how much you would pay per item. Whereas if you go for bulk and buy more the price changes per unit. So that is not a direct proportion. And this is an example of a graph. And this is a non-example of a graph. It's a linear graph that does not pass through the origin. So it's not a direct proportion. Now what that would suggest is, if you have anything missing on your Frayer model, you can pause the video and copy some of this information into yours. And to yours I'll suggest that you do it using different coloured pen, so you know which information was originally yours, which ones you're adding now to improve on it. So if you need to pause the video, please do so. If not, we're going to move on to inverse proportion. And for inverse proportion, I started with this definition, two quantities are said to be inversely proportional If the product of the two quantities is constant. As the quantity as one quantity increases, the other decreases again, at the same rate. We know that xy is equal to constant, the product if I multiply them, and the graph should look like this. Now some examples, the length and width of a rectangle. if the area stays the same, so if I tell you I want a rectangle of an area of 100, you can make one with 100, to 550, and so on. Speed and time you increase the speed the journey takes you less time. Then about number of workers and time so if you want the job get you the number of workers and time, especially if you're looking at a job to get done, whether it's cleaning a classroom, whether it's sorting our garden, building a house, doing painting work, anything that requires number of workers and or number of people doing a job and under time. None-examples could be these graphs. So these graphs are not showing us direct proportion. And we looked at one of these graphs when we did our explore task. I think about two lessons ago. And for non-examples, I just added these graphs here, because these have not examples of inverse proportion. Did you get these right? Do you have this information in your Flayer model? If you do, well done. If not, that's okay. Just grab a pen, pause the video and add information to your Flayer model. Remember, it's okay if your Flayer model is different. These are just my thoughts on it. So you could have done something different and you could have done more examples of less examples. If you would like to add more information to yours you can pause the video now and do that. This brings us to the end of today's lesson and to the end of the unit. you have done some fantastic learning throughout this unit learning about direct and inverse proportion. I would love to see some of your work in particular your Flayer models. So if you'd like to share your work with Oak National please ask your parent or carer to share your work on Twitter tagging at Oak National and hashtag learn with Oak. Please remember to complete the exit quiz to show what you know. This is it from me for today. Enjoy the rest of your learning and I'll see you in another lesson. Bye.

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

Decide whether the scenarios below represent direct or inverse proportion: The speed travelled and the time taken.

The time it takes to build a house compared with the number of builders all working at the same rate.

Is this a direct or inverse proportion question? It takes 6 hours to fill a pool with water coming from three taps. How long would it take if only two taps are switched on?

Is this  a direct or inverse proportion question? It costs £6 for a cinema ticket. Given that there are no current offers. How much would it cost to buy 6 tickets?

Does this scenario represent direct or inverse proportion or neither? 𝑥 represents the number of units of gas used in a house in a month.  𝑦 represents the cost in pence.  There is a charge of £12 each month, even if no gas is used.  Each unit of gas costs 25 pence.

Is this a direct or inverse proportion question? Michael fills a fish tank. Michael uses a 4 litre jug, he will need to use 15 jugfuls. How many jugfuls are needed if he uses a 6 litre jug?

Direct or inverse proportion

In this lesson, we will solve a variety of direct and inverse proportion problems. We will learn how to identify when a problem is about direct or inverse proportion.

Maths

Y8 Maths

Key Stage 3

Year 8

Direct or 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

Inverse proportion in context

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?

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

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

Lesson overview:Direct or inverse proportion?

Core Content

Direct and indirect proportion: