Lesson video

Hello, Mr. Robson here.

Superb choice to join me for maths today, especially seeing as we're problem solving with graphing linear relationships.

And I love problem solving and I love graphing linear relationships.

So let's combine the two and this should be a super lesson.

Let's get started.

The outcome for our lesson today is that we'll be able to use our knowledge of graphing linear relationships to solve problems. Keywords that you are going to hear throughout the lesson.

I'll be saying rate of change and gradient quite a lot.

The rate of change is how one variable changes with respect to another, if the change is constant, there is a linear relationship between the variables.

The gradient is a measure of how steep a line is, it is calculated by finding the rate of change in the Y direction with respect to the positive X direction.

Look out for rate of change and gradient throughout this lesson.

We're going to start with problems involving linear graphs.

A scenario for you, some pupils are collecting sweets.

Aisha starts with 6 sweets and gains 1 more per day.

Jacob starts with 0 sweets, but gains 2 per day.

Izzy starts with 14 sweets, but eats 2 per day.

Who has the most sweets? Pause and tell the person next to you.

The truth is we don't know the answer to that question, it depends on the number of days.

In order to tell you who has the most sweets, I need to know how many days we're talking about.

Who has the most sweets after three days? You can work that one out, pause and do so now.

Aisha has 9 sweets, that's her 6 sweets and 1 per day.

Jacob has 6 sweets, that's 2 sweets for every day.

Izzy had 14 minus 2 lots of 3, 2 sweets times 3 days, she ate 6 sweets, she's left with 8 sweets.

So after three days, Aisha has the most sweets.

So those are my calculations for three days, but could I substitute any number of days in there? Instead of 3, could I substitute 4, 5, 6, 7 into those expressions? Could I substitute d days into those expressions, where d is a variable? Pause and tell the person next to you, what do you think about those two questions? The answer is yes and yes.

The 6 + 1 x (3), that calculated the 9 sweets Aisha had, we could put any number in there.

We could put seven in there and we'll know how many sweets she has after a week.

But we might not want to know just one week, we might want to know two weeks, or three weeks, or a year, so we can substitute in the variable d, and that expression will work for us.

In that case, I shall have an expression of 6 + 1 x (d) = s, d being the number of days and s being the number of, that's right, sweets.

The days will vary, and as the days vary, the number of sweets will vary.

That expression, sorry, equation, works for Aisha.

For Jacob, we're just taking his number of days, multiply it by two, and gain the number of sweets.

That expression works for Jacob, Izzy's is a little trickier, but not too tricky.

We start with 14, and for every day we're taking away two, so we minus two multiplied by d, and that gives us the amount of sweets.

In mathematics, we like to write things as simply as possible, so we won't leave those equations in that form.

6 + 1 x (d) = s, we'll simplify that to s = 6 + d.

That's an equation to express how many sweets Aisha has on any given day.

For Jacob's, a nice simplification, s = 2d.

For Izzy, that simplifies to s = 14 - 2d.

You'll see now that they are linear relationships and we could plot them.

In the past you might have plotted linear graphs, like y = 14 - 2x.

We're gonna plot these in the same way, but instead of X and Y, we're going to use D and S on the axes.

First, we'll need a table of values to plot some coordinates.

Aisha's table of values for s = 6 + d for days one to eight.

Calculate the number of sweets she has and then do the same for Jacob, the same for Izzy, you need three tables of values before we can plot this.

And you pause this video and fill in those tables of values now.

For Aisha, we should have, after one day, seven sweets, after two day, eight sweets, and then nine sweets, 10 sweets, 11 sweets, 12 sweets, 13 sweets, 14 sweets.

For Jacob, the number of sweets he has is double the number of days, so we start with one day, two sweets, two days, four sweets, and that goes all the way up to eight days, 16 sweets.

For Izzy, her sweets went 12, 10, 8, 6, 4, 2, 0, and we can't have a negative number in day eight, because she won't have a negative amount of sweets, she'll just end at zero on day seven.

So, we can read who's got the most sweets when from these tables, but there's a lot of numbers there, there's a lot of information.

It's easier to see this if we graph those lines.

That's what one of the lines will look like.

That's Aisha's sweets.

The relationship s = 6 + d.

You'll notice, on the horizontal axis we've got d for days, and on the vertical axis we've got s for sweets, and that line s = 6 + d, describes or shows us that relationship, and we can read at any point how many days in we are and how many sweets Aisha has.

If we do the same for Jacob, you can see how many sweets he has at any given point, and the same for Izzy.

Izzy's going to look slightly different, why? Because she had a decreasing number of sweets, hence her line has a negative gradient, she started with 14 and then ate two per day, her number of sweets decreased.

From here, what key moments can you see? Now that we've graphed these relationships, can you pick out a couple of key moments that describe the relationship between Aisha, Jacob, and Izzy's amount of sweets? Pause and pick out a few key moments now.

You might have said, by day four, Izzy already had the least number of sweets.

You might also have said, at day six, Aisha and Jacob are equal, they have the same number of sweets, 12 after six days.

A very interesting observation is what happens beyond what we can see in this region of the graph.

Beyond day six, Jacob's got a faster rate of change, which means he'll always have more.

If I said, who's got the most sweets after 10 days, you can see that it's Jacob here.

If I said, who's got the most sweets after 100 days, whilst we can't see it on this graph, you know it's Jacob, because his line has the fastest rate of change.

Quick check that you've got that now.

In this scenario, who has the most sweets after two days? Two lines on the graph, again, days is a horizontal axis, sweets is a vertical axis, who's got the most sweets after two days? Is it Aisha? Is it Izzy? Or did they have the same? Pause and tell the person next to you.

The answer was Izzy.

After two days, she had more sweets than Aisha.

You can see that from those two points there.

Aisha has five after two days, Izzy has seven.

How many sweets did Izzy begin with? Is it five? Is it one? Is it six? Pause, tell the person next to you.

I hope you said five.

You might call that, or you might have seen that called the Y intercept in the past, on this graph, it'd be the S intercept, or the coordinate zero, five.

On zero days, Izzy had five sweets.

How many sweets does Aisha gain every day? Slightly trickier question this, but again, the answer can be seen in that graph.

How many sweets does Aisha gain? Is it four? Is it one? Is it two? Pause, tell the person next to you.

I hope you said two, because you can read that from the graph.

That's a gradient of two on Aisha's line there.

For every increase of positive one in days, the number of sweets she had is positive two.

In this scenario, who has the most sweets after 200 days? Is it Aisha? Is it Izzy? Or, it's not on this graph so we can't tell.

Pause and have a think about that now.

It's Aisha.

Aisha's faster rate of change means she will always have more sweets than Izzy beyond day four.

The gradient of her line is steeper, she's gaining sweets at a faster rate.

Beyond that crossing point at day four, Aisha will always have more sweets.

Many real life scenarios can be modelled by linear graphing.

Some pupils are discussing saving money.

Lucas starts the year with £50 and saves £2 per week.

Laura starts the year with £0, but saves £5 per week.

Sam starts the year with £100 in savings, but spends £1 per week.

Constant rates of change make linear relationships, and these scenarios have constant rates of change.

In Lucas's case, there's a constant rate of change of £2 per one week.

In Laura's case, £5 per one week, her rate of saving will be constant.

In Sam's case, slightly different, a spend of £1 per week, a decrease of £1 for every increase of one in weeks.

But these relationships are constant rates of change, they'll be linear relationships, so we can solve the problem with linear graphing.

We will graph for Lucas, s = 50 + 2w.

What do you think the s stands for? What do you think the w stands for? Well done, s for savings, w for weeks, and they're both italicised in that equation, because w, the weeks, will vary, and as that varies, the savings will vary too.

The 50 represents the 50 that Lucas starts with, and the 2, the coefficient of the w, represents that constant rate of change, that £2 per week.

So, for Laura, we'll plot s = 5w.

That's the week multiplied by five to tell us how much she has in savings.

Sam's will be slightly different, Sam is spending.

Sam will start with 100 and lose one per week.

S = 100 - w, will be the line we'll plot for Sam's savings.

In order to plot, we're going to need a table of values.

I'd like you to pause and fill in this table of values, one for Lucas, one for Laura, one for Sam, pause now.

Hopefully it didn't take you too long to fill in those tables of values.

Lucas's went 52, 54, 56, 58, 60.

Laura's went 5, 10, 15, 20, 25, and Sam's went 99, 98, 97, 96, 95.

However, there's a problem, before we plot now, we should notice, looking at those tables of values, we're five weeks in, and those values are nowhere near each other.

Sometimes it's useful to change the scale.

If you look at the values we've got, Sam started with the most money by quite a way.

Sam still has quite a way more money than Lucas and Laura, so let's change the scale up and see if we can't mix up who has the most and who has the least amount of money.

Let's use this scale for weeks instead.

That's how our table of values read, 0 weeks, 5 weeks, 10 weeks, 15 weeks, 20 weeks, 25 weeks.

Can you pause now and fill in the table of values, one for Lucas, one for Laura, one for Sam.

Lucas's table should go £50, £60, 70, 80, 90, 100.

Laura should start at 0, and then go 25, 50, 75, 100, 125.

Sam starts with 100, and then it goes to 95, 90, 85, 80, 75.

Just pause and check that your table of values matches mine.

Once we've got those tables of values, we can graph these relationships.

The graph will look like that.

You can see Sam's line starting at 100 and decreasing as they spend.

You can see at the other end, Laura starting at 0 and going up quite quickly, because Laura is saving £5 per week.

The graph makes the comparisons a lot clearer and easier than just looking at the word description of their savings or the tables of values of their savings.

You can notice that during the 17th week, there was that moment when they all had the exact same amount of money.

Towards the end, you can see Laura ends up with the most, she's saving the most, she's a faster rate of change.

Provided these linear relationships continue, Laura will always be saving the most money.

Quick check you've got that now.

For this graphical representation, which statements are true? Sam has the greatest savings.

In week 30, Lucas has £40 more.

A scale of 10 is appropriate for the weeks axis.

In week 100, Lucas will have more.

Which of those statements are true? Pause and have a think about that now.

The first statement was not true.

To say Sam has the greater savings is only true for part of the time, it's only initially true, but once we hit 20 weeks, they are equal, and beyond that, Lucas has more.

In week 30, Lucas had exactly £40 more, is that true? Yes, that small arrow there shows you two steps on my grid, what is each step? Well, if we look at the vertical axis for savings, between 0 and 100, there's five steps.

100 divided by five is 20, each of the steps is 20, so that gap between Lucas and Sam's savings there represents two lots of £20 or £40.

C, a scale of 10 is appropriate for the weeks axis.

Absolutely right, if we tried to do this at a scale of one, it would not have been very efficient or very clear.

D was also correct, in week 100, Lucas will have more, provided the same linear relationships continue.

They keep going at that rate of change, Lucas is saving at a faster rate of change, therefore will always have more savings.

Practise time now.

Question one, a different scenario again, this time some pupils are filling buckets of water.

Andeep and Jun fill a bucket each from two different taps.

This is graphed here.

You'll see on the graph the horizontal axis reads seconds, the vertical axis reads volume in millilitres.

Part A, who started with some water in their bucket already? Part B, who had the smaller bucket? Part C, whose tap has the faster rate of flow? And D, when was Andeep's bucket half full? Pause and answer those questions now.

For question two, in your town there are three taxi companies.

Company A charge a flat rate of £4, then £2 for every mile travelled.

Company B charge a flat rate of £8, then £1 for every mile travelled.

Company C have no flat rate charge, but they then charge £4 for every mile travelled.

I'd like you to fill in a table of values for each company and graph the cost of their taxi firms. For part A, we're filling in three tables and plotting three lines.

For part B, you're going to write a paragraph, a few sentences, and explain the conditions in which each firm is best.

Pause and do that now.

Feedback time now.

Andeep and Jun were filling buckets, who started with some water in their buckets already? The graph shows Jun starting at 0, 0 millilitres after 0 seconds, but you can see after 0 seconds, Andeep has 300 millilitres.

For part B, who had the smaller bucket? The question said, Andeep and Jun fill a bucket.

You can see that Jun's bucket gets all the way up to 1,000 millilitres or a litre, whereas Andeep's buckets only gets up to 800 millilitres, so Andeep is full at 800 millilitres, he has the smaller bucket.

Whose tap has the faster rate of flow? You can see Jun's graph is steeper, it has a greater gradient.

That means Jun has a faster rate of flow.

His flows at 200 millilitres per 5 seconds, which you can read from any moment on his straight line.

For part D, when was Andeep's bucket half full? Well, if it's fully full at 800 millilitres, it's half full at 400 millilitres, and that moment came 5 seconds in.

For part two, we need a table of values for each of the taxi companies.

Company A, for company B, and C.

Write a paragraph explaining the conditions in which each firm is best.

If you've correctly plotted your graph, it should look just like mine.

You might wanna pause and just check that your coordinates match mine, your lines match mine, and then we'll look at picking out the key moments, the moments in which each firm is best.

You might have written, initially, company B are the most expensive, and company C the cheapest.

But after 2 miles, C become more expensive than A.

From 2 miles to 4 miles, A was the cheapest firm.

And then at 4 miles, company A becomes more expensive than company B, who are now the cheapest, and will remain cheaper for any further journeys, because their rate per mile or rate of change is lowest.

Onto the second half of the lesson now.

We're going to interpret more complex linear graphs.

Aisha runs 100 metres at sports day.

Below is a table of values of her performance.

Why will this not make a linear graph in the way that all the scenarios we've seen so far in this lesson made linear graphs? Pause this video, have a good look at that table of values and think about that problem.

I hope you noticed that there's not a constant rate of change for both variables.

The time had a constant rate of change, but the distance did not have a constant rate of change.

In order to have a linear relationship, both variables need a constant rate of change.

Whilst not linear, the graph does have useful features.

If I graph those moments, it would look like that.

Time on the horizontal axis, distance on the vertical axis, and those moments, 0 times 0 distance, 3 seconds, 8 metres, et cetera.

I could join lines between those points like that to emphasise some useful features of this graph.

And then you notice that the line is getting steeper.

The gradient is increasing as the race goes on.

What does that mean? It means Aisha was running faster at the end.

She's covered a greater distance in the same time period.

If you think about that, covering a greater distance in the same amount of time must mean you are moving faster.

So Aisha was running faster at the end of this race than she was at the beginning.

Let's check you've got that, true or false, in another race, Aisha speeds up at the end of this 400 metre race.

Hmm, is that true? Is it false? After declaring it true or false, I'd like you to justify your answer with one of these two statements.

A change in gradient means she has sped up.

A decrease in gradient means she has slowed down.

Pause and have a go at that problem now.

It was false.

A decrease in gradient means she has slowed down.

Sam runs the 200 metres, and I've represented that on this graph.

Time on horizontal axis, distance on the vertical axis.

What do you notice about Sam's race? Pause and make a few observations.

Did you notice it's a straight line, a constant rate of change? That tells us that Sam was running at a constant speed.

You can read that constant speed, for every 5 seconds, Sam travels 25 metres.

You can see that change of 5 in seconds and 25 metres in time wherever you look on that straight line.

Alex also ran, but what do you notice about their race? Pause and make a few observations about Alex's race and how it was different to Sam's.

Did you notice these things? A 50 metre head start, a stop between 10 and 20 seconds, a fast finish and victory for Alex.

The 50 metre head start you saw there, on 0 seconds, Alex was already at 50 metres.

The stop you saw there, on 10 seconds or between 10 seconds and 20 seconds, the distance doesn't change.

If the distance isn't changing, Alex is not moving anywhere.

That moment when the gradient becomes 0 on this time distance graph means Alex has made a stop.

A fast finish and victory, we know it's a fast finish, because the gradient at the end of Alex's race is steeper than at the start, and we know it's victory, because Alex reached 200 metres in less time than Sam did.

Let's check you've got that.

On this time distance graph, we see Izzy cycling to school.

What happens after 4 minutes of her cycle journey? She speeds up, she slows down, or she stops.

Pause and tell the person next to you.

I hope you said she stops.

0 distance covered from 4 to 6 minutes shows Izzy having a stop in that journey.

Sticking with that same scenario, Izzy cycling to school, represented on this time distance graph, a different question, when is she going at her fastest? Is it A, between 0 and 4 minutes, B, between 0 and 800 minutes or C, between 6 and 10 minutes? Pause and have a think about that problem.

You should have said option A, between 0 and 4 minutes.

Between 0 and 4 minutes, we saw the fastest rate of change, the steepest gradient, either of those tell us that she's going fastest in that moment.

If you went for option B, you might have accidentally been reading the vertical axis, the distance axis rather than the time axis, and all of our units in that question were about time.

Practise time now, and a lovely practise exercise for you.

Question one, and I only have one question for you in this practise, because it's such a wonderful question.

I'd like you to do a little writing.

I'd like you to write a commentary to describe this race.

We see another time distance graph.

I see Sofia, Andeep, and Laura having a race.

I'd like to hear a commentary from you or see you write one please.

The distance units are in metres, the time is in seconds.

I'd like you to make sure you keep your commentary in chronological order.

That means what happens first, write first, what happens next, and then next, and then next, and then next until the end.

Stay in chronological or time order.

I'd like to pause and get writing now.

Feedback time, there were a lot of things that you could have observed about that race on that graph.

You might have written something along the lines of, "And they're off.

Andeep and Sofia start at a constant speed, Sofia with her 40 metre headstart.

2 seconds in and Sofia has passed the 50 metre mark, but Laura hasn't even started yet, what is she doing? Finally, after 4 seconds, Laura starts, and wow, does she start.

That's the fastest speed we've seen yet in this race.

Within a second of her starting, she overtakes Andeep, within 2 seconds she's passed Sofia, but then she stops.

She must think the race is over." I've only covered half the race there, up to the 6 second point, I hope in your commentary, you picked out a few key features.

For example, Sofia having a 40 metre headstart.

The idea that Laura had not started until 4 seconds had elapsed.

The fact that Laura started incredibly quickly, and quickly overtook the other two.

Hopefully you made those key observations.

Let's look at the second half of the race now.

You might have written after 6 seconds, "8 seconds in, Sofia's still moving at the same constant speed.

She's overtaken Laura, who looks like she might be asleep, but Andeep speeds up slightly in an attempt to try and catch Sofia, will it be enough? It's not, after 10 seconds, Sofia hits the finish line for the gold medal.

Andeep joins her 2 seconds later in silver position.

Laura finally woke up and started running again, but didn't even get to the finish line." So a few key moments there.

The fact that Laura had stopped should have been observed, once she reached the lead, she stopped.

You should have noticed that on 8 seconds, Andeep's speed changed, his line became a little steeper, the gradient increased, the rate of change increased, he sped up to try and catch Sofia.

You should absolutely have identified that Sofia was the winner in a time of 10 seconds, that's a fantastic time.

And you should have noticed that Andeep finished after 12 seconds and that Laura did not finish, she didn't reach the 100 metre mark.

If you got all that, that's awesome.

Sadly, that's the end of the lesson now, but we've had fun and we've learned that we can use our knowledge of linear relationships to solve real life problems and interpret what is happening in graphical models of real life situations, for example, the impact of gradient in a time distance graph.

I hope you've enjoyed this lesson, and I hope to see you again soon for more mathematics.

Goodbye for now.