Lesson video

Hello, Mr. Robson here.

Well done.

Maths, you made the right choice today, especially seeing as we're plotting relationships, which is awesome.

Let's get on with the learning.

Our outcome for today is that we're going to use a graphical representation to show all of the points, within a given range, that satisfy a relationship.

Key words I will use, substitute and substitution.

Substitute means to put in place of another.

In algebra, substitution can be used to replace variables with values.

Two parts to today's lesson.

The first part, we're gonna check our understanding of plotting a relationship.

Andeep and Laura are talking about numbers.

Andeep says, "You think of some numbers and I'll do something to them." Laura says, "6." Andeep replies, "3." Laura says, "4." Andeep replies, "2." Laura says, "10." Andeep replies, "5." Laura says, "-2." Andeep replies, "-1." What's the relationship between Andeep and Laura's numbers? Pause this video and tell the person next to you.

I hope you said something along the lines of, "Andeep's number is always half of Laura's." We can represent this relationship graphically.

We'll need the axes, the horizontal axis to read Laura's number and the vertical axis to read Andeep's number, and then we can plot the coordinates.

When Laura said 6, Andeep said 3.

That's the coordinate 6, 3.

When Laura said 4, Andeep said 2.

It's coordinate 4, 2.

And so on.

You'll notice it's a linear relationship.

Those points align.

For these sets of coordinates, there is a relationship that links the x-coordinate to the y-coordinate, just as we had a relationship between Laura's number and Andeep's number.

There are relationships here.

Can you identify any of these relationships? Pause this video, make a few suggestions to the person next to you.

For set A, you might have noticed that the y-coordinate is the x-coordinate plus 2.

If I put those arrows on, you can see that relationship.

For B, you might have noticed the y-coordinate is the x-coordinate minus 2.

Those arrows show you that relationship.

For C, the y-coordinate is the x-coordinate multiplied by 2.

Can you see that relationship? And for D, the x and y-coordinates sum to 10.

0 plus 10 makes 10.

2 plus 8 makes 10.

The two coordinates sum to 10.

The next challenge I have for you is can you write these as equations using mathematical notation? Can you turn those worded relationships into mathematical notation? Pause this video and make some suggestions to the person next to you.

For the first one, the y-coordinate is the x-coordinate plus 2.

In mathematical notation, we'd describe that as y = x + 2.

For the second one, the y-coordinate is the x-coordinate minus 2.

Well, that's y = x - 2.

For the next one, the x-coordinate times 2, y = x X 2, well done, we'd simplify that to read y = 2x.

The last one, x and y-coordinates sum to 10, x + y = 10, it's that straightforward.

We can plot these relationships to see the nature of them.

If I plot 0, 2, 3, 5, 5, 7, 8, 10, it looks like that.

A linear relationship.

The points align.

Will this set of coordinates do the same? Yes, we get another linear relationship.

How about this one? Instead of an additive relationship, this is a multiplicative relationship.

Let's plot these four coordinates.

Ah, another linear relationship.

D was very different, x + y = 10, that looks different to the rest of the relationships we plotted so far.

What do you think? Linear again? Absolutely.

We plot the coordinates 0, 10, 2, 8, 3, 7, 7, 3, we get another linear relationship.

Quick check to see that you've got that.

What relationship do you see in these coordinates? 3, 6, 0, 3, -3, 0.

There's three descriptions for you.

Pause this video, read them, and try to decide which one of those three fits those coordinates.

I hope you said A, the y-coordinate is the x-coordinate plus 3.

The y-coordinate being double the x-coordinate, that was only true for the first coordinate, 3, 6.

The x-coordinate and the y-coordinate sum to 3, well, that was only true for the second coordinate, 0, 3.

Whereas the coordinates fitting the relationship the y-coordinate is the x-coordinate plus 3, that's true for all of them.

Next, which equation describes the relationship "The y-coordinate is double the x-coordinate minus 1"? Which one of those three is it? Pause this video and make a suggestion to the person next to you.

It was C, y = 2x - 1, 2x being x being doubled or multiplied by 2.

We can work the other way around too.

We can use a relationship to generate coordinates.

The y-coordinate is three times the x-coordinate then minus 5.

If I try and find the y-coordinate to match these x-coordinates, I'll need to multiply the x-coordinate by 3 and then minus 5.

If the x-coordinate is 1, 3 lots of 1 minus 5 will give me -2.

What are the remaining missing y-coordinates? Pause this video, see if you can work out those four missing y-coordinates using that same rule, the y-coordinate is three times the x-coordinate minus 5.

We should have got -5 for the y-coordinate if the x-coordinate is 0.

We should have also got the coordinate 5, 10, coordinate 3, 4, coordinate -2, -11.

Each of those fit this rule, the y-coordinate being three times the x-coordinate then minus 5.

We can plot these coordinates to get a graphical representation of this relationship.

It would look like so.

It's a linear relationship again, and if we wrote it in algebraic notation, 3 times the x-coordinate minus 5, well, that's 3x - 5.

That graph represents the linear relationship y = 3x - 5.

Can be useful to use a table of values to plot such a relationship.

Typically, you'll see a table of values with consecutive integer values of x, consecutive integers 0, 1, 2, 3, 4, 5, you'll typically see tables like that.

We need to input that x value into the relationship to generate the y-coordinate.

So, when x equals 0, we'll take 5 lots of 0, add 3, and we'll get 3.

When x equals 1, 5 lots of 1 plus 3, that gives us 8.

If you kept going for x values of 2, 3, 4, 5, you'll generate the y-coordinates of 13, 18, 23, 28.

Plotting consecutive integer values of x allows us to more easily see patterns in the y-coordinates generated by this relationship.

It's a linear relationship because it's going up in a straight line.

Can you see it going up in constant steps of +5? Sometimes it won't be as simple as 0, 1, 2, 3.

We'll have to substitute negative numbers from a table of values.

I'd highly recommend you start at the simpler values to substitute in.

So start with zero and the positive x values and then see if you spot a pattern.

Once I've substituted it in, x equals 0, 9 minus 4 lots of 0, well, that's 9, I'll substitute in 1.

9 minus 4 lots of 1, that's 5.

And I keep going and I spot something about those y-coordinates.

9, 5, 1, -3.

Ah, it's linear with steps of -4.

So I could work in the other direction and conclude, well, those y values must be 9 add 4 being 13, 13 add 4 being 17, 17 add 4 being 21.

That's logical, I think it works, but it's always sensible to check these things.

I'll check that first coordinate, x being -3, y being 21, by substituting back in to that relationship.

9 minus 4 lots of -3.

Well, that would be 9 minus -12, which would be 21.

It worked! My table of values is accurate.

That means I'm ready to plot.

Once plotted, it looked like that.

That's the linear relationship y = 9 - 4x.

Quick check that you've got that.

Which are the two missing values from this table of values for the relationship y = 1/2 x minus 1/3? Well, this is tricky.

We're gonna have to use our knowledge of fractions here.

My top tip, the y-coordinates -5/6, 2/3, 7/6, 5/3, is there a common denominator that we could work to? That's my top tip.

Pause this video, see if you can figure this one out.

The two missing values were -1/3 and 1/6.

If you'd turned 2/3 into 4/6 and 5/3 into 10/6, you would've seen it changing by 3/6 or 1/2 each time to give you those two missing values.

Another check, and there's lots of ways you could do this one.

It's a matching exercise.

We've got four tables of values and four relationships.

Which one matches to which? So many ways you could do this.

I won't spoil the fun by giving you too many hints.

I'll just say pause this video and see if you can work this out.

Lots of ways you could work this out.

I know that that first relationship y = x + 5 matches that bottom table of values through substitution, our keyword for today, because if I'm substituting -4 for my x value, <v ->4 + 5 gives me +1,</v> <v ->3 + 5 gives me 2,</v> <v ->2 + 5 gives me 3.

</v> I can test that they all work by substitution.

That's how I then know the top table of values is the relationship y = 5 - 2x.

The next relationship, whether x and y-coordinate sum to 5, works for that table.

And the last relationship, y = 2x - 5, works for that table of values.

Practise time now.

Question one, Laura says, "Whatever number Andeep chooses, I'm going to subtract 4 from it." I've got a coordinate grid there, but instead of x and y labels on the axes, I've labelled the horizontal axis as Andeep's number and the vertical axis as Laura's number.

Could you plot some points to graphically represent this relationship? Pause this video, plot some points, and see what happens.

Question two.

I'd like you to complete the tables of values and plot these relationships on a coordinate grid.

The first relationship, y = 2x - 3, the second relationship, x + y = 4, third relationship, y = 1/2 x + 3, and the fourth relationship, y = 4 - 2x, tables of values, and then plot the coordinates.

Pause this video and do that now.

For question three, complete the table of values for this relationship, y = 1.

5x - 0.

8.

Table of values is slightly different here, they're not consecutive integer values of x.

We're going to use decimals.

0, 0.

2, 0.

4, 0.

6, 0.

8, and 1.

Because we're not using consecutive integer values of x, you're gonna have to make a decision about what would be a suitable axis to graphically represent these coordinates.

Pause this video for the table of values and graphically represent the coordinates that come out of the table.

Feedback time.

I left you a lot of freedom on this one.

I didn't say you had to do certain values.

Laura just said, "Whatever number Andeep chooses, I'm gonna subtract 4 from it." You could have chosen any particular number.

You would've got these coordinates.

So Andeep's number's 8, Laura's is 4.

Andeep's number is 7, Laura's is 3.

6, 2, 5, 1, et cetera, et cetera.

You would've noticed that whichever values you chose, they would've fitted on that line.

Did you have to do all the values? Absolutely not.

If anything, you could have just chosen two.

It's a linear relationship, so if Andeep's number is 4, Laura's is 0.

If Andeep's number is 0, Laura's is -4.

You'd still get that same line when you join those two coordinates.

Question two.

Four tables of values to fill in and four relationships to plot.

Your table of values for y = 2x - 3, the y values in that table should have read -7, -5, <v ->3, -1, 1, 3, and 5.

</v> And when you plotted them, they should have looked like this.

And you should have identified that linear relationship.

Part B was the relationship x + y = 4.

Your table of value should have read in the y row, 6, 5, 4, 3, 2, 1, 0.

When you plot those coordinate pairs, it looks like that and you identify that linear relationship.

For part C, we should have got those values.

I've written 2, 2.

5, 3, 3.

5, 4, 4.

5, 5.

If you wrote them in fraction form, that's absolutely fine.

Instead of 2.

5, you might have 5/2.

For the 3.

5, you might have 7/2.

That would be perfectly acceptable.

Once plotted, the coordinates will align like so.

Another linear relationship.

For part D, 4 - 2x, we should have got the y values 8, 6, 4, 2, 0, -2, -4, which would plot like that and give us that straight line.

Question three, we should have got the y values of -0.

8, -0.

5, -0.

2, 0.

1, 0.

4, 0.

7, which when plotted will give us another straight line.

If you look at my axes, I've only labelled the 0.

5 and the 1.

I've crucially included 10 squares to separate 0 and 1, so each of those squares represents a step of 0.

1.

That enabled me to accurately plot those coordinates, and you see I've got a truly straight line going through them.

Hopefully, your drawing looks a little like that.

Onto the second part of the lesson now, securing your understanding of plotting a relationship.

Andeep and Laura are talking about numbers again.

Andeep says, "You think of some numbers and I'll do something to them." Laura says, "6." Andeep says, "36." Laura says, "4." Andeep says, "16." Laura says, "10." Andeep says, "100." Laura says, "0".

Andeep says, "0." Interesting.

Is there a relationship? If there is, can you spot it? Pause this video and have a think about that.

I hope you spotted something about 36, 16, 100.

Andeep's number is the square of Laura's.

36, 16, 100, they're square numbers.

6 squared being 36, 4 squared being 16, 10 squared being 100, 0 squared being 0.

Andeep's number is always the square of Laura's.

We can represent this graphically.

I'll put Laura's number on the horizontal axis, Andeep's number on the vertical axis, and I plot the coordinates, 6, 36 is there, 4, 16 is there, and so on.

And something's different.

When I draw a straight line through those two coordinates, I miss the other two coordinates.

And a straight line through those two coordinates and I've missed the other two coordinates.

A straight line through those two and I've missed the other two coordinates.

So, this isn't a straight line relationship.

We'd call it a non-linear relationship.

We now have a curve instead of a straight line.

If we did all the numbers from 0 to 10 inclusive, or sorry, all the integers from 0 to 10 inclusive, we'd get all of those points.

And all those points will satisfy the relationship Andeep's number equals Laura's squared.

Laura could have said 3, Andeep would've said 9.

Laura could have said 7, Andeep would've said 49.

Laura could have said 10 and Andeep would've said 100.

Andeep's number is Laura's number squared.

We could write a rule for that using algebraic notation, A = L squared.

Lucas is going to save some of his pocket money.

"On the first day of next month, I will save 1p, but then double the amount each day.

So on the second day, I'll save 2p.

And on the third day, I'll save 4p, and so on." If we make a table of values and plot it, do you think we'll see a pattern? What does your mathematical intuition tell you? Well, we could do a table of values.

There's the first three days.

Pause this video and work out the next few days, if that number keeps doubling.

Lucas doubles the amount he saves every day.

Day one is 1, 2, 4, double that, 8, double that, 16, double that, 32, 64, 128, 256, 512.

Did you get those values? If we plot that, it's very difficult to fit all those points on a grid.

I've only actually plotted the first seven days because these numbers are accelerating really rather quickly.

I'd have to change the scale to plot the rest of them.

If I plot the rest of them, I'm gonna notice it's another non-linear relationship.

It's another curve.

If I plot all the way up to day 10 and the value of 512 that Lucas is now saving, it looks like that, another non-linear relationship.

Let's check you've got that.

True or false? All relationships we plot are linear.

Is that true or is it false? And could you justify your answer with either: "We always get a common difference and that plots a straight line" or "Some relationships are linear, some are non-linear and have other shapes like curves rather than straight lines." Pause this video, tell the person next to you your answer.

I hope you said false and some relationships are linear, some relationships are non-linear and will give us a curve instead of a straight line.

So, we've got these two examples of non-linear relationships.

They're both curves.

Lucas, Laura, and Andeep ask a very sensible question.

"Do all non-linear relationships always form an upward curve?" What do you think? All non-linear relationships form an upward curve.

Let's explore a different example, see if we can find an answer to that question.

Izzy says, "We've been given 60 cookies and I'm told to share them, but I'm the only one here.

Yay! I get all 60." Uh-oh! Sam walks in, says, "Sorry, Izzy, you're going to have to share them now.

60 divided by 2 equals 30.

We'll get 30 each." Still not a bad haul of cookies.

Uh-oh! Jun walks in.

"I love cookies! No way I'm missing out.

60 divided by 3 equals 20.

We now have 20 each." What does this relationship look like? And can we plot it? We can do it with a table of values, 60 cookies shared by the number of people in the room.

Our table of values is going to look like this.

The number of people, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

That's curious.

I didn't go consecutively in those numbers.

There are gaps at times.

I wonder why.

We know a few values because 60 cookies shared by one person is 60 cookies.

60 shared by two people, that's 30 cookies each.

60 shared by three, that's 20 cookies each.

Can you fill in the rest of that table? Pause this video, work out those missing values.

You should get these values here.

How might we describe this relationship? Well, 60 divided by 4 is 15 cookies each.

60 divided by 10 people, that's 6 cookies each.

60 divided by 30 people, that's 2 cookies each.

There's a pattern to all these calculations we're doing.

60, dividing it by the number of people to work out how many cookies we get.

Oh, so we can describe the relationship as 60/p = c.

But how will this relationship look when it's plotted? Well, we need an axis that looks like this, with a horizontal label of people and a vertical label of cookies, and then we can plot those values.

1 person, 60 cookies.

2 people, 30 cookies.

3 people, 20 cookies.

And you get those coordinates.

It's another non-linear relationship.

It's another curve, but it's not an upward curve.

Let's check you've got that.

Which words might you use to describe this relationship? 60/p = c.

Would we use the word curve, decreasing, or linear? Which of those words might we use? Pause this video, tell the person next to you.

We could call it a curve, and it's absolutely decreasing.

As we get more people, we get fewer cookies.

But what we can't call it is linear.

We can model all sorts of relationships in maths, but they won't all be linear.

Practise time now.

Question one.

This time, Andeep is squaring Laura's number and adding 10.

Could you complete the table of values and plot the relationship? Explain why the coordinate 5, 25 is not on this curve.

And write the relationship algebraically.

Pause this video, give that a go now.

Question two.

This time, the Oak pupils are sharing 24 apples.

There's a table of values, a graph.

Could you complete the table of values and then plot this relationship? Then, could you write this relationship algebraically? A hint for you is you will use the letter p to represent people and will need to use the letter a to represent apples.

Pause this video and try that now.

Feedback time.

Question one, Andeep is squaring Laura's number and adding 10.

We should get those values in the table.

You might just wanna pause and check that you've got the same values as I have.

Once plotted, you'll notice they make that shape.

Again, it's a non-linear relationship.

We've got a curve.

Explain why 5, 25 is not on this curve.

Because it doesn't fit the relationship.

Andeep is squaring Laura's number and adding 10.

When we do 5 squared and add 10, we don't get 25.

That's why that coordinate is not on the curve.

Writing the relationship algebraically, we would say Andeep's squaring Laura's number and adding 10, well, that's L squared + 10 being equal to A.

Question two, the Oak pupils are sharing 24 apples.

Table of values.

Did you notice I didn't write 1, 2, 3, 4, 5, 6, 7, 8, 9 for the number of people? I only used the factors of 24 for the number of people.

That enabled us to fill in the table of values like so.

Did you notice something pretty in those values? 1 and 24, 2 and 12, at the other end, 24 and 1, 12 and 2.

Huh, that's a nice relationship.

When we plot it, we get a curve like so.

The more people there are, the fewer apples each individual gets.

Writing that relationship algebraically is a tricky one.

Or is it? Maybe you conquered this one.

We would say, 24 divided by the number of people tells us how many apples each person gets.

Algebraically, that's 24/p = a.

Sadly, that's the end of the lesson, but it's been a good lesson.

We've learned we can use graphical representations to show all the points that satisfy a mathematical relationship.

Sometimes the relationship is linear, sometimes it's non-linear, however, all points on the line or curve satisfy that relationship.

I hope you've enjoyed today's lesson, and I hope to see you again really soon for some more mathematics.

Bye for now.