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Hello, Mr. Robson here again.
Well done for making the right decision and joining me for some maths today.
Coordinates with non-integer values.
I'm looking forward to it, let's get going.
So our outcome for this lesson is to describe and plot coordinates including non-integer values in all for quadrants.
Integer is a keyword which I'll be using a lot during this lesson, so it's important we define it.
An integer is a positive or negative whole number or zero.
Examples, negative two, zero, 153.
They're all examples of integers.
This is a lesson on plotting non-integer coordinates.
We're gonna start by looking at coordinates with non-integer values and then go to plotting shapes with non-integer value coordinates.
Let's start with coordinates with non-integer values.
Here are four coordinates with integer values.
The coordinate one, one, coordinate two, one, coordinate two, two and the coordinate one, two.
My question for you, "Is there a coordinate that exists between these four coordinates?" Pause this video and answer that question to the person next to you, to your teacher, or maybe to me on screen.
The answer's yes, we could put a coordinate right in the middle of those four coordinates like so.
But what's that coordinate? I'm gonna start by looking at that coordinate in terms of the X-axis, is halfway between one and two in the X direction.
The same thing for the Y direction.
It's halfway between one and two in the Y direction.
That coordinate then must be 1.
5, 1.
5.
It's halfway between one and two in the X direction, halfway between one and two in the Y direction.
So what we've got here is a non-integer value coordinate.
Having an axis with grid lines between the integers helps us plot non-integer coordinates.
If I said plot the coordinate 1.
5, 1.
5 on these axes, it's easier because of the grid lines.
Where would that coordinate be? Point at the screen.
There it is.
1.
5 in the X direction, 1.
5 in the Y direction.
How about the corner at 0.
5, 2.
5.
Could we plot that one? Point to the screen.
Show me where it is.
It'd be there 0.
5 in the X direction, 2.
5 in the Y direction.
Let's check you've got that.
Which of these is the coordinate for the point shown on this graph? Is it five, three, two, one, or 2.
5, 1.
5? Pause this video, tell the person next to you.
It is 2.
5, 1.
5.
People who got this wrong might have said it's five, three, it's five squares along in three squares up, but those squares aren't ones.
The squares are point fives in this case, we're taking leaps of point fives.
So when we leap one square past two in the X direction, we get to 2.
5, 3 leaps up, gets us to 1.
5 in the Y direction.
So the coordinates 2.
5, 1.
5.
Another coordinate grid with one labelled on the X axis and one labelled on the Y axis.
So what would these blank labels be? That's right, 0.
5 and 0.
5.
What if I do that? Can we add any more labels now? Pause this video and tell the person next to you what other labels could we write? That's right, we could have labelled 0.
25 and 0.
75 on both axes.
We're starting to get really rather accurate now with these grid lines and with our labels, but does it stop there or can I keep going? Could I add those grid lines? Those grid lines? By adding these grid lines, we can go to an incredible level of accuracy with our coordinates.
There is no limit to the level of accuracy we can reach here.
In fact, going to higher levels of precision is really important uses for mathematics, science and many other subjects.
You'll see this a lot in future learning.
This one's unusual.
I've given you a label of one on the X axis, one on the Y axis again, but I've changed the grid lines to point something out.
What's that coordinator make a suggestion to the person next to you, to your teacher, to me on the screen and that suggestion may or may not be right.
It's absolutely fine.
We welcome all wrong answers in the math classroom.
I'm sure you had lots of suggestions.
The best way to communicate this coordinate is to call it one third, one third.
Where one third the way to one in the X direction where one third the way to one in the Y direction.
Now you've might have tried to communicate that as a decimal but it's not necessary.
One third is incredibly concise to write and it's incredibly accurate.
In fact, it's easier to communicate this coordinate in a fractional form rather than decimal form.
It's important you are aware of that.
We can express non-integer coordinates in fractional form as well as in decimal form.
There'll be times when we want them as a decimal.
There'll be times when we want them as a fraction.
Let's check you've got that.
Which of these represent the coordinate shown on the graph? One fifth, one fifth, one half, one half, 0.
2, 0.
2, which is it? Pause this video, tell the person next to you.
So we could have said one fifth, one fifth.
We are indeed one fifth the way to one on the X axis.
One-fifth the way to one on the y axis.
We could also have said 0.
2, 0.
2 because one fifth and 0.
2 are the same numerical value.
The change now, we're not just looking in the positive X and positive Y direction.
The coordinate 1.
5, 0.
5 lives in that first quadrant, but we can plot non-integer value coordinates in all four quadrants.
Can you read those three coordinates? Can you tell me what they are? Pause this video, write them down.
Non-integer coordinates can be plotted in any quadrant such as the coordinate, negative 1.
5, 0.
5, negative 1.
5, negative 1.
5, 1.
5, negative 1.
5.
Let's check you've got that.
Which one of these coordinates is negative 0.
5, 1.
5? Is it coordinate A, B or C? Pause this video, tell the person next to you.
It was coordinate B.
When negative 0.
5 in the X direction and 1.
5 in the positive Y direction.
So we end up at coordinate B in that second quadrant.
Some practise now.
I've got some coordinates for you to plot.
I'd like you to plot coordinate A, 0.
5 negative 2.
5, and label it A.
Coordinate B, I'd like you to plot it negative 0.
5, 2.
5, coordinate C, I'd like you to plot it negative 2.
25, one half and then coordinate D.
I'd like you to plot it negative three over two, negative seven over four.
You might think of those fractions as negative three halves, negative seven quarters.
Pause this video and try and plot those coordinates.
Question two is a matching task.
There are five coordinates on a coordinate grid and I've given you the coordinates for A, B, C, D, and E, but which is which? Coordinate A is at 1.
8, 2.
2.
I'd like you to label one of those five coordinates as A.
Pause this video, try and match all five coordinates.
Question three.
June says, "The point is just past negative three so this is probably the coordinate negative 3.
1,4." June's wrong.
Can you write a sentence justifying why.
Pause this video write me a sentence.
Okay, some feedback.
Plotting coordinates 0.
5, negative 2.
5 would be there in the fourth quadrant, negative 0.
5, 2.
5 is in the second quadrant, negative 0.
5 in the next direction, 2.
5 in the Y direction.
Quadrant C is a little bit tricky and negative 2.
25 is important to spot each of the little squares on the grid is worth 0.
25.
So that's gonna require us going one little square past negative two in the X direction away from the origin to get to that moment there.
Negative 2.
25 and then a half in the positive Y direction.
For coordinate D, negative three over two or negative three halves.
We're gonna have to do a half leap, negatively three times in the X direction and then seven quarter leaps, negatively in the Y direction.
Getting us to that point there.
Pause video, just double check that your coordinates are in exactly the same position as mine and that you've labelled them correctly.
Question two, matching the coordinates to the point on the graph.
Coordinates A and B I'll start with.
On account of they're both positive X coordinates, positive Y coordinates so they must be these two coordinates in the first quadrant.
The difference between A and B is, A has an X coordinate of 1.
8, B has an X coordinate of 2.
1 so we know the difference between them because coordinate A has not quite reached two in the X direction, whereas coordinate B goes past two in the X direction.
Similarly, we could compare their Y directions.
Coordinate A is 2.
2 in a positive Y direction.
It's gone beyond two and a positive Y direction whereas coordinate B, 1.
7 in the Y direction, it doesn't quite reach two in the Y direction.
That's how we can tell those two apart.
In the case of coordinate C and D, negative 2.
1, 1.
6, negative 1.
6, 1.
6, they share the same Y coordinate.
It's the X coordinate that differs.
Point C must be the one on the left because it's gone beyond negative two in the X direction.
It's gone all the way to negative 2.
1 in the X direction whereas coordinate D only reaches negative 1.
6 in the X direction.
That's how we tell those two apart.
And then the coordinate E, 1.
6, negative 1.
6.
You could see from the fact it was a positive X direction, negative Y direction.
It was gonna be all on its own in our fourth quadrant.
Question three, and this is a common misconception when you go to read that coordinate, you think it's just past minus three, so thinking it's minus 3.
1, well that's very logical but the truth is what direction are we travelling? We're travelling in the X direction and when we start that journey we start at the origin.
So we haven't quite reached negative three.
Rather than passing it, we haven't quite reached it, so your feedback might have included.
From the direction of the origin, the coordinate is still approaching negative three, so it's more likely to approximate as negative 2.
9,4.
Okay, moving on now to shapes with non-integer value coordinates.
Let's start by plotting four coordinates and then I'll ask you what you notice.
Pause this video, plot those four coordinates and then write down an observation.
We should have those four coordinates and you should have noticed that they form a rectangle.
Can you see the moments when the X and Y coordinates match in our rectangle? Those two coordinates share an X coordinate of negative 0.
5.
You can see these two vertices and that X direction of negative 0.
5 there.
These two coordinates share a Y direction of negative 1.
5.
You can see that negative 1.
5 direction on those two vertices there.
What that enables us to do is resolve all sorts of problems without the need for grid lines or labels on our axes.
If I told you this coordinate is 1.
5, 0.
8 and I tell you that the rectangle sides are parallel to the X and Y axis, you could finish that coordinate for me.
If I told you the coordinate, the vertices of the top left as a coordinate of negative 1.
1, something, you could tell me that missing Y coordinate, how? Because you know it's 0.
8 in the Y direction, just like the vertices to its right.
Once we've got those two coordinates, if I were to give you a little bit more information about this vertices and tell you it's got a Y coordinate of negative 0.
9, could you tell me the X coordinate? Of course you could.
It's negative 1.
1 in the X direction just like the vertices above it.
So that coordinate must be negative 1.
1, negative 0.
9.
Once we got all that information, that fourth coordinate should flow easily.
We know it's 1.
5 in the X direction and it's negative 0.
9 in the Y direction giving us a coordinate of 1.
5, negative 0.
9.
Let's check you've got that.
What would the fourth coordinate of this square be? Is it negative 1.
5, negative 0.
5? Negative 0.
5, negative 1.
5? Or negative 1.
5, negative one? It's negative 1.
5, negative one.
Negative 1.
5 in the X direction just like the vertices above it.
Negative one in the Y direction just like the vertices to its right.
If we take away those grid lines, you no longer have the option to read what this coordinate is.
You have to look at the other, the coordinates of the other vertices.
When you do that, can you tell me the fourth coordinate of this rectangle? Is it 0.
95, negative 0.
2, 0.
95, negative 1.
3 or is this negative 0.
95,1.
3? Pause this video, tell the person next to you.
It was A, 0.
95, negative 0.
2.
You could see it was 0.
95 in the positive X direction, like the vertices above it and like the vertices to its left, it's zero point negative two in the Y direction.
Next, a square.
And if I tell you that this square sits on the XY axis and I tell you one of the coordinates that being the top right vertices 1.
5, 1.
5, can you tell me the coordinates of the other three vertices? In fact, why tell me? Tell the person next to you or tell your teacher.
Pause this video and do so.
Directly below the known vertices on the X axis must be 1.
5,0.
1.
5 in the X direction, No travel in the Y direction, 1.
5,0.
The origin, zero, zero.
That's the origin, zero, zero.
Don't forget the importance of that word origin and to finish the coordinate, zero, 1.
5.
Zero in the X direction, no travel in the X direction, 1.
5 in the Y direction.
What if I take that square and do that? These four squares now sit between the origin and the coordinate 1.
5, 1.
5.
What would that coordinate be? To figure this one out, we need to consider the fact it's half the journey in both directions.
Halfway to 1.
5 on the X axis, halfway to 1.
5 on the Y axis.
So we need to know half of 1.
5, that's 0.
75.
So this coordinate must be 0.
75, 0.
75.
What if we go even more complex again? Nine squares sit between the origin and the coordinate two, two.
Can you find the coordinate A? This is a tricky problem.
So I'm gonna ask you, just give it a go and if you get it wrong, it doesn't matter.
If we examine all the wrong answers we suggest, we will eventually get to the right answer.
That's the way maths works.
Don't be afraid to make a mistake.
So pause this video, suggest to the person next to you, suggest your teacher or write down in your book.
Why do you think that coordinate would be for point A? Things to consider.
It's one third of the journey to two in both directions.
I'll repeat that 'cause it's important.
It's one third of the journey to two.
So I need to find one third of two which I can do by taking two and dividing it by three.
Two divided by three.
We can write as a fraction two over three, which you would say is two thirds and that is our coordinate, two thirds, two thirds.
You'll notice in this moment it is easier to leave it communicated in fractional form.
We could write that as a decimal, but it's just more concise and incredibly accurate to leave it as two thirds.
In fact, in mathematics it's really important we can move fluently between fractions in decimal and recognise when it's most appropriate to communicate using one or the other.
Let's check you've got that.
What's the coordinate of point A? Leave your answer in decimal form.
Is it 0.
1, 0.
1, 0.
25, 0.
25 or one, one? The coordinate was 0.
25, 0.
25.
So quarter of the way to one in both the X and Y direction.
If I leave that coordinate there and label it 0.
25, 0.
25, can you tell me the coordinate B? I'd like you to leave your answer in decimal form again.
Is it 1,3, 0.
25,0.
75 or 0.
75,0.
75? Pause this video, tell the person next to you.
It was 0.
25,0.
75, 0.
25 in the X direction, just like the coordinate below it, but three of those leaps in the Y direction, 0.
25,0.
5,0,0.
75, hence the coordinate 0.
25,0.
75.
This problem looks different.
We have an isosceles triangle and we note two of the coordinates in full.
Two coordinates sitting on the X axis, one at one, zero and one at five, zero.
I've deliberately left the X coordinate of the vertices at the top blank.
I've given you the Y coordinate, but I left the X coordinate blank.
Can you find that missing x coordinate? Pause this video, make a suggestion to the person next to you, to your teacher or to me on the screen.
To resolve this problem, you might think of it as half of the leap from one to five.
When I leap from one to five, that's a leap of four, but I only want half the leap, so I'm leaping two.
So I start at one and I add two and it's three.
So this coordinate must be three, six.
That felt like a lot of working out to find that three.
Was there a better way? Absolutely, we could think of it as finding the midpoint of one and five and we can do that more quickly by adding one and five and then halfing the result, one plus five is six, half of that is three.
Similar looking problem.
It's an isosceles triangle.
We've got two coordinates on the X axis, one, zero and four, zero, but I've left the X coordinate blank in the vertices at the top of our isosceles triangle.
Can you find that missing x coordinate? Pause this video, give it a go.
It is a more difficult problem this time because it's not a whole number answer.
In fact, it's a non-integer answer.
A non-integer solution will look like so.
Finding the midpoint of one and four, the most efficient method is to add them and half that result.
One plus four is five, half that is 2.
5.
So (indistinct) coordinate, we could call 2.
5,6.
You could have called the X coordinate five halves or five over two instead of 2.
5, but 2.
5 is quite a clean and easy decimal to communicate and use.
So we might well call this coordinate 2.
5,6 as a decimal rather than leaving it in fractional form.
Let's check you've got that.
These three coordinates form an isosceles triangle.
Find the missing X coordinate.
Pause this video.
Take a minute to give this problem a go.
To find that missing X coordinate, we're finding the midpoint of 11 and 30 because of those known coordinates on the X axis, 11, zero and 30, zero, the vertices at the top of this isosceles triangle must be halfway between 11 and 30.
11 plus 30 is 41, half that is 20.
5.
This must be the coordinate 20.
5,6.
If you rate your answer as 41 over two, six, you are mathematically correct.
I think we'll just more commonly see this one communicated in decimal form.
Okay, so in practise now.
Question one.
I'd like you to work out the missing coordinates of these rectangles.
On the left hand side we've got our coordinates communicated in decimal values and I've left out an X coordinate and two y coordinates on the right hand side.
The coordinates are in fractional form as a variety of X and Y coordinates missing.
Pause this video and see if you can fill in the blanks.
Question two.
Work out the missing X coordinate values in these isosceles triangles.
There are two isosceles triangles, one's in the first quadrant, one's in the fourth quadrant.
The one in the first quadrant has the points three, zero and eight, zero on the X axis.
The one in the fourth quadrant has the points eight, zero and 0.
5,0 on the X axis.
With all that information, you should be able to identify the two missing X coordinates.
Pause this video and give it a go.
A third and last question, and I've saved the best for last.
I love this problem, I think it's beautiful.
Four identical squares are drawn on this graph.
Work out the coordinates of A and B.
We know the origin, we know the coordinate one, one, and we've got to find those coordinates A and B.
Pause, give it a go.
Okay, some feedback.
On the left hand side, 0.
4 is the X coordinate, the Y coordinate is 0.
9, why? Because it's 0.
9 in the Y direction, just like the vertices to its left.
Talking of the vertices to its left, the X coordinate there is negative 1.
1, why? Because it's negative 1.
1 in the X direction, just like the vertices below it.
Talking of the vertices below it, negative 1.
1 in the X direction and negative 2.
3 in the Y direction.
The same as the vertices to its right.
And there we know all four quad, all four coordinates on that rectangle.
For the rectangle on the right, the coordinates are communicated in fraction form.
Similar problem just communicated differently.
We know the coordinate in the first quadrant, seven over three, 11 over five, and that helps us to find the coordinate in the second quadrant because it's the same Y direction.
We must be 11 over five in the positive Y direction.
Now knowing that coordinate in the second quadrant helps us to find the coordinate in the third quadrant because they've got the same X direction, negative 20 over three.
And then once we know those coordinates, we can find the X direction of seven over three, that's the positive X direction, seven over three, just like the vertices above it.
And then the negative nine over four Y direction, just like the vertices to its left.
Just double check that you've got the same answers as me.
The second problem.
Finding the missing X coordinates in these isosceles triangles.
The isosceles triangle in the first quadrant.
On the X axis we've got the coordinates of three, zero and eight, zero, and then a blank X coordinate and a Y coordinate of eight.
That blank X coordinate must be the midpoint of three and eight.
To find the midpoint of three and eight, we'll add them together in half it.
That'll give us 11 over two or 5.
5 because it's a nice clean decimal to communicate.
I think we're likely to see this answer written as 5.
5,8, but you might have written 11 over two, eight and that would be fine.
For the next triangle, we need the midpoint of eight and 0.
5.
That's why I love this method.
I don't want to have to think what is the leap from zero point 0.
5 to eight, what's half that leap? Add half that leap to 0.
5, it's too much.
But to find the midpoint, we can add those two together and half it.
Add those two to get 8.
5, half it to get 4.
25.
I think you'll see that coordinate called 4.
25, negative eight.
Question three, my favourite.
Four equally sized squares sitting between the origin and the coordinate, one, one.
It's useful to think of.
That's one quarter of the journey to one that's a jump of 0.
25 each time.
So that coordinates 0.
25,0.
25.
That helps to spot coordinate A.
Coordinate A is 0.
25 in the X direction and two lots of 0.
25 in the Y direction.
So coordinate A is gonna be 0.
25, 0.
5.
Coordinate B is three of these leaps in the X direction and three of those leaps in the Y direction.
So coordinate B must be 0.
75,0.
75.
Okay, we've reached the end.
And in summary, coordinates can be read and plotted using integers, fractions and decimals.
Reading the scales and the axes is especially important when dealing with non-integer coordinates and this becomes especially useful for solving geometry problems on coordinate grids.
I hope you've enjoyed today's lesson, I certainly have.
I look forward to seeing you again for more maths very soon.
