Lesson video

Hello, Mr. Robson here.

Great choice to join me for maths today, especially because we're plotting coordinates using technology.

This is going to be awesome.

A learning outcome is that we'll be able to use technology to quickly represent a set of coordinates graphically.

Keywords I use throughout this lesson: arithmetic and linear.

An arithmetic sequence, also known as a linear sequence, is a sequence where the difference between successive terms is a constant.

For example, 6, 11, 16, 21 has a constant difference of positive 5.

Therefore it is arithmetic, or it's linear.

By contrast, the sequence 1, 2, 4, 8 does not have a constant difference.

Therefore, it's not arithmetic, it is nonlinear.

Two parts of today's lesson: we're gonna get started with plotting sequences and rules using technology.

We can represent sequences and rules graphically by writing tables of values and plotting the coordinates generated.

So if I said the the values for the sequence 2n + 1, you'd give me a table of values and you'd find the terms 3, 5, 7, 9, 11 and you'd plot them like so where the horizontal axis is the term number n, and the vertical axis is the term value t, as a graphical representation of the sequence 2n + 1.

We could do that for Cartesian coordinates in the form of xy.

We'd have a table of values, we'd have some x inputs into that rule.

We'd get some y outputs and we'd get something like that.

Remembering to draw the line this time because we can have an infinite number of x inputs, it's not limited to positive integer values.

We can have any inputs there.

So we do draw a line for that one as a graphical representation of the rule y = 2x + 1.

The challenge when calculating tables of values and drawing these grids on paper is it's very time-consuming and we're limited to the range of values that we've calculated.

By using dynamic software, we can plot far more points and we can do it far quicker.

That's to our advantage as mathematicians.

To get you started, can you open up a web browser and go to desmos.

com? When you open that website, you're looking for the button that says "graphing calculator." It'll be in the centre of your screen.

It should bring up a screen looking like this.

Pause this video and just catch up so that your screen looks like mine.

We're gonna start by plotting sequences, so we're gonna need to change our axes a little bit.

Click on the "graph settings" menu in the top right of your screen, that's the button up there, the little spanner icon that says "graph settings." The graph settings dropdown menu should look like this.

A few changes I'd like you to make to this, I'd like you to turn off minor grid lines by clicking there.

I'd like to type in "Term number (n)" in the x-axis label box there.

I'd like you to change the lower bound of the x-axis to negative -0.

5 by typing -0.

5 in there.

I'd like to type in term value bracket (T) in the y-axis label box.

Finally, change the step on both axes to 1 by typing a 1 there and there.

Pause this video and just check that your graph settings are the same as mine.

Your screen should now look like this.

Term number on the horizontal axis, term value on the vertical axis.

Because term values are likely to change faster than term numbers, we need to change the scale on that vertical axis.

I'd like to hold shift and hover your mouse over the vertical axis.

An arrow will appear.

Whilst holding the shift button down on your keyboard, drag that arrow, and you'll be able to stretch the axis.

Pause this video and try that.

By stretching the axis, you could make it look like that.

This will enable us to show term values that are way higher than the value of the term number itself.

You can also drag your screen to see other parts of your graph.

Can you click and hold your mouse button to drag to there? Pause this video.

See if you can make your screen look like mine.

After dragging, you can recenter from anywhere by pressing the home button.

That's that button there, which looks like that.

Press that button.

It should bring you right back to the centre.

However, we're gonna drag so that we don't see the negative n values.

Term numbers are usually positive integers.

so make your screen look like that.

Quick check that you've got everything I've said so far.

True or false, the axes are always labelled "x and y." Is that true? Is it false? Can you justify your statement with either; we can relabel axis to suit the context of what it is we're plotting.

In the case of arithmetic sequences, we'll plot n and T; or, the horizontal axis is the x-axis.

The vertical axis is the y-axis.

Pause this video and have a think about that question.

I hope you said false, and we can change the labels on the axis to suit our context.

They won't always read x and y, especially not when we're plotting arithmetic sequences.

We need them to read n and T for that context.

Another check: to plot a rapidly increasing or decreasing sequence, we're gonna need to change our axes.

Can you make your axes look like this? Remember, hover over the axes, press the shift button on the keyboard and drag your mouse.

Pause this video, see if you can make your screen look like mine.

I hope you've got that now.

With our axes ready, we can now plot some coordinates.

Type (1, 2) in there.

Be sure to remember that coordinates are written inside brackets and separated by a comma.

Can you make your coordinate look just like that? Pause this video and type that in.

This should appear: the coordinate (1, 2).

Whilst it's acceptable to plot our coordinates as points, we mathematicians prefer to plot them as crosses for the accuracy that gives us.

If we want to do that, we'll need to make some amendments to how that coordinate is presented.

I'd like you to press that button once to hide or reveal your coordinate and then I'd like you to press and hold and you should see that.

Press and hold just there and make sure your screen comes up just like mine.

Once your screen looks just like this, we can change the presentation of that coordinate.

I'd like you to change the point to a cross by clicking there.

I'd like you to resize the cross to make it a little larger by typing 16 there and then I'd like you to recolor the coordinate to make it a black cross.

Plotting crosses instead of dots is a little more accurate.

I'd like you to plot this whole set of coordinates: (1, 2), (2, 4), (3, 6), (4, 8), (5, 10).

Before you do that, a little shortcut.

If you just type them in as we typed in (1, 2), you'll have to reformat every coordinate.

We can bypass that by clicking the "edit list" icon.

That's that little button there just above where you typed the coordinate (1, 2).

Once you've done that, it enables you to copy the coordinate that you've created.

If you click that copy icon 5 times, you'll get 5 copies of the coordinate (1, 2).

That mean that all formatted as a black cross on your grid.

However, we don't want 5 copies of the coordinate (1, 2).

I'd like you to change those coordinates so that they read as follows: (2, 4), (3, 6), (4, 8), (5, 10).

Pause this video and get that done.

Your screen should look like this with those 5 coordinates plotted, and you'll notice that they form a straight line.

They form a straight line because there's a relationship between n and T.

Remember that we're plotting an arithmetic sequence here, so we're plotting the term number versus the term value.

That's n versus T.

Can you see the relationship between our n coordinates and our T coordinates? I hope you are shouting at the screen it's multiplied by 2.

Absolutely.

Double 1 makes 2, double 2 makes 4, double 3 makes 6 and so on.

We can express this relationship as n multiplied by 2 makes T.

The n coordinate times 2 gives us our T coordinate.

We could simplify n multiplied by 2 to 2n, so what we've actually done is generate the first 5 terms of the sequence 2n: 2, 4, 6, 8, 10.

They're the first 5 terms of that arithmetic or linear sequence.

The coordinates form a straight line.

That's why we call it a linear sequence.

I'd like you to repeat that trick for this sequence.

The sequence 40 - 7n.

I'd like to calculate the first 10 terms of the sequence.

You'll do that with a table of values and then you'll input for the first term number n = 1, 40 - 7 lots of 1.

What term value comes out then? For the second term number n = 2, 40 - 7 lots of 2.

What's the term value there? Can you repeat that for the first 10 terms, then plot the coordinates on Desmos.

Scale your axes appropriately and show me that arithmetic sequence.

Pause this video and do that.

We should have got something like this: The term values 33, 26, 19, 12, et cetera, and then we'll need the coordinate (1, 33), (2, 26), (3, 19) and you'll need to scale your axes appropriately so that you can see all of those coordinates.

That's a graphical representation of the sequence 40 - 7n.

However, I bet that took you a while.

Is there a way to do it quicker? Of course there is, and that's the benefit of using technology to do this bit of mathematics.

Desmos has got a table function.

It's gonna enable us to plot multiple terms far quicker than that which we just did.

You'll find the table function here.

In the top left of your screen, click the add item icon.

From the dropdown list, I'd like you to add table.

At first, your screen will appear like this.

I'd like you to change the headers on the table so that they read n and 5n- 8.

Then I'd like you to type the numbers 1 to 10 in your term numbers column.

Pause this video and type 1 to 10 in the n column.

I hope you were delighted to notice what happened.

We input 1, 2, 3, 4, 5, all the up to 10 in the n column, and in the right hand column, Desmos has done the work for us.

It's automatically calculated the term values for the sequence 5n - 8.

Let's just check that it's done that.

The 10th term n = 10, 5 lots of 10 is 50, minus 8 is 42.

Oh, wonderful.

It's generated the term values for us.

That's far quicker than what we had to do when we drew that table and calculate the term values manually ourselves.

This is the benefit of using technology.

It will plot those coordinates for us as well, so in seconds we've managed to generate the sequence 5n- 8 or a graphical representation of the sequence 5n- 8.

If you click and hold here, you can change your coordinates to crosses if you've not already done so, shift and drag to resize your axes so you can see all the coordinates, and then wonderful.

We've got a quick graphical representation of the arithmetic sequence 5n- 8.

Let's just check you've got that.

True or false, we have to manually type the coordinates to plot arithmetic sequences.

Is that true? Is it false? I'd like to justify your answer with either yes, type them individually in the form (n, T) or we can generate more terms more quickly using a table.

Pause this video.

Say your answer to the person next to you.

I do hope you said false and we can generate more terms far more quickly using a table.

Which sequence is graphically represented here? Look at my Desmos screen.

Is that a representation of the sequence n or 4n- 3, or is it impossible to tell? Pause this video.

Tell the person next to you.

I hope you spotted it's a sequence 4n- 3.

That's the heading in the second column of the table.

Desmos can also be used to plot cartesian coordinates in the form of (x, y), not necessarily in the form (n, T) when plotting arithmetic sequences, just cartesian coordinates in the form (x, y).

You'll want to change your axis labels back to x and y.

I'd like to type in the coordinate (1, -3) and then I'd like you to type the coordinate (2, 2).

Then I'd like you to type the coordinate (3, 7).

Now we can type those by doing individual coordinates, but if I said type this family of coordinates, it's much quicker to use the table function again.

If you'd added a table and instead of typing individual coordinates of bracket 1 comma negative 3 close bracket, you could have just typed in the x column 1 and in the y column negative 3, in the x column 2, the y column 2, that gives you the coordinate (2, 2), (3, 7), et cetera.

You can plot a whole family of coordinates in a table like that.

Here's some more coordinates that fit that same rule.

You know it's gonna be a linear rule because look at the coordinates on the grid.

They are aligning, they're forming a straight line.

There's a linear rule going on here.

I'd be impressed if you'd spotted the rule.

I happen to know that it's take the x coordinate, multiply it by 5 minus 8 and you get the y coordinate.

That's the rule that links the x coordinate and the y coordinate consistently in this table.

We could express that as y = x multiplied by 5 minus 8.

I could simplify that to y = 5x - 8.

Unlike term numbers for specific sequences where we're plotting n values, the variable x can have non integer values, so we could plot x inputs of 0.

5, 1, 1.

52, 2.

53, 3.

54.

Notice I've changed my table, so instead of having to calculate each coordinate individually, I'm just gonna ask my right hand column to calculate for me the rule 5x - 8.

I don't just have to stick to steps of null 0.

5, I could make it steps of null 0.

25 and get that set of coordinates.

I could go even further.

I could plot it in steps of null 0.

1, but then there's a lot of typing into that table to be done.

The quicker way of doing that, I'd like to start the table with the headers x and 5x - 8 and in the x column type 0, 0.

1, 0.

2, 0.

3.

Can you pause this video and draw that table please? I'd like you to leave your cursor here in that particular box so that that box is flashing.

Then you can press enter and don't just press it once, press it multiple times and see what happens.

Pause this video and try that.

Desmos automatically generates a whole load more coordinates for you.

All these coordinates fit that same rule, 5x - 8, but the graph looks a little messy with those plotted as crosses, so in this moment it will be easier for us to change those crosses back to points so you can see what this rule is doing.

You can do that by clicking and holding here just at the top of that second column, just next to where you've got the rule 5x - 8, change your points to crosses and the rule becomes a little clearer.

Could we make the increments smaller than null 0.

1? If we went back and did that table again, yes, we could go in steps of null 0.

05.

Could we make increments smaller than null 0.

05s? Absolutely.

We could go in steps of 0.

02.

The purpose of doing this is just show you what is happening with this rule.

If we continue to decrease the size of those increments, the size of those steps forever and ever, we'll end up with a straight solid line.

The line is a graphical representation of the rule y = 5x - 8, but we don't need a table in order to plot the infinite points that exist on that line y = 5x - 8.

You can just type the rule in there, pause this video and give it a go.

That should happen.

The line y = 5x - 8 should appear.

You can now resize your axes to appreciate the infinite nature of this rule.

If I zoom out significantly, you can see that that line is just gonna continue infinitely in both directions.

Quick check now, true or false.

To plot a rule like y = 9 - 4x, you have to create a table of values.

Justify your answer with one of these two statements: Be sure to make the incremental steps in your table really small or we could generate the whole line of infinite points by just typing y = 9 - 4x into Desmos.

Pause this video, tell the person next to you your answer.

I hope you said false, and we can generate the whole line of infinite points by just typing y = 9 - 4x into Desmos, and there it is, the line y = 9 - 4x.

Practise time now.

Question 1, I'd like you to use the table function in Desmos to generate the first 10 terms of the arithmetic sequence, 20 - 4n.

Once plotted, can you resize your axes so that all 10 terms are visible? Question 2, I'd like to use a table function to generate the first 10 terms of the sequence 2.

65n- 17.

5 and then use your graphical representation to identify the first positive term.

That might sound unusual, but once you've graph that sequence you'll see what it means.

Pause this video and try those questions.

Question 3, I'd like you to use Desmos to explore if the sequence is 5n- 40 and 56 - 3n share a common term.

Tricky problem that, but like I said, explore it using Desmos.

Question 4, I'd like you to use Desmos to graphically represent the rules, y = 6 1/2 - 1/4x and y = 3.

26x + 18.

7 and then identify the quadrant in which they intersect.

Pause this video and give those 2 questions a go.

Feedback time.

Use the table function in Desmos to generate the first 10 terms of the arithmetic sequence 20 - 4n.

Your table should have looked like that and then you should have scaled your axes so that whole sequence is visible on your screen.

Just pause this video and check that your screen on Desmos looks exactly like mine does.

Question 2, use the table function to generate the first 10 terms of the arithmetic sequence, 2.

65n- 17.

5.

Then identify the first positive term, so the sequence should have looked like this.

That should have been your table of values and then the first positive term, that would be the 7th term there.

All the terms before it were below the x axis, i.

e.

they had a negative term value.

Question 3, use Desmos to explore if there's a common term in these 2 sequences.

You'd have to use a table function to plot multiple terms and then you'll notice that the sequences do cross and they share a common term.

If you zoom in on that moment and then click that coordinate, you'll notice it's the coordinate (12, 20).

That would be an n value of 12 and a t value of 20, so that's the 12th term with a value of 20.

20 is a shared term in those 2 sequences.

Question 4, you can plop them into Desmos exactly like that.

Desmos will let you type in a fraction, a mixed number, a decimal, so you can type those 2 rules into Desmos like so.

It'll generate those 2 lines for you and then you might have had to zoom out a little bit to identify that point at which they intersect.

That's in the second quadrant.

Next, we're gonna use the dynamic software to identify if points fit a rule.

All of these coordinates fit a rule except one.

That's a difficult problem on pen and paper with calculator, drawing an axis and plotting by hand, that would take a long time.

Technology will make it a lot quicker for us, a lot more efficient, and we mathematicians like efficiency.

Put these coordinates into a table in Desmos and plot them.

My top tip for you is when you're putting them into the table, can you put the x coordinates in order of lowest x coordinates to highest x coordinates? Pause this video, put those coordinates into a table now.

The table should have looked like that.

Notice my x coordinates are in order.

The reason for doing that is because I can see that coordinate looks slightly out of line.

If they all fit the same rule, they would align perfectly.

That one looks out of line.

How can we be sure that that's the one that doesn't fit the rule? We can do that by clicking and holding there to bring up that box and then turning on lines.

When you do that, your screen should now look like this and our suspicions were correct.

You see a perfect line and then it wobbles a little bit.

Those points are not quite aligned.

You can see the moment our linear pattern is broken, so I'm gonna further test this coordinate (1, 2).

I suspect that's the one that's wrong.

I'm gonna remove it from the table and plot it on its own.

If you do that, this should happen.

You take it out of your table, the 4 coordinates remaining in your table, provided you've got those x coordinates in order, those 4 coordinates will still align and you'll see that (1, 2) does not quite fit that line, so it is indeed the one that does not fit the rule.

All of these coordinates fit the rule y = 1.

7x - 7.

6 except for one.

Again, pen and paper, it's gonna be tricky drawing that grid.

I don't fancy that.

Technology's gonna enable us to do this really quickly.

Testing it using substitution.

The first coordinate fits, but testing them all is going to be very time consuming.

Plot the coordinates using the table function in Desmos and then I'd like you to plot the line y = 1.

7x - 7.

6.

Pause this video and give that a go.

There's your coordinates in a table and then when you add the line y = 1.

7x - 7.

6.

You can see the coordinate that doesn't fit.

(1, -5.

3) does not fit the rule.

Quick check that you've got that.

Which term is not in the same arithmetic sequence as the others? Is it the first term, the second term or the third term? Pause this video.

Tell the person next to you.

It's the second term.

You can just see that it's out of line.

If you take it out of your table of values and plot it separately, there's a justification that it does not fit that same arithmetic or linear sequence.

Practise time now.

Use Desmos to find which coordinate does not fit the same rule as the others.

Pause this video, give that problem a go.

Question 2, which coordinates fit the rule y = 7/4x - 2/3? Pause and give that a go.

Question 3, use Desmos to graphically represent the rules y = 3x - 13 and y = 17 - 2x.

Identify which of these 3 coordinates fits both rules.

Pause this video and try that problem.

Feedback time.

Question 1.

I asked you to use Desmos to find which coordinate does not fit the same rule as the others.

You plot them using a table to speed things up and then you'd switch on lines and it looks like so, I hope you remember to put the x coordinates in order.

That will enable the line to be drawn.

We can see the one that looks clearly out of line, (4, 4) doesn't look like it fits, so we take it out of the table, plot it separately and when we do that we can see the rest of those coordinates aligning and (4, 4) clearly not fitting the same rule.

For question 2, which of these coordinates fit that rule? Use the table function to create those coordinates and then draw the line of that rule by typing in y = 7/4x - 2/3.

You can see the first coordinate does, second coordinate does, third coordinate, it was the coordinate (2/3, 2) that did not fit the rule.

For question 3, using Desmos to graphically represent those 2 rules and identifying which of those 3 coordinates fits both.

You should have had those 2 lines and it's that point of intersection that we're interested in where the intersect is the coordinate (6, 5).

That coordinate fits both rules.

That's the end of the lesson I'm sad to say.

Today we've discovered that we can use software like Desmos to rapidly plot coordinates, arithmetic sequences such as 7 - 3n and rules such as y = 7 - 3x.

We can also use Desmos to see more quickly if a point does or does not fit a rule.

Hope you enjoyed that mathematics today.

I hope to see you again soon for more.