Lesson video

Hello, I'm Mrs. Lashley and I'm looking forward to working with you as we go through this lesson.

I really hope you're looking forward to it and you're ready to try your best.

So our learning outcome today is to be able to construct a set of points that satisfy a set of conditions.

So we have a new keyword for today's lesson, which is locus and the plural is loci.

So read the definition by pausing the video here and we'll be making use of that during the lesson.

Other keywords that we'll be using are on the screen.

These are previous words that you've met before in your learning, but again, you may wish to pause the video to read those definitions, familiarise yourself before we make a start.

So today's lesson on constructing loci is gonna be split into three learning cycles and you can see them on the screen.

We are gonna make a start on the first one, which is looking at the loci from a fixed point.

So the Oak pupils asked to stand two metres from the base of a tree, and this is where they put themselves.

Jun is two metres from the base of the tree.

So is Sophia, so is Izzy, so is Jacob.

All the possible points that they could have stood at follow this rule and they form a circle.

The radius of the circle is the two metres and the centre is the base of the tree.

A locus is a set of points that follow a given set of conditions.

This set of points can be constructed by following a given rule or set of rules.

So the locus of all possible points that are an equal distance from a fixed point forms a circle.

And we just saw that with the Oak pupils and the base of the tree.

So here is a check for you.

There is a coffee shop 150 metres away from the bus stop.

Andeep draws this diagram to show the locus of points 150 metres from the bus stop.

He doesn't think the diagram looks correct, explain what he has done wrong.

So pause the video and when you're ready to check, press play.

So the locus of points would be a circle with a radius of 150 metres because the bus stop is a single point.

This radius would be three centimetres on the scale drawing and we can use the scale to help us with that.

So this is what Andeep should have drawn.

So the Oak pupils are now asked to stand two metres or less from the base of the tree and they're allowed to move if they want to.

Laura says, "We don't need to stand the same distance from the tree to follow this rule." So the locus of all possible points will form a region up to and including the circumference of the circle.

So they were asked to stand two metres or less.

So we can see that Alex, Aisha, Jacob, and Jun are still two metres from the base of the tree, whereas everybody else is less than two metres.

They're now asked to stand less than two metres from the base of the tree.

So the locus of all possible points will form a region up to, but not including the circumference.

So Alex, Aisha, Jun, and Jake move inside the circumference of the circle.

Laura says the locus is the shaded region.

The Dotty line shows that the circumference is not included.

You may recognise that idea from other parts of your maths education.

So here is a check.

True or false, this diagram shows the locus of points that are five centimetres or less from point A? And then justify your answer.

Pause the video and when you're ready to check, press play.

So it was false, and both of the justifications are valid.

So the circumference of the circle should be included in the locus, represented by a solid line, and the region inside the circumference should be included in the locus represented by shading the region.

And that's because we were looking for five centimetres or less.

So five centimetres would be the circumference, and less would be within the circle.

So that needs to be shaded 'cause it's a region and a solid line for the circumference because it's included.

Another check for you.

Match up the diagrams with the correct description of the loci.

So pause the video and then we'll go through the answers when you press play.

So diagram a would match with description h, the locus of points four metres from point P.

There's no shaded region, so it is just the locus of points, which is the circumference and it's solid, so it includes them.

The diagram b matches with description e, the locus of points up to and including four metres from point P.

So we've got the circumference as a solid line and a shaded inside.

So up to and including.

Diagram c matches with g, the locus of points less than four metres from point P because the circumference is a dashed line, which leaves d and f, the locus of points more than four metres from point P.

So outside of that circle.

So here we're looking at context of a new "Park and Ride" car park is going to be built between four and six kilometres from the town centre.

So the town centre is a fixed point.

Four kilometres from the town centre would give us our locus of points, and six kilometres would give us another locus of points.

The "Park and Ride" is going to be within because it says it's going to be built between four and six kilometres from the town centre.

The shaded region shows the locus of points that satisfies these conditions.

So here we have Andeep looking at this problem.

So find the loci of points that are three centimetres from point A and four centimetres from point B.

So we've got a scale there to show us that one square is one centimetre.

So point A, we can draw a circle with a radius of three centimetres.

We can then draw a circle with a radius of four centimetres from point B.

So which points are three centimetres from A and four centimetres from B? Well, there are two points that meet these conditions and we can mark those with crosses.

If it now changes to the conditions being find the loci of points that are within three centimetres from point A and four centimetres from point B, then we are now thinking about being inside.

So this loci will be the intersecting region of the two circles.

We can show this Loci by shading the region.

So it's this region here.

So it's within three centimetres of A and also within four centimetres from B.

So here we have a map that shows the location of three mobile phone masts.

The range of these masts is 50 kilometres and our scale this time is one square is 10 kilometres.

So we can draw circles, but in this case they are arcs that are five squares as a radius 'cause that will be our 50.

And we can do that around each of the three points.

So mobile phone signals can be used to determine the location of somebody.

Laura's mobile phone signal is detected by all three masts.

So Laura must be within this region, this region that is shaded because it's being picked up by each of the masts.

Laura says this is the locus of points that are within 50 kilometres from each mast.

So we're now onto the first task of the lesson.

So question one, here is a target for an archery game.

Write a sentence to describe the loci for each score.

So you can see the target on the right hand side and the widest diameter is 120 centimetres.

And we got an example for a score of 15.

So a locus of points that is between 15 centimetres and 30 centimetres from the centre of the target.

So use that to support you when you're writing the other ones.

Press pause whilst you do question one and then when you press play, we'll move on.

Here is question two.

So Laura and Andeep are walking their dogs in the park.

Each dog is on a lead which can extend up to 10 metres.

If Laura and Andeep stay where they are, find the locus of points where both dogs can go.

So, again, there is a scale where one square is two metres.

So pause the video, think about which region you will be shading for this question.

And when you press play, we're gonna move to question three.

Question three, so we've got a map here and it says, use the map to find which towns are less than 60 kilometres from Bristol, less than 50 kilometres from Oxford, but more than 40 kilometres from Cheltenham.

So first of all, locate Bristol, Oxford and Cheltenham.

Think about what the locus of points for each of those will be.

And then you try and find towns that satisfy those three conditions.

Press pause, and when you're finished with that, we're gonna go through the answers to task A.

So question one on task A was writing descriptions of loci.

So the score of 15 had been done as an example.

So score of 20, you would write something such as a locus of points that is less than 15 centimetres from the centre of the target.

We could use the description for the score of 15 to know that that centre circle has a radius of 15.

For the score of 10, it's a locus of points that is between 30 centimetres and 45 centimetres from the centre of the target.

For a score of five, a locus of points that is between 45 centimetres and 60 centimetres from the centre of the target.

And a score of zero would be a locus of points that is more than 60 centimetres from the centre of the target.

Question two, it was Laura and Andeep walking their dogs.

Laura and Andeep stayed still, their dogs are in extendable leads that go up to 10 metres and we needed to show where both dogs could go.

So the locus of points is represented by the shaded region.

So you draw your locus of points that is exactly 10 metres from each individual.

So you can see that with the arcs there.

And then the overlapping region is where the dogs, both dogs could go.

On question three, you are using the map.

So firstly, you needed to look for Bristol, which is sort of bottom left, and you needed to draw a circle using the scale that was 60 kilometres as a radius.

So using that scale, that's our 60.

Bristol is there.

Draw our circle.

Because of the section of the map, it's actually an arc.

So we are looking for towns that are less than 60 kilometres from Bristol, so they've got to be within that region.

Then we need to find Oxford.

And then you need to draw a circle around Oxford, which is 50 kilometres.

Again, we're gonna use the scale on the map and draw our circle.

And we wanted to find towns that were less than 50 kilometres from Oxford.

So again, within that circle.

And then lastly, we needed them to be more than 40 kilometres from Cheltenham.

So find Cheltenham, set up your pair of compasses to the 40 kilometres, draw our circle, and they need to be more than.

This bit here is where we are looking for the towns and we can zoom in there just to show you.

And so Swindon is within that shaded region.

You may have also said Marlborough, depending on the accuracy of your drawing.

So if you look where Marlborough is, but Swindon is the main town or city that is within those three constraints.

So we're now up to the second learning cycle, which is looking at loci from a fixed point on or within a shape.

So it's sort of extending our first learning cycle.

In a previous example, we were constructing loci which were 50 kilometres from each mobile phone mast.

We could not draw the complete locus of points for each mast as we ran out of room on the diagram.

So if you remember, it looked like this.

So we used arcs instead of the full circle.

And that region indicates where all three masts are picking up the mobile signal.

Zoom out, we can see that complete locus of points that are 50 kilometres should be full circle.

So if our diagram was zoomed out, we had more space, then the full circles could be drawn to scale.

And the shade of region shows the locus of points that are within 50 kilometres from all three of the masts.

So the locus of points that are an equal distance from a fixed point forms a circle.

We've seen that quite a bit.

There may be an additional constraint that means that the locus will be an arc instead of a full circle.

So this could be due to the size of the diagram like we just saw, or it could be due to another barrier, such as edges of a shape.

Find the locus of points inside the hexagon that are two units or less from the point P.

So we would set up our pair of compasses to a radius of two units.

We could use the scale there and we'd have to draw an arc because it's inside the hexagon.

If it's two units or less, then we would shade inside as well.

So the locus of points forms a sector with a radius of two units.

If we now look to find the locus of points outside of the hexagon that are two units or less from point P, then again we'd get a sector.

Find the locus of points outside the hexagon that are four units from point P.

So now we've got to set our pair of compasses to four units.

Is it true to say that the locus of points forms a sector with a radius of four units? Well, let's have a look.

One end of a string four units long is attached to the hexagon at that point P.

A pencil is attached to the other end of the string.

And Andeep says that this extra constraint will change the shape of the locus of points.

We would get this sector when we meet the edge.

When our string would meet that edge, then something needs to change, it would become taut along the outside of that hexagon.

And using the scale, we can see that that's three units of length.

So that means that there is one unit left.

And so the edge of the hexagon is three units.

"And so I need to draw arcs with a radius of one unit, says Andeep." So now our arc, our sectors will change.

We have a sector with radius of four and then we also have sectors with a radius of one.

So here is a check.

A goat is attached to an eight metre rope.

One end of the rope is attached to a point on the fence of a rectangular pen.

The pen is a 10 metre by four metre rectangle.

Sketch the loci of points and, hence, shade the area of grass that the goat can graze on inside the pen.

So there's three for them to do.

So pause the video, reread what the scenario is, and then press play when you're ready to check.

So if the goat is tethered attached to an eight metre rope inside the pen at that corner point, then we would draw the arc using our pair of compasses, the radius of eight metres and we would shade within that region because it's on the rope, so it means it could graze up to and including that boundary.

On the second one, the point on the fence where it is attached is halfway along the length of the rectangular pen and actually an eight centimetre radius circle will cover the whole pen.

So if it's attached there, it can graze all of the pen.

So on the last one, the end of the rope is attached halfway along the width of the pen, eight metres is the length of the rope, which is not the full length of the pen.

So we know there will be some area that the goat cannot graze.

And again, you're gonna draw an arc which would be from the circle with a radius of eight metres.

So it's the same scenario, but this time we are going to sketch the loci of points and shade the area of grass that the goat can graze on outside the pen.

So pause the video, think about that.

And then when you're ready to check, press play.

So, if we look at the first one where it was tied to the corner.

Well, this would be the sector with an eight metre radius.

But once it's gone along that edge, we know that the radius will change.

And so that will be four metres leaves you with four metres.

So it'll be a sector with a radius of four metres this time.

So this will be the region that the goat can graze for this one.

The second one here, we can draw our semicircle, which is the largest sector with a radius of eight metres.

But once we hit the edge, then we know our radius will change.

So they are three metres left on either end.

So it'll be a radius of three metres.

And lastly that halfway.

So we can get a semicircle with a radius of eight metres.

There'll be six metres of the rope left at either side.

So the goat can graze in a sector with radius six metres until it meets the fence of the pen.

So this would be the shape of the locus.

So now we're onto the second task of the lesson.

So in this task, there's only the one question, and here it is.

This is a plan of a garden.

A hose 10 metres long is attached at the tap.

You can see the tap there.

Draw the locus of points in the garden that can be reached by the hose.

Press pause and then when you are ready for the answer to this question, press play.

So 10 metres is the length of hose.

From the tap would be able to get this area.

At the point at which it hits the sort of bend in the fencing, we know that three metres of the hose would've been used, so there'll be seven metres left, so we can get this sector.

And then seven goes along that five metre wall or fence of the garden, so there would be two metres left.

So we'd have this sector with a radius of two.

So it's this area that can be reached by the hose.

So we're now up to the final learning cycle of the lesson.

And this is looking at loci from a fixed line or shape.

So the Oak pupils are asked to stand two metres from a fence.

You can see where Laura, Andeep, Jun, Jacob, and Sam have stood.

Laura says the locus of points form a line that is parallel to the fence.

So we can use our feather marks to show that they are parallel.

Andeep says, but we also could stand on the other side of the fence.

So you can see that Jacob and Andeep are now on the other side of the fence, but still two metres from it and this would cause another parallel line.

So the locus appoints form two lines that are parallel to the fence.

Laura says, "What happens to the locus appoints at the ends of the line segment?" So maybe have a think about that yourself.

How could you be two metres from the end of the fence? So Aisha, Lucas, and Alex have now stood themselves two metres from the end of the fence, and we can see that they are on an arc.

That arc forms a semicircle.

And that's because if we think about it, the end of the fence is a single point and we know that the locus of a single point is a circle.

Sophia and Izzy have stood at the other end and there would be another arc, another semicircle.

Laura says, "The end of the line is like a point, so the locus forms an arc." So first check, match up each diagram with the correct description of the loci.

So pause the video, read the descriptions, and match them up to the correct diagram.

Press play when you're ready to check.

So diagram a matches with h, the locus of points four centimetres from the line segment PQ.

B matches with description f, the locus of points four centimetres or more 'cause we're shaded on the outside from the line segment PQ.

Diagram c matches with g, so the locus of points up to and including four centimetres from the line segment PQ.

So the shading is within.

And lastly, d matches with e.

The locus of points less than four centimetres from the line segment PQ.

It's a dashed line.

So now let's look at shape.

So construct a locus of points inside the rectangle that is two units from the closest point on the perimeter.

So the perimeter is made of many points, but we are looking for points that are two units from the closest point on the perimeter.

So we've seen that the locus of points a given distance from a line forms a parallel line.

And this quadrilateral is made from four lines.

So if we look at two units away from the line segment AD would be this parallel line.

Similarly, parallel line two units away from BC is this from AB and from DC.

So those four lines are parallel to the line segments that create this shape.

If we take point E, this is two units from the side AD, but is less than two units from the side AB.

So its closest point on the perimeter is less than two units.

So, therefore, point E is not part of the locus of points.

What about F? So point F is two units from the side CD because it's on that line that is parallel to it, but it's less than two units from the side BC, and that would be the closest point of the perimeter.

So point F is not part of the locus of points.

So the locus of points forms a rectangle that is similar to the rectangle ABCD.

Each point along that similar rectangle is two units away from its closest point on the perimeter.

Let's have a look at constructing the locus of points outside the rectangle that's two units from the closest point on the perimeter.

So the locus of points inside was a rectangle, but is that the same on the outside? Well, if we take point E here, that is more than two units from the rectangle ABCD.

So it is not part of the locus of points.

Its closest point on the perimeter is more than two units.

And so these corners need to change.

And if we think about those four vertices that make up the rectangle ABCD as points in a similar way to how we think about the end of a line segment, then we get these sectors.

So point F is two units from C, which is a vertex of the rectangle, so it's part of the locus of points.

That is the closest point on the perimeter to F and it is two units away from it.

So here's a check.

A bus stop needs to be placed 50 metres from a park.

Andeep and Laura draw diagrams to show the locus of points 50 metres from the closest point of the park.

Which diagram is correct? So pause the video and then when you're ready to check, press play.

So Andeep's is correct.

All points need to be 50 metres from the park.

So we need to draw the arcs at some vertices.

So this is 50 metres.

This is more than 50 metres.

So onto the final task of the lesson.

So question one, construct the locus of points that are two units from the polygon, and one unit is one square.

So pause the video and then when you're ready for the next question, press play.

Question two, I'd like you to construct the locus of points that are one unit from the polygon.

So we've got a hexagon and a triangle.

Press pause and then when you're finished with that question and you're ready for question three, press play.

So here is question three, which is the last question of task C.

Andeep is going to plant an apple tree in his garden.

The tree must be planted more than 2.

5 metres from the house, three metres or more from other trees, and within eight metres from the tap so it can be watered with the hose.

Shade the region in which Andeep can plant the apple tree and label that region R.

So pause the video, work through those different constraints and criteria, and then when you're ready to check all your answers to tasks C, press play.

So here we're gonna construct the locus of points.

So one unit is one square and we were looking for two units, so it'd be two squares.

And at the vertices we're gonna use the sectors so that we have a fixed length of two units from that point.

And so this is the shape that you should have constructed.

On b, we're gonna have the vertices with the sectors and parallel lines to the line segments.

So just check your answer.

Question two, we need to do the locus of points that are one unit from the polygon, and we've got a scale line there so you knew what one unit was.

And so we'd have parallel line segments to the edges.

But at each vertex we need to draw a sector or an arc using a pair of compasses to ensure that that distance is fixed at one unit.

And then onto question three where Andeep was planting his apple tree.

So if we look at each constraint, it has to be more than 2.

5 metres from the house.

So one metre was one square, so we're gonna go parallel to the house two and a half squares, and it needs to be more than, so we're gonna use a dashed line to indicate it's not including that part.

Then it needed to be three metres or more from other trees.

And there is a pear tree planted in his garden already.

So three metres is gonna be a radius of three squares and we're gonna draw a circle because it is a single point, then it needs to be within eight metres from the tap so it can be watered with the hose.

So we're going to use the tap as a single point.

Because of the constraint of the diagram and the garden, the eight metres, which is eight squares, will be an arc.

Now we need to think about the region, the shaded region.

So it has to be more than two and a half metres from the house.

So it'll be above the dashed line.

It has to be three metres or more from the other tree.

So it has to be outside of that circle, but it has to be within eight metres from the tap.

So that's within, so inside our final arc.

So it is this region here that Andeep could plant his apple tree.

So to summarise today's lesson on constructing loci, we've looked at constructing a locus of points that are a set distance from a given point would produce a circle.

A pair of compasses should be used for this construction of this locus.

Constructing a locus of points that are a set distance from a line produces parallel lines, and constructing a locus of points that are a set distance from a vertex of a shape or the end of a line segment produces an arc.

A pair of compasses should be used for this construction as well.

Really well done today and I look forward to working with you again in the future.