# S4 Coordinates and transformations 1

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S4 Coordinates and transformations 1
KS3 Mathematics The aim of this unit is to teach pupils to: Use coordinates in all four quadrants Understand and use the language and notation associated with reflections and rotations Recognise and visualise transformations and symmetries of 2-D (and 3-D) shapes Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp , S4 Coordinates and transformations 1

S4 Coordinates and transformations 1
Contents S4 Coordinates and transformations 1 A S4.1 Coordinates A S4.2 Reflection A S4.3 Reflection symmetry A S4.4 Rotation A S4.5 Rotation symmetry

Coordinates We can describe the position of any point on a 2-dimensional plane using coordinates. The coordinate of a point tells us where the point is relative to a starting point or origin. For example, when we write a coordinate (3, 5) (3, 5) (3, 5) x-coordinate y-coordinate the first number is called the x-coordinate and the second number is called the y-coordinate. the first number is called the x-coordinate and the second number is called the y-coordinate.

Using a coordinate grid
Coordinates are plotted on a grid of squares. 4 y-axis The x-axis and the y-axis intersect at the origin. 3 2 origin The lines of the grid are numbered using positive and negative integers as follows. 1 x-axis –4 –3 –2 –1 1 2 3 4 –1 Stress that when using coordinates we are numbering the lines and not the squares. We should label the x-axis with an x and the y-axis with a y. Every coordinate tells us where each point is relative to the origin. –2 The coordinates of the origin are (0, 0). –3 –4

Quadrants The coordinate axes divide the grid into four quadrants. y x
4 y second quadrant first quadrant 3 2 1 x –4 third quadrant –3 –2 –1 fourth quadrant 1 2 3 4 Establish that the x-coordinate and the y-coordinate of points in the first quadrant are always positive. Points in the second quadrant will have a negative x-coordinate and a positive y-coordinate. Points in the third quadrant will have a negative x-coordinate and a negative y-coordinate. Points in the fourth quadrant will have a positive x-coordinate and a negative y-coordinate. –1 –2 –3 –4

Coordinates The first number in the coordinate pair tells you how many units along from the origin the point is in the x-direction. A positive number means the point is right of the origin and a negative number means it is left. The second number in the coordinate pair tells you how many units above or below the origin the point is in the y-direction. A positive number means the point is above the origin and a negative number means it is below. Remember: Along the corridor and up (or down) the stairs.

Plotting points Plot the point (–3, 5). y (–3, 5) x 7 6 5 4 3 2 1 –7
–6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 Explain that to find (–3, 5) start at the origin and move left 3 and up 5. –2 –3 –4 –5 –6 –7

Plotting points Plot the point (–4, –2). y x (–4, –2) 7 6 5 4 3 2 1 –7
–6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 Explain that to find (–4, –2) start at the origin and move left 4 and down 2. –2 (–4, –2) –3 –4 –5 –6 –7

Plotting points Plot the point (6, –7). y x (6, –7) 1 2 3 4 5 6 7 –1
1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7 x y Explain that to find (6, –7) start at the origin and move right 6 and down 7. (6, –7)

Where could we add a fourth point to make a parallelogram? 1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7 x y (–5, 4) (3, 2) Ask pupils to justify their choice of point before revealing it. The example shows the point (3, 2), but there are two more possible points. Challenge pupils to find them. They are (3, –8) and (–13, 6). (–5, –1) (3, –3)

Where could we add a fourth point to make a square? 1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7 x y (2, 6) (–2, 2) (6, 2) Ask pupils to justify their choice of point before revealing it. (2, –2)

Where could we add a fourth point to make a rhombus? 1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7 x y (–2, 4) (–7, 2) (3, 2) Ask pupils to justify their choice of point before revealing it. (–2, 0)

Where could we add a fourth point to make a kite? 1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7 x y (2, 2) Ask pupils to justify their choice of point before revealing it. (5, –1) (–7, –1) (2, –4)

Where could we add a fourth point to make an arrowhead? 1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7 x y (6, 6) (0, 6) (3, 3) Ask pupils to justify their choice of point before revealing it. (3, –2)

Where could we add a fourth point to make a rectangle? 1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7 x y (2, 7) (–6, 3) (5, 1) Ask pupils to justify their choice of point before revealing it. (–3, –3)

Don’t connect three! The object of this game is to place counters on the grid without getting three in a row horizontally, vertically, or diagonally. Divide the group into two teams, red and blue. Decide who will start and take alternate turns. The game gets more difficult as the grid fills up. Make the game more challenging by only allowing teams to change one of their coordinates each turn (either the x- or the y-coordinate).

Finding the mid-point of a horizontal line
Two points A and B have the same y-coordinate. A is the point (–2, 5) and B is the point (6, 5). ? 8 A(–2, 5) B(6, 5) M(xm, 5). ? What is the coordinate of the mid-point of the line segment AB? Establish that for a horizontal line, the mid-point between two points on that line must have the the same y-coordinate as the other two points, in this case 5 (this is the line y = 5). Let’s call the mid-point M(xm, 5). xm is the point half-way between –2 and 6.

Finding the mid-point of a horizontal line
Two points A and B have the same y-coordinate. A is the point (–2, 5) and B is the point (6, 5). 8 ? A(–2, 5) M(xm, 5). ? B(6, 5) Either, xm = –2 + ½ × 8 or xm = ½(–2 + 6) Talk through the two methods for finding a point half-way between two others. Links: N2 Negative numbers – mid-points. D3 Representing and interpreting data – calculating the mean and finding the median. = –2 + 4 = ½ × 4 = 2 = 2 The coordinates of the mid-point of AB are (2, 5).

Finding the mid-point of a line
If A is the point (2, 1) and B is the point (8, 5), what is the mid-point of the line AB? Start by plotting points A and B on a coordinate grid. y 7 The x-coordinate of the point A is 2 The x-coordinate of the point A is 2 and the x-coordinate of the point B is 8. 6 B(8, 5) 5 4 The x-coordinate of the mid-point is half-way between 2 and 8. 3 Explain to pupils that when we are finding the mid-point between two points with different x-coordinates and different y-coordinates, the x-coordinate of the mid-point will be half way between the x-coordinates of the end points. The y-coordinate of the mid-point will be half way between the y-coordinates of the end points. We can find the point that is half-way between 2 and 8 by finding the mean of 2 and 8 or by looking at the position of 2 and 8 on the x-axis. A(2, 1) 2 1 2 + 8 2 x = 5 1 2 3 4 5 6 7 8 9 10

Finding the mid-point of a line
If A is the point (2, 1) and B is the point (8, 5), what is the mid-point of the line AB? Start by plotting points A and B on a coordinate grid. y 7 and the y-coordinate of the point B is 5. The y-coordinate of the point A is 1 6 B(8, 5) 5 4 M(5, 3) The y-coordinate of the mid-point is half-way between 1 and 5. 3 We can find the point that is half-way between 1 and 5 by finding the mean of 1 and 5 or by looking at position of 1 and 5 on the y-axis. A(2, 1) 2 1 1 + 5 2 x = 3 1 2 3 4 5 6 7 8 9 10 The mid-point of AB is (5, 3).

Finding the mid-point of a line
We can generalize this result to find the mid-point of any line. If the coordinates of A are (x1, y1) and the coordinates of B are (x2, y2) then the coordinates of the mid-point of the line segment joining these points are given by: B(x2, y2) A(x1, y1) x y x1 + x2 2 , y1 + y2 x1 + x2 2 , y1 + y2 Talk through the generalization of the result for any two points (x1, y1) and (x2, y2). x1 + x2 2 is the mean of the x-coordinates. y1 + y2 2 is the mean of the y-coordinates.