Structures 3 Sat, 27 November 2010

1. Structures 3Sat, 27 November 2010

2. 9:30 - 11:00 Straight line graphs and solving linear equations graphically 11:30 - 13:00 Solving simultaneous equations: using algebra using graphs 14:00 - 15:30 Investigating quadratic graphs

3. Starter Activity Bring on the maths! Solving equations (KS3) Find this site at http://www.kangaroomaths.com/botm.php

4. Activity 1 • Card sets E and C Match them up (in two columns) • Card set B Added to the columns above (3rd column) • Card set D Add to the columns above (4th column) Talk to your colleagues and explain your choices.

5. Different representations of the same concept- A splurge diagram The description of the equation in words Table of values y=2x+5 The algebraic expression of the equation of the graph A graph

6. Plotting graphs of linear functions(handout) x –3 –2 –1 0 1 2 3 y = 2x + 5 Given a function, we can find coordinate points that obey the function by constructing a table of values. Suppose we want to plot the function y = 2x + 5 We can use a table as follows: –1 1 3 5 7 9 11 (–3, –1) (–2, 1) (–1, 3) (0, 5) (1, 7) (2, 9) (3, 11)

7. Plotting graphs of linear functions x –3 –2 –1 0 1 2 3 y = 2x + 5 –1 1 3 5 7 9 11 For example, y to draw a graph of y = 2x + 5: 1) Complete a table of values: y = 2x + 5 2) Plot the points on a coordinate grid. 3) Draw a line through the points. x 4) Label the line. 5) Check that other points on the line fit the rule.

8. Activity 2 • For each set of functions, draw their graphs on the same set of axis:

9. Omnigraph for sets of graphs

10. Graphs parallel to the x-axis y x What do these coordinate pairs have in common? (0, 1), (4, 1), (–2, 1), (2, 1), (1, 1) and (–3, 1)? The y-coordinate in each pair is equal to 1. Look at what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the x-axis. y = 1 Name five other points that will lie on this line. This line is called y = 1.

11. Graphs parallel to the y-axis y x What do these coordinate pairs have in common? (2, 3), (2, 1), (2, –2), (2, 4), (2, 0) and (2, –3)? The x-coordinate in each pair is equal to 2. Look what happens when these points are plotted on a graph. All of the points lie on a straight line parallel to the y-axis. Name five other points that will lie on this line. This line is called x = 2. x = 2

12. Gradients of straight-line graphs y y an upwards slope a horizontal line a downwards slope y x x x The gradient of a line is a measure of how steep the line is. The gradient of a line can be positive, negative or zero if, moving from left to right, we have Positive gradient Zero gradient Negative gradient If a line is vertical, its gradient cannot be specified.

13. Finding the gradient from two given points the gradient = (x2, y2) (x1, y1) change in y change in x x y2 – y1 the gradient = x2 – x1 If we are given any two points (x1, y1) and (x2, y2) on a line we can calculate the gradient of the line as follows, y y2 – y1 Draw a right-angled triangle between the two points on the line as follows, x2 – x1

14. Exploring gradients

15. The general equation of a straight line The general equation of a straight line can be written as: y = mx + c The value of m tells us the gradient of the line. The value of c tells us where the line crosses the y-axis. This is called the y-interceptand it has the coordinate (0, c). For example, the line y = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

16. Activity 3Match the equation activity

17. If the gradients of two lines have a product of –1 then they are perpendicular. If two lines have the same gradient they are parallel.

18. Activity 4: Card matching activity Malcom Swan (2007) Standards Unit: Improving learning in mathematics

19. Activity 5: Straight line graphs • Give me an example of a line that has gradient 4. • Give me an example of a line that is perpendicular to y = 3x – 2. • Show me the equations of two lines that are perpendicular. • Find possible equations to make this shape: