1. Structures 3 Sat, 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.phphttp://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 (3 rd column) Card set D Add to the columns above (4 th column) Talk to your colleagues and explain your choices.

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

6. Plotting graphs of linear functions (handout) 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 = 2 x + 5 We can use a table as follows: x y = 2 x + 5 –3–2–10123 (–3, –1) 1357911 (–2, 1)(–1, 3)(0, 5)(1, 7)(2, 9)(3, 11)

7. Plotting graphs of linear functions to draw a graph of y = 2 x + 5: 1) Complete a table of values: 2) Plot the points on a coordinate grid. 3) Draw a line through the points. 4) Label the line. 5) Check that other points on the line fit the rule. For example, y = 2 x + 5 y x x –3–3–2–10123 1357911

8. Activity 2 For each set of functions, draw their graphs on the same set of axis: Set A y = 2x y = 2x-1 y = 2x+3 Set B y =- 3x+2 y = -x+2 y = -2x +2 Set C y = 0.5x+1 y = -2x +3 y = -2x -4

9. Omnigraph for sets of graphs

10. Graphs parallel to the x -axis 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. x y All of the points lie on a straight line parallel to the x -axis. This line is called y = 1. y = 1 Name five other points that will lie on this line.

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

13. Finding the gradient from two given points If we are given any two points ( x 1, y 1 ) and ( x 2, y 2 ) on a line we can calculate the gradient of the line as follows, the gradient = change in y change in x the gradient = y 2 – y 1 x 2 – x 1 x y ( x 1, y 1 ) ( x 2, y 2 ) y 2 – y 1 Draw a right-angled triangle between the two points on the line as follows,

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 -intercept and it has the coordinate (0, c ). For example, the line y = 3 x + 4 has a gradient of 3 and crosses the y -axis at the point (0, 4).

16. Activity 3 Match the equation activity

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

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: