Structures 3Sat, 27 November 2010

9:30 - 11:00 Straight line graphs and solving linear equations graphically

11:30 - 13:00 Solving simultaneous equations:using algebrausing graphs

14:00 - 15:30 Investigating quadratic graphs

Starter Activity

Bring on the maths!

Solving equations (KS3)

Find this site at http://www.kangaroomaths.com/botm.php

Activity 1

• Card sets E and CMatch them up (in two columns)

• Card set BAdded to the columns above (3rd column)

• Card set DAdd to the columns above (4th column)

Talk to your colleagues and explain your choices.

Different representations of the same concept- A splurge diagram

y=2x+5

Table of values

A graphThe algebraic expression

of the equation of the graph

The description of the equation in words

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 = 2x + 5

We can use a table as follows:

x

y = 2x + 5

–3 –2 –1 0 1 2 3

–1

(–3, –1)

1 3 5 7 9 11

(–2, 1) (–1, 3) (0, 5) (1, 7) (2, 9) (3, 11)

Plotting graphs of linear functions

to draw a graph of y = 2x + 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 = 2x + 5

y

x

x

y = 2x + 5

–3 –2 –1 0 1 2 3

–1 1 3 5 7 9 11

Activity 2

• For each set of functions, draw their graphs on the same set of axis:

Set A

y = 2xy = 2x-1y = 2x+3

Set B

y =- 3x+2y = -x+2y = -2x +2

Set C

y = 0.5x+1y = -2x +3y = -2x -4

Omnigraph for sets of graphs

Graphs parallel to the x-axisWhat 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 = 1Name five other points that will lie on

this line.

Graphs parallel to the y-axisWhat 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

Gradients of straight-line graphsThe 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.

Finding the gradient from two given points

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,

the gradient =change in y

change in x

the gradient =y2 – y1

x2 – x1

x

y

x2 – x1

(x1, y1)

(x2, y2)

y2 – y1

Draw a right-angled triangle between the two points on the line as follows,

Exploring gradients

The general equation of a straight lineThe 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 = 3x + 4 has a gradient of 3 and crosses the y-axis at the point (0, 4).

Activity 3Match the equation activity

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.

Activity 4: Card matching activity

Malcom Swan (2007) Standards Unit: Improving learning in mathematics

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 toy = 3x – 2.• Show me the equations of two lines that are perpendicular.• Find possible equations to make this shape: