Download - Structures 3 Sat, 27 November 2010
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Structures 3Sat, 27 November 2010
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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
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Starter Activity
Bring on the maths!
Solving equations (KS3)
Find this site at http://www.kangaroomaths.com/botm.php
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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.
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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
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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)
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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
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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
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Omnigraph for sets of graphs
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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.
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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
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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.
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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,
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Exploring gradients
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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).
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Activity 3Match the equation activity
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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.
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Activity 4: Card matching activity
Malcom Swan (2007) Standards Unit: Improving learning in mathematics
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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: