Integrating ICT into mathematics at KS4&5
Integrating ICT into
mathematics at KS4&5
Tom Button
tom.button@mei.org.uk
www.mei.org.uk/ict/
Integrating ICT into mathematics at KS4&5
Integrating ICT into mathematics at KS4&5
This session will detail the ways in which
ICT can currently be used in the teaching
and learning of Mathematics and how best
to exploit these at GCSE and A level.
Topics covered will include software,
hardware, teaching ideas and suggestions
for ways to encourage the use of ICT
within a department.
Integrating ICT into mathematics at KS4&5
Why integrate ICT into mathematics teaching?
Appropriate use of ICT can:
• Demonstrate ideas/links between topics more
effectively
• Allow students to generalise more easily
• Calculate/plot quickly without numerical errors
• Engage students in mathematics
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Integrating ICT into mathematics at KS4&5
Essentials
• Software
– Graph plotter
– Spreadsheet
• Hardware
– Data projector
– Occasional access to a computer suite
Integrating ICT into mathematics at KS4&5
Autograph
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Constant controller
Dynamic text-boxes (v3.2)
Scribble tool
Other useful tools
– 3D Vectors
– Differential equations (C4 and DE)
For more examples see: www.autographmaths.com
Integrating ICT into mathematics at KS4&5
Geogebra
• Slider
• Dynamic text
• Other useful tools
– Slope
www.geogebra.org
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Integrating ICT into mathematics at KS4&5
Excel
• Spreadsheet algebra
• Labelling cells
• Sliders/scroll bars
NB 200% view is better for class presentations
Integrating ICT into mathematics at KS4&5
Other software
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Interactive activities for GCSE Mathematics
Interactive AS Core Mathematics
Omnigraph
TI-nspire
IP Mechanics – www.fable.co.uk/ipmaths
GSP/Cabri – www.chartwellyorke.com
Fathom
Lindo
Maxima
Tarsia
Integrating ICT into mathematics at KS4&5
Hardware
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Data projector
Interactive Whiteboard
Wireless graphics tablet
Graphical calculators
Handhelds/palmtops/laptops
Data-logger
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Integrating ICT into mathematics at KS4&5
Teaching ideas
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Pure – Ideas for AS Core and A2 Core
Differential Equations – Autograph: 1st and 2nd order DEs
Numerical Methods (inc. C3) – Excel
Mechanics
– IP Interactive Maths
– Flash interactive animations
– Geogebra
• Statistics
– Use 1D graphs on Autograph
– Encourage students to use Excel to find statistics long-hand
• Decision – PowerPoint is useful
Integrating ICT into mathematics at KS4&5
Teaching ideas – GCSE
• Percentages
– Calculating percentages and reverse percentages with a spreadsheet.
• Straight line graphs
– Investigate straight line graphs with Autograph/Geogebra
• Inequalities
– Investigate inequalities with Autograph
• Transformations
– Transformations with Autograph/Geogebra
• Trigonometry
– Using a spreadsheet to investigate Pythagorean triples
– Geogebra/GSP for basic trigonometry and sine/cosine rule
• Circle theorems
– Geogebra/GSP for circle theorems
• Averages
– Spreadsheet for averages/moving averages
Integrating ICT into mathematics at KS4&5
Encouraging the use of ICT
• Have a member of staff whose specific
role it is to promote the use of ICT –
research indicates this is essential
• If you have an IWB ensure all members of
staff regularly teach with it
• Have some PCs in communal/drop-in
areas
• Demand the use of IT rooms
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Suggested ideas for using ICT in AS Core
Algebra
Quadratics
Graphic calculators or Autograph/Geogebra can be used to think about the link between:
•
the factorised form of a quadratic and
the axes intercepts of the graph:
4
y = ( x + 2) −1
2
1× 4 = 4
y
the completed square form of the
quadratic and the turning point:
y = ( x − 1)( x − 4 )
y
5
4
3
2
x
1
2
3
4
1
5
-4
-3
-2
-1
x
-1
( −2, −1)
•
Similarly, a graphical approach to simultaneous equations will prepare the students for
Chapter 2 and give students a visual insight into why, with linear and quadratic
simultaneous equations, it is necessary to substitute back into the linear equation.
Graphs
•
There are also opportunities to investigate a range of other questions supported by a
graph-plotting device:
o Can a quartic equation have three roots?
o Given a straight line y = px + q and a quadratic y = ax 2 + bx + c what
can
(
)
you tell about the axes intercepts of y = ( px + q ) ax 2 + bx + c ?
o
Plot the graph of y = x 2 . Investigate the each of the following in turn, using
several values of a and explain the effect it has :
y = x 2 + a,
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In Autograph, the
y = ( x + a) ,
2
y = a × x2 ,
y = ( ax )
2
button allows you to define a function, say f ( x ) = x 2 . You can
then enter equations y = f ( x ) + k , y = f ( kx ) etc. The slow plot mode
allows you to
compare these graphs.
In Geogebra functions can be entered directly into the Input bar, e.g. f(x)=x^2. Then a
slider (Named “a”) can be added and equations such as y=f(x+a) and y=a*f(x) can be
inputted.
o
If the curve y = f ( x ) touches but does not cross the x − axis at the point x = a
what do you know about the factors of f ( x ) ?
o Draw the graphs of y = ( x 2 + 3x + 5 ) ( x − 3) and y = f ( x ) = x3 − 4 x − 6 . Explain
the link between these graphs and the value f ( 3)
5
y
x
-2
2
-5
-10
-15
-20
o
Choose another non-linear function f ( x ) and plot the graph of y = f ( x ) . As
above investigate each of the following in turn:
y = f ( x ) + a,
y = f ( x + a),
y = a f ( x ) . y = f ( ax )
Inequalities
•
Graphic calculators can be used
to illustrate inequalities such as
x −1 < 5 − x2 …
… which can be compared
with the simplified version
x 2 + x − 6 < 0 (or ( x + 3)( x − 2 ) < 0 ) …
y
… whereas Autograph shades
the unwanted values of x :
x
-6 -5 -4 -3 -2 -1
1
2
3
4
5
6
Coordinate Geometry
Equation of a line
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Use a graph-plotter or graphical calculator to plot a line through two points, then work
out the equation of the line and check if it is correct.
On the TI-83Plus, LINE (2,4,5,3) will draw a line through the points (2,4) and (5,3).
Circles
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You can use a graph-plotter or graphical calculator to illustrate the affect a, b and r
have in the equation of the circle ( x − a ) + ( y − b ) = r 2 , or analyse the points of
2
•
2
intersection of a line and a circle.
http://www.meidistance.co.uk/pdf/circlesspreadsheet.xls provides a dynamic approach to
teaching circles.
Also on the TI-83Plus experiment with circle from the draw menu:
What features of Circle(3,4,5) meant that the circle passed through the origin?
Describe what Circle(-3,-4,5) would look like then check using your calculator.
Find another circle which passes through (a) the origin, (b) the point (3,2).
Draw two circles which touch but not at the origin:
Draw other touching circle patterns:
in Autograph also gives the student a dynamic understanding of the
The constant controller
role played by coefficients and constants in the equations of curves. See
http://www.mei.org.uk/files/pdf/Const_Control_Autograph.pdf.
Sliders in Geogebra can also be used in a similar manner.
Sequences and Series
Introduction to sequences and series
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Sequences and series can be entered into Excel to reinforce concepts such as
Σ-notation.
Binomial
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Students could be encouraged to write an excel spreadsheet to generate Pascal’s
triangle (cell B2 = Cell A2 + Cell B1):
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
1
3
6
10
15
21
28
1
4
10
20
35
56
1
5
15
35
70
1
6
21
56
1
7
28
1
8
1
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Link with Statistics:
http://www.meidistance.co.uk/pdf/Binomial.xls could be used to discuss the general
shape of the binomial distribution. This also gives a very visual way of displaying the
answers to Statistics textbook questions.
APs and GPs
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http://www.meidistance.co.uk/pdf/ass2ap.xls illustrates the way the first term and
common difference affect the graphs of the terms of the A.P. and the sum to n terms
against n.
http://www.meidistance.co.uk/pdf/ass3gp.xls repeats this for G.P.s. clearly illustrating the
importance of r < 1
Trigonometry
Graphs of trigonometric functions
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In Autograph 3 under File>New extras page>Trigonometry the trigonometric functions
are plotted from the unit circle. You could also plot y = sin 2 x , y = cos 2 x , highlighting
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and explaining their similarities, and then y = sin 2 x + cos 2 x .
Graphic calculators can be used to draw the trigonometric graphs from the unit circle.
For example on the TI-83Plus you could briefly discuss parametric equations and then:
Graphs of trigonometric functions and solving equations
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The spreadsheet http://www.meidistance.co.uk/pdf/Circfns.xls generates the
trigonometric graphs and shows how to solve basic trigonometric equations. This is
highly recommended!
Transformations of graphs
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Graphics calculators and Autograph/Geogebra are useful when investigating
transformations of graphs:
3
y
y = sin x + 2
2
y = sin x + 1
1
x
-360
-270
-180
-90
90
-1
-2
180
270
360
y = sin x
y = sin x − 1
y
y
1.5
1.5
y = cos ( x − 60D )
1
0.5
1
y = cos 3 x
0.5
x
x
150
-90
90
-0.5
-0.5
-1
-1
180
button define f ( x ) = sin x and then enter the equations y = f ( x ) + k ,
Using the
y = f ( kx ) , etc. Using the slow plot mode
allows you to compare these graphs.
NB In Geogebra the degrees sign is used to indicate angles are measured in degrees. If
it is omitted the graphs will be drawn in radians.
e.g. y=sin(x°) will plot the graph in degrees and y=sin(x) will plot it in radians.
Logarithms and Exponentials
Logarithms
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In Autograph the reflection in y = x button:
allows you to compare the reflection of
y = log x with y = 10 :
x
3
y
2
1
x
2
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Use the
4
button define f ( x ) = log x and then enter the equations y = f ( x ) + log 2 and
y = f ( 2 x ) .Using the slow plot mode
allows you to compare these two graphs. This
can be used to demonstrate the laws of logs.
Reduction to linear form
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Excel can be used to generate a table of data and the scatter graph and line of best fit
can be added to the spreadsheet.
Differentiation
Finding the gradient of the tangent to a curve at a point
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On Autograph plot a curve then add a point to the curve. Select the point, right-click (or
Object menu) and select Tangent. The point can then be moved along the curve and
the values of the gradient of the tangent to the curve at the point can be seen in the
equation of the tangent.
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On Geogebra plot a curve then add a point to the curve. Add a tangent to the curve at
the point (4th menu) and then add a Slope (7th menu) at the point. The point can then be
moved along the curve and the values of the gradient of the tangent to the curve at the
point can be seen from the gradient triangle.
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On the TI-83Plus
Using 6 on the CALC menu … will draw the tangent to
the curve at the chosen
5 on the DRAW menu…
point giving its equation:
will give the gradient of a
curve at a chosen point:
Gradient functions
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Autograph allows you to enter a function ( y = x 3 − 4 x in red) and using the
button it
will plot the gradient for each value of x (in blue) thereby generating the gradient function
curve ( y = 3 x 2 − 4 ).
y
2
x
-2
-1
1
2
-2
-4
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To plot a derivative on Geogebra input y=f’(x) .
Stationary points
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On Autograph draw a curve, select it and then right-click. Solve f’(x)=0 will give you the
stationary point.
On the TI-83Plus:
Draw the graph
From the CALC
to find the minimum
2
y = x − 2x + 5
menu choose 3
value of y :
Integration
Finding areas
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In Autograph you can enter a curve ( y = 3 x − x 2 in red) then using the integral function
button
click on a point on the curve and it will draw the integral curve passing through
this point. In this way it solves the differential equation
dy
= 3 x − x 2 passing through a
dx
specific point.
y
6
4
2
-4
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-2
x
2
-2
-4
4
6
Also in Autograph you can left click on a curve, then right click and choose the option
‘find area’. After selecting the area required it will be displayed in the status bar.
y
6
4
2
x
-1
1
2
3
Numerical integration
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On a TI-83Plus
Showing
∫
4
2
x 2 dx =
56
.
3
Showing
Autograph
Further examples of the use of Autograph can be seen at:
http://www.autograph-math.com/inaction/
∫ (x
3
1
2
− 2 x + 5 ) dx =
32
.
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Suggested ideas for using ICT in A2 Core
Core 3
Chapter 2: Natural logarithms and exponentials
The exponential function
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Differentiating y = ax with autograph
The value of e can be approximated by plotting y = ax then adding the gradient function.
As the value of a is changed with the constant controller students should observe that
dy
= k × ax
dx
the gradient function is of the form
where the value of k depends on the value
of a. (For a = 2, k ≈ 0.7; a = 3, k ≈ 1.1; a = 4, k ≈ 1.4: these can be returned to later as
k = lna). The students should then be able to observe that there exists a value of a
between 2 and 3 such that k =1, i.e. a function whose derivative is itself. Scaling in on
the axes and the constant controller leads to a value of approximately 2.72.
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Differentiating y = ax with Geogebra
1. Add a slider (which should be "a")
2. Input y=a^x (you may want to vary a
at this point)
3. Add a New Point on the curve (this
should be "A")
4. Add the tangent at A
5. Add the slope at A
As an alternative to points 3-5 you
can Input:
Derivative[f(x)]
or
f’(x)
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Expressing ex as the sum of terms with Excel.
Ask students to suggest an infinite polynomial
which when differentiated term-by term would
remain unchanged. They will hopefully(!) come
x x 2 x3
up with e x = 1 + +
+
+ ...
1! 2! 3!
Enter a value for x in Cell:A1, start with 1.
Start 2 columns labelled “n” and “term”.
Enter the numbers 0, 1, 2, 3, … in the n column.
In the first row of the term column enter
=$A$1^B2/FACT(B2)
”Fill-down” the term column with this formula.
Sum the term column.
The value of x can then be changed and their formula verified.
Chapter 3: Functions
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Investigating transformations of functions with Geogebra
Functions can be inputted directly into the Input bar. e.g. f(x)=x^3+2
Parameters can be defined by adding sliders.
Graphs can then be plotted to investigate y = f(x), y = af(x), y = f(bx), y = f(x) + c,
y = f(x + d), y = −f(x), y = f(−x) and the values of the parameters can be altered with the
sliders.
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Investigating transformations of functions with Autograph
The function button
allows functions to be entered for f(x) and g(x).
Graphs can then be plotted to investigate y = f(x), y = af(x), y = f(bx), y = f(x) + c,
y = f(x + d), y = −f(x), y = f(−x) and the values of the parameters can be altered with the
constant controller.
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Investigating inverse functions with Autograph
Functions can be entered as above and the graphs of y = f(x) and x = f(y) can be
compared.
Alternatively for a graph entered directly (i.e. not as a function), the reflection button
will display the line y = x and the graph reflected in y = x.
Students can use this to investigate inverse functions.
,
Chapter 4: Techniques for differentiation
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Derivative of sin and cos by observing the tangent on Autograph
The tangent to y = sinx can be displayed by adding a point to the graph and then right
clicking on the point and selecting “Tangent”.
Students should then observe that gradient varies from 1 (at 0) to 0 (at π/2) etc. which
suggests cosx. Students can then investigate cosx, asinx and sin(bx).
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Derivative of sin and cos by observing the tangent on Geogebra
The gradient of the tangent to y=cos(x) can be displayed by adding a curve to the line,
drawing the tangent to the curve at the point and then adding the slope at that point.
NB y=cos(x) will plot the graph with the angle measured in radians and y=cos(x°) will
plot the graph with the angle measured in degrees.
Chapter 6: Numerical solution of equations
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Excel and Autograph can be used extensively for this – see the online resources.
Core 4 (MEI):
Chapter 7: Algebra
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Use of Excel for evaluating (1 + x)n for small x.
Set students the task of produce a spreadsheet that will calculate the first few terms in
the expansion of (1 + x)n for small x and identifying how many terms are needed for the
sum to be accurate to a certain number of d.p.
To do this:
Label 6 columns: x, n, r, coeff, term and sum
Enter 0.1 in Cell A2.
Enter 0.5 in Cell B2.
Enter the values 0, 1, 2, … in Cells C2, C3, C4…
Enter 1 in Cell D2.
Enter =B2 in Cell D3.
Enter =D3*($B$2-(C4-1))/C4 in Cell D4 and “Fill-down” the rest of this column with this
formula.
Enter =D2*$A$2^C2 in Cell E2 and “Fill-down” the rest of this column with this formula.
Enter =E2 in Cell F2.
Enter =F2+E3 in Cell F3 and “Fill-down” the rest of this column with this formula.
The values in the cumulative sum column can be compared with the “true” value of
(1 + x)n and both x and n can be varied.
Chapter 8: Trigonometry
Identities
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Double angle formulae in Geogebra
Define points A:(0,0), B(1,0) and C: (-1,0).
Use these to draw the unit circle.
Add a point on the circle D.
Add a point E:(x(D),0)
Draw segments AD, BD, CD and ED.
Add angles ACD and BAD
The resulting diagram can be used to prove
the double angle formulae.
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Graphs – Use of Autograph/Geogebra to discover/display identities
(e.g. can the graph of y = cos2x be transformed into the graph of y = cos2x by means of
a stretch and a translation?
Use the constant controller/slider to find values of a and b so that the graph of
y = asinx + bcosx is the same as y = Rcos(x + 30°)
Small angle approximations
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Use of Autograph/Geogebra to find small angle approximations for sin, cos and tan:
x, 1 −
x2
and x
2
respectively can be demonstrated/discovered by drawing the tangents
to the curves at x = 0 and the range for which they are valid can be demonstrated.
Chapter 9: Parametric Equations
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Parametric equations can be entered into Autograph using the parameter t and a comma
to separate the x and y equations.
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Parametric curves can be entered into Geogebra using the syntax:
Curve[expression for x, expression for y, parameter, lower limit, upper
limit]
e.g. Curve[2t, t^2, t, -5, 5]
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Parametric equations can also be entered into most graphical calculators
Chapter 10: Further techniques for integration
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Autograph – Volumes of revolution
See: http://www.autograph-math.com/inaction/td/Volumes.htm
Chapter 11: Vectors
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Vectors on Autograph
See “C4 Vectors with Autograph”:
http://www.mei.org.uk/files/pdf/C4_Vectors_Autograph.pdf
Chapter 12: Differential equations
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Differential equations on Autograph
1st order DEs can be displayed on Autograph by entering
dy
= ...
dx
this then gives a tangent field.
Clicking on the screen at a point gives a particular solution through that point.
The tangent fields can be used to explain why
dy
= ky
dx
generates an exponential curve and
dy
= kx
dx
generates a parabola.
Autograph
Further examples of the use of Autograph can be seen at:
http://www.autograph-math.com/inaction/
Further Information
For further information, including advice about the use of ICT in AS Core see:
www.mei.org.uk/ict/