Integrating ICT into mathematics at KS4&5

Transcription

1 Integrating ICT into mathematics at KS4&5 Tom Button This session will detail the was in which ICT can currentl be used in the teaching and learning of Mathematics and how best to eploit these at GCSE and A level. Topics covered will include software, hardware, teaching ideas and suggestions for was to encourage the use of ICT within a department. Wh integrate ICT into mathematics teaching? Appropriate use of ICT can: Demonstrate ideas/links between topics more effectivel Allow students to generalise more easil Calculate/plot quickl without numerical errors Engage students in mathematics

2 Essentials Software Graph plotter Spreadsheet Hardware Data projector Occasional access to a computer suite Autograph Constant controller Dnamic tet-boes (v.) Scribble tool Other useful tools D Vectors Differential equations (C4 and DE) For more eamples see: Geogebra Slider Dnamic tet Other useful tools Slope

3 Ecel Spreadsheet algebra Labelling cells Sliders/scroll bars NB 00% view is better for class presentations Other software Interactive activities for GCSE Mathematics Interactive AS Core Mathematics Omnigraph TI-nspire IP Mechanics GSP/Cabri Fathom Lindo Maima Tarsia Hardware Data projector Interactive Whiteboard Wireless graphics tablet Graphical calculators Handhelds/palmtops/laptops Data-logger

4 Teaching ideas Pure Ideas for AS Core and A Core Differential Equations Autograph: st and nd order DEs Numerical Methods (inc. C) Ecel Mechanics IP Interactive Maths Flash interactive animations Geogebra Statistics Use D graphs on Autograph Encourage students to use Ecel to find statistics long-hand Decision PowerPoint is useful 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 Trigonometr Using a spreadsheet to investigate Pthagorean triples Geogebra/GSP for basic trigonometr and sine/cosine rule Circle theorems Geogebra/GSP for circle theorems Averages Spreadsheet for averages/moving averages 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 ou have an IWB ensure all members of staff regularl teach with it Have some PCs in communal/drop-in areas Demand the use of IT rooms 4

5 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 aes intercepts of the graph: the completed square form of the quadratic and the turning point: ( )( 4) = 4 5 = + = ( ) Similarl, a graphical approach to simultaneous equations will prepare the students for Chapter and give students a visual insight into wh, with linear and quadratic simultaneous equations, it is necessar to substitute back into the linear equation. Graphs There are also opportunities to investigate a range of other questions supported b a graph-plotting device: o Can a quartic equation have three roots? o Given a straight line = p+ q and a quadratic = a + b + c what can = p + q a + b + c? o ou tell about the aes intercepts of ( )( ) Plot the graph of =. Investigate the each of the following in turn, using several values of a and eplain the effect it has : ( ) ( ) = + a, = + a, = a, = a In Autograph, the button allows ou to define a function, sa ( ) then enter equations = f ( ) + k, f ( k) compare these graphs. (, ) f =. You can = etc. The slow plot mode allows ou to In Geogebra functions can be entered directl into the Input bar, e.g. f()=^. Then a slider (Named a ) can be added and equations such as =f(+a) and =a*f() can be inputted. o If the curve f ( ) what do ou know about the factors of f ( )? o Draw the graphs of = ( + + 5)( ) and ( ) the link between these graphs and the value f ( ) = touches but does not cross the ais at the point = a = f = 4 6. Eplain

6 o Choose another non-linear function f ( ) and plot the graph of f ( ) above investigate each of the following in turn: = f + a, = f + a, = af. = f a ( ) ( ) ( ) ( ) =. As Inequalities Graphic calculators can be used which can be compared to illustrate inequalities such as with the simplified version < < 0) + < (or( )( ) whereas Autograph shades the unwanted values of : Coordinate Geometr Equation of a line 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-8Plus, LINE (,4,5,) will draw a line through the points (,4) and (5,). Circles You can use a graph-plotter or graphical calculator to illustrate the affect ab, and r a + b = r, or analse the points of intersection of a line and a circle. provides a dnamic approach to teaching circles. Also on the TI-8Plus eperiment with circle from the draw menu: have in the equation of the circle ( ) ( )

7 What features of Circle(,4,5) meant that the circle passed through the origin? Describe what Circle(-,-4,5) would look like then check using our calculator. Find another circle which passes through (a) the origin, (b) the point (,). Draw two circles which touch but not at the origin: Draw other touching circle patterns: The constant controller in Autograph also gives the student a dnamic understanding of the role plaed b coefficients and constants in the equations of curves. See Sliders in Geogebra can also be used in a similar manner. Sequences and Series Introduction to sequences and series Sequences and series can be entered into Ecel to reinforce concepts such as Σ-notation. Binomial Students could be encouraged to write an ecel spreadsheet to generate Pascal s triangle (cell B = Cell A + Cell B):

8 Link with Statistics: could be used to discuss the general shape of the binomial distribution. This also gives a ver visual wa of displaing the answers to Statistics tetbook questions. APs and GPs illustrates the wa the first term and common difference affect the graphs of the terms of the A.P. and the sum to n terms against n. repeats this for G.P.s. clearl illustrating the importance of r < Trigonometr Graphs of trigonometric functions In Autograph under File>New etras page>trigonometr the trigonometric functions are plotted from the unit circle. You could also plot = sin, = cos, highlighting and eplaining their similarities, and then = sin + cos. Graphic calculators can be used to draw the trigonometric graphs from the unit circle. For eample on the TI-8Plus ou could briefl discuss parametric equations and then: Graphs of trigonometric functions and solving equations The spreadsheet generates the trigonometric graphs and shows how to solve basic trigonometric equations. This is highl recommended! Transformations of graphs Graphics calculators and Autograph/Geogebra are useful when investigating transformations of graphs: = sin = sin + = sin = sin -

9 .5 ( ) = cos 60.5 = cos Using the button define f ( ) = sin and then enter the equations f ( ) f ( k) - = + k, =, etc. Using the slow plot mode allows ou 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. =sin( ) will plot the graph in degrees and =sin() will plot it in radians. Logarithms and Eponentials Logarithms In Autograph the reflection in = button: allows ou to compare the reflection of = log with = 0 : 4 Use the button define f ( ) = log and then enter the equations f( ) f( ) = + log and =.Using the slow plot mode allows ou to compare these two graphs. This can be used to demonstrate the laws of logs. Reduction to linear form Ecel 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 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.

10 On Geogebra plot a curve then add a point to the curve. Add a tangent to the curve at the point (4 th menu) and then add a Slope (7 th 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. On the TI-8Plus 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 Autograph allows ou to enter a function ( = 4 in red) and using the button it will plot the gradient for each value of (in blue) thereb generating the gradient function curve ( = 4) To plot a derivative on Geogebra input =f (). Stationar points On Autograph draw a curve, select it and then right-click. Solve f ()=0 will give ou the stationar point. On the TI-8Plus: Draw the graph From the CALC to find the minimum = + 5 menu choose value of :

11 Integration Finding areas In Autograph ou can enter a curve ( = in red) then using the integral function button click on a point on the curve and it will draw the integral curve passing through d this point. In this wa it solves the differential equation = passing through a d specific point Also in Autograph ou can left click on a curve, then right click and choose the option find area. After selecting the area required it will be displaed in the status bar Numerical integration On a TI-8Plus Showing 4 56 d=. Showing ( ) + 5 d=. Autograph Further eamples of the use of Autograph can be seen at:

12 Suggested ideas for using ICT in A Core Core Chapter : Natural logarithms and eponentials The eponential function Differentiating = a with autograph The value of e can be approimated b plotting = a then adding the gradient function. As the value of a is changed with the constant controller students should observe that d k a the gradient function is of the form d = where the value of k depends on the value of a. (For a =, k 0.7; a =, k.; a = 4, k.4: these can be returned to later as k = lna). The students should then be able to observe that there eists a value of a between and such that k =, i.e. a function whose derivative is itself. Scaling in on the aes and the constant controller leads to a value of approimatel.7. Differentiating = a with Geogebra. Add a slider (which should be "a"). Input =a^ (ou ma want to var a at this point). 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 -5 ou can Input: Derivative[f()] or f ()

13 Epressing e as the sum of terms with Ecel. Ask students to suggest an infinite polnomial which when differentiated term-b term would remain unchanged. The will hopefull(!) come up with e = !!! Enter a value for in Cell:A, start with. Start columns labelled n and term. Enter the numbers 0,,,, in the n column. In the first row of the term column enter =$A$^B/FACT(B) Fill-down the term column with this formula. Sum the term column. The value of can then be changed and their formula verified. Chapter : Functions Investigating transformations of functions with Geogebra Functions can be inputted directl into the Input bar. e.g. f()=^+ Parameters can be defined b adding sliders. Graphs can then be plotted to investigate = f(), = af(), = f(b), = f() + c, = f( + d), = f(), = f( ) and the values of the parameters can be altered with the sliders. Investigating transformations of functions with Autograph The function button allows functions to be entered for f() and g(). Graphs can then be plotted to investigate = f(), = af(), = f(b), = f() + c, = f( + d), = f(), = f( ) and the values of the parameters can be altered with the constant controller. Investigating inverse functions with Autograph Functions can be entered as above and the graphs of = f() and = f() can be compared. Alternativel for a graph entered directl (i.e. not as a function), the reflection button, will displa the line = and the graph reflected in =. Students can use this to investigate inverse functions.

14 Chapter 4: Techniques for differentiation Derivative of sin and cos b observing the tangent on Autograph The tangent to = sin can be displaed b adding a point to the graph and then right clicking on the point and selecting Tangent. Students should then observe that gradient varies from (at 0) to 0 (at π/) etc. which suggests cos. Students can then investigate cos, asin and sin(b). Derivative of sin and cos b observing the tangent on Geogebra The gradient of the tangent to =cos() can be displaed b adding a curve to the line, drawing the tangent to the curve at the point and then adding the slope at that point. NB =cos() will plot the graph with the angle measured in radians and =cos( ) will plot the graph with the angle measured in degrees. Chapter 6: Numerical solution of equations Ecel and Autograph can be used etensivel for this see the online resources.

15 Core 4 (MEI): Chapter 7: Algebra Use of Ecel for evaluating ( + ) n for small. Set students the task of produce a spreadsheet that will calculate the first few terms in the epansion of ( + ) n for small and identifing how man terms are needed for the sum to be accurate to a certain number of d.p. To do this: Label 6 columns:, n, r, coeff, term and sum Enter 0. in Cell A. Enter 0.5 in Cell B. Enter the values 0,,, in Cells C, C, C4 Enter in Cell D. Enter =B in Cell D. Enter =D*($B$-(C4-))/C4 in Cell D4 and Fill-down the rest of this column with this formula. Enter =D*$A$^C in Cell E and Fill-down the rest of this column with this formula. Enter =E in Cell F. Enter =F+E in Cell F 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 ( + ) n and both and n can be varied. Chapter 8: Trigonometr Identities Double angle formulae in Geogebra Define points A:(0,0), B(,0) and C: (-,0). Use these to draw the unit circle. Add a point on the circle D. Add a point E:((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.

16 Graphs Use of Autograph/Geogebra to discover/displa identities (e.g. can the graph of = cos be transformed into the graph of = cos b means of a stretch and a translation? Use the constant controller/slider to find values of a and b so that the graph of = asin + bcos is the same as = Rcos( + 0 ) Small angle approimations Use of Autograph/Geogebra to find small angle approimations for sin, cos and tan:, and respectivel can be demonstrated/discovered b drawing the tangents to the curves at = 0 and the range for which the are valid can be demonstrated. Chapter 9: Parametric Equations Parametric equations can be entered into Autograph using the parameter t and a comma to separate the and equations. Parametric curves can be entered into Geogebra using the snta: Curve[epression for, epression for, parameter, lower limit, upper limit] e.g. Curve[t, t^, t, -5, 5] Parametric equations can also be entered into most graphical calculators Chapter 0: Further techniques for integration Autograph Volumes of revolution See: Chapter : Vectors Vectors on Autograph See C4 Vectors with Autograph :

17 Chapter : Differential equations Differential equations on Autograph st order DEs can be displaed on Autograph b entering d... d = 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 eplain wh d k d = generates an eponential curve and d k d = generates a parabola. Autograph Further eamples of the use of Autograph can be seen at: Further Information For further information, including advice about the use of ICT in AS Core see: