MEI Conference Teaching numerical methods using graphing technology

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1 MEI Conference 2017 Teaching numerical methods using graphing technology Jo Sibley

Casio CG20 Task Numerical Methods: Change of sign 1. Add a new Graphs screen: p5 5. Generate the Table: u 2.

You can now investigate further by changing the set-up of the table so that it goes from 1.6 to 1.7 in steps of 0.01.

In this example you can see that the root lies between x = 1 and x = 2. 3. Now go into the Table mode: p7 4.

2 Casio CG20 Task Numerical Methods: Change of sign 1. Add a new Graphs screen: p5 5. Generate the Table: u 2. Plot the graph: y = x³ x 3 (Y1 = x³ x 3): f^3$-f-3lu In this example there is a change of sign between x = 1.6 and x = You can now investigate further by changing the set-up of the table so that it goes from 1.6 to 1.7 in steps of (Alternatively you can do this by overtyping x-values in the table or inserting new rows). In this example you can see that the root lies between x = 1 and x = Now go into the Table mode: p7 4. Using SET y set the Table Start to 1, End to 2 and Step to 0.1: 1l2l0.1ld 7. You can check your answer in Graphs mode using G-Solv > Root: p5uyq Try using your calculator to find the roots of other equations using the change of sign method.

Casio CG20 Task Numerical Methods: Fixed point iteration 1. Add a new Graphs screen: p5 5. Using SET y set the Table Start to 1, End to 5 and a 0 to 1: 1l5l1ld 2.

3 Casio CG20 Task Numerical Methods: Fixed point iteration 1. Add a new Graphs screen: p5 5. Using SET y set the Table Start to 1, End to 5 and a 0 to 1: 1l5l1ld 2. Plot the graph: y = x³ x 3 (Y1 = x³ x 3): f^3$-f-3lu 6. Generate the Table: u In this example you can see that the root lies between x = 1 and x = Now go into the Recursion mode: p Add the recurrence relation x n+1 = x n + 3 L^3$w+3l In this example after 5 iterations the approximation is accurate to 3 decimal places. 7. You can check your answer in Graphs mode using G-Solv > Root: yq$$$$ Try using your calculator to find the roots of other equations using a fixed-point iteration.

4 Casio CG20 Task Numerical Methods: Newton-Raphson 1. Add a new Graphs screen: p5 5. Using SET y set the Table Start to 1, End to 5 and a 0 to 2: y1l5l2ld 2. Plot the graph: y = x³ x 3 (Y1 = x³ x 3): f^3$-f-3lu 6. Generate the Table: u In this example you can see that the root lies between x = 1 and x = Now go into the Recursion mode: p8 4. Add the recurrence relation x n+1 = x n f(x n) f (x n ) rw-jw^3$- w-3kmj3ws- 1kld In this example after 5 iterations the approximation is accurate to more than 3 decimal places. 7. You can check your answer in Graphs mode using G-Solv > Root: yq$$$$ Try using your calculator to find the roots of other equations using the Newton-Raphson method.

Desmos Task Numerical Methods: Change of sign 1. Plot the graph of y = x³ x 3 In this example you can see that the root lies between x = 1 and x = 2.

5 Desmos Task Numerical Methods: Change of sign 1. Plot the graph of y = x³ x 3 In this example you can see that the root lies between x = 1 and x = With the curve selected press Edit and Convert to table : 3. In the x-column enter the values 1.1, 1.2, 1.3, until you observe a change of sign in y. In this example there is a change of sign between x = 1.6 and x = You can now investigate further by entering x-values from 1.6 to 1.7 in steps of You can check your answer by zooming-in and selecting the point of intersection with the x-axis. Try using your software to find the roots of other equations using the change of sign method.

GeoGebra Task Numerical Methods: Change of sign 1. Enter the function f(x) = x³ x 3: f(x)=x^3-x-3 In this example you can see that the root lies between x = 1 and x = 2.

6 GeoGebra Task Numerical Methods: Change of sign 1. Enter the function f(x) = x³ x 3: f(x)=x^3-x-3 In this example you can see that the root lies between x = 1 and x = Enable the spreadsheet: View > Spreadsheet 3. In cells A1 and A2 enter the values 1, 1.1 and then fill down to 2 (double click bottom RH corner of cell). 4. In cell B1 enter f(a1) and then fill down. In this example there is a change of sign between x = 1.6 and x = You can now investigate further by changing the column A so that it goes from 1.6 to 1.7 in steps of You can check your answer in the Graphics view by finding the point of intersection of the function and the x-axis (the number of decimal places can be set with Options > Rounding). Try using your software to find the roots of other equations using the change of sign method.

GeoGebra Task Numerical Methods: Fixed point iteration 1. Enter the function f(x) = x³ x 3: f(x)=x^3-x-3 In this example you can see that the root lies between x = 1 and x = 2.

7 GeoGebra Task Numerical Methods: Fixed point iteration 1. Enter the function f(x) = x³ x 3: f(x)=x^3-x-3 In this example you can see that the root lies between x = 1 and x = Enable the spreadsheet: View > Spreadsheet 3 3. To add the recurrence relation x n+1 = x n + 3, enter the function 3 g(x) = x + 3: g(x)=cbrt(x+3) 4. In Cell A1 enter 0 and in Cell B1 enter In Cell A2 enter A1+1 and in cell B2 enter g(b1) and fill down. 6. You can check your answer in the view by finding the point of intersection of the function and the x-axis (he number of decimal places can be set with Options > Rounding). Try using your software to find the roots of other equations using a fixed-point iteration.

9 Teaching numerical methods using graphing technology mei.org.uk/technology

10 mei.org.uk/technology Learning objectives matrix or The things I ve planned which we might fit in to the session but might end up being homework Method Task Casio Desmos GeoGebra v5 Change of sign Handout and demo. Handout and partial demo. Fixed point iteration Handout and over-the-shoulder help. Nice example and over-the-shoulder help. Handout and overthe-shoulder help. Newton- Raphson Any of the above Handout and over-the-shoulder help. Over-the-shoulder help. Nice example and over-the-shoulder help. How I Learned to Stop Worrying and Love the Slider

11 mei.org.uk/technology Dynamic Examples Fixed Point Iteration (Desmos) Newton Raphson (Desmos) Newton Raphson (GeoGebra)

12 About MEI Registered charity committed to improving mathematics education Independent UK curriculum development body We offer continuing professional development courses, provide specialist tuition for students and work with employers to enhance mathematical skills in the workplace We also pioneer the development of innovative teaching and learning resources

MEI Casio Tasks for A2 Core

MEI Casio Tasks for A2 Core Task 1: Functions The Modulus Function The modulus function, abs(x), is found using OPTN > NUMERIC > Abs 2. Add the graph y = x, Y1=Abs(x): iyqfl 3. Add the graph y = ax+b, Y2=Abs(Ax+B): iyqaff+agl 4.