Using Graphical Calculators in the new A level Maths. Tom Button Simon May

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2 Using Graphical Calculators in the new A level Maths Tom Button tom.button@mei.org.uk Simon May simon.may@nulc.ac.uk

3 Plot these curves

4 Answers

5 Ofqual guidance for awarding organisations The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A level mathematics.

6 A series of tasks, each in 4 parts: Construction Exploration Question Extension Classroom tasks

7 Full set of classroom tasks Classroom tasks mapped to the new A level curriculum Trialled by teachers and improved based on their feedback mei.org.uk/casio-networks

8 Ofqual guidance on calculators Calculators used must include the following features: an iterative function the ability to compute summary statistics and access probabilities from standard statistical distributions the ability to perform calculations with matrices up to at least order 3 x 3 (FM)

9 Specimen question (Calculator use) OCR A level Paper 3 A market gardener records the masses of a random sample of 100 of this year's crop of plums. The market gardener models the distribution of masses by N~(47.5,10²). Find the number of plums in the sample that this model would predict to have masses in the range (a) 35 m < 45, [2] (b) M < 25. [2]

10 Experiences of using graphical calculators For A level Ease of use/availability Graph Transformations Trigonometry Numerical Methods Solution Checking

11 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

12 MEI Conference 2017 Using Graphical Calculators in the new A level Maths Tom Button tom.button@mei.org.uk Simon May simon.may@nulc.ac.uk

13 The Factor Theorem 1. Go into Table mode: p7 2. Add Y1 = x³ 2x² x + 2 : f^3$-2fs-f+2l 3. Use SET to set the table to Start: 5, End: 5, Step: 1: yn5ld 4. Display the table: u 5. Go into Graph mode and plot the graph of this function: p5u Questions How do this table and graph confirm that x³ 2x² x + 2 = (x + 1)(x 1)(x 2)? Can you find the factors of the following cubics: y = x³ + 4x² + x 6 y = x³ 4x² 11x + 30 y = x³ x² 8x + 12 y = x³ 7x² + 36 Problem (Try the question with pen and paper first then check it on your calculator) Show that (x 2) is a factor of f(x) = x³ + 4x² 3x 18. Hence find all the factors of f(x). Further Tasks Find examples of cubics that only have one real root. Investigate using the factor theorem for polynomials of other degrees, e.g. quadratics or quartics. Investigate the polynomial solver: pafw. Page 2 of 6 mei.org.uk/casio-networks

14 Quadratic Inequalities 1. Add a new Graphs screen: p5 2. Add the curve Y1=(x A)(x B) : jf-afkjf-agkl 3. Plot the curves using modify: y The range of values for which the curve lies below the y-axis is the solution to the inequality (x a)(x b) < 0. Questions for discussion If the product of two numbers is negative what does this tell you about the numbers? Will you always be able to find x-values for which a quadratic is negative? What would the solution to (x a)(x b) > 0 look like? Problem (Try the problem with pen and paper first then check it on your software) Sketch the graph of y = 2x² x 6 and hence solve the inequality 2x² x 6 0. Further Tasks Find the range of values for k such that x² 4x + 3 = kx has two distinct roots. Investigate y > mx + c and y > ax² + bx + c graphically. Page 3 of 6 mei.org.uk/casio-networks

15 Solutions of Trigonometric Equations (Degrees) 1. Select Graphs mode: p5 2. Check the angle type is set to degrees: SHIFT > SET UP and scroll down to Angle. 3. Enter the graph Y1=sin x : hfl 4. Enter the graph Y2=0.5 : 0.5l 5. Set the View-Window to TRIG: Lewd 6. Draw the graphs: u 7. Use G-Solve to find the points of intersection: Lyy You can use the cursor (!/$) to move between the points of intersection. Try finding the points of intersection for other values of Y2 (e.g. Y2 = 0.75 or Y2 = 0.3). Questions What symmetries are there in the positions of the points of intersection? How can you use these symmetries to find the other solutions based on the value of sin -1 x given by your calculator? (This is known as the principal value.) Problem (Try the question just using the sin -1 function first then check it using the graph) Solve the equation: sin x = 0.2 ( 360 x 720 ) Further Tasks Investigate the symmetries of the solutions to cos x = k and tan x = k. Investigate the symmetries of the solutions to sin 2x = k. Page 4 of 6 mei.org.uk/casio-networks

16 Functions The Modulus Function The modulus function, abs(x), is found using OPTN > NUMERIC > Abs 1. Add a new Graphs screen: p5 2. Add the graph y = x, Y1=Abs(x): iyqfl 3. Add the graph y = ax+b, Y2=Abs(Ax+B): iyqaff+agl 4. Plot the curves using modify: y Questions What transformation maps the graph of y = x onto the graph of y = ax+b? Where is the vertex on the graph of y = ax+b? Where does the graph of y = ax+b intersect the y-axis? Problem (Try the question with pen and paper first then check it on your calculator) Sketch the graph of y = 3x+2 3 and find the points of intersection with the axes. Further Tasks Investigate the graphs of o y = f(x) o y = f( x ) for different functions f(x), e.g. y = sin(x) or y = x³ x². Investigate the solutions to the inequality x + a +b > 0. Page 5 of 6 mei.org.uk/casio-networks

17 Sum of an Arithmetic Progression 1. Go into Table mode: p7 2. Add y = 2x+1 and display the table. Y1=2x+1 : 2f+1lu 3. Go into Run-Matrix mode: p1 4. Find 5 x1 2x 1: Math Σ( ruw2f+1n!f$1b5l Now find the sum of the terms of some other arithmetic progressions. You can also try entering the sequence in Recursion mode p8 Questions Why will the terms of bn+c (for n = 1,2,3, ) be an arithmetic progression (AP)? How can you express the link between the terms of an AP and its sum? Problem (Try the question with pen and paper first then check it on your calculator) What is the first term of an arithmetic progression if the 3 rd term is 11 and the sum of the first 10 terms is 185? Further Tasks Investigate arithmetic progressions with the same sum, e.g. how many APs can you find that have a sum of 100? Express the n th triangle number as the sum of an AP. Investigate whether the n th hexagonal number can be expressed as the sum of an AP. Page 6 of 6 mei.org.uk/casio-networks