Using Graphical

Calculators in the

new A level Maths

Tom Button

tom.button@mei.org.uk

Simon May

simon.may@nulc.ac.uk

Plot these curves

Answers

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.”

Classroom tasks

A series of tasks,

each in 4 parts:

• Construction

• Exploration

• Question

• Extension

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

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)

Ofqual guidance on calculators

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]

Experiences of using graphical calculators

For A level

• Ease of use/availability

• Graph Transformations

• Trigonometry

• Numerical Methods

• Solution Checking

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 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

Page 2 of 6 mei.org.uk/casio-networks

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 3 of 6 mei.org.uk/casio-networks

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 4 of 6 mei.org.uk/casio-networks

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

-1x 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 5 of 6 mei.org.uk/casio-networks

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 6 of 6 mei.org.uk/casio-networks

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

1

2 1

x

x :

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 3rd 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 nth triangle number as the sum of an AP. Investigate whether the nth hexagonal number can be expressed as the sum of an AP.

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