Use of technology in the 2017 A level Maths & Further Maths

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.” This is the guidance from Ofqual to the awarding organisations. Emphasise that this is about students’ experiences in the maths classroom over 2 years of study and not explicitly about the examinations BUT the exams should be designed so that students who used technology are better prepared for them. Two other points to emphasise: “… mathematical and statistical graphing tools …” means that it’s not just for stats. This session focusses on how to use mathematical graphing tools in pure maths. “… must permeate the study …” is referring to student use of technology (not just the teacher stood at the front using it). This session is about student tasks for investigating with technology that are designed to enhance students’ understanding.

Your experience of using graphing tools Give an example of where using graphing tools has enhanced students’ understanding of mathematics. Brief discussion. Ask teachers in groups to discuss this and then feed the answers back. This should give you a good gauge of the level of experience of using technology of the teachers. It is useful to draw out how it enhanced understanding and not just say “I used it for graph transformations”.

Starter activity Plot the graph of y = x³ + ax + 1 What questions can you ask about this function? This is best done in GeoGebra, though Desmos or the Casio emulator are fine. You can right-click on the slider and switch the animation on. Ask the teachers to discuss in groups and then feedback what questions they could ask. When they have done that stop the animation add a tangent to the curve at a point. By moving this you can observe that the tangent to the curve has a minimum gradient at (0,1) that is equal to a. Adding f’(x) emphasises what is going on at the point of inflection. Non-stationary points of inflection are a new topic on the A level. Emphasise how this 5-minute activity demonstrates the power of graphing tools to enhance students’ understanding of mathematics – this is what is intended by the Ofqual guidance.

Approach of awarding organisations Graphing questions with parameters Strong emphasis on linking graphs and algebra Questions involving the output of spreadsheets (MEI only) Use of calculator to find probabilities Spreadsheets encouraged for large data set These are examples of the types of things seen in the SAMs. The guidance from Ofqual was about the learning, and not the assessment; however, the specifications should be encouraging the use of technology and the examples above show the awarding organisations are doing this, albeit implicitly in some cases.

Sample Assessment questions How will you use technology to ensure that students are better prepared for these type of questions? These are examples of exam questions featuring graphs with parameters. The first one is a particularly good example to demonstrate that graphing tools can give students an appreciation of families of curves.

Sample Assessment question (MEI) If time give teachers an opportunity to try this question from the MEI paper. Currently MEI is the only specification that has output from spreadsheets in the questions but all specs are meant to encourage the use of spreadsheets throughout the course and this is an example how they can be used in Pure Maths.

SAM question (MEI) – solution Print out p10 of MEI Spec?

Ways to use technology During A level Maths courses students will benefit from: Exploring using graphing technology Experiencing investigating data with statistical tools (especially the large data set) Using spreadsheets for Statistics, Pure Maths (functions) and modelling in Mechanics Using CAS software to investigate algebraic relationships.

Using spreadsheets/statistical tools Students should experience using spreadsheets and statistical tools for investigating the large data set The statistical functions in GeoGebra are easy to use No need to spend much time on this if it is being coupled with session 3.

Graphing tools – GeoGebra/Desmos GeoGebra and Desmos are free graphing tools that are available online or as an phone/tablet app from: www.geogebra.org www.desmos.com/ No need to spend much time on this if it is being coupled with session 2. For classroom tasks see: mei.org.uk/integrating-technology

Graphical Calculators Graphical Calculators are a very effective way of ensuring student use of technology permeates the study of A level. Support with using Casio Graphical Calculators is available via: mei.org.uk/casio-networks Highlight here if there is a local network session supported by a Casio teacher. For classroom tasks see: mei.org.uk/integrating-technology

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)

SAM questions (Calculator use) Edexcel A level Paper 3 A machine cuts strips of metal to length L cm, where L is normally distributed with standard deviation 0.5 cm. Strips with length either less than 49cm or greater than 50.75cm cannot be used. Given that 2.5% of the cut lengths exceed 50.98 cm, (a) find the probability that a randomly chosen strip of metal can be used. [5] OCR AS level Further Mathematics Pure Core Paper You are given two matrices, A and B, where 𝐀= 2 1 5 1 1 3 −1 2 2 and 𝐁= −4 8 −2 −5 9 −1 3 −5 1 Show A−1 = kB where k is a constant to be determined. [2] If there is time allow teachers to try these on a calculator. The markschemes are on the next slide.

SAM questions (mark schemes) Edexcel A level Paper 3 A machine cuts strips of metal to length L cm, where L is normally distributed with standard deviation 0.5 cm. Strips with length either less than 49cm or greater than 50.75cm cannot be used. Given that 2.5% of the cut lengths exceed 50.98 cm, (a) find the probability that a randomly chosen strip of metal can be used. [5] OCR AS level Further Maths Pure Core Paper You are given two matrices, A and B, where 𝐀= 2 1 5 1 1 3 −1 2 2 and 𝐁= −4 8 −2 −5 9 −1 3 −5 1 Show A−1 = kB where k is a constant to be determined. [2] The key issue to emphasise here is there are now processes that it is expected that students will do on a calculator, such as normal probabilities and inverse matrices. They will only get 1 mark for these (B1 in both of the examples above).

OCR & MEI “non-calculator” questions Questions that include the instruction: Any calculations in your written solution to this question must use only elementary functions. “… This is not a restriction on a learner’s use of a calculator when tackling the question, e.g. for checking an answer …” OCR and MEI are aware that some of the more advanced calculators, especially the graphical ones, are very powerful. This statement above is used when the candidates need to show how they have arrived at an answer not just used a button to calculate it directly. One example of this would be definite integration.

Reflections What technology do you want your students to have in the exam: scientific or graphical calculator? How does this impact on your use of technology in the classroom? The key issue for teachers to leave this session with is to be considering how using technology in the teaching and learning of A level Maths can help enhance their students’ understanding. To fully realise this potential it is necessary for the students to be using technology, not just the teacher.

Further Support Ideas for integrating technology: mei.org.uk/integrating-technology Large Data Set: integralmaths.org/ Getting started with GeoGebra: mei.org.uk/geogebra#getting-started Support with using graphical calculators (including free 1- year trial of the emulator): education.casio.co.uk/

The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics. The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students. To find out more please visit www.furthermaths.org.uk