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

3. 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 a repeat of a slide from session 1. Go over this quickly if using this session after session 1.
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.

4. GeoGebra
GeoGebra is a free graphing tool that is available online from geogebra.org or as an Android/iPhone/iPad app
GeoGebra is now available as a phone app. There will be an opportunity to discuss the issues relating to the use of phones in class towards the end of this session.
This session is not training on how to use GeoGebra for teachers. More guidance on this and a self-study course are available via mei.org.uk/geogebra
Classroom activities are available at: mei.org.uk/geogebra

5. Desmos
Desmos is a free graphing tool that is available online from desmos.com or as an Android/iPhone/iPad app
Desmos is available as a phone app. There will be an opportunity to discuss the issues relating to the use of phones in class towards the end of this session.
Classroom activities are available at: mei.org.uk/icttasks#desmos

6. Exploring the basics www.geogebra.org/
Get teachers to open software or visit website.
Select Geometry
Exploring the basics

7. Select a point
Put on three points and then lines
Select move tool and move the points
Add an extended line
Add two angles and the external angle

9. Click on new and don’t save
Add two points and a line between them.
Add circles around these points
Use the hand tool to pulsate these!!
Now add a line by selecting the intersection of the circles
Go back to pulsating and what can we say about the intersecting line?

10. Challenge:
Create dynamic examples of the basic circle theorems for angles at GCSE

11. Exploring algebraic graphs www.geogebra.org/
Select Graphing calculator
Exploring algebraic graphs

12. A=
Se;ect slider B=
Select slider
(y-a)2+(x-b)2=16
PLAY R=
Change equation to include r2

15. Challenge:
Create an alternative dynamic examples of a circle, using the equation format
y2 +ky + x2 + qx = s
What exploration activity could students engage in to better understand the relationship between the two formats of the equations?

16. Student tasks for using GeoGebra/ Desmos in A level Maths
The focus of this session is about using GeoGebra and/or Desmos for student tasks in the new A level.

17. SAM question (MEI paper 1)
Determine the values of k for which part of the graph of y = x² − kx + 2k appears below the x-axis.
Get the teachers to enter this into their software and explore varying the k, then ask them to answer the question.
Discussion:
Does having explored the family of curves in the software help with this question?
Will students be better prepared for these type of questions if they done similar activities in lessons (used technology to vary parameters)?

18. Why use technology?
Strengthens the links between graphical and algebraic representations
Easy to vary parameters and view the effects
Students can change things to ask “What if …?” questions
More realistic way of doing mathematics
The graphs will always be accurate! …
These are a few ideas which can be expanded on and the teachers can suggest their own.

19. Example Student Task Design of student tasks: Construction
Discussion question(s)
Problem
Extension
Work through the Differentiation – Exploring the gradient of a curve task.
A useful question to prompt the discussion is “what things would you like you students to notice about the relationship between the gradient function and the curve”. Also useful to highlight the point of inflection on this curve is where the gradient function has a minimum value – this will be where the tangent crosses the curve.

20. Further tasks
Opportunity to try one of these further tasks. Examples of both are given for GeoGebra and Desmos.
Task 5: Functions – Transformations
For the problem students should expand the function using either a binomial expansion or by multiplying out the brackets.
The graph of the function is the graph of f(x)=x^3 translated by (-2,3).
Be careful with second further task (horizontal stretches) – they can look like vertical stretches for many functions but this is an excellent discussion point. Students should also take care with the scales on the axes here as these can cause confusion. sin(x) or x^3-x are good functions to use for this.
Task 13: Quadratic Inequalities
The software shades the region that satisfies the inequality as a vertical strip.  
Problem solution: x ≤ -3/2 or x ≥ 2

21. Break

22. Integrating into your scheme of work
This page on the MEI website gives suggested student tasks for (almost) all topics in the new A level. There are versions of most tasks designed for Desmos and GeoGebra (and Casio).
mei.org.uk/integrating-technology

23. How to integrate technology
Suggested ideas
Make frequent reference to graphing tools in your lessons – e.g. when teaching quadratics use technology to vary a, b and c in y = ax² + bx + c
Use the list of tasks to schedule a technology lesson into your scheme of week
Set (optional?) technology homework tasks
These are a few suggestions – the teachers might suggest some more. The emphasis here is on student use of technology.

24. Use of devices in class
Are computers available for you to use in lessons?
Do students use tablets in your lessons?
What is your school/college policy on use of phones in class?
This is an opportunity to discuss hardware. We strongly advise teachers not to do anything counter to their school/college’s disciplinary policy; however, this might be an opportunity for them to discuss it with their senior leaders. There are many places that allow sixth form students to use their phones in class for lessons where this doesn’t cause issues.

25. 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.

26. More GeoGebra Materials
Lots of prepared files are available to view and download from:
Teachers don’t need an account to view/download files. Files can be viewed directly on phones. If teachers have an account they can upload hidden files so they can give their students the link but it is not searchable.
If time show:
How they can download files.
How they can access a hidden file with the link (e.g. is hidden but they can view with the link)

27. More Desmos Materials
Lots of activities are available to view and download from: teacher.desmos.com/
Teachers don’t need an account to view/download files. Files can be viewed directly on phones.

28. Teaching with Large Data Sets
You can download the datasets from
This session will demonstrate some of the techniques and then give teachers an opportunity to try them.

29. Data Sets: Key Points
Each awarding body will supply one or more datasets; questions will be set which assume familiarity with the dataset.
Students must be familiar with both the data and its context; they need to know the origin of the data and how it might have been collected.
Technology must be used to explore the data set.
Increased emphasis on interpretation over calculation and being able to select the correct representation or model
Emphasise that students will be better prepared for the exam if they are familiar with the context and the best way to do this is through exploring the datasets with technology.

30. The Data Sets Body Format Description of data Lifetime Edexcel
Spreadsheet with multiple sheets.
Met Office weather for 5 different stations over 2 different time periods along with 3 international weather stations
Until further notice
MEI
Single sheet spreadsheet
2012 Olympics Medals and demographic data by country
3 data sets will be used in rotation. Other 2 sets available in June.
OCR
Spreadsheet with 4 sheets
Methods of Travel by Local Authority
AQA
Selected data from the family food survey by region.
Teachers might find it useful to explore the datasets for other awarding organisations – these can form a bank of useful contexts for teaching statistics.

31. In the exam:
Students should be expected to interpret output from a spreadsheet or statistical software.
Students should be expected to interpret and explain terminology which has been introduced via the data set.
Students should be asked to explain what effect missing data would have on a model that has been derived.
Students should be asked to explain how they would collect data and to describe the drawbacks and advantages of particular sampling methods.
The style of questions in statistics is changing more than in mechanics. The next few slides feature a question from the OCR Sample materials that demonstrates this.

32. The Edexcel Data Set
If we look at the Edexcel data files in excel. Questions we might ask:
Does it need to be cleansed? If so how?
Does it include the source of the data so that learners can understand how it was collected?
Is there a glossary to help learners understand the data and associated terminology?
Is it clear whether it is (essentially) a population, or a sample from a larger population?
Little cleansing needed although we need to decide what to do with a trace of rainfall and the day when no wind recordings were made. If we work out the average wind speed excel will ignore the n/a and divide the total by 183 data items which is fine as that will give a realistic mean value. However if we do the same for the rainfall, it will ignore the trace values, of which there are many and we will a value which is too high. The data is probably best regarded as a subset of a larger population of weather data.

33. How might students use technology?
Sorting and searching; identifying outliers; cleansing;
Producing and using summary statistics to make conclusions about the data;
Producing graphs such as histograms and box-plots that allow comparisons to be made;
Producing scattergraphs and modelling using trend lines or curves;
Selecting random samples and comparing them to the population to illustrate variation;
Checking to see if data fits a particular model or distribution; hypothesis testing.
This is just a few ideas. The main aim is for students to become familiar with statistical ideas and the context of their data set through exploring it with technology.

34. Spreadsheets and statistical software
What are the options?
Advantages and Disadvantages?
Excel
GeoGebra
This presentation uses only Excel and GeoGebra. Other technology is available. Gnumeric is a free spreadsheet which has better statistical functions than excel. This session concentrates on using GeoGebra for Statistics. There is a lot more help for using GeoGebra in pure maths at

35. Sorting and Filtering: The Data Set
Sorting can help establish if there are rogue values or outliers
Filtering can help focus on a subset of the data
In the next few slides (14-21) various techniques that can be demonstrated. Spend about mins demonstrating and then let teachers work on the workbook. The slides show what can be demonstrated and this should be done live. The workbook also shows how to do this.
Here show how to Sort and Filter on a few different fields using Excel, including a multiple level sort.

36. (Not) Drawing Graphs Using Excel
Drawing meaningful graphs using Excel is very difficult.
Show how graphs can be drawn on Excel. Or can’t be – it’s pretty hopeless. Might be ok for line graphs/time series. The add-in can used but need to set up bins first and filter out n/a as excel can’t cope with this. Excel 2016 has more built-in features such as box-plots.

37. GeoGebra Spreadsheet View!
GeoGebra has its own spreadsheet view: You can paste from an Excel file into it.
Copy and Paste some maximum temperature data from the Excel into a couple of columns. E.g. for Camborne 1987 and Heathrow 1987

38. GeoGebra: One Variable Analysis
Select the first column and click on one variable analysis and then Analyse.

39. GeoGebra: Charts
Show the different graphs that can be plotted and how they can be changed by the use of the slider.

40. GeoGebra: Multiple Variable Analysis
Select both columns and then Multiple Variable Analysis. 2 box plots can be drawn side by side. What do these show?

41. GeoGebra: Using the statistics box
By clicking on the sigma x icon we can get the statistics box (demo this).

42. Making Comparisons
Using the statistics box I have summarised the Weather in Camborne in October 1987 and in October 2015.
What do you notice?
Why was Oct 1987 different? What happened?

43. Correlation and Regression
Both Excel and GeoGebra can calculate correlation coefficients. GeoGebra has tools to plot regression ‘lines’.
Select some bivariate data (can use the 2 sets of max temps at different locations). Click on the Two Variable Regression Analysis option and also reveal the statistics box. Geogebra which will fit different functions and work out y-values based on the x-values (can also swap x and y roles and get x on y regression line instead of y on x). Students are expected to model with different functions, not just linear. Change the model and explore which curve is the best fit. The SSE gives a measure of the squared deviations from the curve.

44. Workbook
Using the dataset of your choice, work through the activities in the workbook. You’ll need:
Laptop or tablet
The Dataset
Spreadsheet and GeoGebra
Workbook for your dataset.
Allow 40 minutes for delegates to work through activities and float round and answer questions. You may want to refer back to earlier slides.

45. Next Steps…
Lead a discussion of what teachers might do from here. What do they need to consider in planning? Hardware? (What sort of devices, can students use their own) Software? Costs? (Should be none for Software). How to integrate use of tech with LDS into their SOW? Most of the stats could be taught through the LDS.

46. Resources Awarding body SAMs Awarding body PD and supporting materials
Integral 2017 has resources to support the teaching of statistics through the LDS, including videos
will have applets to demonstrate particular statistical concepts
Anything else?

47. 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