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# Graphs workshop style lesson

One class I teach has a number of students taking the GCSE exam at the higher level along with students entered at the foundation level. Meeting the needs of this group has lead me to use workshops for extended parts of the lesson.

Extra graph topics at Higher include graphical solution of simultaneous equations, graphical inequalities and puzzle questions asking you to show that two straight lines are parallel. There is more emphasis on the formula based straight line graph questions compared with the foundation level. Foundation level has functional maths questions about phone tariffs and distance time graphs. Students have to be careful to actually answer the question.

I decided to run a workshop style lesson with students working in level based groups from a series of worksheets while I coached. It is a fairly small class (common in adult education settings because of room size limitations), so there was enough time to talk to individuals. I needed to provide worksheets and activities that students could access directly, so that I could coach rather than explain. I planned to round the session off with a pairs activity where students sort cards holding examples of straight line graphs.

## The resources

• Some exam questions based on mobile phone tariffs taken from the AQA Modular Maths Unit 2 Practice Paper sets
• A four page higher level summary of straight lines. This was the only new resource needed for the lesson.

A set of five handouts for the students entered at foundation level

Finally, a set of graph cards and a question sheet about the cards.

## Tariff questions To kick off, I used some GCSE practice paper questions about mobile phone tariffs. I made up the graph above to use later with an interactive whiteboard. The exam questions included

• What is the monthly fee for tariff A and B?
• How many free minutes for tariff A and B?
• What is the cost per minute for extra minutes for tariff A and B?
• For what range of minutes per month is tariff A cheaper than tariff B?

Most students looked at the cost of 100 or 200 extra minutes to find the cost per extra minute. That lead to a link with steepness and cost per unit. Formulating the answer to the question about when tariff A is cheaper than tariff B leads on to inequalities and the idea that there may not be a 'cheapest' company.

## Distance time graphs I made up the distance time graph above for an interactive whiteboard slide. The exam questions involved

• Identifying the phases of the journey
• How long were the stationary periods
• Which section of the outward journey was the fastest (and how do you know?)
• H: what was the average speed of the outward journey?

I had to mention decimal hours, e.g. 3.45 hours is 3h 27 minutes and 3:45h is 3.75 hours. Scale reading is important as well, scales seen on exam questions include 10, 15 and 20 minutes per square.

Again making a link between steepness of the graph and speed helps to make the idea of the gradient of a line familiar and gives practice in finding the values using triangles drawn on the graph. The average speed question did raise the idea of just taking the 20km and dividing by the total time, 4.75 hours.

## Workshop phase

The students worked through the various activities for about an hour. I sat the higher level students together and let them loose on the handout. Some useful peer interactions did occur. I was able to move about and coach through the tasks, spending most of my time with the foundation students. Students were drawing directly onto the prepared grids on the handouts saving time, and many were making notes about things they found hard on the sheets.

A few points that stuck in my mind about the new higher level handout

• The formula m = (y2 - y1)/(x2 - x1) was ok for the students after some explanation. I need to link it more with the triangle on the graph line
• Getting the formula from the coordinates two points by solving the equations went well, two students plotted the points on squared paper
• The words gradient and intercept should be earlier in the handout - perhaps an overview right at the beginning with a labelled sketch
• The m1 × m2 = -1 test for perpendicular lines needs more explanation. This is probably beyond the syllabus and I might drop it for more work on parallel lines.

The foundation level handouts are a bit wordy and need editing. The sticking points involve symbolising the results of plotting the graphs...

• All can find the midpoints by eye, but need coaching on calculating the coordinates of the midpoint. Making a link to midpoints in e.g. mean from grouped frequency table helps. I use a vertical layout with the coordinates of the end points written underneath each other. Then just total the X 'column' and divide by 2 and total the Y 'column' and divide by 2
• Special lines activity needs a re-write, I think I'll just define two lines like Y=2 and X=5 and then ask the students to plot a few others
• The next three activities could probably be collapsed into a couple of exercises based on a graph with positive gradient and another with negative gradient. I'll probably have less 'discovery' and more 'telling' around the formula as that seems a low confidence area

## Card sort

The graph cards are printed 4 to a side and laminated so each pair of students has a set of 8 cards. The first four cards are positive quadrant and the last four are all four quadrants. The questions are designed to help students organise their knowledge of the straight line graph. I think I'll just give the cards out first and ask students to classify them into four classes as