Starter: y = x + 1 B) y = x – 1

Real Life Graphs Objective: To understand how graphs are used to show relationships between variables Must:

Graph of monthly mobile phone charges Explain that graphs can be used to illustrate any function or formula containing two variables. This graph compares two types of tariff offered by a mobile phone company: The ‘Pay as you go’ tariff (shown in blue) has no monthly charge and charges a fixed amount per minute. Drag the moving point on the graph to help pupils work out the cost per minute. This is £0.10 per minute. The ‘Monthly’ tariff (shown in red) has a fixed monthly charge and charges calls at fixed amount per minute. Drag the moving point on the graph to help pupils work out the monthly charge and the cost per minute. This is £7.50 for the monthly charge and £0.05 per minute for calls. Ask pupils for the significance of the point where the two lines cross. This is the point where the two tariffs cost the same amount for the same amount of time spent on calls. Ask pupils which tariff is cheaper. Establish that if we spend less than 150 minutes (2½ hours) on calls in a given month, the ‘Pay as you go tariff’ is cheaper. If we spend more than 150 minutes on calls in the month, the monthly tariff is cheaper. (150 minutes a month is about 5 minutes a day.) If we include another month we would have to pay another £7.50 on the ‘Monthly’ tariff. Discuss whether or not the intermediate points have any practical significance. Suppose we could zoom in on the graph to read off the charge in pence and the time in seconds. Since calls are rounded to the nearest penny and the time spent on the phone is rounded to the nearest second, intermediate points between pennies and seconds, although they have a practical significance in theory, would have to be rounded up. Ask pupils to suggest a formula to describe each tariff. For example, for the ‘Pay as you go’ tariff: Cost in pence = 10 × number of minutes (or C = 10n) and for the ‘Monthly’ tariff: Cost in pence = 5 × number of minutes + 7.5 (or C = 5n + 7.5).

Plotting graphs – using a table of values When we plot a graph we usually start with a table of values. The values in the table usually come from a formula or equation or from an observation or experiment. For example, a car hire company charges £30 to hire a car and then £25 for each day that the car is hired. This would give us the following table of values: Number of days, d Cost in £, c 1 2 3 4 5 Ask pupils if they can give you a formula linking c, the cost, with d, the number of days (c = 25d + 30). Ask pupils to tell you what kind of graph this will produce (a straight-line graph). Explain that when we are plotting a graph it is very important to know which variable depends on the other. This tells us which variable will go along the horizontal axis and which variable will go along the vertical axis. When we are plotting graphs of functions, for example, the value of y depends on the value of x. 55 80 105 130 155 The cost of the car hire depends on the number of days. The number of days must therefore go in the top row.

Plotting graphs – choosing a scale The next step is to choose a suitable scale for the axes. Look at the values that we need to plot. Number of days, d Cost in £, c 1 2 3 4 5 55 80 105 130 155 The number of days will go along the horizontal axis. The numbers range from 1 to 5. A suitable scale would be 2 units for each day. The cost will go along the vertical axis. Discuss suitable scale for the range of units. Explain that when the range is small (as for the days) we usually choose 2, 4, 5 or 10 units (squares) for each whole one. When the range is large (as for the cost) we usually choose each unit to represent 2, 5, 10, 50, 100, … If the range starts at a large number, the scale can start at a number other than 0 but this must be shown by a zigzag on the axes. The cost ranges from 55 to 155. A suitable scale would be 1 unit for each £10. We could start the scale at £30.

Plotting graphs – drawing the axes We then have to draw the axes using our chosen scale. We will need at least 10 squares for the horizontal axis and 13 squares for the vertical axis. When the scale does not start at 0 we must show this with a zigzag at the start of the axis. 30 40 50 60 70 80 90 100 110 120 130 140 150 Number the axes. Cost (£) Label the axes, remembering to include units, if necessary. 1 2 3 4 5 Number of days

Plotting graphs – plotting the points Use the table of values to plot the points on the graph. Number of days, d Cost in £, c 1 2 3 4 5 55 80 105 130 155 Cost of car hire 150 It is most accurate to use a small cross for each point. 140 130 120 110 If appropriate, join the points together using a ruler. 100 Cost (£) 90 80 Tell pupils that it is most accurate to use a small cross when plotting points on a graph. Stress that when the points do not lie in a straight line we have to decide whether to use a line of best fit, a smooth curve through the points or to join the points together using straight lines. The one we choose depends on the context from which the graph is generated and whether intermediate points have any significance. 70 Lastly, don’t forget to give the graph a title. 60 50 40 30 1 2 3 4 5 Number of days

Science experiment A group of pupils are doing an experiment to explore the effect of friction on an object moving down a ramp. They attach weights of different mass to the object and time how long the object takes to reach the bottom of the ramp. They put their results in a table and use the table to plot a graph of their results. Discuss how we can produce a table of values from an experiment. Before revealing the table ask pupils whether the time or the mass will go on the top row of the table, and hence along the x-axis of the graph. Establish that the time taken for the object to move down the ramp depends on its mass. It is therefore the mass that goes at the top of the table. The mass is the variable that we are choosing or controlling. Mass of object moving down ramp (grams) Time taken for object to move down ramp (seconds) 100 4 150 7 200 12 250 17

Science experiment Mass of object moving down ramp (grams) Time taken for object to move down ramp (seconds) 100 4 150 7 200 12 250 17 We can join the points using straight lines. 20 16 Do the intermediate points have any practical significance? Time taken (seconds) 12 Discuss the significance of the intermediate point in this context. Since the variables are both continuous, we can assume that the intermediate points have a practical significance. We can estimate, for example, how long a mass of 175 g would take to slide down the ramp. Discuss the following: How the points should be joined. Should each point be joined together using straight lines or should we use a line of best fit? Would it make sense for the line to meet the horizontal axis? How are the variables related? (The heavier the object the longer it takes to move down the ramp.) How long it would take for an object of mass 225 grams to slide down the slope? How long it would take for an object of mass 300 grams to slide down the slope? What mass we would use if we wanted the object to side down the slope in 10 seconds? End the discussion by suggesting that although we may expect a straight line graph in theory, when we do an experiment many factors including human error and rounding can produce points that do not lie on a straight line. The graph would be more accurate if more points were plotted. Any inaccurate readings would then stand out more easily. 8 4 How could we make the graph more accurate? 50 100 150 200 250 300 Mass of object (grams)

Class Work Frameworks (9) Page 75 Read Exercise 5b- Question 1,2,3 Pg 77 &Exercise 5c 1 is a must