Q How do you help students to sketch straight line graphs without resorting to any plotting? They seem to forget that the "c" in y = mx + c is the point where the line crosses the "y" axis. Then they aren't sure how to use the gradient to create the line. I have an overhead projector. Have you any ways that you have used successfully to help them?
A The Mathagony Aunt column of June 16 spoke about how to introduce straight line graphs. A good way to learn about the attributes of the general formula is for students to interact with ICT exploring different relationships via Excel or Autograph or graphical calculators. But there are schools I know that don't have access to a set of laptops for the classroom and the computer suites are not always available when you want them. An OHP is great for this, but this exercise could also be done without an OHP if you can create a large copy of a Cartesian graph. I have created a slide with a copy of a Cartesian graph that you could copy onto acetate for use in the classroom as well as some graph labels that might be handy. These can be copied onto card to indicate which graph to sketch and onto acetate for labelling the actual line. This is a great way to get the class to select a graph to sketch at random. You will also find it useful to have some bamboo skewers. For those readers who will use the poster then skewers will be useful, as well as some Blu-Tack.
I think the fact that the examples can be randomly selected is very helpful. One of the reasons that students aren't able to connect the y-intercept (the point where the line crosses the y-axis) is because they haven't had enough experience of seeing what happens at the point x = 0 via substitution into the equation of the line. I also suggest that they can use the fact that crossing beings with a "c", which suggests that it is a crossing point.
In the general formula y = mx + c, the "m" represents the gradient.
Demonstrate this with a line. To use this method of sketching the linear equations would have to be arranged in the same format. I have chosen y = 2x + 3 to demonstrate the sketching of a line. Ask where the line crosses the y axis. Mark the point (0, 3), the y-intercept. Then explain to students that the line can be created by using the given gradient to move horizontally and vertically from this point. In our case the gradient is 2 (the number in front of the x variable), but to be able to draw the gradient you need two numbers, one indicating the horizontal and one the vertical measure. I suggest that you write m = 2 = 2Z1 as every whole number can be written as a fraction with 1 underneath, and this leaves the value unchanged. This fraction tells us that there you move one across and two up from the intercept. This can be repeated from the point on which you land. Join up these points - straight line sketched! Place the skewer on the line.
Now that the students have a known intercept, rotate the skewer to give a new gradient. Ask students for the equation of this new line. Invite a student to show the rest of the class the triangle that provides the value of the gradient. In the example, the green triangle shows that the gradient is m = 2Z3
For those students who cannot remember which numbers in the gradient show the changes in x and y, I tell them that the gradient is the difference in y over the difference in x. By being this way up y can give x a cuddle - using y's tail! This picture has appeared in a previous column. If you would like a copy then please email me.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
Email your questions to Mathagony Aunt at firstname.lastname@example.org Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX