In recent examiner reports it is noted how important it is for students to understand the properties of a parabola when plotting quadratic graphs on Cartesian axes. Students who have a secure understanding of parabolas can use them to correct miscalculated values in their table of results and are more likely to attain full marks on this Grade 5 topic.
To help students draw a quadratic graph they need to understand the properties of a parabola. A parabola is a smooth, symmetrical curve with a clearly defined maximum or minimum turning point.
To demonstrate the shape of a parabola I stand at one end of the classroom and ask for a volunteer to stand at the other. I throw a whiteboard pen up in the air towards the volunteer and ask the students to sketch on their mini-whiteboards the path of the pen as it travels under the force of gravity.
Gravity is a force pushing the pen vertically downwards, which is working against the pen moving upwards. This decelerates the pen until zero velocity at the maximum point and then the pen accelerates back towards the ground. The horizontal motion is due to throwing the pen in that direction. Because gravity has a constant acceleration of 9.8 m/s2 the pen falls at the same rate as it climbed. This is what makes the path symmetrical.
When plotting quadratic graphs on Cartesian axes students are expected to complete a table of results to help calculate the y value for changing values of x. I like to include an additional row in the table for each calculation involved in the equation. The x and calculated y value form a series coordinate pairs which are plotted on a Cartesian grid. A common mistake when completing a table of results from a quadratic equation is to incorrectly calculate -32 as -9. In my experience this is more likely to happen if students use a calculator.
It is important for students to understand the table of results are used to generate a series of coordinate pairs involving x and y. A common mistake here is to complete the table but not know how to use it to plot the graph.
To help the pace of the lesson students are provided with a pair of correctly scaled axes. As learning progresses I expect them to choose their own scale and draw the grid on A4 graph paper.
After we have plotted the coordinates I share some tips for drawing the parabola which passes through the points. It is often easier to draw the parabola with your writing hand inside the curve. Turning the paper to draw the parabola is more comfortable than rotating their hand.
The parabola does not have a clearly defined minimum or maximum point.
A coordinate pair is either calculated or plotted incorrectly.
Drawing line segments between each coordinate pair suggest the relationship between them is linear which it is not.
The graph is does not pass through each of the coordinate pairs to form a clear defined and smooth parabola.
When students have completed plotting the first equation I ask them hold up their graphs for me to check. To feedback we have a short discussion about whether the graph they have drawn obeys the discussed properties of a parabola. The students who had made mistakes go back and correct them.
In the next lesson we progress on to using parabolas to solve quadratic equations graphically. Here, we talk about whether an equation has one to two solutions depending on where the linear graph cuts the quadratic. We also look at whether a quadratic and linear will have any solutions depending on the intercept value and gradient of the linear graph.
In this blog I will share some practical tips for using mini-whiteboards in a mathematics lesson. I use mini-whiteboards nearly every lesson because they help the students show me the progress they are making. When I understand what the misconceptions are I am able to address them in subsequent examples as part of my feedback. […]
Demonstrating student progression during a mathematics lesson is about understanding the learning objective and breaking that down into explicit success criteria. Using Success Criteria Take, for example, a lesson on calculating the area of compound rectilinear shapes. The intended learning objective was written on the main whiteboard. Success criteria were used to break down the individual […]
Plotting and interpreting conversion graphs requires linking together several mathematical techniques. Recent U.K. examiner reports indicate there are several common misconceptions when plotting and interpreting conversion graphs. These include: drawing non-linear scales on the x or y axis, using the incorrect units when converting between imperial and metric measurements, taking inaccurate readings from either axis not […]