y = mx + c (part 3)

The link at the base of my first post should have landed you at the GeoGebra Tube applet page (y = mx + c). Geogebra applets are great for students to experiment with or for teachers to use who aren't that confident with GeoGebra. They allow access to a GeoGebra sketch without opening the full program so although they offer limited functionality compared with opening the sketch in the program, they are very simple to use and display.

To try out the ideas in this post you will need to open up my sketch in the full program (the Chrome App or GeoGebra Applet do not offer full functionality of the second sketch I will share). One way to to do this is to click on the share or copy link at the bottom of the applet page. You then have the option to download the sketch as a GeoGebra file:

I should add at this point that if you haven't already done so you will need to download GeoGebra to open this file; you can do this here.

Ok, so after recapping some of the questions I asked in the first lesson on y = mx + c this is how I would develop students knowledge through questioning whilst using the following sketch;

Click on the background then hold down shift and use the right/left keys to zoom in/out on the x-axis only (up/down controls the y axis zoom). Is the gradient of the line still the same? Zoom the axes back to how they were.

Can you give me the equation of a line parallel  to this one? - Use the input bar to enter an equation and check. I have included a "clear" button in the sketch to quickly delete any additional lines (this is especially useful if the sketch is accessed as an applet as you can't just click on the line and press delete).

Before the next question hit clear, hide the axes labels and change the original equation using the sliders.

Can you give me the coordinates of a point on the line? - Enter the coordinates in the input bar; multiple points can be entered and the 'Show Point Coordinates' button used to, well you guessed it ... show coordinate points but it also colours incorrect points red for clarity. When you want to change the equation use the "clear" button to declutter.

When we're ready to move onto perpendicular lines I'd click the line tool, select an integer point on the line and then invite a student to select a second point such that the line drawn will be perpendicular - I'd check the gradient using the slope tool ask; Is their a link between the gradients of the two lines? 

                

Time to introduce another sketch; 

I'd explain to the students that this time I have defined the red line such that it’s always perpendicular to the blue line. I'd demo this by changing m₁. This sketch uses the spreadsheet view of GeoGebra to record the gradients of both lines so that the relationship between them can be examined. To toggle record to spreadsheet 'on/off', click both of the grey/red buttons at the top of columns one and two.

I like to start from m₁ = 1, record up until m₁ = 8 then toggle record off. I'll then ask the students to look for a rule connecting m₁ and m_2. Next I'd toggle record off and move the gradient slider to -1 and then toggle on and record through to m₁ = -8; does your rule hold for negative gradients? A further more challenging question is when won't this rule work and why?

Once you have the rules for parallel and perpendicular lines established you can explore a range of problems problems;

Can you give me the gradient of a line perpendicular to:

$y = 4x + 7$

$y = -\frac{x}{2} -2$

$y = -\frac{3x}{5}$

To show the last two lines using the sliders you will have to first change the increment setting for them. Right click on the slider and click Object Proprieties. You then have the option to change the increment. I originally set it as 1 to avoid generating an excessive number of values when recording, if you make it too fine its difficult to control using a small slider, 0.1 works well for what we are doing here. Incidentally I usually explain what I am doing to students when building or editing a sketch on the fly - I want them to see that the models we are using are dynamic and it's not just a series of pre-programmed tricks I am showing them. Often I will have to tweak something in a sketch or build a new one to respond to a question from a pupil.

Can you give me the equation of a line that is perpendicular to: 

$y=\frac{2x}{5}+2$

How many answers are there to this question?

Ok then how about the equation of a line that is perpendicular to the line 

$y=\frac{x}{2}+4$ 

that passes through the point (0,-3) ... How many answers are there to this question?

What about one that passes through the point (1,2).

As students propose answers to these questions I will use GeoGebra to check them at the board. You will notice that I have the sketch set-up to show the equations in decimal form (it is much more of a faff to show them with neat fractions embedded). I like to present the equations in the questions as fractions as this is an interesting discussion point and they understand that they can be displayed in both forms.

As always and comments, further ideas or questions are most welcome.