INforM – Interactive Notebooks for Mathematics
Book 8 – Straight line graphs through the origin
Straight line graphs through the origin
The activities are related to the work in Ma2 Number and algebra:
Sequence, functions and graphs
6 Pupils should be taught to:
Functions
e use the conventions for coordinates in the plane; plot points in all four quadrants; recognise (when values are given for m and c) that equations of the form y = mx + c correspond to straight-line graphs in the coordinate plane; plot graphs of functions in which y is given explicitly in terms of x [for example, y = 2x + 3 ], or implicitly [for example, x + y = 7 ].
The SMART Notebook file is designed to support lesson materials which are available for download from the Practical Support Pack area of the DfES’ TeacherNet website at:
http://www.teachernet.gov.uk/supportpack/module.aspx?t=2&s=10&y=36&p=&m=18911
These materials include support for the use of graphical calculators – but since they were written, there is now a free web-based resource designed by the Mathematical Association and programmed in Flash by the Skoool team at Intel. This Mathematical Toolkit won the BETT 2006 award for Key Stage 3/4 mathematics. You and your pupils can download it from the Skoool area of the London Grid for Learning at:
http://lgfl.skoool.co.uk/content.aspx?id=657
The Mathematical Toolkit is now included in the Mathematics resource area in the latest Gallery for SMART Notebook at: http://education.smarttech.com/ste/en-US/Ed+Resource/Software+Resources/Notebook+collections/Mathematics/General+resources/Mathematical+Toolkit+-+interactivity.htm
Organisation of the materials
The Smart Notebook file is saved as `straight line graphs.notebook’.
It consists of 9 pages of which the first is the title page, shown above.
There are 7 pages to support the activity and its extension. Page 8 is a blank page.
Page 9 contains teacher notes which are amplified here.
The first activity
The background is a screen-grab from the Coordinates and Graphing area of the Toolkit.
The image below shows the layout of Page 2. There are three points already created in the graphing area to the left of the screen. Their coordinates are shown in the table to the right of the Toolkit area. At the extreme right of the board are three coloured line segments and a variety of possible equations for lines.
Ask pupils to imagine three lines through the origin, each passing through one of the three points. Which point will lie on the green line, the orange line, the blue line? Ask them for some words to describe how the shapes of the three lines differ. Drag the line segments from the edge of the board to join the origin to each of the three points.
The coordinates of point1 are (2,4) – which is the x-coordinate and which is the y-coordinate? Which of the 10 equations are satisfied by x=2 and y=4? Repeat for points 2 and 3. Drag the equations onto the graph-paper to label the line segments. Use the line tool, or draw free-hand lines, to make the point that the graph is a line which extends to infinity. Ask for the co-ordinates of 3 more points which lie on the orange line. Repeat for the other lines.
The main task
Page 3 of the notebook carries on from the starting activity. Each of the three bold coloured segments is now shown as forming the hypotenuse of a right-angled triangle. The horizontal (x) distance from (0,0) to each point is called “across” and the vertical (y) distance is called “up”. Make sure that pupils are aware that it is the sign of the “up” measurement which determines whether the segment slopes upwards or downwards.
The space to the right of the board is there for you to record the “up” and “across” values for each of the three points. Explain how the slope is calculated as “up” divided by “across” and represented by a signed decimal number, or equivalent fraction. Ask pupils to calculate the slopes – and where sensible to give both the decimal and fractional form.
Now move to page 4 which shows the equations of the three graphs plotted on the graph-paper. The table shows values for the third, green, graph through point3. Explain how the slopes just calculated are related to these equations.
Now move to Page 5. You will now see the Macromedia Flash player running the live version of the Mathematical Toolkit. Click on the Coordinates and Graphing icon at the top of the screen, and you will be presented with a dialogue box to set up the axes for the graphing area. Just click the Draw Grid button to use the default ranges from -10 to 10 on each axis. Then click on the Proceed button to get started.
Now you can enter points of your own on the grid. As you move your pen, or finger, or mouse over the graphing area the X: and Y: entries in the Coordinates column change. When you click these, are entered in the table under the Graphing tab and a point is labelled on the grid. You can’t move the point by dragging it, but you can edit the X and/or Y values in the table and then click on Update Points.
When you have plotted some points you can click on the Graphing tab to enter equations of the form y = mx for lines to plot through the origin.
Of course you can use this page for doing any kind of graphing – so, for example, you can extend the ideas to plotting graphs of the form y = mx + c.
The extension task
Using the ideas and tools met in this Notebook so far, it is easy to set up an extension activity to explore the relationship between the slopes of two lines through the origin which are at right-angles (perpendicular) to each other.
Open page 6, and proceed as before to start the Coordinates and Graphing with the standard graph-paper. You will now see that the Notebook tools have been used to drag some lines and text over the graphing area. But no points or equations have yet been entered – the Toolkit is now ready for you to use in whatever way you choose!
Finally page 7 just uses an image of the graph-paper for you to set up an exploration of the particular case of B (4,8) and B′ (-8,4) – with two right-angled triangles shown. So you can explain how the “ups” and “acrosses” are related by seeing, for example, that the orange triangle is a rotation of the blue triangle through 90˚ about the origin O (0,0).
SMART specific issues
Page 9 (after the blank page 8) refers to the sources of useful supporting materials. The lesson plan from the TeacherNet site is attached to the Notebook as the file: `straight_line_graphs.doc’. The cropped screen-grab of the graphing region of Toolkit is also attached as `graphpaper.jpg’.
You should note that when you are using the live flash Toolkit you cannot use the pens to draw or write over the part of the board used by the Toolkit. This is because the Toolkit software assumes you are using the pen (or finger, or mouse) to interact with software. If you want to annotate on top of graphs it is best to resize the active flash area so that you have room to create some text or lines at the side or below the area. Then you can drag them over the area and use the Select tool to change their properties and to lock them in place.
INforM Page 5 of 6 August 2006