Exploring 3-D graphs using Autograph
Math 11 AC
Outcomes: C12 / E1 / E2 / C13
Introduction:
Students will use Autograph to explore the characteristics of planes in 3 dimensions thereby developing a better understanding of planes and their graphical representation.
Prior knowledge:
· Orientation of the x,y,z axes ;
· An equation in three variables describes a plane;
· Graphing of planes using the intercept method.
Projected timeline for this activity:
· One class (75 minutes).
Equipment requirements:
· Grade 11 roll-out laptops with Autograph software installed.
Procedure:
See handout for students (pages 2 and 3).
Assessment:
No formal assessment for this activity. This is a hands-on visual learning experience to help clarify previous classroom instruction on this topic.
Instructions for Students
Step 1: Launch the Autograph program by double-clicking the Autograph Icon on the Desktop.
An X-Y grid should now be on your screen.
Step 2: Click on the “New 3-D Graph Page” icon on the toolbar below Edit.
A box with the x - y - z axes should now be on your screen.
Step 3: With your cursor near or inside the box, hold down the left mouse key and move cursor using the track pad.
What is happening now? ______…
To bring the box back to its original orientation, click on the x-y-z Orientation icon .
Step 4: To label the axes inside the box, click on “Axes” on the Menu Bar and chose “Edit Axes…” on the drop-down menu. The box that opens up allows you to adjust the axes for your graphs. Now click on the “Options” tab and, under the Axes heading, click on the “Always Outside” box to clear it. Now click “OK” to go back to your graph.
Your labels are now on the Axes.
Step 5: You will now draw a Plane by entering an equation. Click on “Equation” on the Menu Bar. Chose “Enter Equation…” from the drop-down menu. Enter the following equation: 2x + 2y – 4z = 4. Click “OK”.
What do you see now? ______…
Rotate the box to see the Plane from various angles.
Step 6: Click on the “x-y-z Orientation” icon to bring your graph back to its original orientation. To display the points of intersection, we will enter the three sets of coordinates by clicking on “Data” on the Menu Bar and choosing “Enter Shape…” from the drop-down menu. In the box that opens up, enter the coordinates for the three intercepts from your equation. Make you click the “Select” box for each of your entries. Now click on the “Add” button to the right of your entries. Also check that the “Triangles” and “Fill Shape” options are chosen. Click “OK”. Does the resulting shape look familiar?
Intersection of two Planes
Step 7: Click on the “New 3D Graph Page” icon . Follow the directions in Step 4 above. Using “Equation” from the Menu Bar, choose “Enter Equation” from the drop-down menu. Enter the following equation: 2x + 2y – 4z = 4. Click “OK”. Repeat the above process to enter equation 2, which is: 6x-3y+4z = 12. You should now see 2 intersecting planes. To highlight the line of intersection, click on one of the planes and notice the colour change. Holding down the “Shift” key, click on the 2nd plane. Move the cursor outside the box and right-click the mouse to open a drop-down menu. On this menu, click on “Intersection Line”. Click outside the box to restore original colours. To make the faint line of intersection more prominent, click on the line (it should turn white), then click on the “Thick Line” icon on the Tool Bar and choose a 4.5 or 6 pt line on the drop-down menu. Now the line of intersection should be more prominent.
Can you find the point of intersection of 3 planes, if you enter this 3rd equation: -2x+3y+3z = 6 ? (Hint: watch the bottom of the screen for the result) J
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