We did this the day after a GeoGebra lab, in which students used Jennifer Larson Silverman's Drawing an Ellipse GeoGebra to experiment with drawing ellipses using virtual string. I had them keep string length constant while moving the endpoints, then vice versa. It was concluded that:
- When you keep the string length constant, the distance between the ends of the string influences the shape but only in the length of the smaller axis - the longer axis remains the same.
- When you keep the string ends constant, changing the length of the string influences the shape of the ellipse in both directions
- the length of the longer axis is always equal to the length of the string.
1. (My voice in bold) Draw an ellipse with major axis length 6 and minor axis length 4.
|(After some minor corrections, and with the names changed of course)|
Do all of these have the right major and minor axis length? Yes How are they the same, how are they different? All are same shape but oriented differently. If you had had to draw these yesterday, would you all have had the same string length? Yes. And end points? No. Julie and Bridget's ends would be vertical. What would be the same for all endpoints? distance between them. If you had to find their rules, would they all be the same? Horizontal ones will all have same rule and it will be different from the vertical ones.
2. Draw the foci in their approximate locations.
What part of yesterday's lab do the foci represent? ends of string What do you notice about the foci? Lots here: horizontal ones have foci on x axis, vert on y, all hor foci should be in exact same locations and at same distance from each other. Vert ones will be at same dist from each other as the hor but on y axis (based on yesterday that moving the string ends changed the ellipse). Everyone's foci have origin as midpoint. Personality bonus - Julie and Bridget think a bit differently than most - and that's cool.
3. Draw any point on your ellipse in purple.
What from yesterday's lab does your purple point represent? pencil tip What do we call Heather's point? Covertex. Anyone else picked a covertex? Julie. And who picked a vertex? Bridget and Bob. (In both classes, someone picked those key points, fortunately. If they hadn't I would have had to draw my own.)
4. Draw the focal radii for your point.
What do the focal radii represent? the string Who has one focal radius longer than the other? Everybo- no wait. everybody except Julie and Heather. What about Julie and Heather? Theirs are the same length. About how long is Susie's short focal radius....and her long one.....hard to tell huh. But how much must they add up to? 6. Because it's the string, and the major axis is the same length as the string. Who else's d1 + d2 add up to 6? Everyone! Even Julie and Bridget? yup! Why? Bc they have major axis length 6 also, so their strings are 6 units long. So that means how long is Julie's d1? and Heather's d1? 3.
5. I drew that on Heather's:
What other side length of the green right triangle do we already know? The leg is 2, because it's the semi-minor axis length. So how far from the origin must this focus be? After Pythagoras-ing - it's 2.2. Who else's focus is 2.2 units from the origin? Julie's. Right. How do you know? Because it's the same ellipse just drawn vertically, so it's string ends were the same distance apart. Right. Anyone else whose foci are 2.2 units from the origin? Long pause. OH! Everyone's! Really?!? How do you know that? Not everyone picked a covertex but! Doesn't matter, all foci are at the same distance from centre.
Next we went to Desmos and did this activity to develop the rule. At the end of that, I asked them to type into Desmos x²/9 + y²/4 = 1 and look at the graph. Does it look familiar? Yup we all just drew it. Everyone? Well no, actually, Julie and Bridget didn't. Hm any guesses what we could type in to get their ellipse, which has the same dimensions but is rotated? They got it. x²/4 + y²/9 = 1.