I've always enjoyed showing students how to derive the quadratic formula. I don't test them on it, so the stress level is lower. And it's late in the term, so they appreciate a break from the pressure, and most really do try to get it. I get a few making those appreciative sounds that happen when the lightbulb goes on, and that makes it especially fun.
But it's hard slogging through some of the weird steps. Here's the standard derivation, if you haven't done it in a while. Check it out, and imagine trying to explain it to people who are pretty fragile around math.
So the math education gods were smiling on me last week, and the day before I brought this topic to my students, I interviewed James Tanton, who (out of the blue) showed us his twist on this. (Thanks, James, for helping me with my lesson plans!)
If you just can't find the time to watch the video, it goes something like this:
But it's hard slogging through some of the weird steps. Here's the standard derivation, if you haven't done it in a while. Check it out, and imagine trying to explain it to people who are pretty fragile around math.
So the math education gods were smiling on me last week, and the day before I brought this topic to my students, I interviewed James Tanton, who (out of the blue) showed us his twist on this. (Thanks, James, for helping me with my lesson plans!)
If you just can't find the time to watch the video, it goes something like this:
ax2+bx+c=0
We want a perfect square in the first term,
so we multiply both sides by a: a2x2+abx+ac=0
We want the second term to have a factor of 2, and to keep the first term a perfect square,
so we multiply both sides by 4: 4a2x2+4abx+4ac=0
We almost have what we see in the box above, but we want b2 and not 4ac,
so we do a little adding and subtracting: 4a2x2+4abx+b2 = b2-4ac
Now factor the left side:
(2ax+b)2 = b2-4acTaking the square root of both sides steals away a solution, so we include a plus or minus:
2ax+b = �v b2-4ac Subtract b from both sides and divide both sides by 2a, and you've got it. Much prettier than the standard derivation, I think.
I used this in class last Thursday, and I think it went much more smoothly than the standard version. I always do it twice, so I did it again on Monday. Students said they got it, and some liked it. I haven't spent enough time on completing the square (our days together are numbered, at this point in the term), so I don't expect their understanding goes very deep, but it's a start.
I'll do this again next semester in my intermediate algebra course, with a better grounding in completing the square. I look forward to it. Maybe all our conics problems will be easier, with James Tanton's brilliant help.
I used this in class last Thursday, and I think it went much more smoothly than the standard version. I always do it twice, so I did it again on Monday. Students said they got it, and some liked it. I haven't spent enough time on completing the square (our days together are numbered, at this point in the term), so I don't expect their understanding goes very deep, but it's a start.
I'll do this again next semester in my intermediate algebra course, with a better grounding in completing the square. I look forward to it. Maybe all our conics problems will be easier, with James Tanton's brilliant help.
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