Explain to an eighth grader...?

Question

Here's a conversation:
"How do you find how much stuff is under a graph?"
"You mean, an area under a curve - a function"
"Ye"
"Well, you pretend there are lots of rectangles underneath the curve, and add their areas. You'll notice that the more rectangles I have, the closer I am getting to a limit, a determined value" ... But after that, I just loose him. I try to explain how to get to the basic intg procedure, I even let him try out a mathematica simulation... Anyone know how I can explain the concept of the integral to an eighth grader - with really simple math? He /at least/ understands the concept of Riemman sums, but where do I go from there to reimman integrals and all the other fun stuff without supplementing summations and limits ?

Answer 1

If he doesn't understand what you tried to explain to him, he is just not ready to learn these advanced topics. Math all builds on itself, and one must have a nearly perfect foundation in algebra, geometry, etc. in order to learn advanced topics such as calculus.

I think my high school calculus teacher put it the best: "At some point, we all reach our mathematical maturity." Whoever you are teaching will reach a point where he has a good enough basis in the fundamental concepts of math that he can learn about the area under a curve. Your explanation was simple, concise, and fairly accurate. If he doesn't understand it now, give him some time.

As an aside, if he understands Riemann sums, he must understand the area under a curve. Something is not right about what you said. Are you trying to teach this kid real analysis or something?

Good luck.

Answer 2

My approach would be to use two sets of rectangles like so.

Draw your curve (I would use a parabola opening downwards) so it intersects the x-axis a two widely spaced points (you need area to work with). From these two point draw two rectangles, one that has its top tangent to the top of the parabola --and thus has area both above and under the curve; the second should be completely under the parabola --and thus omits part of the area under the parabola. Find the area of each. Note their difference.

Then, ask, "Is the area under the parabola less than the large rectangle, and larger than the smaller one?"

Yes, it is between the two.

OK, lets use 4 rectangles.

Now, you use your method to subdivide each set of rectangles noting the areas they cover and the difference in their areas. You gently use the "sigma" to represent the sums. One sigma is the upper rectangle set, and the other sigma is the lower (or inner) rectangle set. Their difference is just SigmaUpper - SigmaLower. That difference will get smaller and smaller as the number of rectangles increases. Yet, the area under the curve is always between the two. It is always true that:

SigmaLower < Curve Area < SigmaUpper

So, then, when do we have the area under the curve? When we have so many rectangles that their difference (upper ones minus the lower ones) can be made as small as we wish. We get the area under the curve by squeezing it between two areas which we can always compute and which get ever closer to each other.

That lets you move from the sigma notation for summation, to the ess curve of the integral notation.

Well, that would be my first try. Probably won't work! :)

HTH

Charles

Answer 3

You don't need to know anything besides basic algebra to understand calculus.

Start off with derivatives and talk straight to integrals.

Just say an integral is the area under a curve and is an antiderivative.

Derivatives aren't all that hard of a concept at the beginning, so you can probably give him a sip of knowledge from there and skip the rest about derivatives.

Honestly, I didn't learn integrals with limits or the summation. I just knew it's the opposite of a derivative =D.

If you don't want to get into limits with derivatives it'll be extremely hard. It wouldn't be hard to teach limits anyway. Only a small amount and you can bring him to the so called "fun stuff."

Start with LINES. Curves will just make it harder to teach him.
_____________________________

I'd be interested to see him =D.

Answer 4

well if the graph is a curve i have noidea how to do it simply, im going in yr 11 and we just did that in adv maths. but if its a straight slope graph just pretend its a triangle with the formula
A=1/2 lb
OR
Area=(length x breath )(divided by) 2

wich for a graph would be

Area under= (rise x run) divided by 2

Answer 5

Over tax wealthy whites to grant to destructive welfare recipients-- Spend everydime we've on socilaistic nazis ideals-- start up a clean Civil conflict by using taxing us without representation-- placed all small companies out of employer willhe bails out coorupt companies--his frineds in low places-- ruin our great usa!! he's a nightmare waitng to ensue--God help u.s. now. with BO on the helm.

Answer 6

im an 8th grader and 8th graders do pre algebra and algebra.not that stuff. we are dumb.