Question

Any intuitive examples to convince an 8th grader that x + x^2 can be approximated to :

1) x for small values of x
2) x^2 for large values of x

?
I have tried comparing graphs and table values, but nothing seems to get into her head...

Answer 1

How many lines does she see in this picture? https://www.desmos.com/calculator/wkjqtktqlb

Answer 2

How many parabolas does she see in this picture? https://www.desmos.com/calculator/wmurfattvi

Answer 3

I have tried that and the table thingy
http://www.wolframalpha.com/input/?i=Table%5Bx%2Bx%5E2,+%7Bx,0,0.01,0.001%7D%5D

she says she understands, but when i ask relevant questions she blanks out

Answer 4

Maybe she's lying cause she's embarrassed. Maybe she doesn't know how to graph it, doesn't know what an exponent means, or she's rote memorized some values of functions, maybe she doesn't know how to draw a graph herself, or some other kinda fundamental thing that she's never had to apply in this kinda way before.

Answer 5

I think she has a real trouble seeing how at x=1 the value of y=x^2 changes suddenly from decreasing to increasing compared to y = x

Answer 6

so i though i'd spend some time on this myself and get back to her with the magic words that makes her see whats going on

Answer 7

You might want to show her that when

0 < x < 1

x^2 is smaller than x

and when x > 1

x^2 is greater than x

Answer 8

Yeah any clever ideas on how to teach how/why x^2 behaves like that in 0<x<1

I have percentages and few other things in mind

Answer 9

Use fractions

\[x = \frac{1}{10} = 0.1\]

\[x^2 = \left(\frac{1}{10}\right)^2 =\frac{1}{100} = 0.01\]

Answer 10

Using fractions is easier to show how the number is squared, and then you can convert it to decimal to show that 1/x^2 is indeed smaller than 1/x.

Answer 11

Nice, squaring seems to change the units from length to area

Answer 12

@Kainui she knows coordinate grid and all that but gets confused easily with graphs

Answer 13

She can plot a given point on the coordinate grid, but can't draw graphs yet

Answer 14

Drawing graphs is all about connecting the points

Answer 15

Maybe she thinks that there are only integers "on the graph" and doesn't understand that you can plug in all the points and the grid lines are just convenient markers for some convenient points.

I'd be curious to see what she does if you relabel the coordinates on the x-axis with 0, 1/3, 2/3, 1, 4/3,... and see what she labels the y-coordinates of her graph if she's left to do it on her own

Answer 16

Completely agree; she needs to play with fractions a lot to see why x^2 behaves like that in (0, 1).

Answer 17

I'm thinking of stressing on below examples

1) taking half of a half cake gives you a small piece
2) taking twice of a two cakes give you a large cake

Answer 18

very nice examples :)

Answer 19

thinking of convincing her with simple examples of her level.. i really don't want to teach her too many new things at this point haha

Answer 20

Well, zooming into a graph until the graph looks straight, I think is a very intuitive way to show the first order Taylor polynomial approximation for any small neighbourhood of any \(x_0\)... although this probably isn't what you are asking

Answer 21

The best way to show that the \(x^2\) term dominates when \(x\) large, is to either give examples with some really, really big values of x... this will clearly show the domination of the \(x^2\) term and the insignificance of the \(x\) term. Another way is to zoom out from a graph, lots and lots, and then try graphing both \(x^2+x\) and \(x^2\) and you'd notice almost no difference at all.

Answer 22

there is something you should be careful about, that x^2 is not decreasing in the interval 0<x<1, its increasing slowly that is all...
1/3-->1/9
1/2 -->1/4
2/3-->4/9
.....ect