Tuesday, June 09, 2009

Math Questions

What image pops in your head when you see the following?



What image pops in your head when you see 2 x 3?


Can you give a concrete example of how 1/2 x 1/3 might look?

12 comments:

Susan said...

First thought that comes to mind is a third of a pie (circle) cut in half to make a slice that's 1/6.

I had a teacher who always phrased these questions like "1/2 • 1/3" as "1/2 of 1/3". I find that phrasing very useful for visualizing things.

Also useful was this analogy:

"1/2 • 3 = 3/2"

Hope that makes sense.

The Glasers said...

Susan, you had an excellent math teacher. In my mind, I use a ribbon to demonstrate the same idea!"

I had some interesting math conversations and I am trying to see if people view math conceptually or simply memorized and forgot what they learned!

In the world, not of it said...

The image that pops in my head is the answer. . . what does that say about me?

theresa said...

OK, at first it didn't occur to me that the image for X squared would have anything to do with something squared... I just thought of it as some sort of general "mental associations" question.

For 2X3, no image pops into my head except the written number that is the answer, although I understand that two sets of three items would be the "right" answer.

Realizing what this exercise was about, I did pretty easily start to visualize the last one as a half of an amount sectioned away from the other half, although I don't think it even occurred to me that I had to visualize the original amount as being one-third "of" something else till I started reading the comments!

Signed, a "word person" who got mediocre grades in high school math along with a reasonably high math SAT score.

Susan said...

I definitely learned math conceptually. Not sure if that is just because that's the way I think or my teachers' or parents' doing. It's also the way that I teach - which is probably what leads to all my frustration! ;)

Steve Glaser said...

x^2 = x * x

2 x 3 = 2 * 2 * 2

1/2 * 1/3 = 0,1666667

Queen Mum said...

i know from talking w/ someone working w/ dd on fractions..that they do NOT visualize...at least at their school. they just teach how to get the answer. ie: the teach procedure... i am not there yet in teaching DS however, when we were doing fractions ..generally...i found myself explaining things visually..ie: we had 1/4 of a piece of ... Read Moreplastic pie...and I said what happens if we take HALF of this... 1/2*1/4. that way when we get to procedure the concept will be there. on dividing..i still can't get my own brain around it. (and then there is FRED who Andrew is loving...maybe that is how fractions should be taught.)

C said...

i get ^ meaning to the power of (when typed). i get 2x3 spread out. but what is the fraction....meaning 1/6 which is .1666667.....is the , instead of the decimal?

P said...

I don't get any of it!

K said...

Question 1 - X squared or x times x.

Question 2 - 2 sets of 3 added to 6
but as the numbers get larger, I rely on memory not sets.

Question 3 - the word concrete has me confused. Concrete verses solving is two different things at least to me regarding fractions. Solving would be
(1/2 x 3)X(1/3 X 2=6/36 divided 6=1/6
since you always bring it back down to the lowest common denominator. I've never got how "concrete" this information is. Probably why I finally had to
quit trig. I'd bluffed through as far as I could go.

Tammy's Mom the Quilter said...

Draws a square with her fingers and says a square!

The Glasers said...

X and 2 or X times X are the typical answers. This is a reflection on how we were taught math. About 30 attended my workshop on math at the ChildLight USA conference. Only ONE person had X squared conceptualized! On other person emailed me privately and she got all the answers.

Anyway, the conceptual way of
viewing x squared is a square with sides of an unknown length that you could call X.

If you line up an array of objects to show 2 x 3, it forms a rectangle. Conceptually, 2 x 3
is a rectangle with length 3 and width 2 . . .

You can think of 1/2 x 1/3 as "1/2 of 1/3". Take a ribbon and cut it in thirds. Grab one of the pieces (which is a third) and cut that in half . . .