Can a math whiz explain this to us?

[Julie in MN]

Is there a math whiz around here who can explain this to us? I tried getting answers at various math sites online but they all want to tell me "how" and we want to know "why"?!

 

Why do you have to un-cube a cube-to-cube conversion?

 

Say we are converting "meters cubed" to "centimeters cubed"...

So there are 100 centimeters in a meter.

But if you are doing cubes, then you multiply by a million because you have to do the 100 cubed (100 x 100 x 100 = 1,000,000).

 

However, both my son and I are stuck on the idea that the centimeters are ALSO CUBED so why would you "un-cube" the meters by multiplying them out, when your answer also needs to be cubed? I mean, doesn't the little 3 there *imply* 100 x 100 x 100? ----this is the part that we aren't seeing----

 

I'm sure there's something I'm not thinking of here, because no one else seems to have this question (lots of sites ask how to convert but no one seems to be confused about why it works that way).

 

I was looking at this visual

and something started moving in my brain... maybe you *would* need a million centimeters cubed. But then... nothing. I couldn't explain it because my brain just didn't quite go all the way there.

 

Julie

Edited October 11, 2010 by Julie in MN

[kiana]

If you want visuals, it might be easier to look at it in two dimensions rather than three, and with smaller numbers.

 

For example, if you draw '1 square meter', each side is 100 centimeters, right? But the area of the square is length*width = 100cm*100cm=10000cm^2. The area of the square is also 1m^2.

 

In a more algebraic way, if you have 1m^3, think of this as 1*m*m*m. You need to convert *each* of those meters to centimeters.

 

This is one reason why I prefer to think of unit conversions as fractions. We know that there are 100 centimeters in a meter, so we can write that as 100cm/1m. When we multiply (1*m*m*m)*(100cm/1m), we only get to cancel one of the 'm's, so we're left with meters^2 * cm instead of cm^3. In order to cancel all of the 'm's, we need to multiply by (100cm/1m)*(100cm/1m)*(100cm/1m).

[regentrude]

A cubic meter is a cube of side length 1 m. So measure out three feet and imagine the cube.

A cubic centimeter is about the size of a sugar cube.

It will take 100 sugar cubes to line one EDGE of the big cube.

It will take 100*100 sugar cubes to line the bottom surface of the big cube.

It will take 100 layers, stacked on top of each other, of 100x100 sugar cubes to fill the volume of the big cube:

100*100*100 sugar cubes (cubic centimeters) equals the big cube (cubic meter)

Edited October 11, 2010 by regentrude

[Julie in MN]

Okay, my son has testing this morning so I may not get back to you right away, but I think you are helping! Sugar is definitely something I can visualize, too :)

[Julie in MN]

A cubic meter is a cube of side length 1 m. So measure out three feet and imagine the cube.

A cubic centimeter is about the size of a sugar cube.

It will take 100 sugar cubes to line one EDGE of the big cube.

It will take 100*100 sugar cubes to line the bottom surface of the big cube.

It will take 100 layers, stacked on top of each other, of 100x100 sugar cubes to fill the volume of the big cube:

100*100*100 sugar cubes (cubic centimeters) equals the big cube (cubic meter)

 

Okay, I think I can visualize this. It doesn't seem so much like "un-cubing" as just stacking the cubes in all the dimensions.

 

I will try this on ds tomorrow. I have to say a big THANK YOU tonight, though!

 

Julie