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Week 3 Math Task

Complete the Growing Rectangles activity on pages 237 – 238.
Share your questions, thoughts, sketches, process, mistakes, or
solutions in the discussion board.

Comments

ecappleton's picture

Hi all,

I like the growing rectangles problem because of how open it is. I explored rectangles, triangles, pyramids, cones and spheres. It seems to me that increasing 2-dimensional figure by a scale factor of 2 results in an area 4 times the original.

Here's a 4x5 rectangle scaled up by a factor of 2 to be a 8x10 rectangle:

         
         
         
         

---> scaled up by factor of 2 --->

                   
                   
                   
                   
                   
                   
                   
                   

If the scale factor is k, then the area of the scaled-up figure will be k2 times bigger than the original figure. 

Increasing a 3-dimensional figure by a scale factor of 2 results in a volume 8 times the original. If the scale factor is k, then the volume of the scaled-up figure will be k3 times bigger than the original figure.

This makes sense if we consider that multiplying two dimensions (length and width) times a scale factor k results in (k * length) * (k * width) or k2 * length * width. Multiplying three dimensions (length, width and height) times a scale factor k results in (k * length) * (k * width) * (k * height) or k3​ * length * width * height. I haven't tried enough plane figures and solids to know if these generalizations are always true, but it seems like they would be. Even if we have to multiply by 1/3 to get the volume of a pyramid, the scale factor is still going to be cubed in the volume formula.

Patricia Helmuth's picture

Eric,

I haven’t played around with the 3D figures yet, but I got similar results to what you did when increasing a 2-dimensional figure by a scale factor of 2 – the area would be 4 times that of the original. Also, when increasing by a scale factor of 3 – the area would be 9 times that of the original. At least that’s what I got with what I’ve done so far.

I say this because I was looking for other connections that seemed to work sometimes, but not all the time. For example, if a rectangle has a width of 2 and you increase that rectangle by a scale factor of 2, the enlarged rectangle will have a length that is equal to the area of the smaller rectangle. But that only seems to work with 2. I played with other numbers and didn’t get similar results but maybe I need to expand my research. Will it only work with 2?

Then, I tried it with triangles. I started with a triangle that had a height of 2 and a base of 3 and increased it by a scale factor of 2. I did not find the same connection there that I did with the rectangle.

After that, I played with an isosceles triangle with legs that have a length of 4 and increased that by a scale factor of 2.  I found that the legs of the enlarged triangle (8) were equal to the area of the original triangle! However, if I scaled the original triangle down or the enlarged triangle up, it didn’t work. So again, will this only work if the legs of an isosceles triangle are 4?

 - The ramblings of Patricia

ecappleton's picture

Hi all,

Our math circle looked at Growing Rectangles yesterday. Here's the board work and a link to a writeup of the meeting.

-Eric

http://nyccami.org/growing-rectangles/

 

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