Comparing rectangles by their area and by their diagonal

The other day I was in someone’s office waiting for some paperwork to be completed.
It was a small room, about 12 feet by 12 feet.
The back wall was orange, and its lower part was horizontally covered by a desk.
In the middle of the orange wall there was a picture of a green car. I thought that was an interesting color combination.
Looking at the picture on the wall, I wondered how bigger the visible part of the wall (including the picture) was, in area, in relation to the area of the picture.
The horizontal length of the wall was about two-and-a-half times the length of the picture, while the hight of the visible part of the wall was about three times the height of the picture.
Putting

b = length of the picture
h = height of the picture
a = area of the picture
B = length of the wall
H = height of the visible part of the wall
A = area of the visible part of the wall (including the picture)

we have, approximately

B = 5b/2, and H = 3h

therefore

A = BH = (5b/2)(3h) = 15bh/2 = 15a/2

So the proportion between those two areas is 15/2, 0r 7.5

Then I wondered what the proportion would be between the diagonals of those same two rectangles. As it turns out, when the two rectangles (the wall and the picture) do not have the same shape, the proportion between their diagonals depends not only on how many times the sides of the wall are on relation to the sides of the picture, but on the precise shape of the reference rectangle, as well.
If we use

d = measure of the picture’s diagonal
D = measure of the diagonal of the visible part of the wall

Then we have

D = sqrt(B² + H²)
 = sqrt((5b/2)² + (3h)²)
 = sqrt(25b²/4 + 9h²)

while

d = sqrt(b² + h²)

The coefficients of b² and h² inside the square root in the expression for D are not the same, so we cannot simplify that expression completely just in terms of d.

If we introduce a new constant, k, to represent how many times the height of the picture equals its length, then we can write

b = kh

In that case, the diagonals of the two rectangles can be expressed as

d = sqrt((kh)² + h²)  =
 = sqrt(k²•h² + h²) = 
 = h•sqrt(k² + 1)

D = sqrt(25(kh)²/4+ 9h²) = 
 = sqrt(25k²h²/4 + 9h²) = 
 = h•sqrt(25k²/4 + 9)

This shows the proportion between D and d being dependent on k, the proportion between h and b, because

D/d = sqrt(25k²/4 + 9) / sqrt(k² + 1) 

and this is a function of k that is not constant.

The Square of a Sum

An arithmetic and geometric approach to Algebra

A very common mistake test takers make when they have been out of school for a while is that they automatically try to expand the square of a sum as if it was the same as the plain sum of the squares of the individual terms. For example, sometimes some students, when presented with say, (x + 5)², they wrongly expand it as x² + 25, leaving out the middle term 10x.
They forget they have to use the foil method to multiply the given sum times itself. Since typographically the sum of the squares looks like something that could equal the square of the sum, they guess they can expand the square that way, as if it was a real math rule, and then they get the problem wrong.
I try to preemptively address the possibility of this mistake, because both sums of squares and squares of sums show up a lot in the math section of standardized tests. For a long time I have used a drawing, with two little squares and two congruent rectangles filling up a larger square, to show the geometric interpretation (in terms of area) for the algebraic formula (a + b)² = a² + 2ab + b². I emphasize the middle term, 2ab, in this expansion, and I point to the corresponding two rectangles in the drawing.
The above approach is useful but many students forget the formula when it comes to actually applying it. Most do not realize it is a universal pattern where to plug any other expressions in.
So, lately I have been experimenting with a slightly different approach: I present two or three numerical examples first, before drawing the squares and rectangles. I ask the student to choose two numbers, and then I guide them through a sequence of calculations that allows them to actually compute and see the numerical difference between the square of the sum, and the sum of the squares. Then I ask them to multiply the original numbers, then to double this product. By asking all the questions in the right order, I have gotten comments from them like: “Wow!” or “How weird!” or “That is so funny!” or “This is very interesting!” or “Does it always work like that?” or “How can that be?” or “What is going on?” or “I’m sure there is a pattern here!” Then I draw the squares and rectangles with all the numbers in their proper places. So, creating the perception of a “mystery” with the numbers, and then explaining it away with the picture seems to work well. It makes sense for the students. They usually say: “Oh, right! Now I get it.”