## Homework 20 Squares And Scoops Math

We describe a nice way to do it, unfortunately in words. It really needs a picture.

Put down your cardboard rectangle, one corner at the origin, the long side along the positive $x$-axis. So the corners of your cardboard rectangle are at $(0,0)$, $(0,45)$, $(45,20)$, and $(0,20)$.

Draw a $30\times 30$ square, with corners $(0,0)$, $(30,0)$, $(30,30)$, and $(0,30)$.

Draw the line that joins $(0,30)$ to $(45,0)$.

This line will meet the top side of your cardboard rectangle at $P=(15,20)$, and will meet the right side of the square at $Q=(30,10)$. Let $R=(30,0)$ and $S=(45,0)$.

All set up! Use a **razor knife** to cut along the line $PS$. That will slice a substantial triangle from the cardboard. Leave it in place for now.

Use the razor knife to cut straight down along $QR$. This slices off a smallish triangle from the cardboard.

**Slide** the big triangle upward until its top side agrees with the top line of the square. It will.

Slide the little triangle way up so that it fills in the top left corner of the square. It will.

Done, two cuts.

It is a very pretty construction, works uniformly for **all rectangles** that are not too skinny. If the rectangle is very skinny, a not too hard adjustment can be made.

You will have to *prove* that this works. Straight coordinate or similar triangle geometry.

**Remark:** This construction is one of the steps in the proof of the Bolyai-Gerwien Theorem (which, as is so often the case, was proved a number of years earlier by at least two other people). The result is that if $A$ and $B$ are **any** polygonal regions with the same area, then $A$ can be cut into a finite number of polygonal pieces that can be reassembled to maske $B$.

This week’s challenge was suggested by Jack Dieckmann, a math teaching expert at Stanford University. Jack is a core member of the popular YouCubed, a Stanford-based online resource for K-12 math teachers founded by math teaching superstar Jo Boaler and her colleague Cathy Williams. YouCubed’s mission is to help ignite students’ natural enthusiasm for math, which may have been crushed by worksheets, drills and homework.

Homework! That’s something I thought was a good thing. At least a little homework. How else can you really learn? But studies are mixed on the value of doing schoolwork at home, with opponents claiming it produces little, if any, gains in learning at the cost of family time. (For more from this perspective, check out *The Homework Myth* (Ch. 2 here) or Race to Nowhere.).

I visited Jack in his office recently and he shared the following problem, which he thought might be fun to try out on Numberplay. I agreed completely. Let’s try —

#### Lines and Boxes

Here are 10 straight lines and 17 squares.

Here are 9 straight lines and 20 squares.

Find the smallest number of lines needed to make exactly 100 squares.

Once you’ve done this, investigate further.

How many different ways can you make a particular number of squares?

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