Here’s a nice summer-days math project …
I just happened to be looking at the NM Highway signs page online a couple of days ago when I saw this nice little list of signs, just below:
NM Highway Signs
I couldn’t help but notice that there are quite a few recognizable geometric figures on this page, and I thought, “This would be a cool thing to show to kids who either have studied, or are studying geometry.”
My suggestion: Show this to your children and ask them how many geometric figures they can recognize.
Math in the movies … if there ever was a cool way to explore math, this has to be it. And if you missed my earlier posts on this, check them out here and here.
I was looking through the links to movies with math themes, and a question came up.
On the site showing the movies, the text says that there are “mathematical themes and patterns motivated by math” in the introduction scene for the James Bond movie, Casino Royale, this clip:
I’ve watched the clip a few times, and I have my own ideas as to mathematical themes and patterns.
Would you say that the square root of two is an important number in math? Hmmm … and would you agree that the square root of three, while perhaps not quite so important, is still a quantity whose value students should be able to estimate?
Why not, right? After all, these numbers play key roles in the 30-60-90 and 45-45-90 “special triangles.” And therefore they both appear a lot in geometry, and a great deal in trig. And on top of that, root two, widely believed to be the first irrational number discovered, shows up in a wide range of other math contexts as well.
square root of 2 w/ "parent" triangle
Answer to problem about the circular and square pegs and holes.
The “fit” for each situation is the following ratio:
(Area of Inner Figure) ÷ (Area of Outer Figure)
For the square peg in a round hole —
Call the radius of the circle r.
Then the diagonal of square “peg” = 2r
Notice that by slicing the square along its diagonal,
we get a 45-45-90 triangle, with the diagonal being
the hypotenuse and the sides being the two equal legs.
Using the proportions in a 45-45-90 triangle,
side of square peg = r times the square root of 2
Multiplying this side of the square by itself gives
us the area of the square, which comes out as:
2 times the radius squared
This being the case,
Area of square is: 2 times radius squared, and
Area of circle is: Pi times radius squared, and so …
Cancelling the value of the radius squared, we get:
Ratio of (Area of square) to (Area of circle) is:
2÷Pi = 0.6366
For the round peg in a square hole —
Call radius of the circle r.
And since the diameter of the circle is the same length as
the side of the square, the side of the square = 2r
Multiplying the side of the square by itself to get the
area of the square, we find that the area of the square
is given by: 4 times radius squared.
This being the case,
Area of circle is: Pi times radius squared
Area of square is: 4 times radius squared, and so …
Ratio of (Area of circle) to (Area of square) is therefore:
Pi ÷ 4 = 0.7854
Of the two ratios, the ratio of the circular peg in a square hole
is greater than that of the square peg in a circular hole.
Therefore we can say that the circular peg in a square hole
provides a better fit than a square peg in a circular hole.
And that is the answer!
From time to time I will post interesting math problems.
Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit!
I will post the answer to the problems two days later, after people have had time to respond.
To post your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.
The problem: Which provides the fuller fit? Putting a circular peg in a square hole, or putting a square peg in a circular hole? To get credit, show all work, and justify your answer by expressing each “fit” as a percent.
A few term-clarifications, to help you do this correctly:
a) By “fit,” I mean the ratio of the smaller shape to the larger shape, expressed as a percent. For
example, if a ratio is 4 to 5, that would represent a “fit” of 80 percent.
b) For the circular peg in the square hole, assume that the diameter of the circle equals the side of the
square. For the square peg in a circular hole, assume that the diameter of the circle equals the diagonal of the square.
c) By “fuller fit,” I mean the larger of the two ratios.
What provides a better fit? A square peg in a round hole, or a round peg in a square hole? How well does each peg fit in its respective hole?
To provide a complete answer, express the “fit” of each plug as a percentage. To find
this percentage, divide the area of the peg by the area of the hole.
Saying that one of these creates a “better fit” means that the peg fills up a greater percentage
To get credit, you must not only provide the correct.
answer, you must also show your work.
Send answers to: info@SingingTurtle.com
Make the subject POTM.