Posts tagged ‘geometry’

4 Folds, 40 Theorems, and Chinese New Year

If you asked a mathematician how to make an origami coin purse for Chinese New Year, you might receive the following instructions:

  1. Rotate a square piece of paper 45°.
  2. Reflect an isosceles right triangle about a diagonal of the square so that congruent isosceles right triangles coincide.
  3. Reflect the lower left vertex of the isosceles right triangle so it coincides with the opposite leg, such that the base of the isosceles right triangle and one side of the newly formed isosceles acute triangle are parallel.
  4. Reflect the other vertex of the isosceles right triangle to coincide with a vertex of the isosceles triangle formed in the previous step.
  5. Fold down a smaller isosceles right triangle from the top.
  6. Fill the coin purse with various denominations of currency, such as a $2 bill, a $1 gold coin, a quarter, or a penny.

Don’t like those directions? Perhaps these pictures will help.

Coin Purse Instructions
To be truly authentic for Chinese New Year, you’ll want to use red paper instead of white.

The Common Core State Standards for Mathematics (CCSSM) have added rigor to geometry by requiring students to prove no fewer than a dozen different theorems. Further, the geometry progressions document formulated by Dr. Hung-Hsi Wu suggests that a high school geometry course should use a transformational approach, basing a majority of the foundational axioms on rotations, translations, and reflections.

What struck me about the origami coin purse was the myriad geometry concepts that could be investigated during this simple paper-folding activity. Certainly, the idea to use origami in geometry class in nothing new, but this one seems to have a particularly high return on investment. Creating the coin purse requires only four folds, and though perhaps I lied by claiming forty theorems in the title, the number of axioms, definitions, and theorems that can be explored is fairly high. (Claiming “forty theorems” wasn’t so much untruthful as much as it was archaic. The word forty has been used to represent a large but unknown quantity, especially in religious stories or folk tales, as in Ali Baba and the 40 Thieves, it rained “40 days and 40 nights,” or Moses and the Israelites wandered the desert for 40 years.)

For instance, the fold in Step 3 requires parallel lines, and the resulting figures can be used to demonstrate and prove that parallel lines cut by a transversal result in alternate interior angles that are congruent.

Alternate Interior Angles

No fewer than four similar isosceles right triangles are formed, and many congruent isosceles right triangles result.

The folded triangle in the image above is an isosceles acute triangle; prove that it’s isosceles. And if you unfold it from its current location, the triangle’s previous location and the unfolded triangle together form a parallelogram; prove it.

The final configuration contains a pair of congruent right triangles; you can use side-angle-side to prove that they’re congruent.

Isosceles Right Triangles

But maybe you’re not into proving things. That’s all right. The following are all of the geometric terms that can be used when describing some facet of this magnificent creation.

  • Parallel
  • Congruent
  • Similar
  • Alternate
  • Angle
  • Square
  • Reflection
  • Rotation
  • Translation
  • Transversal
  • Bisect
  • Coincide
  • Right Angle
  • Perpendicular
  • Isosceles
  • Vertex
  • Triangle
  • Trapezoid
  • Parallelogram
  • Pentagon

And the following theorems and axioms can be proven or exemplified with this activity.

  • Sum of angles of a triangle is  180°.
  • Alternate interior angles formed by a transversal are congruent.
  • Any quadrilateral with four congruent sides is a parallelogram.
  • Congruent angles are formed by an angle bisector.
  • Triangles with two sides and the included angle congruent are congruent.

There is certainly more to be found. What do you see?

February 5, 2016 at 11:50 am Leave a comment

What’s Your Problem?

Problems in the MathCounts School Handbook are presented “shotgun style,” that is, a geometry problem precedes a logic puzzle and follows a probability question. (I worked for MathCounts for seven years and then served as a writer and chair of their Question Writing Committee, so I’m not unbiased.)

By comparison, textbooks often present 50 exercises on the same topic, each one only minimally different from the previous one. That tips the hand to students, methinks, and makes them realize, “Oh, I just need to do the same thing.” I prefer the MathCounts approach, where students have to dig into their bag of tricks to find a viable solution strategy.

With that in mind, here are a few problems I’ve encountered recently, each one not like the others.

Problem 1. The simple polygon is made from 73 squares, connected at their sides. What is the perimeter of the figure?

Thing Puzzle

A simple polygon made from 73 unit squares.

Problem 2. What is the expected number of times that a six-sided die must be rolled to get each number 1–6?

Problem 3. A wall is to be constructed from 2 x 1 bricks (that is, bricks that are twice as long in one direction as the other). A strong wall must have no fault lines; that is, it should have no horizontal or vertical lines that cut entirely through a configuration, dividing it into two pieces. What is the minimum size of a wall with no fault lines? The figure below shows a 3 × 4 wall that has both horizontal and vertical fault lines.

Fault Line - 3 x 4 Wall

A 3 × 4 wall with a fault line.

Please share great problems you’ve recently encountered in the Comments.


No answers, but here are some hints.

Problem 1. Look for a pattern.

Problem 2. Check out this simulation for the Cereal Box problem.

Problem 3. The smallest arrangement without a fault line is larger than 3 × 4 and smaller than 10 × 10.

December 1, 2014 at 6:54 am Leave a comment

Questions and Answers

I’ve been teaching my sons how to tell time on an analog clock. The following is a recent conversation:

Me: The little hand is the hour hand, and the big hand is the minute hand.
Eli: What’s the third hand for?
Me: That’s the second hand.
Alex: Why is the third hand called the second hand?

I had no idea how to respond.Fibonacci Clock

Here are some questions that I’ve been asked recently for which I did know to respond…

How good are you at algebra?
Vary able.

Why is simplifying a fraction like powdering your nose?
It improves appearance without changing the value.

What do you get when you cross geometry with McDonald’s?
A plane cheeseburger.

June 6, 2011 at 1:19 pm 3 comments

Observations

Some things I’ve noticed…

  • Algebra is x sighting.
  • Rational people are partial to fractions.
  • Geometricians like angles… to a degree.
  • Vectors can be ‘arrowing.
  • Calculus teachers can go on and on about sequences.
  • Translations are shifty.
  • Complex numbers are unreal.
  • Most people’s feelings about integers are positive.
  • On average, people are mean.

May 31, 2011 at 1:17 pm 2 comments

Polyhedra Play Things

This morning at the Battlefields of Northern Virginia Council of Teachers of Mathematics (BNVCTM) spring conference, I had the pleasure of meeting Art Stoner of A+ Compass. Art was wearing a t-shirt with the following joke:

ALGEBRA: It’s only fun till someone loses an i.

Art introduced me to Magformers, winner of the 2007 Oppenheim Toy Portfolio Platinum Award. Magformers are plastic polygons with small magnets along the middle of each edge, which can be used to make Archimedean solids, Johnson solids, houses, tanks, stars, and various other polyhedra.Magformers

I bought a set of 18 squares and 20 triangles. If my sons like them, I’ll buy more. (I know, I know… who am I kidding? Whether I buy more depends mostly on whether I like them.)

Talking about geometric objects reminds me of two geometry jokes…

  • Without geometry, life would be pointless.
  • Think outside the tesseract.

April 9, 2011 at 5:20 pm Leave a comment


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The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

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