Thursday, February 14, 2019

of words and frogs

Inspired by Lillian Ho's article on using origami with adult ESL learners (see references below), I decided to build a lesson for high school ESL students around the hopping frog model.

The basic activity went like this:
1. Students were shown how to fold the model without being given any verbal instructions. 
2. Students were given a version of the instructions with all written instructions removed. 
3. In groups of 3 or 4, students were asked to provide written instructions on chart paper to go along with the diagrams. 
4. The written instructions were shared by placing the chart papers up around the class, and students were asked to identify the important words that were used in the instructions. 
5. As a whole class, we folded the frog again, noting the mathematical ways we could describe each step.

If you try an activity along these lines, I expect that should be split over two or three sessions. We did step 1 at the end of another lesson, steps 2 and 3 on a second day, and steps 4 and 5 on a third day.

In thinking about the sort of descriptions that the students should be guided towards, it's helpful to note some of the observations provided in an article by Koichi Tateishi (see references): (1) the written words are not a replacement for the diagrams, but should be thought of as complementary, and (2) we should avoid technical origami terms (mountain fold, squash fold, etc.).

When one group of students uncovered a useful word, it would get written up on the board for all groups to share, so as we went we developed a list of helpful words. With reference to the steps on the instruction page, some words that the students used included or discussed at each step were:

step 1: paper, card, square, rectangle, fold, half, long, short, side, middle, open, line;
step 2: diagonal, top, side, edge;
step 3: do again, repeat, other side;
step 4: turn over, flip, corners, intersection, lines crossing;
step 5: push, pinch, squash, together, peak;
step 6: tip, up;
step 7: center, in, inwards;
step 8: flaps, outside, inside, arms;
step 9: bottom, nose, legs;
step 10: down, knees;

These words were used in the context of providing instructions and directions - a very important type of speech act that students are routinely confronted with.

In the appendix to her article on the benefits of origami lessons for middle school students, Norma Boakes (see references below) provides a great sample mathematical dialog for the jumping frog model (step 5 of our lesson plan). Like Boakes suggests, in our mathematical discussion we talked about rectangles, squares, quadrilaterals, right angles, bisectors, 45 degree angles, triangles, pentagons, right-triangles, parallel and perpendicular lines. Each (now very familiar) step of the frog construction providing a concrete model for the concept we were talking about.

Through this lesson, we learned and reviewed a lot of everyday language around the giving and receiving of instructions that involve everyday spacial terms, and were also able to then apply a mathematical lens to deepen our understanding of what we were doing.


Boakes, N.J. (2009). "The Impact of Origami-Mathematics Lessons on Achievement and Spacial Ability of Middle-School Students." In Lang, R.J. (Ed) Origami 4: Fourth International Meeting of Origami Science, Mathematics and Education. Natick, MA: A K Peters Ltd.

Ho, L.Y. (2002).  "Origami and the Adult ESL Learner." In Hull, T. (Ed.) Origami 3: Third International Meeting of Origami Science, Mathematics and Education. Natick, MA: A K Peters Ltd.

Tateishi, K. (2009). "Redundancy of Verbal Instructions in Origami Diagrams." In Lang, R.J. (Ed) Origami 4: Fourth International Meeting of Origami Science, Mathematics and Education. Natick, MA: A K Peters Ltd.

Tuesday, February 12, 2019

card puzzles

In these puzzles, each suit is given a value. The value of the card is the face value of the card multiplied by its suit value. Aces may be low (face value 1) or high (face value 11).  Face cards (Jack, Queen, King) have face value 10. Other cards have face value equal to their number.

Puzzle 1 (warm up)
In this puzzle, black cards (spades and clubs) have suit value 2 and red cards (diamonds and hearts) have suit value 3. What is the card value of each card shown?

Puzzle 2 (introducing sets)
When cards are put next to each other, we add their values. In this puzzle, spades have suit value 1, clubs have suit value 2, diamonds have suit value 3, and hearts a have suit value 4. Aces are high. What is the total value of each set?

Puzzle 3
In this puzzle, spades are worth 2. Each set is worth 20. What are the values of the other suits?

Puzzle 4
In this puzzle, Aces are low. The value of each set is shown below it. What is the value of each suit?

Puzzle 5
In this puzzle, Aces are high. The value of each set is shown below it. What is the value of each suit?

Puzzle 6
In this puzzle, Aces are low. The value of each set is shown below it. What is the value of each suit?

These puzzles can be solved by modeling the cards algebraically and then solving by substituting in known values.

The first two puzzles require you to evaluate by substituting in the known value of the suits. The five of diamonds is represented by 5d. We know that d = 3, so our card is worth 5(3)=15. Puzzle 3 tells you the value of spades (s), and requires you to find the other values by substituting in known values and solving for unknown values. The remaining puzzles require you to find the value of one of the suits by solving a one-step equation, then find the others by repeated substitution and solving.

puzzle 1: 15, 30, 20, 30.
puzzle 2: 45, 64, 36.
puzzle 3: = 1, = 3, = 1.
puzzle 4: s = 1,  h = 5, c = 7, d = 4.
puzzle 5: s = 2,  h = 6, c = 10, d = 5.
puzzle 6: s = 11,  h = 7, c = 5, d = 2.

Thursday, January 31, 2019

Farey Sequences and Ford Circles in JavaScript

Like the phyllotaxis spiral, a nice mathematical figure to draw in code is the sequence of Ford Circles.

A while back I tried generating these using Fathom and Processing - now for fun I've tried them in JavaScript. A page to play with them is here, and the source code is in a Github repo.

On the page, you can control the level of the Farey sequence used to generate the circles - you start off with just 0 and 1:

Using the buttons provided, you can increase the number of terms in the sequence and the corresponding number of circles.

After a certain point, the page does not list the entire sequence associated with the circles.

Wednesday, January 30, 2019

an origami surprise

For a recent origami-based math activity, I gave students printed instructions for two origami models: a pinwheel, and an open-top masu box (both from Origami USA).

They were to learn how to fold the models and answer some questions about the results:
Assuming that the paper has length of one unit, without measuring can you determine the perimeter and area of the pinwheel, and the volume and surface area of the box? 
I could honestly tell them: I did not know the answers, so they would have to explain to me how they found the results.

Opening up the finished models to reveal the pattern of folds provided a good strategy for getting to the answers. Considering how the folds divided the paper (into sixteen squares) and using the Pythagorean Theorem to calculate the lengths of diagonal folds allows you to get all the lengths you need.

The multiform pinwheel has a crease pattern like this:

And the masu box has the following crease pattern:

When it came time for the answers to the math problems I had posed, I had a mild surprise: two of the quantities that I had asked for came out to the same value - the area of the pinwheel and the outside surface area of the box were identical (3/8 units - interestingly just under half of the paper is exposed, the rest is folded in).

Taking another look at the crease patterns, you can see how the image of the box can be transformed into the pinwheel, demonstrating without calculations that the areas are the same:

Wednesday, December 19, 2018

Tweedledee and Tweedledum

Illustration by John Tenniel

Here is another logic puzzle by Raymond Smullyan, this time from his book Alice in Puzzle-Land:
Just then Alice practically stumbled on Tweedledum and
Tweedledee, who were grinning under a tree right by their house.
Alice looked carefully at their collars to see which was marked
"Dum" and which was marked "Dee," but neither collar was
"I'm afraid I can't very well tell you apart without your embroidered
collars," remarked Alice. 
"You'll have to use logic," said one of the brothers, giving the
other an affectionate hug. "We were expecting you to come around
these parts, and we have prepared some nice logic games for you.
Would you like to play?" 
"As you see, this is a red card. Now, a red card signifies that the
one carrying it is telling the truth, whereas a black card signifies that
the speaker is telling a lie. Now, my brother there [he pointed to the
other one] is also carrying either a red card or a black card in his
pocket. He is about to make a statement. If his card is red, he will
make a true statement, but if his card is black, he will make a false
statement. Then your job is to figure out whether he is Tweedledee
or Tweedledum." 
"Oh, that sounds like fun!" said Alice. "I'd like to play!"  
...Well, Tweedledee [and Tweedledum] went into the house, and both
brothers came out shortly after. They look more alike than ever! thought
Alice. Well, one of them—call him the first one—stood to Alice's left,
and the other—call him the second one—stood to Alice's right. They
then made the following statements:  
FIRST ONE: My brother is Tweedledee, and he is carrying a black card. 
SECOND ONE: My brother is Tweedledum, and he is carrying a red card.  
Which one is which?

If you think you may have a solution, you can try it out against the online version here.

After solving a puzzle (or reading the solution) in one of Smullyan's books, I am often left wishing there were more like it. Why not try generating some more?

Let's say there are 8 simple statements each brother could make.
  • My name is [Tweedledee|Tweedledum]. 
  • I am carrying a [red|black] card.
  • My brother's name is [Tweedledee|Tweedledum].
  • My brother is carrying a [red|black] card.
And 16 more compound statements:
  • My name is [Tweedledee|Tweedledum] [and|or] I am carrying a [red|black] card.
  • My brother's name is [Tweedledee|Tweedledum] [and|or] he is carrying a [red|black] card.
This gives us 24 possible statements each brother could make, so a total of 24^2 = 576 possible puzzles.

But some of these puzzles will lead to contradictions. For example, if either brother makes the statement "I am carrying a black card." We end up with the liars paradox: the brother must be lying, but if he is, he is telling the truth (and vice versa). Some of these also lead to multiple solutions.

A good puzzle must have a unique solution, and running these through a logic-puzzle solver yields 168 good puzzles (you can check out a Python notebook that generates and verifies the puzzles here). Really, only half of these puzzles are unique, as having the anonymous brothers "first one" and "second one" switch statements gives us essentially the same puzzle. So, ignoring the order of the statements/brothers there are 168/2 = 84 unique puzzles based on these statements.

The distribution of names and cards is uniform in these 84 puzzles (the brothers are as likely to be telling the truth as they are to be lying). In the graphs below the brothers are referred to as bro0 and bro1, and the counts are based on the set of 168 puzzles where the 'reverse' of each puzzle is also included.

cards held by both brothers

cards held by each brother

Try your puzzle solving skills against the brothers here. Other Smullyan-inspired puzzles can be found on the puzzle page of this blog.