## 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.

References

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?

Solutions
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.