Fun with Folding. led by Thomas Clark. October 7, Take a strip of paper, fold it in half, and make a good crease at the midpoint position.

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1 Preliminary Folding Investigation Fun with Folding led by Thomas Clark October 7, Take a strip of paper, fold it in half, and make a good crease at the midpoint position. 2. Open up the strip, and mark the crease Midpoint. 3. With the strip unfolded, fold the left end over the meet the Midpoint. Make a new crease halfway between the Midpoint and the left end of the strip by folding the left end of the paper. 4. Open up the strip again, and mark new crease with a With the strip unfolded, make a new crease halfway between fold 1 and the right end of the strip by folding the right end of the paper to meet the crease marked fold Repeat, alternating left and right folds, with each fold made to the most recent crease mark. Mark each successive crease with 3, 4, The sequence of crease marks seems to converge to two positions on the strip, what are they? 8. With a new strip of paper, repeat the experiment, except this time make the initial crease mark anywhere on the strip, not at the midpoint. Does the sequence of crease marks converge as before? 9. Why is that? 1

2 Mathematical Analysis Let s consider this folding problem from a mathematical perspective. Suppose the strip is one unit long, and the initial crease is at arbitrary position x (0 < x < 1 measured as a fraction of the length of the strip ) as in the last step above. 1. Then a left fold creates a new crease 1 at what position? (Express algebraically in terms of x.) 2. Now a right fold creates a new (even-numbered) crease at what position? (Express algebraically in terms of x.) 3. You can think of taking a point x and doing a left fold and then a right fold as a function. If x is the initial location and f(x) is the location of the new crease (after first folding left and then right), what is the formula for f(x)? (hint see 2.) 4. Are there any points that return to the same spot? That is, is there a number x for which x = f(x)? This is called a fixed point. 5. Repeat the previous two questions with doing a right fold and then a left fold. 6. What do your answers to 4. and 5. have to do with the folding experiment? Back to the Base(2)ics Now let s see if we can see why the behavior we ve seen above is happening. In base ten arithmetic, the decimal 0.abcd... represents a 10 + b c d , where a, b, c, d,... are digits between 0 and 9. Now what would this look like in base-2? 2

3 1. In base two arithmetic, 0.abcd... would represent what? Answer: a 2 + b 4 + c 8 + d We can represent every real number x, 0 x 1 in base two as 0.abcd..., where a, b, c, d,... are digits between 0 and 1. (a) What is 3 4 in base two? (b) What does represent in base two? (c) What is 1 3 in base two? 3. If x = a 2 + b 4 + c 8 + d that is x = 0.abcd..., then what is the binary representation of 2x? 2x = a + b 2 + c 4 + d Describe what multiplying by 2 does to the decimal point. (Really it s a binary point!) 5. In base two, if: x = a 2 + b 4 + c 8 + d so x = 0.abcd..., then what is the binary representation of x 2? x 2 = a 4 + b 8 + c 16 + d Describe what dividing by 2 does to the binary point. 3

4 Putting it All Together Now we connect the base two representation of a number and the paper folding activity: 1. If the initial crease is at x = 0.abcd... (in binary) then a left fold puts a new crease at x 2. What is the binary representation of the new point? 2. If the initial crease is at x = 0.abcd... then a right fold puts a crease at x. What is the 2 binary representation of the new crease? (Hint: recall that 1 2 = in binary. ) 3. Starting with a point x = 0.abcd... in binary, what is the binary representation of the new crease resulting from: (a) A left fold, followed by a right fold. (b) A left, right, left, right fold. (c) 5 sets of left then right folds. 4. Starting with a point x = 0.abcd... in binary, what is the binary representation of the new crease resulting from: (a) A right fold, followed by a left fold. (b) A right, left, right, then left fold. (c) 5 sets of right then left folds. 5. With more and more folds, what values will the crease marks be approaching? 6. Do you see the connection between fixed points and the binary representations? 4

5 Classroom Connections 1. Are there any connections between the math we ve done here and what you teach in you classroom? 2. Is there an area of your curriculum that might benefit from something we ve done here tonight? 3. How would you have improved the activity for this context or for your classroom? I ve included below a few standards from the CCSS that we have touched on in one way or another tonight. Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 7.EE.4: Solve real life and mathematical problems using numerical and algebraic expressions and equations. 8.F: Define, evaluate, and compare functions. F-IF1, F-IF2: Understand the concept of a function and use function notation. F-BF1: Build a function that models a relationship between two quantities. F-BF3: Build new functions from existing functions. The CCSS Modeling standard is intended to be developed in the context of the the other standards. An activity like this that begins in a concrete object which prompts questions and then uses mathematics to move toward abstraction and deeper understanding of the underlying structures is at the heart of mathematics. 5