Mathematics Galore! (Classroom Resource Materials)
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Mathematics Galore! Showcases some of the best activities and student outcomes of the St. Mark s Institute of Mathematics and invites you to engage the mathematics yourself! Revel in the delight of deep intellectual play and marvel at the heights to which young scholars can rise. See some great mathematics explained and proved via natural and accessible means.
Based on 26 essays ( newsletters ) and eight additional pieces, Mathematics Galore! offers a large sample of mathematical tidbits and treasures, each immediately enticing, and each a gateway to layers of surprising depth and conundrum. Pick and read essays in no particular order and enjoy the mathematical stories that unfold. Be inspired for your courses, your math clubs and your math circles, or simply enjoy for yourself the bounty of research questions and intriguing puzzlers that lie within.
of a braid by ±4, or not at all. Thus only those braids whose index is a multiple of four have hope of being untangled. (Of course, in addition, the ends of the strings must be tied to the spoon in the correct order for the hope to continue!) This result rules out hope that one particular type of braid can be untangled: Hang a spoon from a ceiling with three untangled strands and give it one full turn of rotation. It is impossible to untangle this braid the braid that results by holding the spoon
and promote creative mathematical thinking, awareness, and enjoyment of the subject. I meet these goals by offering workshops, courses and activities for students (both from St. Mark’s School and from the wider Boston community), professional development and graduate courses for mathematics teachers (with accreditation from Northeastern University), and public lectures, activities and written materials for the general public. Three Institute products are particularly popular: the mathematics
4 Mathematics Galore! √ Pythagoras’s Theorem and some algebra show that x = 1 + r 2 = y and z = √ √ (1 − r )2 + (1 + r )2 = 2 + 2r 2 = 2 · x. This establishes that the central triangle is a 45-9045 triangle. Thus θ = 45◦ and arctan (r ) + arctan 1−r 1+r = 45◦ . (Put in r = 1/2 to see a familiar result.) Here are more intriguing angle tidbits: 2 arctan 4 arctan 1 1 − arctan = 45◦ 2 7 1 1 1 − arctan + arctan = 45◦ 5 70 99 2 arctan 12 arctan 12 arctan (Herman, ca. 1706) (Euler, 1738)
glance, but actually the polynomial is easy to understand. Personalized Polynomials 125 There are six terms, one for each appearance of the six individual values a0 , a1 , a2 , a3 , a4 , a5 . Each term has a numerator designed to vanish at all but one of the values x = 0, 1, 2, 3, 4, 5. The denominator of each term cancels with the numerator that arises when the numerator does not vanish so that the fraction has value 1. Multiplying by ai causes the polynomial to have value ai in that case.
spread themselves across the entire circle in steps less than 0.001 units, or 0.00000001 units, and so forth. So there we have it! The multiples of log 2 ﬁll up the circle in an evenly distributed way and so log 4 − log 3 = 12.5% of them correspond to powers of two that begin with three. By the same method, log 8 − log 7 = 5.8% of the inﬁnitely many powers of two begin with 7 (answering the opening puzzler), log 78 − log 77 ≈ 0.56% of the powers of two begin with 77 and log 778 − log 777 ≈ 0.056%