Stony Brook Mathematics Department MAE 301 Foundations of Secondary School Mathematics

Index Announcement Syllabus & Homework

We will cover material in High School Mathematics: An Advanced Perspective. by Usiskin, Peressini, Marchisotto and Stanley.

Week 1 (Sept 4) Chapter 1. What Is Meant by `An Advanced Perspective'?
    Students will take part of the NYS Math III Regents Exam in class on Wednesday Sept 11.

Homework 1 (due Sept 9)  p 16 nos 1, 7, 8

• Team project. List some topics in the high-school curriculum on which you now have a more advanced perspective. Topics listed by students included:
  • geometrical proofs (from MAT 200)
  • limits
  • reading math textbooks
  • learn theorems by using them
  • more tools, more confidence
  • experience with n dimensions
  • better understanding of exp, sin, log, etc
  • deeper understanding of SOHCAHTOA
  • algorithms
  • use of matrices in solving linear equations
  • use of technology for visualization
  • how to implement Newton's method, etc "by hand"
  • logic beyond "P or Q"
  • solution of word problems

Week 2 (Sept 9 & 11) Chapter 2. Real Numbers
omit sections 2.1.3 and 2.1.4 (except for proof of irrationality of e)

Homework 2 (due Sept 17) p 30 nos 5, 6, 12
                         p 41 nos 1, 4, 7

Week 3 (Sept 17 & 18) Note that Tues Sept 17 follows a Monday schedule. Chapter 2 (end) Complex Numbers; Chapter 3 Functions

• Team activity. Data: the US Tax Schedule for 2000. Graph, as a function of x your adjusted income, f(x) your marginal tax rate and g(x) the total tax you owe. How are these two functions related? (Marginal tax rate is the percent you pay on the last dollar you earn).

Homework 3 (due Sept 23) p 80 nos 1bd, 4, 11
                         p 98 nos 3, 5
                         p 103 nos 4, 6-7ab
                         p 113 nos 1, 2, 7, 8

• Team activity: An unknown function f(x) is in a box. It is either an exponential A a^cx, a rational function p(x)/q(x) where p(x) and q(x) are polynomials with no common factors (q nonzero, but this includes the special case where q is 1 and f is a polynomial) or a trigonometric function, say sin kx or cos kx for any nonzero k. Find a scheme to determine what kind of function it is by asking two yes-or-no questions about its behavior at + and - infinity
• Team activity: given f(x) = (2x+3)/(x-1), calculate f^-1.

Homework 4 (due Oct 2) p 124 nos 1, 4, 7
                       p 132 nos 1, 5, 8
                       p 139 nos 2, 6
                       p 149 nos 1bdfh, 4, 6, 7

• Team activity: Solve with calculator:
  x⁵ + x + 1 = 0
  x + sin x + 1 = 0
• Team activity: Calculate the square root of 3 by Newton's method with x₁=1; do three iterations.

Homework 5 (due Oct 9) p 194 nos 12, 15
                       p 199 no 4
                       p 204 nos 2, 3
                       p 217 nos 1, 3, 5

Week 7 (Oct 13 and 15) Monday: Midterm. Wednesday: Chapter 5. Integers and polynomials, beginning.

• Team activity: give the proof by induction that the sum of the integers from 1 to n is n(n+1)/2, and that the sum of their squares is n(n+1)(2n+1)/6.

Homework 7 (due Oct 22)

     1. Prove, using the definition C(n,k) = n!/(k!(n-k)!),
        that C(n+1,k+1) = C(n,k) + C(n,k+1).

     2. Use this fact to write a complete proof of the Binomial Theorem:
        (a+b)n = C(n,0)an + C(n,1)an-1b + ... + C(n,n)bn.
    
     3. Also use this fact to show how the binomial coefficients C(n,k) can be
        calculated in a triangle ("Pascal's Triangle")

                             C(0,0)
                   
                         C(1,0) C(1,1)

                      C(2,0) C(2,1) C(2,2)

                    C(3,0) C(3,1) C(3,2) C(3,3)

                              etc.

          where the entries along the edges are all ones, and where
          each interior number is the sum of the two directly above it.



      4. Prove that the sum of the elements in the nth row
         of Pascal's Triangle is exactly 2n and
         that their alternating sum is 0.

      5. The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ...;
         each one is the sum of the two preceding ones. Prove
         that the nth DIAGONAL sum of the entries in Pascal's
         Triangle is the nth Fibonacci number:

             E.g.  C(0,0) = 1
                   C(1,1) = 1
                   C(1,0)+C(2,2)=2
                   C(2,1)+C(3,3)=3
                   C(2,0)+C(3,2)+C(4,4)=5
                   C(3,1)+C(4,3)+C(5,5)=8
                   C(3,0)+C(4,2)+C(5,4)+C(6,6)=13


         Hint: use 1. and induction.

• Team activity: Use the "row reduction" method to express (6341, 7272) as an integral linear combination of 6341 and 7272.

Homework 8
(Due October 29)

       1. Explain in your own words the "row reduction" method 
          for finding the g.c.d. (a,b) of integers a and b, and 
          why it works. Specifically, show how going by integral 
          row operations from


                      1  0  a
                     (       )
                      0  1  b

           to         x  y  d
                     (       )
                      z  w  0
                      

           gives d = (a,b) = x a + y b.

                             p  q  r
Hint Show that the entries  (       )  in each of the matrices in the process
                             s  t  u

satisfy p a + q b = r   and  s a + t b = u, by induction.

• Team activity: What does it mean to say "we use base 10"? Multiply 110111 (55 in binary) by 10001 (17 in binary) using the standard multiplication algorithm, and check that you get 935.

Homework 8 (due Nov 6) 

       1. Write out the proof of the statement: if a prime divides the product
          of two integers, it must divide at least one of the factors.

       2. Work out the binary long division 1010101/11111 (341/31) to
          five binary digits past the "decimal" point, and convert your 
          answer into decimal notation.

• Team activity:
• Calculate the coordinates of the image of (1,4) after rotation by 60^o about the point (2,1).

No homework assigned for this week. 

Week 11 (Nov 11) Chapter 7: Isometries (cont.)

• Team activity: Calculate the coordinates of the image of (x,y) after rotation by 30^o about the point (1,2).

  Calculate the coordinates of the image of (x,y) after rotation by 45^o about the point (-1,3).

  Calculate the coordinates of the image of (x,y) after reflection in the line 2x+3y=5.

Homework 10 (due Nov 20) 
                 

          1. Prove "Jason's Formula:" T-P RP,theta = R0,thetaT-P. Translate this formula into
             words.

          2. Let F: R2 ---> R2 be defined by  F = R(1,3),pi/6 (i.e. rotation by pi/6 about 
             the point (1,3)). Write F(x,y) in terms of x and y.

          3. Let F: R2 ---> R2 be defined by  F = r{y=2x-5} (i.e. reflection in the line 
             y=2x-5). Write F(x,y) in terms of x and y.