1. Patterns in numbers

1.Counting  by .. 10's, 2's, 2 1/2's, .. 

a. Add 10 (43+10=? use 100 card if necessary), take away 10, add 20, take away 20..

b. count in binary, and base 3 with the Guess My Number game..  and Kaitlin's 6 "Magic number cards"

Counting how many pieces of a size make the whole cake, to name a fraction of the cake in chapter 1, or cookie in chapter 2. This is a key idea which many students are not aware of and causes difficulty in all their math courses!

Counting the number of rows of hexagonal cells on a pineapple in Chapter 7.

Counting squares on a geoboard to find the area within a shape.

Counting squares under a curve which leads to the integral. See chapter 13 and Geoff's and Grace's work.

Counting squares on graph paper to find the square numbers (see Tara's work).

Counting up, in looking at the differences in the output of a function (guess my rule- chapter 6). Also see Sheri's work (#25).

Counting small cubes that make cubes and pyramids, in chapter 13 and see Sheri's work.

Counting the moves to interchange the pegs in the Shuttle Puzzle (or Peg Game) in chapter 6. See also KatieR finds a pattern.

Counting the minimum # of moves to move the discs in the Tower Puzzle in chapter 6. See also Sheri's work.

Counting the number of little triangles and the number of edges in the Snowflake curve to obtain sequences to find its area and perimeter. See chapter 4 and Emily's work.

Counting the number of images in the hinged mirrors to obtain a function (chapter 6).

Counting squares and cubes to find the Surface area/Volume ratio of rods and why rodents are nocturnal animals

Counting cubes to find the Volume of the dog that was "doubled" in size, by Genny

2.  29 ways to look at multiplication

      i. as repeated addition 3+3=2x3

      ii. as intersections

                 6 intersections

      iii. as combinations: 2 shirts (s1 , s2) to go with 3 pair of pants (p1, p2, p3) = 6 outfits (s1 , p1), (s2 , p1), (s1 , p2), (s2 , p2), (s1 , p3), (s2 , p3)

      iv. as area of a rectangle A=l*w

      v. as a pattern or shape of numbers (square numbers)

      vi. as a mapping

      vii. as a function  f(x) =2x

      viii. in open sentences:  2y=8

      ix. as related to other operations a/b = a*1/b; 2x=x+x

      x. with different kinds of numbers- whole numbers, fractions, decimals, imaginary numbers, complex numbers

      xi. on one number line

      xii. on 2 parallel lines

xiiia. on 2 crossed number lines; the graph of 2x=y (graph of 2x table)

xiv. The 2-times table on a 72-dot circle:   x -> 2x; 1->2, 2->4, 3->6, ..

xv. Multiples of a number -> LCM

xvi. factors of a number -> GCF

xvii. Graph of l vs w of rectangles of area 6 or xy=6

xviii   Units digits of the multiples of 3 on a 10-point circle

xix. As patterns in a 100-chart- fill in the rest of the multiples of 3

xx. As patterns in the multiplication tables -the 3x table below


-and the square numbers below:

xxi. The 3D Multiplication Tables

xxii. “unusual” multiplication tables to fill in

xxiii. Use a calculator to do 27x5, say 10 different ways

xxiv. Using Napier’s “Bones” to multiply 26x3. The 2 times table follows down under the 2 and the 6 times table follows down below the 6, each number written with two digits, and each marked with a diagonal line between the two digits. The rod at the right is the multiplying number. The answer is read to the left of the 3 along the diagonals: 0, 1+6, 8 or 078 = 78. Made out of wood, each multiplication ‘rods’ can have 4 tables, and when put together, they sound like “bones” clinking together, hence the name, Napier’s “Bones”.

xxv. Using repeated multiplication 3x3x3x3= 34 to get exponents

xxvi. multiplication of signed numbers

See Sheri's work on graphing parabolas where she does this.

xxvii. In growth and form in nature (length, area and vol.)

xxiii. Using the distributive property a(b+c)=ab+ac

xxix. Applications of multiplication

xxx. The language of multiplication - times, product, factor, common multiple, least common factor, distributive property, law for 0: a*0=0, multiplicative identity x*1=x

2. Even numbers  2n

3. Odd numbers - 2n + 1, difference of squares

4. Prime numbers

5. Composite numbers- prime factorization

6. Figurate numbers

a. Triangular numbers ...

How do these number increase? What is the rule to get the nth triangular number?

        b. Square numbers (see Tara's work )

7. Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... How does one get the next one? see Chapter 7

8. Fractions

      a.  Adding fractions

         i. Chapter 1 using student-made pictures

        ii. Jocelyn's patterns

        iii. Lizzy and Cheryl's work (also using Jeff's great idea)

        iii. Chapter 2- Cookie-sharing to infinite series (mixed numbers and adding)

        iv. Chapter 5- the Harmonic series

        v. Sheri's work on 9/13-14/00 (#8) 1/A + 1/B and A/B + C/D

    b. Multiplying fractions

        i.. Jay multiplies mixed numbers    3  ways!

        ii. Multiplying infinite series!! - Grace's work

    c. Division of Fractions- Sheri's work of 9/20  (#11)

    d.  Fractions as division; fractions to decimals- see Maggie P's work (can you predict the number of  repeating digits in the decimal?)

    e. Continued fractions: see chapter 8 and Jesse's work

    f. Fractions <-> Decimals <-> Percents see Daly's work

9.  Ratios are fractions  (comparing 2 numbers by division)      

a. Find the ratio of consecutive Fibonacci numbers, taking the larger to the smaller. Here's what you get:

What patterns do you see in the fractions? in the mixed numbers? in the decimal?

        b. The ratio of the of the Cuisenaire©  rods and why rodents are nocturnal animals  & Sheri's work

       c. Ratio of   ® Golden  mean

       d. Chapter 10 - the ratio of  ® p  

        e. Chapter 14- using ticker-tape in finding velocity as a ratio of

         f. Trig functions are ratios

         g. Percent is a ratio. see Daly's work for beginning % and Chapter 11 compound interest  to e and i and .

         h. Ian finds Patterns in the ratios and differences of powers of powers, and gets e.

10.  Chapter 2: Changing fractions to decimals and bimals (Magi's work+)

11. Chapter 2: Changing infinite repeating decimals (and bimals) to fractions; Grace's work (Oct. 7, 2001)

11 a.

12. Chapter 14- The slope of a line is a ratio, slope of a curve -> the derivative

13. Chapter 8: Complex fractions

14. Chapter 8: Infinite continued fractions. See this page.

    a. in solving quadratic equations

    b. for  4/p

    c. for

   d. for the golden mean

15. Decimals are fractions  .75= 75/100

16. Division of

a.      Whole numbers

i.  With young people: How many 2’s make 6?  2/6 ; 60?  2/60  2/600    2/6000  2/6 sin x  2/-6

ii. using cookie- sharing (chapter 2) (writing remainder as a fraction, not as R1!)

b.  fractions : see Sheri's work of 9/21/00 and Jesse's work

c. decimals   2/6    .2/6   .02/6 

d. divisibility rules

17. Comparing numbers by Ratios and Differences (guess my rule in Chapter 6 and Ian's finding a pattern for e)

18. Exponents in chapter 1 (also see Sheri's work , in finding the rule for The Tower Puzzle, and she used negative exponents as well in the graph of this rule).

    18a. See Geoffrey's work on Polypowers in which he uses fractions!

19. Logarithms -patterns!! (exponents) in chapter 13 (questions and answers and see Kevin's work 

20. 2x2 Matrices: the 2x2 matrix corresponds to 1 (acts like 1). See Don's book "Changing Shapes With Matrices"

21. Complex numbers

a. in See chapter 11  of Don's worksheet book and Abe's graph and work to get i1/2

b.  iii...See the IES java applet  inspired by Don's chapter 11 (11a) to get Fractals

c. Multiplying two complex numbers, by Ian at age 12

    To multiply two complex numbers like (4 + i) * (2 + 3 i), just multiply their lengths and add their angles. See the diagram below.

            The length of  4 + i =

                        The length of  2 + 3 i  =  and 

The angles are 14°  and 56°  and 14°  + 56° = 70°. So (4 + i) * (2 + 3 i) is a complex number whose length is  and angle is  70°. Using the distributive law (4 + i) * (2 + 3 i) = 4* (2 + 3 i) +  i * (2 + 3 i) =  8 + 12 i +  -3 + 2 i  =  5 + 14 i.

On the graph 4* (2 + 3 i) has length 4 times (2 + 3 i) and  i * (2 + 3 i) takes (2 + 3 i) and rotates it 90° counter-clockwise. Then you add these. The length of   5 + 14 i  =  and its angle is 70°   (tan 70° » 14/5).

    The 2x2 matrix   corresponds to i and a rotation of 90° CCW.

    d. See Don's work on polypowers with and the IES wonderful java applet.

22. Pascal's triangle from area and volume, people pieces, routes.. See Chapter 9 . Here also is Ian's method of finding any number in Pascal's triangle!

23.  Probability (fractions <= 1)see chapter 13

 24. Trig functions: See Don's new , unfinished book, "Trig Functions for Young People" (Based on ratios of lengths in a circle).

24a. See the IES Java applet using Don's idea to show the 6 trig functions in one diagram!

25.  p  See Chapter 10 in Don's book

26.  e from compound interest See Chapter 11 in Don's book and Ian's function using differences and ratios of powers of powers to get e.    

        a. The famous statement which has in it  the five most important numbers in mathematics e , i, p, 1 and 0 :    eip + 1 = 0  WOW!

        b. Writing e as an infinite series -> eix = cos x + i sin x

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