I started Paul with the problem 
Counting and operations in arithmetic
Paul, as a 3rd grader, divides by ZERO
I started Paul with the problem
-which means how many quarters make one-half? Like in the problem 12 divided by 3 = ? Or what times 3 makes 12? We worked our way up the page, and he caught on to the pattern. I asked him to make up some problems like these. This is where the interesting part comes in! He starts to divide by zero!
We discussed 3/0 = ? as what times 0 = 3? And he said there is no number times 0 = 3. His notation was N.A. for no answer. The we talked about 0/5 = ? What times 5 = 0? and of course this was 0. What about 0/0=? and he came up with the fact that any number times 0 = 0. So 0/0= Any Number (A.N.); this is equivalent to AN x 0 = 0.
17 June 2008 Don received this note from Paul, who came to Don from 3rd grade through 12th grade:As of this writing in 2010, Paul is a sophomore in college.
Counting
by .. 10's, 2's, 2 1/2's, ..
The language
of counting: With young children I
use 'one-ty two' instead of twelve, 'two-ty three' instead of twenty
three. Notice when we get to 42 we say forty two. And everything is fine. A good
aexample of this is a 5 yo who added 8+8 to get 61. What he meant was 16, but
because the sound of sixteen, he thought the 6 should go first! I told him to
say one-ty six, instread. I use
the same language for fractions; I say 'one twoth'
for 1/2 because the 'twoth'
tells you how many pieces are in the whole. Saying 'one half'
doesn't tell you that information.
The other thing to realize is that counting is not always a simple task, like
trying to count the rows on a pineapple, below.
Make
copies of the 100 card below. Add 10 (count
up 10): put your left pointy finger on 45, count up 10 with your right
pointy finger, where do you end up? 55. Do more like that; what pattern to you
see? When you add 10, you just have to go up one row. Take away 10, add 20, take
away 20. Do these without the card, in your head.
Counting by .. 10's, 2's, 2 1/2's, ..
Counting how many pieces of a size make the whole cake, to name a fraction of the cake in chapter 1 (infinite series), or cookie-sharing 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 or the number of rows of seeds on a sunflower head to get Fibonacci numbers - an infinite sequence (see Map) and Chapter 7.
Counting squares on a geoboard to find the area within a shape, and leads to a 3D graph and function.
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 patterns in 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).
Counting small cubes that make cubes and pyramids, and other shapes (Genny doubles the size of a dog), in chapter 13 and see Sheri's work.
Counting the moves to interchange the pegs in the Shuttle Puzzle (or Peg Game) and get a function, in chapter 6.
Counting the minimum # of moves to move the discs in the Tower Puzzle in chapter 6. See also Sheri's work (see Map-functions, non-linear).
Counting the number of triangles and the number of edges in the Snowflake curve to obtain infinite series 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 and Map).
Counting squares and cubes to find the infinite sequence of Surface area/Volume ratios of rods and why rodents are nocturnal animals (also see Map).
29 ways to look
at multiplication
i. as repeated addition
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 a POWERFUL idea
v. as a pattern or shape of numbers
vi. as a mapping
vii. as a function
viii. in open sentences
ix. as related to other operations
x. with different kinds of numbers
xi. on one number line
xii. on 2 parallel lines
xiiia. on 2 crossed
number lines
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- now called "lattice arithmetic". 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
See Sheri's work on graphing parabolas where she does this.
xxvii. In growth
and form in nature (length, area and vol., in Chapter 6)
xxiii. Using the
distributive property
xxix. Applications of
multiplication
xxx. The language
of multiplication
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 (see MAP at Infinite series).
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. Chapter 4- The area and perimeter of the Snowflake curve and on the MAP
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!! - see MAP
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 decimals?
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 - On MAP,
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 and MAP
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 e
d. for the golden mean (see MAP)
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
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 Kavi's and Kevin's work , and on the MAP
20. 2x2 Matrices: the 2x2 matrix
corresponds to 1 (acts like 1). See Don's book "Changing
Shapes With Matrices" and Lori's Mathematica Player Demo
21. Complex numbers
corresponds to i and a rotation
of 90' CCW.
and
the