### The important math your child will work on in Don's books (note, all of Don's materials come from work he has done with young people!):

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

a. Add 10, -10, +20, -20..

b. count in binary, and base 3..  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 squares on graph paper to plot points

Counting the number of rows of hexagonal cells on a pineapple in Chapter 7-> Fibonacci numbers->The golden mean

Counting the number of leaves and no. of times they go around a sunflower stalk-> same as above

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/down, 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

Renaming numbers: 4/6 is a name for 2/3 as is 1/2 + 0/4 + 1/8 + 0/16 + ...!

Fractions- within the context of interesting mathematics:

 addition (and infinite series)    multiplication   equivalent fractions   complex fractions continued fractions  infinite continued fractions- in     solving quadratic equations division & fractions- cookie sharing changing fractions to infinite     decimals and bimals changing infinite repeating decimals and bimals to simple fractions fractions in graphing x+y=5 from area on a geoboard sequence of partial sums in an infinite  series  SA/Vol ratio of white to orange rods to show why rodents are nocturnal animals slope of a line->slope of tangent to a curve-> the derivative ratio of (diagonal of a regular pentagon/side of a reg. pentagon)-> Golden mean ratios of consecutive Fibonacci numbers ratios of (perimeter of inscribed polygon/diameter of circle)->Pi velocity as ratio of distance/time rate of interest,% sine function is a ratio

Guessing Functions (guess my rule):

 linear quadratic exponential finite differences from #nails and area on a geoboard from sequences, like (SA/Vol) ratio of rods from the ratio of the Fibonacci numbers the shuttle puzzle the Tower puzzle from hinged mirrors the slope of a line from slopes of tangents to a curve from area under curves

Infinite series

• from coloring in squares
• from finding the area of patterns in squares
• convergent, divergent, limit of ('goes to')
• as a name for a simple fraction
• as a name for an infinite repeating decimal
• for p, ex, eix, sin(x), cos(x)
• to find the distance a ball travels in bouncing, from being dropped until it stops bouncing
• Don's interpretation of Archimedes' method to find the area under a parabolic segment
• For the area and perimeter of the snowflake curve

Infinite sequences:

• partial sums of infinite series
• Fibonacci numbers
• ratios of consecutive Fibonacci numbers
• to get the Golden angle
• from solving quadratic equations by iteration
• from finding square roots (using different methods)
• convergent, divergent, limit of ('goes to')

Infinite continued fractions

• for Pi, e, phi, square root of 2, from solving equations, graphs of, finite continued fractions

Graphing:

• linear graphs->fractions, negative nos., slope, intercept
• quadratic->moving parabolas, from distance/time experiment, length vs. area of rectangles of constant perimeter of 20; quadratic equations, iterations; circles, ellipses, hyperbolas
• partial sums of infinite series
• infinite sequences
• geometric transformations using matrices

Angles:

• from whorling leaves on a sunflower stalk and other plants
• central angle in a circle->Pi
• sum of angles of a polygon

Changing shapes with matrices

Area, perimeter, and volume

• on a geoboard, the area within shapes, # of squares-> sum of squares
• perimeter of the snowflake curves and Sierpinski curves (result in divergent series)
• area of the snowflake curves and Sierpinski curves (result in convergent series)
• finding the (SA/Vol) ratio of rods (results in a convergent sequence)
• volume of pyramid, volume of cubes of same base and their ratio
• area under curves (counting squares first!)->integral
• area of rectangles of constant perimeter of say, 20 (when is the area a maximum?)
• similar shapes->how perimeter and area of shapes change

Binomial expansion:

• Pascal's triangle (obtained 3 or 4 ways), Ian's way
• (a+b)^n and its relation to infinite series
• finding the cube root of 2
• from compound interest -> e

Interest:simple->compound->e->e(i*Pi) +1 = 0  (try on a computer LN(-1) and see what you  get!)

Solving equations:

• linear--like 2x+3=17, 2x+3=18, 5x+3=2x+18 -- by guessing first!!, by balance pictures-> to algebraic transformations
• quadratic--by guessing!!, sum and product of roots, by iteration many ways, by graphing, by quadratic formula, using a calculator to hone in on the answer
• cubic--by iteration, by computer

Iterating functions:

• like 5 + x/2, if you start with 4 in for x, what happens to the sequences?
• to solve equations
• to find compound interest
• to find the square root of a number
• to find the square root of the square root..of a number

The use of computers and calculators: computer programs are in almost every chapter, with an appendix in the worksheet book on how to write programs to get infinite sequences and series. Don uses Derive to zoom in on a curve to get the slope of the tangent leading to the derivative. He uses Mathematica to show 100 iterations of a function. (See the use of computers page).

Probability: The area under the normal curve is 1 and is related to probability

Trig functions: sine is used in finding the perimeter of polygons inscribed in a circle to get to Pi and is shown as an infinite series. See also Trig for young people.

To order