The discovery of the year 2010-2011 is David's Rule Generator. We can't show this now because he is sending it in to be published!

[N.B. - the next differences in each case above, are zero.]

You can click here to see how Don figured out the cubic on the right above, without a calculator or computer. See Sheri's work on the Tower Puzzle rule and look at the differences with an exponential function .

Some of Don's students make up very hard rules, like Anna, a 4th grader, did. Anna did give Don a hint that her rule was exponential. Don spent over an hour working on her rule, without success, then he gave her table of numbers to another of his students, Jerry, a 5th grader, who figured it out in about 20 minutes. Don likes it when his students do better than he does! [Anna made a small mistake- the first pair should be (-1, -1)].

Here are a couple of more rules, see if you can figure them out:

                                      

See Jerry's rule, unusual, to say the least!

More FUNCTIONS to explore

 Compiled by Don

     

1.                                         How many Squares can you make

                 'on a 5 x 5 geoboard (5x5 square array of nails/dots)?

                  'on a 4 x 4 geoboard (square array of nails/dots)?

                  ...on a 20 x 20 geoboard (square array of nails/dots)?            

You have a circular pie and make straight line cuts. Try to get as many pieces per cut as you can;  with 5 cuts, how many pieces?

For six equally-spaced points on a circle, how many straight line segments connect  these points? 8 points? 12 points?

How does the area of a picture change with the distance from the projector to the screen?

Height of a burning candle vs. time.

Rise and run of steps on a stairway.

# of diagonals of a polygon vs. the # of sides.

Interior angle of a polygon vs. the # of sides.

Exterior angle of a polygon vs. the # of sides.

Central angle of a regular polygon inscribed in a circle vs. the # of sides.

Rectangles with constant perimeter of 20; length vs. width. 

The area of rectangles with constant perimeter of 20; length vs. area. (see Map- Kelda's work)

Rectangles with constant area of 36, length vs. width.

Perimeter of rectangles with constant area of 36, length vs. perimeter.(see Map-Maya's work)

edge of cube vs. surface area.

edge of cube vs. volume.

SA/Vol ratios of rods (Nanako) and why small animals are noctunal

Celsius vs. Fahrenheit Temperature.

Weight vs. stretch of a spring.

Length of shadow vs. height of object.

Weight vs. value of gold coins.

No. of straight lines vs. max. no. of intersections.

Stopping distance of a car vs. speed (see driver's manual or police dept.).

Weight vs. volume of various size solid objects of the same kind.

Bicycle:   a.  # wheel turns vs. # pedal turns

 b.     Distance wheel moves vs. # pedal turns.  (changing gears.?)

Perimeter of inscribed ploygons vs diameter of circle. (see Map/ chapter 10)

Tower puzzle:  # discs. vs. minimum # of moves to move the pile.

Shuttle puzzle:  # pairs of pegs vs. no. of moves to interchange the color pegs.

Angle of Sun vs. day.

Hours of daylight vs. day.

Temperature vs. hour of day.

Ball:  Height of drop vs. height of bounce.

Use ticker-tape: distance vs. time of brick dropping.(see Map-Distance vs. time)

# kwh used vs. cost of electricity.

Oz. vs. gm. on cans, boxes of food.

Distance between cities vs. airline times/cost.

# miles traveled in car vs. # gallons of gas consumed.

Pendulum:  length of string vs. time for 10 swings.

Resistance of wire vs. thickness.

Pulse beat vs. length of animal.

# coins vs. # ways coins can come  up.  

Triangular nos., square nos., pentagonal nos'.

Polyhedral nos.

Surface area of rods vs length of rod.(see Map)

SA/vol ratio of rods (see Map)

Postal scale function