dansmath4kids
2/03 8/08 Some of you parents, and a few kids, have asked me for math materials that the younger set would enjoy and learn from. I believe math can be enjoyed at any age, but if you can start young, so much the better. (As for me: I could count before I could spell...) What's on this page ? (click the red round buttons to go there \\ / ) Finding Math in the World Patterns & Numbers 
Puzzle Problems Elem, Middle, High (school / level) . Math Activities and Ideas for Parents . . . . Lesson Plans and Ideas for Teachers 
Graphics & Animations view, print, or take! . . .
Finding Math in the World / Patterns & Numbers (back to top) - "As a math teacher,
one of the most-often asked questions I get is:
- What are we gonna use this stuff for? I try to find some area of math that
- applies to their favorite interest, even if it's music, art, film, or dance!" - Dan
- What are we gonna use this stuff for? I try to find some area of math that
-
- Physics: Where would this world be without the laws of physics? Right where it is, but we
- couldn't understand it or the rules that govern its gravity and motion without knowing math!
-
- Advertising: How much should we spend on TV commercials & newspaper ads this year?
- Some ads will increase sales, but too many ads will backfire and some customers will leave.
-
- Music: The chromatic scale (C, C#, D, D#, ... , B, C) is made up of notes defined by frequencies that
- grow in a "geometric progression" whose ratio is the 12th root of 2 ; spanning one octave!
-
- Art: Many of the movements in classical, impressionist, or modern art have been based upon
- geometric principles: Cubism, OpArt of the 1960's, and Digital Illustration rely heavily on math.
-
- Film: Whether in live-action films or 3D animations, math is an 'integral part' of moviemaking.
- Animators need to know coordinates, light values, and camera paths; actors need to 'read lines'!
-
- Dance: George Balanchine of the New York City Ballet was obsessed with geometric patterns and
movements for his dancers. His choreography stressed straight lines and intricate timing. - "As a math teacher,
one of the most-often asked questions I get is:
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Math Activities and Ideas for Parents . . from www.dansmath.com . . (back to top)
- 1. Be positive about math and about learning; kids listen more than you know!
- Most kids will enjoy a subject
if there aren't negative preconceptions; try not to say things
like
- "I was never any good at math," or "math is hard," or "you probably won't need to know that."
-
- 2. Encourage fun and exploration, not just grades and competition.
- Math isn't about A's, drills,
and endless practice; it's a classical subject, a universal language,
- and a beautiful structure of patterns and logic. It's also a place to learn problem solving.
- Achievement in the usual subjects of arithmetic, algebra, geometry, trig, and calculus will
- happen only when students invest mental energy in doing math; so make it relevant and fun!
- and a beautiful structure of patterns and logic. It's also a place to learn problem solving.
- 3. Be sure to provide a good place for your kid to study.
- Have a room, desk, or table
that isn't in the line of sight of a TV set. Math can be done
well
- with or without a computer; web surfing while studying can be distracting. Got three kids in
- a one-bedroom apartment? Carve out a corner of the kitchen and set up a lamp and small
- bookshelf; allow for your kid's concentration time without noise or frequent interruptions.
- with or without a computer; web surfing while studying can be distracting. Got three kids in
- 4. Allow computer exploration on the internet; at least to a list of websites you select.
- There are some amazing places
to see (including dansmath.com), and some of them are at a
- more advanced level than your kid might be, but that doesn't mean they can't go there.
- Try mathworld.wolfram.com or Ask Dr.Math or visit Dan's Favorite Links.
- more advanced level than your kid might be, but that doesn't mean they can't go there.
- 5. Look for mathematical patterns together (numerical and geometric) in the world around you.
- a) Did you know that a soccer
ball is really a geometric solid called a "truncated icosahedron"?
- b) What are triangular numbers, and where can you find them besides bowling?
- c) What's the story of Pi (
=
3.14159265 . . . ) and
why do the decimals go on forever?
- d) What's so special about numbers like 60; why do they have so many divisors?
- b) What are triangular numbers, and where can you find them besides bowling?
- Lesson Plans and Ideas
for Teachers 8/08 (back to top)
- The greatest thing
about my job is that I teach what I love, and (for the most part)
- nobody tells me how to teach it. But what if you want someone to tell you what to do?
- Here are some suggestions you can build on, look up, modify, or ignore! Read on! - Dan
- [ Younger Kids | The Usual Topics | Tangential Topics | Fringe Topics ]
- nobody tells me how to teach it. But what if you want someone to tell you what to do?
-
- Younger Kids: Counting, Primes, Divisors, Proportions - browse my basic math lessons - Dan
-
- Counting . .1. Simple association of objects: count the children in a room by using pennies.
- 2. How many letters in
their last name? Who has the fewest in class? The most?
- 3. How many M & Ms are in a bag? Use snack-pack size or substitute raisins.
- . .You or they can make a chart of the results and discuss totals or averages.
- 4. When I was a kid I'd take a handful of chocolate chips, count them, and draw
- . .that number (say 37) using all the chips. Then eat one; rearrange into a 36. Etc.
- 3. How many M & Ms are in a bag? Use snack-pack size or substitute raisins.
- Primes . . . .1. Have them arrange 10 pennies or beads in a rectangle, then try 9, then 11. Discuss.
- 2. Which numbers from
2 to 20 can be made into rectangles, and which only in lines?
- 3. Do the 'Sieve of Eratosthenes': they write numbers from 1 - 30 (or 50, 60, 100) in rows of
- . .5, 6, 7, or 8 (you choose) then circle the 2, cross out all higher mults. of 2; then 3, etc.
- 3. Do the 'Sieve of Eratosthenes': they write numbers from 1 - 30 (or 50, 60, 100) in rows of
- Divisors . . 1. Have the kids draw rows of dots or find patterns in a grid corresponding to which
- . .numbers go into 12: 2 x 6 , 3 x 4 , 4 x 3 , 6 x 2. What number (under 50) has the most?
- Proportion 1. If half the kids in your class size were girls & half boys, how many of each would there be?
- Counting . .1. Simple association of objects: count the children in a room by using pennies.
- .
.How does that compare
with your actual class roster? Extend this proportion to 100
kids.
- 2. If, in a class of 30, there are three girls for every two boys, how many of each are there?
- (c) 2008 dansmath.com
- The Usual Topics: Arithmetic, Algebra, Geometry, Trigonometry - browse my free math lessons on these topics! - Dan
-
- Arithmetic 1. What is the sum of the first 10 natural numbers? First 20? 100? Find & explain the pattern.
- 2. Introduce different bases, I suggest especially base 2 and base 5. Example: 8 base 10 = 13 base 5.
- Algebra . . 1. Present Dan's Prime Code, where all letters are primes: A=2, B=3, C=5, . . . , Z=101. Each word
- is coded as the product of its letters. Teaches substitution, decoding teaches factoring and anagrams.
- Geometry . 1. How many squares (of any sizes) does it take to cover a 5 x 6 rectangle (w/out overlapping or going over the edge)?
- 2. What design will pack the most circles of diameter 1 into a 10 by 10 square? Spheres in a box?
- Trigonometry 1. The height of a person 5/8 of the way around (or 1/3 etc.) on a ferris wheel of radius 20 meters.
- .
2. Measuring distances
of 'nearby' stars and heights of mountains without going there.
- . .3. Calculating the angles in a 3-4-5 triangle, discovering other 'Pythagorean Triples.'
- . .3. Calculating the angles in a 3-4-5 triangle, discovering other 'Pythagorean Triples.'
- (c) 2008 dansmath.com
- Arithmetic 1. What is the sum of the first 10 natural numbers? First 20? 100? Find & explain the pattern.
- Some Tangential Topics: Number Theory, Tessellations, Polyhedra browse my feature pages on these topics! - Dan
-
- Number Theory 1. Divisors, supercomposite numbers (opposite of prime), their special prime factorizations.
- . . . .2. Abundant, deficient, and perfect numbers, Mersenne Primes
- Tessellations . 1. Which regular polygons will fit around a common point, and how many? Why? Angles?
- . .2. What if we get to use more than one kind of regular
polygon, like squares and triangles?
- . .3. What other common (or uncommon) shapes will tessellate the plane? Rectangles? T-shapes?
- . 4. What is the background pattern on this web page? I call it < 3 , 4 , 6 , 4 > (Why is that?)
- . .3. What other common (or uncommon) shapes will tessellate the plane? Rectangles? T-shapes?
- .
- Polyhedra . 1. Find various polyhedra out in the world (cubes, pyramids, soccer balls, geodesic climbing bars)
- . 2. Construct polyhedra of various types using printed
nets or supplied cardboard polygons.
- . 3. Plan a lesson on geodesic domes and Buckminster Fuller (who I once met when I was 15)
- . 3. Plan a lesson on geodesic domes and Buckminster Fuller (who I once met when I was 15)
- (c) 2008 dansmath.com
- Number Theory 1. Divisors, supercomposite numbers (opposite of prime), their special prime factorizations.
- On The Fringe: Chaos & Fractals,
Math Sounds, Any-Pointed Stars
- Chaos Theory - Discuss the idea that a small change in initial conditions could have a major effect long-term.
- Possible references to
The Butterfly Effect, the delicate balance of nature, evolution,
and world events.
- Fractals - Find pictures of the Mandelbrot Set and Julia Sets, and a computer or iPhone app that zooms in.
- The recursion f(z) = z^2 +
c can generate the Mandelbrot set; a 'complex figure' based on
a simple formula!
- Math Sounds - The connection between Trig and Sine Waves and Sounds; amplitude, frequency, waveform
- See my podcast dansmathcast
#3 for a discussion and sound samples! Compound waves, square
waves, etc.
- Any-Pointed Stars - Using the GCD of n and k to predict how many points @(n, k) will have.
- Method: Draw a circle,
divide the circumference into n equal parts, label the points
0 thru n-1, and connect
- dot 0 to dot k, then k to 2k, etc. For example, @(10, 4) goes 0 to 4 to 8 to 12 (=2) to 6 then back to 0.
- Five points, because the GCD of 10 and 4 is 2, and 10 / 2 = 5.
- dot 0 to dot k, then k to 2k, etc. For example, @(10, 4) goes 0 to 4 to 8 to 12 (=2) to 6 then back to 0.
- (c) 2008 dansmath.com
- Lesson Plans and Ideas
for Teachers
-
Graphics & Animations 7/04 You can view, print, or use these in your class! (back to top)
This is a "3D space curve"
with its shadows on two walls
and floor! (Done in Mathematica)
- It's a soccer ball! No,
it's a spherical
- projection of a truncated icosahedron!
- Here every corner is the meeting place
- of two hexagons and one pentagon.
- projection of a truncated icosahedron!
This is the picture proof of the
famous Pythagorean Theorem;
comparing the yellow areas we
can see that a^2 + b^2 = c^2 .
(I showed this in my job interview!)
- I like to blur the line
- between Math and Art.
- This is a surface generated
- by Mathematica, showing
- the red space curve and a
- progression of curves along
- normal vectors at each point.
- between Math and Art.
This is the simplest
dissection of a rectangle
into incongruent squares
(diff sizes) rectangle is 32 x 33.
- This animation, used in calculus,
explains
- that the (red) secant lines that connect two
- points P and Q approach a limiting tangent
- line (blue) at P, as Q approaches P.
- (Click Reload or Refresh if it's not moving.)
- that the (red) secant lines that connect two
- It's a soccer ball! No,
it's a spherical
(back to top)
