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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! . . .
 
 
 
the rest of dansmath -- [ home | info | meet dan | ask dan | matica | lessons | dvc ]
 
 
 

 

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
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.
 
 
 
 

 

 
 

 

 
 
Number Problems . . . back to top (c) 2003-2008 dansmath.com
N1. a) Find a Fibonacci number, bigger than 1, that's a perfect
square. (E) b) Any more square Fibonaccis under a million? (M)
 
N2. Which numbers from 1 to 100 tie for the most divisors? (M)
 
N3. Find a three-digit number, without any 0's, that equals the
sum of the cubes of its digits. (M)
 
N4. What number (under 1000) leaves a remainder of 3 when divided
by 7, a rem of 4 when div by 11, and rem 5 when div by 13? (H)
 

 


Geometry Problems . . . back to top (c) 2003-2008 dansmath.com
G1. How many circular cookies of diameter 4" will fit onto a
round plate of radius 6"? (E)

 

G2. Copy the square grid at the right onto a piece of paper.

a) How many squares of all sizes can you find in the picture? (E)

 

b) Try the problem with different size grids; look for a relation

between the size of the grid and the number of squares. (M)

 
G3. What's the inside corner angle of a STOP sign (8-sided) ? (M)
 
G4. How do you cut a 4" by 9" rectangle into three easy pieces
that can be rearranged into a square? What size square? (M/H)

 

G5. What's the diameter of the

small circle in the corner? (H)

 
 

 


Train Problems . . . back to top (c) 2003-2008 dansmath.com
T1. Two trains, each going 20 meters per second, approach
each other from 6,000 meters apart starting at noon.
When will they meet? (E)
 
T2. On your vacation, your train, traveling at 45 mph,
passed a train in the opposite direction going 36 mph.
You saw the second train take 6 seconds to pass your eye.
How long was the second train? (mph is miles per hour) (M)
 
T3. You fell asleep on the train halfway to your destination.
You slept until you had half as far to go as you went while
you slept. What fraction of the trip were you sleeping? (H)
 

 


Logic Problems . . . back to top (c) 2003-2008 dansmath.com
L1. My younger niece is 13 years old, but her older sister just
had her fourth birthday. How is this possible? (E)
 
L2. Can you plant ten trees in four rows of five trees each? I can! (M)
 
L3. You see two twins from your class; one always tells the truth, and
the other always lies, and you don't know which one is which. You
want to know if there will be a test today. What one question
can you ask one random twin so you'll know for sure? (H)
 
 
 
 
 

 

 
 

 

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!
     
    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.
     
    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.
     
    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?
 
 
 
 
 
 
 

 

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 ]
 
 
    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.
       
    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.
     
    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?
    . .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.'
       
    (c) 2008 dansmath.com

     
    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?)
    .
    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)
       
    (c) 2008 dansmath.com

     
    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.
     
    (c) 2008 dansmath.com
     
 
 
 

 

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.

        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.

        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.)
 
 
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