dansmath.com > the rest of dansmath -- **[ home | info | meet dan | ask dan | matica | lessons | dvc ]**Patterns & Numbers (back to top)__Finding Math in the World__/*"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.*/ by School or Difficulty Level__Puzzle Problems__**:**(back to top)*(c) 2003-2008 dansmath.com**Problems are labeled: Elementary, Middle, High ; either 'school' or 'difficulty' level!**No answers are provided, that makes it more of a 'treasure hunt' for you.*

**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)**

. . from www.dansmath.com . . (back to top)__Math Activities and Ideas for Parents__

**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?

(back to top)__Lesson Plans and Ideas for Teachers__8/08*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 ]**

browse my basic math lessons__Younger Kids:__Counting, Primes, Divisors, Proportions -*- 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*. .**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* - browse my free math lessons on these topics!__The Usual Topics__: Arithmetic, Algebra, Geometry, Trigonometry*- 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.*

(of any sizes)*Geometry*. 1. How many squares**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?**

(or 1/3 etc.)*Trigonometry*1. The height of a person 5/8 of the way around**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*

browse my feature pages on these topics!__Some Tangential Topics__: Number Theory, Tessellations, Polyhedra*- 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?)

**.**(cubes, pyramids, soccer balls, geodesic climbing bars)*Polyhedra*. 1. Find various polyhedra out in the world**. 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.)

(back to top) **[ home | info | meet dan | ask dan | matica | lessons | dvc ]**This site maintained by B & L Web Design, a division of B & L Math Enterprises