A Home Schooling Notebook: Cool Math

Showing posts with label Cool Math. Show all posts

Saturday, November 8, 2014

How to figure out 36% of 25 or 250% of 20

Quick, what’s 36% of 25? Or how about 250% of 20?

Learn a quick tip to help you calculate all of those pesky percentages in your head.

36% of 25 is the same as 25% of 36.
How does this help?
Well, 25% (or 1/4) of 36 is a whole lot easier to figure out.

In other words: 36/4 which equals 9

25% of 36 must b

AND SO . . . . 36% of 25 is 9. Pretty cool, huh?

250% of 20? Easier to do if you flip it.

20% of 250.
Because 10% is so easy to figure out, I always tell myself that 20% is 10% twice or . . . . 25 + 25 which is 50.
AND SO . . . . 250% of 20 is 50. Pretty cool again.

Math Tricks are fun!

http://www.quickanddirtytips.com/education/math/how-to-quickly-calculate-percentages

t’s helpful to take a minute to see why this seemingly magical trick works. As I mentioned before, 36% is equivalent to the fraction 36/100. Since we can write the fraction 1/100 as the decimal number 0.01, we see that the fraction 36/100 can also be written 0.01 x 36. That means that 36% of 25 must be equal to (0.01 x 36) x 25. Now here comes the cool part: The associative property of multiplication tells us that we can multiply several numbers in any order we’d like. Which means that (0.01 x 36) x 25 = (0.01 x 25) x 36. But 0.01 x 25 is the same thing as 25%, which means that this is equal to 25% of 36. So 36% of 25 is equal to 25% of 36. It ain’t magic, it’s math! Pretty cool, right? - See more at: http://www.quickanddirtytips.com/education/math/how-to-quickly-calculate-percentages#sthash.eIOccEFG.dpuf

36% of 25 is the same as 25% of 36. How does that help us? Well, since 25% is the same as the fraction 1/4, we see that 25% of 36 must be 36/4 or 9. So 25% of 36 is equal to 9, and 36% of 25 must also be 9.
The beauty of this trick is that every time you’ve solved one problem, you’ve actually solved two! And that’s especially useful when one of the problems is much easier to solve than the other—as was the case here.
- See more at: http://www.quickanddirtytips.com/education/math/how-to-quickly-calculate-percentages#sthash.eIOccEFG.dpuf

Monday, September 5, 2011

Math Helps for Little Guys (and maybe adapt for older students, too)

Students who have difficulty memorizing math facts often have different ways of learning. The following activities are designed, not only to be fun, but also to provide opportunities to learn in different styles.

1. Let your child write the facts with water on the sidewalk, using either a plastic squirt bottle or a paintbrush in a bucket of water.

2. Empty a carton of salt into a gift box or shallow baking pan. Allow your child to write the facts with his finger in the salt. (This activity can also be done using yogurt or pudding.)

3. Use your finger to write a math fact on the child’s back. Ask him to identify the fact you have written and give you the answer.

4. Have your child use his finger to write the math fact in the air.

5. Let your child write his math facts on a chalkboard or marker board.

6. Have your child invent a motion for each number in the math fact (or teach him the numbers in sign language). Have him say the math fact out loud, making his motions for the numbers, and then tell you the answer.

7. Have the child do jumping jacks or other exercise while saying his math facts.

8. Cut out pieces of colored paper in different shapes (ex., red triangle, blue rectangle, green circle, yellow square). Do not repeat a shape or a color. Write a number on each shape. Have the child choose two of the numbers and give you the appropriate math fact that uses those numbers.

9. Have your child write the facts in color, writing the first number in red, then using different colors for the other letters.

10. Have your child invent a song for the math facts. (There are also many commercially-produced tapes of math facts set to songs. Public libraries often have these for loan.)

Friday, February 11, 2011

Ten Events in the History of the Universe

Ten events spanning the history of the universe organized by powers of ten.

	Date Range	Date	Event
10¹⁰	tens of billions of years ago	13-14 billion years ago	Universe began
10⁹	billions of years ago	4.5 billion years ago	Earth formed
10⁸	hundreds of millions of years ago	500 million years ago	First fossilized animals
10⁷	tens of millions of years ago	65 million years ago	Extinction of the dinosaurs
10⁶	millions of years ago	2 million years ago	First stone tools made
10⁵	hundreds of thousands of years ago	800 thousand years ago	First fires kindled
10⁴	tens of thousands of years ago	30 thousand years ago	First art
10³	thousands of years ago	5 thousand years ago	First writing
10²	hundreds of years ago	1492	Columbus: east meets west
10¹	tens of years ago	1969	First people on the moon

Memorize these ten dates as a framework for understanding deep time.

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Keith Enevoldsen's Think Zone

Physical Quantities

Quantity	Definition	Formula	Units	Dimensions
Length or Distance	fundamental	d	m (meter)	L (Length)
Time	fundamental	t	s (second)	T (Time)
Mass	fundamental	m	kg (kilogram)	M (Mass)
Area	distance²	A = d²	m²	L²
Volume	distance³	V = d³	m³	L³
Density	mass / volume	d = m/V	kg/m³	M/L³
Velocity	distance / time	v = d/t	m/s c (speed of light)	L/T
Acceleration	velocity / time	a = v/t	m/s²	L/T²
Momentum	mass × velocity	p = m·v	kg·m/s	ML/T
Force Weight	mass × acceleration mass × acceleration of gravity	F = m·a W = m·g	N (newton) = kg·m/s²	ML/T²
Pressure or Stress	force / area	p = F/A	Pa (pascal) = N/m² = kg/(m·s²)	M/LT²
Energy or Work Kinetic Energy Potential Energy	force × distance mass × velocity² / 2 mass × acceleration of gravity × height	E = F·d KE = m·v²/2 PE = m·g·h	J (joule) = N·m = kg·m²/s²	ML²/T²
Power	energy / time	P = E/t	W (watt) = J/s = kg·m²/s³	ML²/T³
Impulse	force × time	I = F·t	N·s = kg·m/s	ML/T
Action	energy × time momentum × distance	S = E·t S = p·d	J·s = kg·m²/s h (quantum of action)	ML²/T
Angle	fundamental	θ	° (degree), rad (radian), rev 360° = 2π rad = 1 rev	dimensionless
Cycles	fundamental	n	cyc (cycles)	dimensionless
Frequency	cycles / time	f = n/t	Hz (hertz) = cyc/s = 1/s	1/T
Angular Velocity	angle / time	ω = θ/t	rad/s = 1/s	1/T
Angular Acceleration	angular velocity / time	α = ω/t	rad/s² = 1/s²	1/T²
Moment of Inertia	mass × radius²	I = m·r²	kg·m²	ML²
Angular Momentum	radius × momentum moment of inertia × angular velocity	L = r·p L = I·ω	J·s = kg·m²/s ћ (quantum of angular momentum)	ML²/T
Torque or Moment	radius × force moment of inertia × angular acceleration	τ = r·F τ = I·α	N·m = kg·m²/s²	ML²/T²
Temperature	fundamental	T	°C (celsius), K (kelvin)	K (Temp.)
Heat	heat energy	Q	J (joule) = kg·m²/s²	ML²/T²
Entropy	heat / temperature	S = Q/T	J/K	ML²/T²K
Electric Charge +/-	fundamental	q	C (coulomb) e (elementary charge)	C (Charge)
Current	charge / time	i = q/t	A (amp) = C/s	C/T
Voltage or Potential	energy / charge	V = E/q	V (volt) = J/C	ML²/CT²
Resistance	voltage / current	R = V/i	Ω (ohm) = V/A	ML²/C²T
Capacitance	charge / voltage	C = q/V	F (farad) = C/V	C²T²/ML²
Inductance	voltage / (current / time)	L = V/(i/t)	H (henry) = V·s/A	ML²/T²
Electric Field	voltage / distance force / charge	E = V/d E = F/q	V/m = N/C	ML/CT²
Electric Flux	electric field × area	Φ_E = E·A	V·m = N·m²/C	ML³/CT²
Magnetic Field	force / (charge × velocity)	B = F/q·v	T (tesla) = Wb/m² = N·s/(C·m)	M/CT
Magnetic Flux	magnetic field × area	Φ_M = B·A	Wb (weber) = V·s = J·s/C	ML²/CT

Note: Other conventions define different quantities to be fundamental.

Mass, energy, momentum, angular momentum, and charge are conserved, which means the total amount does not change in an isolated system.

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Keith Enevoldsen's Think Zone

Numbers

Reds: Real number sets

Blues: Imaginary number sets

Purples: Complex number sets

Real Number Sets

Natural, N

Natural numbers are the counting numbers {1, 2, 3, ...} (positive integers) or the whole numbers {0, 1, 2, 3, ...} (the non-negative integers). Mathematicians use the term "natural" in both cases.

Integer, Z

Integers are the natural numbers and their negatives {... −3, −2, −1, 0, 1, 2, 3, ...}. (Z is from German Zahl, "number".)

Rational, Q

Rational numbers are the ratios of integers, also called fractions, such as 1/2 = 0.5 or 1/3 = 0.333... Rational decimal expansions end or repeat. (Q is from quotient.)

Real Algebraic, A_R

The real subset of the algebraic numbers: the real roots of polynomials. Real algebraic numbers may be rational or irrational. √2 = 1.41421... is irrational. Irrational decimal expansions neither end nor repeat.

Real, R

Real numbers are all the numbers on the continuous number line with no gaps. Every decimal expansion is a real number. Real numbers may be rational or irrational, and algebraic or non-algebraic (transcendental). π = 3.14159... and e = 2.71828... are transcendental. A transcendental number can be defined by an infinite series.

Real Number Line

Real Number Venn Diagram

N ⊂ Z ⊂ Q ⊂ A_R ⊂ R

Complex Number Sets

Imaginary

Imaginary numbers are numbers whose squares are negative. They are the square root of minus one, i = √−1, and all real number multiples of i, such as 2i and i√2.

Algebraic, A

The roots of polynomials, such as ax³ + bx² + cx + d = 0, with integer (or rational) coefficients. Algebraic numbers may be real, imaginary, or complex. For example, the roots of x² − 2 = 0 are ±√2, the roots of x² + 4 = 0 are ±2i, and the roots of x² −4x +7 = 0 are 2±i√3.

Complex, C

Complex numbers, such as 2+3i, have the form z = x + iy, where x and y are real numbers. x is called the real part and y is called the imaginary part. The set of complex numbers includes all the other sets of numbers. The real numbers are complex numbers with an imaginary part of zero.

Complex Number Plane

z = x + iy, i = √−1

Complex Number Venn Diagram

N ⊂ Z ⊂ Q ⊂ A_R ⊂ R ⊂ C

Properties of the Number Sets

	Natural N	Integer Z	Rational Q	Real R	Algebraic A	Complex C
Closed under Addition¹	x	x	x	x	x	x
Closed under Multiplication¹	x	x	x	x	x	x
Closed under Subtraction¹		x	x	x	x	x
Closed under Division¹			x	x	x	x
Dense²			x	x	x	x
Complete (Continuous)³				x		x
Algebraically Closed⁴					x	x

Closed under addition (multiplication, subtraction, division) means the sum (product, difference, quotient) of any two numbers in the set is also in the set.
Dense: Between any two numbers there is another number in the set.
Continuous with no gaps. Every sequence that keeps getting closer together (Cauchy sequence) will converge to a limit in the set.
Every polynomial with coefficients in the set has a root in the set.

The complex numbers are the algebraic completion of the real numbers. This may explain why they appear so often in the laws of nature.

Infinity, ∞

The integers, rational numbers, and algebraic numbers are countably infinite, meaning there is a one-to-one correspondence with the counting numbers. The real numbers and complex numbers are uncountably infinite, as Cantor proved.

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Keith Enevoldsen's Think Zone

Math Gems - An Assortment of Mathematical Marvels

Fun arithmetic with the number nine.	Fun arithmetic with the number seven.	A magic square. All rows, columns, and diagonals have the same sum.
The ratio of the circumference of a circle to its diameter is pi. Pi is transcendental, i.e., irrational and non-algebraic.	Area and volume formulas. Archimedes solved the sphere.	Pi, expressed as an infinite series and an infinite product.
The sum of the numbers from 1 to n.	The product of the numbers from 1 to n is n factorial.	Stirling's approximation of n factorial. Euler's gamma function gives factorials for integers but has surprising values for fractions.
A prime number is divisible only by one and itself. The sieve of Eratosthenes finds primes.	The prime number theorem of Gauss and Legendre approximates the number of primes less than x.	The zeta function of Euler and Riemann, expressed as an infinite series and a curious product over all primes.
The binomial theorem expands powers of sums. The binomial coefficient is the number of ways to choose k objects from a set of n objects, regardless of order.	Pascal's triangle shows the binomial coefficients.	Proof that the square root of two is irrational.
The quadratic equation defines a parabola.	The Pythagorean theorem. A proof by rearrangement.	The trigonometric functions. Another form of the Pythagorean theorem.
The golden ratio, phi. The ratio of a whole to its larger part equals the ratio of the larger part to the smaller. phi is irrational and algebraic.	The golden rectangle, a classical aesthetic ideal. Cutting off a square leaves another golden rectangle. A logarithmic spiral is inscribed.	The pentagram contains many pairs of line segments that have the golden ratio.
The golden ratio, expressed as a continued fraction.	Each Fibonacci number is the sum of the previous two. The number of spirals in a sunflower or a pinecone is a Fibonacci number.	The ratio of successive Fibonacci numbers approaches the golden ratio. An exact formula for the nth Fibonacci number.
Napier's constant, e, is the base of natural logarithms and exponentials. e is transcendental.	Calculus, developed by Newton and Leibniz, is based on derivatives (slopes) and integrals (areas) of curves. The derivative of e^x is e^x. The integral of e^x is e^x.	e, expressed as a limit and an infinite series.
Euler's formula relating exponentials to sine waves. A special case relating the numbers pi, e, and the imaginary square root of -1.	The Gaussian or normal probability distribution is a bell-shaped curve.	Gibbs's vector cross product. Del operates on scalar and vector fields in 3D, quad in 4D.
The five regular polyhedra. Euler's formula for the number of vertices, edges, and faces of any polyhedron.	The hypercube. Schläfli's formula for vertices, edges, faces, and cells of any 4-dimensional polytope.	The Möbius strip has only one side. The Klein bottle's inside is its outside.
Fractals of Mandelbrot, Koch, and Sierpinski have infinite levels of detail.	Cantor's proof that the infinity of real numbers is greater than the infinity of integers.	Gödel proved that if arithmetic is consistent, it must be incomplete, i.e., it has true propositions that can never be proved.