Thinking about Math

Here are some things I learned about math since homeschooling.

*Zero*
Zero is not merely nothing. Zero is the balance point between positive and negative numbers. Zero itself is neither positive nor negative.

*Multiplication is NOT Repeated Addition*
Multiplication is not repeated addition. Repeated addition works for whole numbers, but not fractions. I like to think of multiplication as groups of objects. For 2 x 3, start with a three, and take two of them: 6. For 1/2 x 3, start with a three, and take half of it: 1 1/2. For 1/2 x 1/3, start with a third and take half of it: 1/6.

*Are there more even numbers or whole numbers?*
I guess this is obvious, but I never thought of it before. There are half as many even numbers as whole numbers. Yet, there are an infinite number of both.

*A Parallelogram IS a Trapezoid*
And along those lines, a square is a rectangle. All the formulas and propertes that apply to a rectangle also apply to a square. All the formulas and properties that apply to a trapezoid apply to a parallelogram.

*Algebra Manipulatives Exist*
There's a host of math manipulatives for early elementary math: addition, subtraction, multiplication, division, place value, fractions, decimals, etc.
I don't remember using them in school, but they are easy to find nowadays. Slightly less common are manipulatives for algebra. My favorite algebra manipulatives are the ones used in Hands-On-Equations, and algeblocks.

*The Distributive Property is Very Useful*
I learned about FOIL (first, outside, inside, last) in algebra class when multiplying things like (x+3)(x+4). The manipulatives for showing (x+3)(x+4) are also ideal for explaining 13*14 = (10+3)(10+4).
traditional algorithm: ((4*3) + (4*10)) + ((10*3)+(10*10))
FOIL: (10*10)+(10*4)+(3*10)+(3*4)

*Cuisenaire Rods shouldn't have notches*
Dear me, other people have written at length about this topic. Suffice it to say that I am firmly in the Cuisenaire Rods should't have notches camp.

*Grading math worksheets is BORING*
I guess this also goes without saying. However, I never had to do it before homeschooling and I now actively look for ways to avoid grading math worksheets.