• 4-Quadrant Graphing (image from http://math.tutorvista.com/geometry/quadrant.html).  Points , also know as ordered pairs, are always written (x, y).

SCALE - the distance between each hashmark on a number line or axis. This is not necessarily the same number that is labeled.

Example 1: scale is 1/4 and labeled by fourths also.

Example 2: scale is 1/4, but label is every 1/2.

Example 3: scale is 1/8, but every 1/4 is labeled.

Changing repeating decimals to fractions:

Method 1

Method 2

Positive Numbers - Numbers greater in value than 0. Positive numbers are to the right or above 0 on number lines.

Negative Numbers - Numbers less in value than 0. Negative numbers are to the left or below 0 on number lines.

**0 is niether positive nor negative**

Opposites - numbers that are the same distance from 0 but on other sides of 0.

Absolute value - the distance a number is from 0. Since absolute value is a distance, it is always positive.

BENCHMARK FRACTIONS/DECIMALS/PERCENTS

Hints for turning fractions to decimals:

Here is a short instructional video to help with long division.

STRATEGIES FOR CONVERTING FRACTIONS TO DECIMALS

More examples:

TO CONVERT DECIMALS TO FRACTIONS  - use place value. Then simplify your fractions.

image from /blog.udemy.com/how-to-use-a-decimal-place-value-chart/

Examples:

Video on converting between different bases

Note scale is 1/8 on strategy #4

Generalizations

Addition is commutative - you can change the order of the addends without changing the sum. This works since it does not matter what order you place chips on a board, the board will have the same value and the same number of zero pairs.

Pos + Pos = pos because you only move right of zero on a number line or you only have black chips (no zero pairs).

Neg + neg = neg because you only move left of zero on a number line or you only have red chips (no zero pairs).

Pos + neg = neg + pos (commutative property) and can have a pos, neg or 0 sum

If absolute value of pos> absolute value of neg, then pos + neg = pos because there are more black than red chips so after the zero pairs only positive values remain or you move more right than left of zero so end on the right side of zero.

If absolute value of pos< absolute value of neg, then pos + neg = neg because there are more red than black chips so after the zero pairs only negative values remain or you move more rleft than right of zero so end on the left side of zero.

If absolute value of pos= absolute value of neg, then pos + neg = 0 because there are the same number of black as red chips so all chips are in zero pairs  or you move right and left of zero the same amount so return to 0. (Opposites add to zero)

1) Combining like terms (cows, common denominators, chips).

2) Direction and distance on a number line - start, move, end.

Subtraction

1) Take away - chip model.

2) Opposite direction of addition on a number line- start, move opposite direction of addition, end.

3) Distance apart with direction (larger - smaller = positive, smaller - larger = negative).

Multiplication

Models for adding and multiplying proper fractions

Multiplying Mixed Numbers