Rational numbers
Integers, fractions, negatives and decimals each came with their own rules. Is there one kind of number that takes them all in, and how do you work with it?
Where you are 10 inside
What this area is about
A rational number is one that can be written \displaystyle \frac{a}{b} with a and b integers and b \ne 0. Integers, fractions, mixed numbers and decimals that stop or repeat all qualify, on both sides of zero. This area joins them into one family on one number line, and does arithmetic with signed fractions and decimals.
How these fit together
What they are. What makes a number rational gives the test. Negative fractions and decimals places the negative ones left of zero.
As decimals. Converting between fractions and decimals divides top by bottom. Terminating and repeating decimals adds the bar for digits that repeat forever, and turning a repeating decimal into a fraction runs it back. A result: a fraction in lowest terms terminates exactly when its denominator has no prime factors but 2 and 5.
Procedures. Comparing and ordering, adding and subtracting, multiplying and dividing, and rational numbers in real situations.
A path through it
The definition and negatives first, then comparing. Converting to decimals comes before repeating decimals, which come before turning them back into fractions; which fractions terminate can follow the converting. Adding and subtracting comes before multiplying and dividing and before the word problems.
Finish here and you can sort a number into its sets and see why \displaystyle \sqrt{2} is not rational.