Place Value and Ordering Numbers on the Number Line

Every digit in a number carries two pieces of information: what it is, and where it sits. The digit 7 in 0.07 sits in the hundredths place, so it is worth 7/100. In 70 it sits in the tens place, worth 7 tens. In 7000 it sits in the thousands place, worth 7 thousands. Same symbol, three completely different sizes, because the place multiplies the digit.

To order fractions, decimals and mixed numbers on one number line, put everything in the same format first. Decimals are easiest to line up: write each number with the same number of digits after the point, padding with zeros, then compare left to right, place by place, starting with whichever digit carries the most weight. Fractions convert to decimals by dividing the numerator by the denominator, or by finding a common denominator so they all compare directly. Once everything speaks the same language, plotting them in order on the line is just a matter of reading off which is smallest to largest.

This skill sits underneath almost every number and quantity item: estimating square roots, locating irrational numbers between consecutive integers, comparing signed numbers, reasoning about absolute value and distance. Get place value solid and ordering any mix of numbers, however they are dressed up, becomes mechanical rather than a guessing game.