Gathering integers, fractions, and decimals into one family — and lining them up by size.
Point 2 of 5 in this lesson: 5.3.2 Sorting the family: by kind and by sign
You have met them all by now: the counting numbers, the integers that reach below zero, the fractions that fall between the whole numbers, and the decimals that do the same with tenths and hundredths. Here is the quiet surprise of this lesson — they are not four separate kinds of number at all. They are one big family with a single family name: the rational numbers. And every last one of them has a home on the number line.
By the end you will be able to say exactly what makes a number rational, sort a messy pile of them by kind and by sign, and compare any two of them at a glance. We keep one steady habit of color: positive numbers are teal, negative numbers are red, zero is slate, and an absolute value — a distance from zero — is amber.
Here is the whole idea in one line: a rational number is any number you can write as a ratio of two integers — one integer over another — as long as the bottom one is not 0. We write that shape as pq, where p and q are integers and q ≠ 0. The word "rational" comes straight from "ratio" — it has nothing to do with being sensible.
The beautiful part is how many old friends fit this one shape:
• Every integer fits — just put it over 1. 5 = 51, −4 = −41, 0 = 01.
• Every fraction already is this shape: −34, 72.
• Every decimal that stops fits: 0.5 = 12, 0.25 = 14.
• Even a decimal that repeats forever fits: 0.333… = 13.
A number is rational exactly when it can be written as pq with p and q integers and q ≠ 0. Integers, fractions, stopping decimals, and repeating decimals all pass this test — and every one of them lands on the number line.
A few special numbers, like √2 = 1.41421356…, never stop and never settle into a repeating pattern, so they can never be written as a clean pq. Those rebels are called irrational, and you will meet them in Stage 6. Everything in this lesson is rational.
Step through a handful of numbers. For each one, see its pq form, which ring of the family it belongs to, and where it sits on the line.
When you have a pile of rationals on the table, two sorting questions tidy them up fast.
Sort by KIND. Is the number a whole-number-sized integer (like 3 or −4), or does it fall between the integers as a non-integer fraction or decimal (like −34 or 2.5)? Integers land exactly on the labelled ticks; everything else lands in the gaps between them.
Sort by SIGN. Is the number positive (right of zero), negative (left of zero), or is it zero itself? This is the split we lean on hardest when comparing size, so it is worth doing quickly and surely.
Sort −7, 0.2, 0, −12, 5 by sign.
Negative: −7 and −12 (both left of zero). Zero: 0. Positive: 0.2 and 5 (both right of zero). The size of the number does not matter for this sort — only which side of zero it lands on.
A number appears. Tap Negative, Zero, or Positive to file it. The number line shows you where it really lives, so you can check yourself.
Once two numbers sit on the line, comparing them needs no arithmetic at all — just look. Here is the one rule that runs all of Stage 5:
The farther right a number sits, the bigger it is. So a number on the right is greater than any number to its left, and a number on the left is less than any number to its right. The line is already sorted smallest-to-largest, left to right.
The two symbols that record this are < ("is less than") and > ("is greater than"). A handy memory trick: the small pinched point always aims at the smaller number, and the wide open mouth gapes at the bigger one. So −2 < 1 reads "−2 is less than 1," and that matches the picture — −2 is to the left.
Move A and B along the tenths. Whichever sits farther right is bigger — watch the correct <, =, or > appear between them.
The "look at the line" rule has three consequences so reliable you can use them without drawing anything:
① Every positive number is greater than 0 — it lives to the right of the origin.
② Every negative number is less than 0 — it lives to the left.
③ So any positive beats any negative, no matter their sizes, because the whole positive side is to the right of the whole negative side — with 0 standing in the middle as the referee.
That is why a tiny positive crushes a giant negative: −100 < 0 < 0.01. The number 0.01 is barely a whisper above zero, yet it still beats −100, which sits a hundred steps to the left. Position on the line, not loudness of the digits, decides the winner.
Positive > 0 > negative — always. You never have to compare the actual sizes of a positive and a negative number; the signs alone decide it. Only when both numbers share a sign do you look closer — which is the next section.
Now the famous trap. Which is bigger, −5 or −2? Almost everyone's first instinct shouts "−5, because 5 is bigger than 2." That instinct is wrong, and the number line shows exactly why.
Recall from Lesson 5.2 that the absolute value of a number is its distance from zero: |−5| = 5 and |−2| = 2. With two negatives, a bigger absolute value means the number sits farther to the left — and farther left means smaller. So:
−5 < −2 because |−5| = 5 is bigger than |−2| = 2
For positives, bigger digits mean a bigger number. For negatives it flips: the bigger the absolute value, the smaller the number. Think of temperature — −5° is colder (lower, smaller) than −2°, even though 5 > 2. When in doubt, trust the number line: more negative means farther left means smaller.
Set two negative numbers. The widget shows each one's absolute value, points them on the line, and tells you which is the smaller — proven by which sits farther left.
Integers, fractions, and stopping-or-repeating decimals are all one family — the rational numbers, every one writable as a ratio pq of integers with q ≠ 0 — and every one lives on the number line. You can sort them by kind (integer vs. in-between) or by sign (positive / negative / zero). To compare any two, just look: farther right is bigger. So every positive beats 0 beats every negative; and when both are negative, the one with the bigger absolute value is actually the smaller number — because it sits farther left.
Now that the whole family is lined up and ordered, you will learn to add and subtract them. The number line stays your guide: adding a positive walks you right, adding a negative walks you left.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson gathers integers, fractions, and decimals into the rational numbers and develops ordering, aligned to the U.S. Common Core grade-6 number-system standards. Students locate rationals on the number line and recognize opposites and sign structure (6.NS.C.6), write and interpret inequalities such as −3 > −7 as statements about relative position (6.NS.C.7a), and interpret order in real-world contexts like temperature — e.g. understanding −8°F < −3°F as "−8 degrees is colder" (6.NS.C.7b). The deliberate emphasis on the number line — rather than symbol rules alone — is what lets students see that any positive outranks any negative and that more-negative means smaller. The single most common misconception is "−5 > −2 because 5 > 2" (whole-number bias applied to negatives). The matching antidote, repeated in Section 5.3.5 and the quiz: more negative means smaller — trust the number line.