Fractions

1Fractional Units and Equal Shares

A fractional unit is one equal part of a named whole: \frac{1}{5} is one of five matching parts. Fair share still means equal first.

Unequal pieces are not fractional units of that whole.

How it works

1. Name the wholeThis bar, this length.
2. Cut equal partsThe unit size.
3. One such part is the fractional unit1/n of that whole.

2Fractional Units as Parts of a Whole

\frac{3}{5} means three of those fifths of the same whole. The denominator names the unit; the numerator counts how many units.

Swapping them names a different share.

How it works

1. Name the unitFifths of this whole.
2. Count how many units you take3.
3. Write count over unit-name3/5.

3Measuring Using Fractional Units

A length can be 2 whole units and 3 fifth-units of the same unit-length. That is a measure, not a new whole.

Mixing a fifth of a short bar with a fifth of a long bar is two units mashed.

How it works

1. Lock the unit-lengthThis span is 1.
2. Count wholes, then fractional units of that spanA measure.
3. Keep the same spanDo not swap bars mid-measure.

4Marking Fraction Lengths on the Number Line

On a number line, \frac{3}{4} sits 3 hops of \frac{1}{4} from 0 toward 1. Equal hops. 1 is four fourth-hops.

A mark that is “about 3/4” without equal hops is a guess.

How it works

1. Split 0 to 1 into n equal hopsThe denominator.
2. Walk k hops from 0The numerator.
3. Land on that markk/n.

5Mixed Fractions

A mixed number is wholes plus a proper fraction: 2\frac{1}{4} = 2 + \frac{1}{4} = \frac{9}{4}. The mix is a writing, not extra chocolate.

Writing 21/4 as 2 1/4 without converting is a mash of digits.

How it works

1. Count the wholes2.
2. Name the leftover proper part1/4.
3. Convert if asked: 2\frac{1}{4}=\frac{8}{4}+\frac{1}{4}=\frac{9}{4}Same amount.

6Equivalent Fractions

Same point on the number line, or same region of the same whole: \frac{1}{2}=\frac{2}{4}=\frac{3}{6}. Multiply or divide numerator and denominator by the same nonzero number.

New writing is not a new extra piece.

How it works

1. Fix the amountThis region or this point.
2. Scale both parts of the writing by the same nEquivalent.
3. Check the region or the point againSame, or you scaled wrong.

7Comparing Fractions

Compare after the wholes match and the writings use a common unit (common denominator), or compare to the same benchmark (½). \frac{2}{3}>\frac{1}{2} of the same whole.

A bigger piece of a smaller whole is a different story.

How it works

1. Lock the wholeSame bar or same 0–1.
2. Rewrite to the same denominator, or use a benchmarkA fair compare.
3. Then read which is largerMore units of that size.

8Addition and Subtraction of Fractions

Add or subtract when the part-size matches: \frac{1}{5}+\frac{2}{5}=\frac{3}{5}. Different denominators need equivalent fractions first. Same whole.

Adding 1/2 of one bar to 1/3 of another is two wholes.

Figure. Add only when the parts match. One-fourth plus two-fourths is three of the same fourths, not a new kind of piece.

How it works

1. Same wholeDo not swap.
2. Same denominator, or rewriteMatching units.
3. Add or subtract numeratorsDenominator stays the unit size.