Fractions

1A fraction names equal parts of one whole

A fraction is coloured equal parts over how many equal parts the whole was cut into. Unequal pieces are not this writing.

A pretty shade without equal parts is a drawing, not a fraction.

How it works

1. Name the wholeThis bar, this roti.
2. Cut equal partsSame amount each.
3. Write coloured over partsThat pair is the fraction.

2The wholes must match before you compare

\frac{1}{2} of this bar versus \frac{1}{2} of a tiny square are two different amounts of chocolate. Compare fractions of the same whole, or say the wholes first.

A bigger piece of a smaller whole can still be a smaller amount.

How it works

1. Lock the wholeSame bar, or name both wholes.
2. Then compare the sharesWhich covers more of that whole?
3. If wholes differ, do not rank the fractions as amountsTwo stories.

3Playing with a grid: shade a named part

A 3-by-4 grid has 12 equal cells. Shading 4 cells can be \frac{4}{12} of that grid. The name follows the equal cells.

Shading any doodle that is not a cell-count is not the named part.

How it works

1. Count equal cellsThe denominator if you shade cells.
2. Shade the asked numberThe numerator.
3. Write the fraction of this gridColoured over 12.

4Equivalent fractions are two writings of the same amount

\frac{1}{3} and \frac{2}{6} of the same bar can cover the same region. The writing changed; the amount did not.

New writing is not a new extra piece.

Figure. One-half and two-fourths fill the same width of the same whole. Equivalent fractions are two writings of one amount, not two amounts.

How it works

1. Fix the wholeSame bar.
2. See two writings1 of 3, or 2 of 6.
3. Check they cover the same regionThen they are equivalent.

5One-third and two-sixths can name the same shade

Split each third into two sixths. One third becomes two sixths of the same bar.

Do not add a new bar when you rewrite.

Figure. One-third and two-sixths shade the same span of one whole. The extra cuts rewrite the name; they do not grow the shade.

How it works

1. Start with thirds3 equal parts.
2. Split each third in two6 equal parts.
3. See 1 third = 2 sixthsSame shade.

6A bigger piece of a smaller chocolate is not a bigger fraction

Half of a tiny bar can be smaller than a quarter of a family bar. Fraction compares shares of a named whole, not raw piece-size across wholes.

Always name the whole before you say “bigger fraction”.

How it works

1. Name whole A and whole BTwo chocolates.
2. Name each share1/2 of A, 1/4 of B.
3. Do not rank the pieces without the wholesTwo amounts.

7Same amount, new writing

You may write \frac{2}{4} or \frac{1}{2} for the same half of one sheet. Pick the writing the question asks for.

Changing writing is not adding chocolate.

Figure. Two-thirds and four-sixths are the same amount written two ways. New writing, same shade — the whole did not change.

How it works

1. Fix the amountThis shaded region.
2. Allow a new pair of numbersEquivalent fraction.
3. Check the region againSame, or you rewrote wrong.

8Adding only when the parts match

\frac{1}{6} + \frac{2}{6} = \frac{3}{6} of the same whole. Different denominators need a rewrite first.

Adding \frac{1}{2} of one bar to \frac{1}{3} of another is two wholes.

Figure. Add fourths to fourths. 1/4 + 1/4 = 2/4 because the parts match. Unlike parts must be rewritten before they may join.

How it works

1. Same wholeDo not swap the bar.
2. Same part-size, or rewriteSixths with sixths.
3. Add numeratorsDenominator stays the part-size.