Same Size Pieces

Break a fraction into pieces, then add, take away and repeat them. The piece never changes size.

Book Equivalent fractions and decimals 4.NF.B.3a4.NF.B.3b4.NF.B.3c4.NF.B.4b ALPACA · Elementary operations

New to fractions? Start here. A short lesson where one bar gets cut up while you watch, so you can see for yourself that one half and two quarters are the same amount. About about 5 minutes.

Show me how fractions work → Why are one half and two quarters the same?

▤ Print this All of 4th grade

For grown-ups

How long: 10 to 15 minutes is plenty. Short and frequent beats long and occasional — the early-numeracy trials with the largest effects ran eight weeks or less.
When they get one wrong: the page shows the working. Read it together rather than moving straight on. Feedback that explains is worth several times more than feedback that only marks right or wrong, and that gap is wider in maths than in any other subject.
Not timed: thinking time is the point of a book. Let silences run.
Print it too: doing the same problems on paper a few days later is worth more than doing twice as many today.

What this practises

Adding, subtracting and multiplying fractions that share a denominator, and decomposing one into a sum.

This closes the one Illustrative Mathematics content unit in the whole K-5 map with no activity against it: 4.NF.B.3 appeared once in the repo, on a grade-5 game, and 4.NF.B.4 only in an SSDD set. The warrant for the bond is the standard itself — 4.NF.B.3b is literally "decompose a fraction into a sum of fractions with the same denominator in more than one way". Fuchs et al. (2013) is cited for exactly one job, and the limit matters: that trial’s winning arm taught the MEASUREMENT interpretation of fractions and its control condition focused more on part-whole work and on procedures, with part-whole improvement explicitly not mediating the effect. So it is the reason the crossing-a-whole stage is anchored on a number line, and it is not a warrant for part-whole work. A book that cited it the other way round would be inverting its own source.

The underlying idea: A fraction with a fixed denominator behaves like a unit that can be counted. Decomposition makes that countability visible, which is what 4.NF.B.3b asks for in so many words: decompose a fraction into a sum of fractions with the same denominator in more than one way.
Where this sits in Illustrative Mathematics: 4th grade Unit 3: Extending Operations to Fractions
Sources: Illustrative Mathematics (2024) · Illustrative Mathematics / Kendall Hunt (2024) · Fuchs et al. (2013) all references →

Related measured skill: ALPACA · Elementary operations. What this means