Fractions of a whole

Activities to try

1. 1

   Cut it fair, then count the unit

   Why it works: a/b is a copies of an equal part whose size is fixed by the denominator; shading the same NUMBER of parts does not give the same amount, because sixths are smaller than fourths of the same whole (so 2/6 < 2/4).

   1. Take one whole bar (a chocolate bar works well) and cut it into equal parts, e.g. 6, checking that the parts are genuinely equal in area before accepting them.
   2. Shade ONE part and name it as a quantity: 'this is one-sixth, 1/6'.
   3. Build 2/6, then 3/6, by copying that single 1/6 piece and counting as you go, saying 'two one-sixths, three one-sixths' and writing the label each time.
   4. Reset the sixths bar to show 2/6, then set an identical bar beside it, cut into 4 equal parts, and shade 2 parts on it (so you have 2/6 next to 2/4).
   5. Slide the two shaded regions over one another to compare how much chocolate each shows, and state which is more and why.
2. 2

   How big is ONE part? (vary the whole and the denominator)

   Why it works: The same fraction names the same relationship to its whole (and the same number) whatever picture is used, but the amount it shows depends on the size of the whole; for any given whole the denominator controls how many equal parts it is cut into (regardless of that whole's shape), and more equal parts of the SAME whole means each part is smaller (1/8 < 1/4).

   1. Build the SAME fraction across different wholes — a bar, a circle, a fraction strip, a length/number line, a set of objects (e.g. 2 of every 6), and Cuisenaire rods with one chosen rod as the whole.
   2. On a number line, make the fraction by repeating a jump of size 1/b, a times, instead of shading an area.
   3. Re-cut one bar into 8 equal parts and name one part (1/8); compare its size to one quarter of that same bar and state which is bigger and why.
   4. Draw a given fraction, e.g. 2/6, of the SAME whole a DIFFERENT way — differently shaped parts but the same amount.
   5. For each model, answer not just 'how many are shaded?' but 'how big is ONE part?'
3. 3

   Predict, then check: comparing same-numerator fractions

   Why it works: With the whole held fixed, more equal parts means each part is smaller, so 2/4 > 2/6 and 3/4 > 3/6 even though the counts match; comparing the AMOUNTS shown by two models is only fair when the wholes are the same size, though the fractions themselves, as numbers, remain comparable (3/4 > 1/2 always).

   1. Before cutting, look at two identical bars. Predict and give a REASON: if 2 parts are shaded on a bar cut into 4, and 2 parts on an identical bar cut into 6, which shows more chocolate — or the same? State the reason out loud BEFORE observing.
   2. Then cut both bars and slide the shaded regions over each other to check.
   3. Revise or confirm your prediction, explaining what the comparison showed.
   4. Build 3/4 from single pieces and say how many quarters and how big each one is.
   5. Tackle the trap question: could 3/4 of a small bar be LESS chocolate than 1/2 of a big bar? State what must be true about the wholes to compare the amounts fairly.