Find unit fractions of sets using addition facts.
Number Framework Stage 5
Using Materials
Problem: “Two birds, Bertie and Beatrice, pulled a large worm out of the ground.
They measured it to be 10 cubes long. Here it is.” Show the students a length of 10
cubes joined in fi ve breaks. “If each bird gets one-half of the worm, how much will
they get?”
At this stage, the students should be able to apply their doubles knowledge (5 + 5)
to solve the problem because halving is similar to adding two equivalent sets. Ask
them how the answer could be checked using the cubes (by splitting off the halves
and aligning them).
Pose related problems so that the students can generalise the equal sharing nature of
fractional numbers and how they can apply additive strategies. For example:
“Four birds shared a 20-cube worm. They each got one-quarter.
How much of the worm was that?” (Vary the number of birds.)
“Three birds caught a 15-cube worm. They ate one-third each.
How much was that?”
“Two birds caught a 12-cube worm. One bird got two-thirds. The other bird got the
rest. How much worm did each bird get?” (Vary the unit fractions.)
Record the answers using symbols, for example, 13
of 15 is 5, or in words, for example, one-third of fifteen is five.
Using Imaging
Shielding and predicting: Make a worm of 18 cubes in fi ve breaks. Show it to the
students and ask them how long the worm is. Expect them to use fi ves and tens to
shorten the counting. Hide the worm under a sheet of paper.
“Along come three birds, and they pull this worm out of the hole. If each bird gets
one-third, how much will they eat?”
Look for the students to use additive strategies such as:
“Five and fi ve is 10. Another fi ve is 15. So they can have fi ve cubes each. That leaves three, so they can have one more each. That’s six altogether.”
“Three and three and three are nine. Nine and nine is 18. So each bird will get three
and three. That equals six cubes.”
Students’ predictions about the equal parts can be tried by splitting the worm under
the masking paper and revealing the parts one at a time.
Record the answer as symbols, i.e., 1/3 of 18 is 6 or 1/3 x 18 = 6.
Pose similar problems with the variations from Using Materials above.
For example:
“There are three birds. The hidden worm is 12 cubes long. One bird gets one-half.
Another bird gets one-third. The other bird gets the rest. How much does each bird
get?”
Using Number Properties
Provide problems that involve finding fractions of a whole number. This encourages
the use of number properties.
These can be presented in story and symbol form, e.g., word stories for 1/5 of 20 is □, 1/6 of 24 is □, 2/3 of 12 is □.
Extensions
For students who make strong progress towards using multiplicative strategies on
the worm and birds problems, provide examples that have remainders.
For example, “Three birds share a worm that is 16 cubes long. Each bird gets one third of the worm. How much is that?”
Students will need to recognise that each bird gets fi ve whole cubes of worm and the
remaining cube is cut into thirds. So 1/3 of 16 is 5 1/3 (one-third of sixteen is fi ve wholes and one-third).
This will strengthen the place of fractions as both operators and as numbers among
the whole numbers.
Other examples might be:
“Four birds each get one-quarter of an 18-cube worm.”
“Five birds each get one-fi fth of a 23-cube worm.”
“Three birds each get one-third of a 14-cube worm.”