Multiplying That Results in Improper Fractions
It may be easier in recipe making, to take a 1/3 measuring cup and fill it 4 times without explanation. They would write the math sentence down as: 2/3 x 2 = 4/3.
To fully show them how they can double 2/3, they should take the 1/3 measuring cup, twice and fill a larger container, the 1 cup. They will be converting 4/3, the improper fraction to 1 and 1/3.
They can take the 1/3 measuring cup and fill a 1 cup container 3 times, then finding they had another 1/3 cup, they can visualize and demonstrate that 4/3 means filling a 1 cup measuring cup once with 1/3 cup measuring cup also.
They may then understand that a fraction can be doubled by multiplying the numerator only.
Multiplying Mixed Numbers
To see that 1 and ½ cups when doubled means 3 cups, they can fill both a 1 cup and a ½ cup measuring cup and put it into a larger measuring cup. When they do it twice, they will see that it makes 3 cups.
They can see also that you can easily double a whole number. So they can double the 1 cup to make 2 cups. Then they can double the ½ cup by filling it up twice and seeing that it makes 1 cup. By writing the math sentence ½ x 2 they can see that it makes 2/2. The numerator, is doubled. This is the only way it can be 1 cup.
Doubling or tripling recipes means taking the fractions and multiplying the numerators. The students will see that it is easy to multiply the whole numbers and then multiply the numerators of the fractions. Doubling 1 ½ cups made 3 cups.
If they would like to show a fraction written in its improper form such as writing 1 ½ cups as 3/2, this may help them to multiply the numerator only. 3/2 doubled will be 6/2. This is also 3 cups and means dividing the 6 by 2. Students will realize that mixed fractions are actually easier to double when you first multiply the whole numbers, then the fractions.