Use proportional reasoning to make reasonable estimates
Understanding proportional reasoning requires students to consider a number or quantity in relative terms. With fractions, students must consider the fraction in context, such as one half of a whole figure. Students use proportional reasoning to partition a whole into unit fractions.
A unit fraction is the base unit of any fraction and always has a numerator of 1; for example,
1 / 4
1 / 5
1 / 23
3 / 4
Consider the fraction one and three-fourths. This number can be decomposed using a unit fraction.
one and three fourths
seven one-fourths units
One and two-fourths can be composed using a unit fraction.
A student may say, “One whole is the same as 4 one-fourth units. I added another 2 one-fourth units to the whole to obtain 6 one-fourth units. So I can see that 6 one-fourth units is equal to one and two-fourths.
Use of unit fractions supports a deeper understanding of quantity. Notice that in the student dialogue above, early understanding of equivalency is being developed, i.e., one and one half is the same as six fourths. Counting by naming the unit fractions helps students to see the parts of the fraction when composing and decomposing. Notice that both counting unit fractions and composing and decomposing fractions are pre-cursors to addition and subtraction. For example, composing 6 one-fourth units is the same as adding 6 one-fourth units together to make one and one half.
Walk the Line
Students actively equi-partition a number line using different fractional units (e.g., halves, fifths) as they place mixed and improper fractions. Students will enjoy walking, jumping or using every day classroom items as a method of kinaesthetically partitioning a number line on the floor. This task becomes increasingly complex based upon the sets of fractions used.