Compare wholes, repartition fractions, add unlike units, and distinguish two meanings of division.
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Use 12 cm wholes. Show 2/3 on strip A and 4/6 on strip B. How many centimeters are shaded on each?
Both wholes are 12 cm. Two thirds and four sixths each measure 8 cm; both shaded endpoints agree.
Watch for: Adding the same number to numerator and denominator.
Use a 12 cm whole for A and a 24 cm whole for B. Shade 1/2 of A and 1/4 of B. How many centimeters are shaded on each?
One half of 12 cm is 6 cm; one quarter of 24 cm is also 6 cm. Different fractions can describe equal lengths when wholes differ.
Watch for: Comparing fraction labels as physical amounts without checking the whole.
Use 12 cm wholes. Shade 1/3 on A and 1/4 on B. What fraction of one 12 cm whole is their combined length?
The amounts measure 4 cm and 3 cm. Together they cover 7 of the twelve 1 cm units: 7/12, not 2/7.
Watch for: Adding denominators as if thirds and fourths were the same-sized parts.
Set A to a 12 cm whole with 1/2 shaded. Share that shaded half equally among 3 children. What fraction of the original whole does each child get?
Half of 12 cm is 6 cm. Three equal shares are 2 cm each: 2/12 = 1/6 of the original whole. Check: 3 × 1/6 = 1/2.
Watch for: Treating the number of children as a serving size.
Set A to a 12 cm whole partitioned into fourths. There are 3 whole strips. How many 1/4-strip pieces can you cut?
Each whole has four quarter-strip pieces. Three wholes have 3 × 4 = 12 pieces. Check: 12 × 1/4 = 3.
Watch for: Sharing three wholes among four people instead of counting fourths.