Two Raised Beds: Comparing Planted Fractions and Buying Compost
Two raised beds behind Rosa Parks Elementary are the same size. Bed A is divided into 8 equal sections: 3/8 planted with beans, 1/8 with peppers, 2/8 with lettuce, and 1/8 with marigolds. Bed B is divided into 4 equal sections, and a parent planted 3 of them with tomatoes.
During recess, one group argues that Bed B holds more plants because '3/4 is bigger than 3/8,' while another group says Bed A has more planted because more sections are in use.
The garden committee also needs compost. Bed A has 5 planting rows, and each row requires 2/3 cup of compost. Compost is sold in 2-cup bags, and the committee buys only whole bags.
Finally, a new Bed C - the same size as Bed A, divided into 8 equal sections - will get a sign reading '6/8 planted,' but the teacher wants the sign to show the simplest fraction that names the same amount.
1. Which bed has more of its area planted? Resolve the disagreement by comparing the planted fractions, and identify the error in each group's reasoning.
Read the worked answer
Bed A planted: 3/8 + 1/8 + 2/8 + 1/8 = 7/8. The sections are all eighths, so add the numerators and keep the denominator: 3 + 1 + 2 + 1 = 7.
Bed B planted: 3 of 4 equal sections = 3/4. To compare with eighths, rename it: 3/4 = 6/8 (multiply numerator and denominator by 2, which splits each fourth into two eighths).
Compare: 7/8 > 6/8, so Bed A has more planted, by 1/8 of a bed.
Errors: the first group compared 3/4 of Bed B with 3/8 of Bed A - but 3/8 is only the bean portion of Bed A, not its total planted area, so the two fractions describe different things. The second group reached the right answer for a shaky reason: counting 'more sections' is not a comparison, because Bed B's sections are twice as large. Fractions can only be compared after both describe the same whole divided into the same-size parts.
2. How much compost does Bed A need in total, and how many 2-cup bags must the committee buy? What is left over?
Read the worked answer
Total compost: 5 rows × 2/3 cup = 5 × 2/3. Multiply the whole number by the numerator: (5 × 2)/3 = 10/3 cups.
Rewrite as a mixed number: 10 thirds = 3 wholes (9 thirds) plus 1 third, so 3 1/3 cups. A quick reasonableness check: each row is less than 1 cup, so 5 rows must be less than 5 cups - 3 1/3 fits.
Bags: one bag = 2 cups, which is not enough. Two bags = 4 cups, and 4 − 3 1/3 = 2/3 cup left over. So the committee buys 2 bags.
Judgment: the decision rule is 'round up to whole bags,' but the rounding must happen after the exact total is known. Buying 2 bags leaves two-thirds of a cup, which is enough for exactly one more row of planting.
3. Bed C will be labeled '6/8 planted.' Write the simplest fraction and justify the equivalence in a way a fourth grader could check with a model.
Read the worked answer
6/8 = 3/4. Divide numerator and denominator by the same number: 6 ÷ 2 = 3 and 8 ÷ 2 = 4.
Model check: draw Bed C as 8 equal sections with 6 shaded. Group the sections in pairs - 4 pairs, of which 3 pairs are shaded. That is 3/4 of the bed shaded. Nothing was added or removed; the grouping simply puts the same shaded area into larger pieces.
So the sign can read 3/4 planted, and both labels describe identical planted area.
Judgment point: simplest form changes only the name of the amount, not the amount. The teacher should confirm that 6/8 and 3/4 give the same answer to 'how much of the bed is planted?' - the test for any equivalent fraction.