Planter Volume and a Line Plot of Seedlings
Ms. Osei's Grade 5 class is in the middle of a three-week garden unit that combines volume and data. Each group builds a rectangular planter with inside dimensions 2 1/2 feet long, 1 1/3 feet wide, and 2 feet deep, and fills it with soil sold in 1 1/2-cubic-foot bags.
Group A's paper: 5/2 × 4/3 × 2 = 40/6 = 6 2/3 cubic feet, so buy 5 bags. Group B's paper: 2.5 × 1.3 × 2 = 6.5 cubic feet, so buy 4 bags, because 4 bags hold 6 cubic feet and 6.5 is 'about 6.'
Each student also measures one seedling to the nearest 1/4 inch for a class line plot. One table's six measurements are 2 1/2, 3, 3 1/2, 3 1/2, 3 3/4, and 4 3/4 inches. The table must report the total height and the height each seedling would have if the total were redistributed equally.
Before the share-out, Ms. Osei wants to know which group's bag count to use as the anchor for the discussion.
1. Verify Group A's volume with fraction multiplication, explain why 1.3 is not an acceptable substitute for 1 1/3, and quantify how far off Group B's volume is.
Read the worked answer
5/2 × 4/3 × 2 = (5 × 4 × 2)/(2 × 3) = 40/6 = 20/3 = 6 2/3 cubic feet. Group A is correct, and the answer is reasonable: the base is a bit over 3 square feet (2 1/2 × 1 1/3 ≈ 3.33) and the depth is 2 feet, so the volume should be a little over 6 cubic feet.
Group B's error is the width. 1 1/3 = 4/3 = 1.333..., whereas 1.3 = 1 3/10 = 13/10. The difference is 4/3 − 13/10 = 40/30 − 39/30 = 1/30 foot, so 1.3 is short by 1/30 of a foot. Multiplied by the other two dimensions, 2.5 × 2 = 5, that shortfall is 5 × 1/30 = 1/6 cubic foot.
So Group B's 6.5 is 6 1/2, which is 1/6 cubic foot too large. The lesson is not that decimals are bad but that thirds have no terminating decimal, so converting to decimals introduces error that can change the answer to a practical question.
2. Using the correct volume, determine how many bags the group must buy and justify the rounding decision. Show why 4 bags cannot work.
Read the worked answer
One bag holds 1 1/2 = 3/2 cubic feet.
6 2/3 ÷ 1 1/2 = 20/3 ÷ 3/2 = 20/3 × 2/3 = 40/9 = 4 4/9 bags.
The exact quotient is between 4 and 5, and bags are sold as whole units, so the decision is between 4 and 5 - not a rounding-to-nearest question, but a 'enough or not enough' question. Test the lower option: 4 × 1 1/2 = 6 cubic feet, which is less than 6 2/3. Four bags leave the planter 2/3 of a cubic foot short. Five bags hold 7 1/2 cubic feet, which covers the planter with 7 1/2 − 6 2/3 = 5/6 cubic foot left over.
So the answer is 5 bags, and the reasoning must be rounding up. Group B is wrong twice over: the volume is 6 2/3 rather than 6.5, and even their own 6.5 exceeds 4 bags' capacity of 6. A check students can run: does bags × 1 1/2 reach or exceed 6 2/3?
3. Find the total of the six seedling heights and the height each would have if the total were shared equally. Explain what that equal share represents, and name a likely student error with a check that catches it.
Read the worked answer
Add whole-number parts and fraction parts separately:
- Whole numbers: 2 + 3 + 3 + 3 + 3 + 4 = 18.
- Fraction parts: 1/2 + 1/2 + 1/2 + 3/4 + 3/4 = 3.
- Total: 21 inches.
Equal share: 21 ÷ 6 = 3 1/2 inches. Redistribution means every seedling is assigned the same height while the total stays at 21, so 3 1/2 is the fair share - the mean of the data. It is a value in the middle of the distribution: the 2 1/2 and 3 inch seedlings would gain height, and the 3 3/4 and 4 3/4 inch seedlings would lose it.
A likely error is dividing by the number of different heights (5) rather than the number of measurements (6), giving 4 1/5; another is dropping the fractional parts. The check that catches both is 6 × 3 1/2 = 21, which confirms the total was preserved. On the line plot, 3 1/2 is one of the taller stacks, which is a good discussion point about what the mean does and does not represent.