Garden Soil Budget: Diagnosing a Unit-Conversion Error
Students plan soil for three raised beds, each 8 ft by 4 ft, filled level to a depth of 10 inches. Soil is sold in bags covering 1.5 cubic feet each and costing $7.25; the class budget is $450.
One student submits:
8 x 4 = 32; 32 x 3 = 96; 96 x 10 = 960; 960 / 1.5 = 640 bags; 640 x $7.25 = $4,640. We are way over budget.
The class concludes the project is impossible and proposes shrinking the beds. The teacher wants to use the work as a teaching case rather than simply correcting it.
1. Diagnose the error in the student work and name the underlying concept, then predict what the same student would likely do with a similar problem.
Read the worked answer
The error: the student multiplied square feet (96) by inches (10) and treated the result, 960, as cubic feet. The computation is arithmetically correct but dimensionally meaningless - 960 ft²·in is not 960 ft³.
Underlying concept: measurement units must be consistent before quantities are multiplied; converting inches to feet (or feet to inches) is a prerequisite, not an afterthought.
Prediction: the student is likely operating procedurally - "multiply all the numbers, divide by the bag size" - without tracking what each number is. Expect the same failure wherever units are mixed: computing speed from minutes and hours, comparing areas and perimeters, converting a recipe scaled between cups and tablespoons. The pattern to look for is numbers without labels in the work.
Diagnostic question that reveals the issue fast: "What are the units of 960?" If the student says cubic feet without hesitation, they are guessing; if they pause, the concept is available to be taught.
2. Carry out the correct computation and decide what the class should do about the budget, including how to handle rounding.
Read the worked answer
Volume: 3 beds x (8 x 4) ft² = 96 ft² of bed area. Depth 10 in = 10/12 ft = 5/6 ft. 96 x 5/6 = 80 ft³ needed.
Bags: 80 / 1.5 = 53.33… bags.
Rounding decision: 53 bags supply 79.5 ft³ - half a cubic foot short of the specified depth. Over 96 ft² that shortfall is only about 0.06 in of depth, so it is nearly invisible, but it is still less than specified. 54 bags supply 81 ft³: enough, with 1 ft³ left over.
Cost: 54 x $7.25 = $391.50, leaving $58.50 of the $450 budget. Even 53 bags ($384.25) fits.
So the class conclusion should flip: the project is affordable, and the "we must shrink the beds" proposal rests on a unit error, not on a real constraint.
The judgment worth surfacing with students: rounding down is a decision about risk, not a rounding rule. Buying 53 bags saves $7.25 and leaves the beds a hair under depth. Either choice is defensible if the class states the tradeoff - which is the reasoning the standard is really asking for.
3. Write two follow-up questions that press the student's reasoning about units, and say what each is designed to reveal.
Read the worked answer
Question 1: "What are the units of the 960 you wrote down?" Designed to reveal whether the student is tracking labels at all. A student who says "cubic feet" is guessing; one who says "I multiplied by inches, so… square-foot-inches?" has the concept within reach and needs practice, not reteaching.
Question 2: "If the beds were filled to 12 inches instead of 10, how would your answer change - and by what factor?" Designed to test whether the student now reasons proportionally rather than recomputing blindly. 12 in = 1 ft, so the volume goes 96 -> 96 x 1 = 96 ft³ rather than a ten-step repeat of the same procedure. A student who sees the 12/10 ratio (96 vs 80) is generalizing; one who restarts the arithmetic from scratch is still procedural.
Optional third press: "Estimate the number of bags before you calculate. Is 640 in the right neighborhood?" A quick mental check - roughly 100 ft² times about a foot deep is about 100 ft³, so about 65 bags - would have flagged the error immediately. Building the habit of estimating first is the durable fix.