The Kraft Paper Roll: Diagnosing a Fraction Multiplication Error
The art teacher has one 6-yard roll of kraft paper for lining display shelves. Each shelf needs 5/8 yard of paper, and there are 7 shelves to line. Before cutting, Maya writes 5/8 × 7 = 35/56 and announces that "the class needs about 0.6 of a yard, so a 1-yard piece is plenty." Devon disagrees and insists, "Seven shelves at almost a yard each means we need more than 6 yards, so we have to buy another roll." The teacher asks the class to settle who is right using a model, not a guess, and then asks a follow-up: after all 7 shelves are lined, how much paper is left on the roll, and how many additional shelves of the same size could still be lined with that leftover. Paper is sold only in 6-yard rolls, so the class needs a clear yes-or-no answer about whether one roll is enough.
1. Evaluate Maya's computation 5/8 × 7 = 35/56. Identify the error, give the correct product as a mixed number, and use a model to show why multiplying by 7 increases the amount.
Read the worked answer
Maya multiplied the denominators (8 × 7 = 56), which is a procedure for fraction × fraction, not fraction × whole number. The 7 is 7 ones, not sevenths.
Correct: 5/8 × 7 = (5 × 7)/8 = 35/8 = 4 3/8 yards.
Model: seven groups of five eighths is 35 eighths. Bundling 8 eighths into each whole gives 4 wholes (32 eighths) with 3 eighths left over.
Her answer, 35/56, simplifies to 5/8 - that is the amount for one shelf, so she described a single shelf while claiming to answer for seven.
Check for reasonableness: 5/8 is a bit more than 1/2, so 7 × 5/8 should be a bit more than 7 × 1/2 = 3.5. The value 4 3/8 fits; 0.625 does not.
2. Use comparison reasoning to decide whether Devon is right that more than 6 yards are needed, then compute the leftover paper after 7 shelves.
Read the worked answer
Each shelf needs less than 1 yard, and there are only 7 shelves, so the total must be less than 7 yards - but that alone does not settle it against the 6-yard roll.
The computed total is 4 3/8 yards. Since 4 3/8 < 6, one roll is enough and Devon is wrong; he overestimated by treating 5/8 as "almost a yard."
Leftover: 6 − 4 3/8 = 5 8/8 − 4 3/8 = 1 5/8 yards.
Additional shelves: 1 5/8 = 13/8, and each shelf uses 5/8. 13 ÷ 5 = 2 remainder 3, so 2 more shelves can be lined, using 10/8 = 1 1/4 yards and leaving 3/8 yard. Three more shelves would need 15/8 = 1 7/8 yards, which is more than the 1 5/8 yards available, so 3 is too many.
3. The teacher writes both 5/8 × 7 and 7 × 5/8 on the board. Explain why the two expressions have the same value and describe what each one could mean in this shelf-lining situation.
Read the worked answer
Both equal 35/8 = 4 3/8 because multiplication is commutative: the same 35 eighths are produced whether you count seven groups of five eighths or five eighths of a group of seven.
- 5/8 × 7 reads naturally as "7 groups of 5/8 yard" - the total paper for 7 identical shelves.
- 7 × 5/8 can be read as "7 yards scaled down to five eighths of that amount," which matches a different picture: if the roll were cut to 7 yards, the shelves would use five eighths of it.
The equal value does not mean the two expressions mean the same thing in every problem. When a problem asks for a count of groups (how many 5/8-yard pieces fit in a length), multiplication by the reciprocal situation changes, so students should match the expression to the story before computing.