Bulletin Board Paper: Deciding How Much Area a Fraction of a Roll Covers
The fifth-grade hallway committee is covering a bulletin board that is 1 1/2 yards long and 2/3 yard wide. They have one roll of paper that covers exactly 1 square yard. Before cutting, a student announces, "The board is bigger than 1 square yard, because it is 1 1/2 times 2/3 - and multiplying always makes the number bigger - so we do not have enough paper." A second student argues the answer is 2 1/6 square yards, having added 1 1/2 + 2/3. The committee also plans a second, smaller board that is 2 1/4 yards long and 1/2 yard wide, and wants to compare the two boards' areas. The custodian has already unrolled the paper and will not cut a second piece, so a wrong estimate means a bare spot on the wall or a wasted afternoon.
1. Find the area of the first board using fraction multiplication, and explain why the product is smaller than 1 1/2 rather than larger.
Read the worked answer
Area = length × width = 1 1/2 × 2/3 = 3/2 × 2/3.
Multiply numerators and denominators: (3 × 2)/(2 × 3) = 6/6 = 1 square yard.
Area-model check: a 1-yard square cut 1 1/2 yards long and 2/3 yard wide holds one full 1-by-2/3 strip (that is 2/3 square yard) plus a final 1/2-yard strip, which contributes 1/2 × 2/3 = 1/3 square yard. 2/3 + 1/3 = 1 square yard. The roll is exactly enough.
The product is smaller than 1 1/2 because 2/3 is less than 1. Multiplying by a number less than 1 scales down, exactly as taking 2/3 of something leaves less than the whole. The first student's rule holds only when the second factor is greater than 1.
2. Find the area of the second board and determine whether the single 1-square-yard roll covers both boards.
Read the worked answer
Second board: 2 1/4 × 1/2 = 9/4 × 1/2 = 9/8 = 1 1/8 square yards.
Estimate check: half of 2 1/4 is a bit more than 1, so 1 1/8 is reasonable.
Together the boards need 1 + 1 1/8 = 2 1/8 square yards. The roll covers 1 square yard, so it covers the first board exactly and falls 1 1/8 square yards short of both. The committee needs a second roll for the second board; a partial roll would be exactly the leftover after board one, which is zero. Answering 2 1/8 for the first board, as the second student did, would have made them think the single roll covered far less than it does.
3. Explain what goes wrong when someone adds length and width to get area, using this board as the evidence.
Read the worked answer
Adding gives 1 1/2 + 2/3 = 9/6 + 4/6 = 13/6 = 2 1/6, which is not an area at all - it is the total of two side lengths, measured in yards, not square yards.
The unit check exposes the error: length + width is a distance, while area counts unit squares. If adding sides gave area, then a 1-yard-by-1-yard square would have area 2 square yards when it plainly covers exactly 1.
A second check is that the answer must be believable for the object. A board 1 1/2 yards long and 2/3 yard wide is narrower than a yard, so its area has to be less than 1 1/2 square yards; 2 1/6 is larger than the whole length, which is impossible. Only length × width, 1 square yard, survives both checks.