Pieces become a whole. A flat record becomes a solid clue.
The old plan shows a 12 m² back room; the current survey marks that corner as inaccessible. Behind a loose board, the group found sketches pointing to a cabinet. Their calculations have checked the records, but nobody has yet seen what is behind the wall.
Two lessons, each planned for approximately 50 minutes. Use graph paper for work. Responses are not saved or submitted.
The cabinet stood beside the back wall. Tess stopped Mara before she tried to pull it forward.
“We’re taking inventory. We’re not moving furniture by ourselves.”
They asked Ada, the town’s workshop caretaker, for help. While she checked the cabinet, Jun spotted a paper template sticking out of its drawer.
“A replacement top,” he said. “Look at the corner cuts. We could measure the whole rectangle and take them away.”
Mara studied it. “Or split what’s left into shapes we already know.”
“Both,” Eli said. “If we’re describing the same surface, the answers should agree.”
Find the area of a triangle with base 10 cm and perpendicular height 8 cm. Then find a parallelogram with the same base and height. Explain the relationship.
The cabinet-top template fits in a 10 dm × 6 dm rectangle. Each upper corner cut is a right triangle with horizontal leg 2 dm and vertical leg 3 dm. The diagram uses one consistent scale.
A composite polygon can be measured by adding non-overlapping areas or subtracting excluded areas. Account for every piece exactly once. Area alone does not determine shape or whether one shape fits inside another.
Worked example: an 8 cm × 5 cm rectangle with one right-triangle corner removed (legs 2 cm and 4 cm) has area 40 − ½ × 2 × 4 = 36 cm².
1. 84 cm². 2. 35 m². 3. The student subtracted the full 2 × 3 rectangle rather than half; correct area 51 cm². 4. Answers vary; an example is a 6 × 6 square with a 2 × 3 rectangular corner removed.
Write a short recommendation: what area should Ada record for the cabinet top, and what else must a carpenter check before cutting?
The area is 54 dm²: 60 − 3 − 3. Addition gives 30 + 18 + 3 + 3. The carpenter must also check the stock’s shape and dimensions, orientation, and cutting allowance.
Ada unlocked the drawer. Beneath the template was a small wooden box and a folded cutting pattern.
“The top measures fifty-four square decimeters,” Jun said. “But that doesn’t mean any fifty-four-square-decimeter board will fit.”
Ada smiled. “Keep that distinction. You’ll need it.”
The cutting pattern showed six rectangles. Eli compared it with the box.
“Six faces,” he said. “We should be able to match them.”
Mara added the three measurements written along the edge. Tess shook her head.
“That gives a length. We’re trying to cover a surface.”
Mara crossed out her sum. “Then let’s count the faces, not just the edges.”
A rectangle measures 5 cm × 3 cm. Find its area and perimeter. Explain why the units differ.
The closed rectangular box has length 6 cm, width 4 cm, and height 3 cm. This net has four faces in a strip, with the top and bottom attached. Each label gives that face’s dimensions.
All face dimensions are in centimeters. Matching colors mark equal-area face pairs; labels also identify each pair.
A net is a flat arrangement of faces that can fold into a solid. Surface area is the total area of its exterior faces. For a closed rectangular prism, add two copies of length × width, length × height, and width × height. Surface area is in square units; volume, which measures interior space, uses cubic units.
Worked example: a closed 5 cm × 2 cm × 3 cm box has surface area 2(10) + 2(15) + 2(6) = 62 cm².
1. 92 cm². 2. 150 cm². 3. 94 cm². 4. That product is the volume, 72 cm³; it measures space rather than the six exterior faces.
Find and justify the closed story box’s surface area. Then explain why a paper wrapper might need more material than this ideal total.
“One hundred eight square centimeters,” Mara said. This time she pointed to all six faces.
The net matched the box’s dimensions and face arrangement. It was supporting evidence, not proof of who had made it. Ada opened the box.
Inside lay an atlas page showing a narrow entrance behind the cabinet. Ada checked it against the building record in her office, then arranged for a maintenance worker to inspect the blocked doorway.
The next morning she returned with a photograph. The room existed. Its shelves held survey notebooks.
Eli unfolded the atlas page. Across its edge were three words: Continue at Stone Bridge.
For the first time, their map led beyond the workshop.
1. 134 cm². 2. Faces: 45, 45, 18, 18, 10, 10; total 146 cm². 3. 40 + 64 + 40 = 144 cm². 4. Both have surface area 22 square units despite different dimensions. 5. Calculations establish model areas; confirmation of the actual room required inspection and records.
Find a closed rectangular prism with whole-number dimensions different from 6 × 4 × 3 but surface area 108 square units. Explain your search.
Next story destination: Stone Bridge. Ratios will help the group interpret the survey notebooks. Further geometry lessons and practice will build on this opening sequence.
Return to the map →