10 Solid geometry · g10-10-01 · 5–10 min
Classify by structure, not by viewpoint
Does rotating a solid change its type?
Predict: Compare a prism,pyramid,cylinder,cone,frustum andsphere; decide which have parallel bases and which have a common apex.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: A solid has two parallel congruent circular bases and a curved side. Name it.
10 Solid geometry · g10-10-02 · 5–10 min
A drawing does not preserve every length
Can equal perpendicular sides look unequal in an oblique drawing?
Predict: Draw a square with its depth lengths halved. Predict the displayed side ratio and angle.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Real front length = 4; Real depth length = 4; Depth scale q = 0.5; Projected depth angle (degrees) = 45 | |
Case 2
Real front length = 4; Real depth length = 4; Depth scale q = 1; Projected depth angle (degrees) = 45 | |
Case 3
Real front length = 4; Real depth length = 4; Depth scale q = 0.5; Projected depth angle (degrees) = 0 | |
Revise: My claim needs the condition… because…
Exit: A depth edge of real length6 is drawn with q=1/2. What is its displayed length?
10 Solid geometry · g10-10-03 · 5–10 min
A cone height is not its slant height
Which length belongs in the lateral-area formula?
Predict: For radius3 and vertical height4, predict the slant length and total surface area of a cone.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Radius r = 3; Vertical height h = 4 | |
Case 2
Radius r = 3; Vertical height h = 4 | |
Case 3
Radius r = 3; Vertical height h = 0 | |
Revise: My claim needs the condition… because…
Exit: Find the total surface area for a cone with r=3,h=4.
10 Solid geometry · g10-10-04 · 5–10 min
The same base and vertical height
Which height determines a prism or cone volume?
Predict: For base area12 and height5, predict the prism and cone volumes.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Base area S = 12; Perpendicular height h = 5 | |
Case 2
Base area S = 12; Perpendicular height h = 5 | |
Case 3
Base area S = 12; Perpendicular height h = 0 | |
Revise: My claim needs the condition… because…
Exit: Find the cone volume when S=12,h=5.
10 Solid geometry · g10-10-05 · 5–10 min
A frustum is a difference of cones
Can the volume use the average of the two end areas?
Predict: Compare the correct volume with hπ(R²+r²)/2 for R=3,r=1,h=4.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Bottom radius R = 3; Top radius r = 1; Height h = 4 | |
Case 2
Bottom radius R = 3; Top radius r = 0; Height h = 4 | |
Case 3
Bottom radius R = 3; Top radius r = 3; Height h = 4 | |
Revise: My claim needs the condition… because…
Exit: Find the volume for R=3,r=1,h=4.
10 Solid geometry · g10-10-06 · 5–10 min
Double a sphere: four or eight times?
Do surface area and volume scale at the same rate?
Predict: Double a sphere radius. Predict its area ratio and volume ratio.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Original radius r = 1; Radius scale k = 1 | |
Case 2
Original radius r = 1; Radius scale k = 2 | |
Case 3
Original radius r = 1; Radius scale k = 0 | |
Revise: My claim needs the condition… because…
Exit: A sphere radius triples. State the area and volume factors.
