Create cylinder- and cone-shaped containers.
Outline: "cutting cylinder s and cone s , and observation of the cut surfaces"

Learning activities ... Learning activities     Teacher's instructions/guide ... Teacher's instructions/guide     Evaluation (expected student responses)   ... Evaluation (expected student responses)   

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[Question] Somebody wants to buy some beads at a shop and there are two containers, A and B, and there a written notice, "One cup from container A is 300 yen, and one cup from container B is 150 yen." The shape of container A is a cylinder whose radius of the base surface is 6cm and whose height is 8cm, while the shape of container B is a cone whose radius of the base surface is 6cm and whose height is 8cm. If somebody wants to buy some beads worth 600 yen, which container w ould be the mo st economical?
[Task] How many cups from cone-shaped container B are equivalent to a cup of cylinder-shaped container A? Measure it by creating the actual objects.

Learning content

Teacher's instructions/guide Show the actual objects, and en sure to allow the students t o visualize it .
Learning activities Think about the difference in size while looking at the actual objects sideways and picking them up in his or her hands.
Teacher's instructions/guide Let the students know that, because the number of the actual objects is small, more objects need to be created.
Teacher's instructions/guide Let the students know that, if developments are created with cardboard paper, the containers can be formed.
Teacher's instructions/guide Instruct that lids also need to be created for the containers.
Learning activities Think about how the developments will appear.
Teacher's instructions/guide Request the students to think about the appearance of the developments. For students who do not understand the size, instruct them to think about, for example, parts that have the same length while the teacher concretely shows and opens the actual objects.
Learning activities Create the containers by drawing each development.
Learning activities Measure the volume (capacity) using the created containers and beads.
Learning activities Summarize that one cup from container A seems to be equivalent to about three cups from container B.
Teacher's instructions/guide When measurement has been carried out to a certain degree, confirm through carrying out demonstration s that one cup from container A is equivalent to three cups from container B.
Teacher's instructions/guide Confirm that the volume can be obtained respectively through the following formula: "a cylinder's volume = the area of the base × its height" and "a cone's volume = the area of the base × its height ÷ 3." Summarize that, in addition to cylinders and cones, the volume of columns and conic solids can be respectively obtained in a similar way.
Evaluation (expected student responses) Understand ing how to obtain the volume of columns and conic solids.
Teacher's instructions/guide Set learning as the surface areas by deciding that the task for next time will be to find out which paper pattern of containers A and B will be smaller.

Expected responses from the students

A When looked at sideways, containers A and B appear to be a rectangle and triangle respectively, the bases have the same length, and the heights are the same. Thus, one cup from container A seems to be basically equivalent to two cups from container B. The results will be probably the same regardless of which container is used for measure ment .
B One cup from container A also seems to be more than two cups from container B.
C The result can be realized if actual objects are created and used for measurement.
D The containers can be definitely created from cardboard if their developments are drawn.
E The cylinder can be created using a rectangle and two circles. How big do the y need to be?
F The size of the developments can be obtained respectively when they are calculated back wards from the size of the containers.
G Each development will be as shown below. cabri3D
H When measurement is actually carried out, the amount that can be included in one cup of container A is more than w hat can be included in two cups of container B.
I When measurements have been carried out a few times, one cup of container A seems to be approximately equivalent to three cups of container B.
J When somebody wants to buy some beads worth 600 yen, it is more economical to us e two cups from container A.
K I was surprised that the cone-shaped container had a smaller amount than I had expected.
L How about the size of the paper models? I would like to think about which w ould be smaller.

In addition, all copy rights of the unit structure, Class development, and worksheets belong to Hitoshi Arai (affiliation: Nagano City Yanagimachi junior high school).

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