Questions to Ask Myself

1. E2 Students are expected to identify, describe, and represent the various cross-sections of cubes and rectangular prisms.

2. Questions to Ask Myself What does a cube look like? How many faces does it have? What do I know about each face? How many edges does it have? How many vertices does it have? Can I create a net for a cube? Remember a net is a pattern or template that can be cut and folded to make a geometric solid. What are the properties of a rectangular prism? Can I create a net for this solid?

3. A cross-section is the 2-D shape of the face produced when a plane cut is made through a solid. What is a cross-section of a solid? How do I identify one? How do I describe it and represent it?

4. Introductory Activity: Let’s create several cross-sections for a circular cone. We can make cross-sections by cutting a cone in several different ways: • parallel to its base • cut down through its vertex • cut in a plane parallel to a plane of symmetry • cut obliquely (slanting, not parallel to its base) towards its base

5. Results: • If the cone is cut in any plane parallel to its base, the face produced is a circle. • If it is cut down through its vertex, the exposed face is a triangle. • If it is cut in a plane parallel to a plane of symmetry, the shape below is produced. • If the cone is cut obliquely – not parallel to its base – the face produced is an oval. Scan shape for plane of symmetry cut

6. Let’s Practice Making Cross-Sections with a Triangular Prism: We can make cross-sections of a prism by: • by making a cut that is parallel to its base • by making a cut parallel to one of its rectangular faces • by cutting obliquely (slanting) towards its base • by cutting obliquely to a rectangular face • You should start at a vertex, as well as, at different points along the edges of the prism

7. Investigating Cross-Sections of a Prism by Making Plane Cuts A plane cut can be made in several ways: • by making a cut that is parallel to its base • by making a cut parallel to one of the prism’s rectangular faces • by cutting obliquely (slanting) towards its base • by cutting obliquely to a rectangular face • You should start at a vertex, as well as, at different points along the edges of the prism

8. Let’s Begin! 1. Work within your assigned group to complete the activity for each material. Your group will be called to complete the Center 2 activity.2. For each activity, visualize the shape that will be made when you make a cross-section of that solid. Record your prediction. . 3. Use one of the solids at your center and the other materials to make the cross-section you visualized. Record your result.4. Now repeat the procedure 1 to 3 for the same solids at the center.

9. Centers: Center 1: Materials: Directions 4 Rice crispie squares Elastic Butter knife Recording sheet Center 2: Materials: Directions 1 clear cube Water Large container Recording sheet Center 3: Materials: Directions 4 Plasticine rectangular prisms Elastic Fishing line Recording sheet Center 4: Materials: Directions 4 Cheese cubes Elastic Butter knife Recording sheet

10. Center 1 Directions: • Decide which plane cut you will make first (parallel to face, oblique, beginning at a vertex, beginning at a different point along the edge of the solid) to your rice crispie squares (rectangular prisms). • Visualize the shape that will be made when you make this cross section. Hint: If you wrap a shape with an elastic band where the cut will be made, it can help you visualize. Record your prediction on the recording sheet using a careful sketch. Remember to draw the shape that you think will be left after the cut is made. Then shade in the face that was exposed by the cut. • Make this cross-section cut carefully with the butter knife. Closely examine the 2-D face that has been left on the solid after the cut. Compare this to your earlier prediction. Record your result using a sketch as above. • Using a different plane cut, now repeat the procedure 1 to 3 for the other two rice crispie squares at your center.

11. Center 2 Directions: • In this center you will examine cross-sections not by cutting, but by using water and a clear cube. You will examine the shapes that can be made by the surface of the water when you tip the cube in different ways. • First, decide which way you will tip the water in the cube. You can still think about the types of plane cuts we discussed earlier to help you here (parallel to face, oblique, beginning at a vertex, beginning at a different point along the edge of the solid). • Visualize the shape that will be made when you make this cross section. Record your prediction on the recording sheet using a careful sketch. Remember to draw the shape that you think will be left after the water is tipped. Then shade in the face that was exposed. • Make this cross-section by tipping the water in the way you chose earlier. Carefully look at the 2-D face that has been left on the solid after the ‘cut’. Compare this to your earlier prediction. Record your result using a sketch as above. • Using a different way of tipping the cube, now repeat the procedure 1 to 4 twice.

12. Center 3 Directions: • Decide which plane cut you will make first (parallel to face, oblique, beginning at a vertex, beginning at a different point along the edge of the solid) to your plasticine rectangular prisms. • Visualize the shape that will be made when you make this cross section. Hint: If you wrap a shape with an elastic band where the cut will be made, it can help you visualize. Record your prediction on the recording sheet using a careful sketch. Remember to draw the shape that you think will be left after the cut is made. Then shade in the face that was exposed by the cut. • Make this cross-section cut carefully with the fishing line. Examine closely the 2-D face that has been left on the solid after the cut. Compare this to your earlier prediction. Record your result using a sketch as above. • Using a different plane cut, now repeat the procedure 1 to 3 for the other two plasticine solids at your center.

13. Center 4 Directions: • Decide which plane cut you will make first (parallel to face, oblique, beginning at a vertex, beginning at a different point along the edge of the solid) to your cheese cube. • Visualize the shape that will be made when you make this cross section. Hint: If you wrap a shape with an elastic band where the cut will be made, it can help you visualize. Record your prediction on the recording sheet using a careful sketch. Remember to draw the shape that you think will be left after the cut is made. Then shade in the face that was exposed by the cut. • Make this cross-section cut carefully with the butter knife. Closely look at the 2-D face that has been left on the solid after the cut. Compare this to your earlier prediction. Record your result using a sketch as above. • Using a different plane cut, now repeat the procedure 1 to 3 for the other two cheese cubes at your center.

14. A cube could be cut to produce these shapes (among others) Scan 5-76 sketches.

15. If geoblocks are available, cubes, square prisms, and rectangular prisms can be built in a variety of ways; thereby, some cross-sections of these prisms can be demonstrated without having to cut. Optional Activity can be incorporated into the lessons.