Lesson 1 Any Way You Slice It Develop Understanding
Learning Focus
Identify shapes formed by slicing a solid with a plane.
How can I visualize the slices formed when a solid object, such as a cube, sphere, or pyramid, is run through a meat slicer? What would the slices look like if the object is oriented in different ways before being sliced?
Open Up the Math: Launch, Explore, Discuss
Students in Mrs. Denton’s class were given cubes made of clay and asked to slice off a corner of the cube with a piece of dental floss.
Jumal sliced his cube this way.
Jabari sliced his cube like this.
1.
Which student, Jumal or Jabari, interpreted Mrs. Denton’s instructions correctly? Why do you think this student is correct?
When describing 3-D objects such as cubes, prisms, or pyramids we use precise language such as vertex, edge, or face to refer to the parts of the object in order to avoid the confusion that words like corner or side might create. A cross-section is the face formed when a 3-D object is sliced by a plane. It can also be thought of as the intersection of a plane and a solid.
2.
Draw and describe the cross-section formed when Jumal sliced his cube.
3.
Draw and describe the cross-section formed when Jabari sliced his cube.
4.
Draw some other possible cross-sections that can be formed when a cube is sliced by a plane.
5.
Describe your strategies for drawing cross-sections of the cube.
6.
What type of quadrilateral is formed by the intersection of the plane that passes through diagonally opposite edges of a cube? Explain how you know what quadrilateral is formed by this cross-section.
Cross-sections can be visualized in different ways. One way is to do what Jumal and Jabari did—cut a clay model of the solid with a piece of dental floss. Another way is to partially fill a clear glass or plastic model of the 3-D object with colored water and tilt it in various ways to see what shapes the surface of the water can assume.
Experiment with various ways of examining the cross-sections of different 3-D shapes.
7.
Partially fill a cylindrical jar with colored water, and tilt it in various ways. Draw the cross-sections formed by the surface of the water in the jar.
8.
Try to imagine a cubical jar partially filled with colored water tilted in various ways.
a.
Which of the following cross-sections can be formed by the surface of the water?
a square
a rhombus
a rectangle
a parallelogram
a trapezoid
a triangle
a pentagon
a hexagon
an octagon
a circle
b.
Which of the following cross-sections are impossible to form by the surface of the water?
a square
a rhombus
a rectangle
a parallelogram
a trapezoid
a triangle
a pentagon
a hexagon
an octagon
a circle
Ready for More?
Explore how cross-sections of a cube are related to the possible 2-D shadows of a cube that can be formed when the cube is oriented in different ways directly beneath a strong light. Can every possible cross-section be found as a shadow of the cube? Does every possible shadow of the cube correspond to a possible cross-section?
Takeaways
A cross-section is formed by
We can visualize a cross-section by:
Vocabulary
- cross-section of a solid
- edge / face / vertex of a 3-D solid
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we identified cross-sections, or slices, of various 3-D shapes, such as cubes and cylinders. Some of the cross-sections we found were obvious, but some were surprising. We also learned how to draw the cross-section on a 2-D representation of the three-dimensional shape.
1.
Assume that in each of the figures, the space between each grid mark is
a.
What is the length of
b.
What is the area of rectangle
c.
What is the volume of the rectangular prism?
d.
On each figure, draw the shape of the unit of measure (
2.
Calculate the area of