Lesson 3

Creating Cross Sections by Dilating

Lesson Narrative

In this lesson, students build on their work with dilations in earlier units. Students create dilations of a rectangle and suspend them to resemble cross sections of a pyramid. They learn that given a pyramid’s base, its cross sections are dilations of the base with scale factors between 0 and 1.

These activities help build the spatial visualization skills and familiarity with cross sections they will need to derive volume formulas later in the unit. For example, students will later use dilations and cross sections to conclude that pyramids of the same height and with bases of equal area have equal volumes, regardless of the particular shapes of the bases.

Through articulating things they notice and things they wonder about dilations, students attend to precision in the language they use to describe what they see (MP6).


Learning Goals

Teacher Facing

  • Comprehend that a pyramid’s cross sections can be thought of as dilations of its base using scale factors from 0 to 1.

Student Facing

  • Let’s create cross sections by doing dilations.

Required Preparation

Be prepared to display an applet for all to see in the synthesis of the activity Pyramid Mobile.

Learning Targets

Student Facing

  • I know that a pyramid’s cross sections are dilations of its base with scale factors ranging from 0 to 1.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • axis of rotation

    A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.
  • cone

    A cone is a three-dimensional figure with a circular base and a point not in the plane of the base called the apex. Each point on the base is connected to the apex by a line segment.

  • cross section

    The figure formed by intersecting a solid with a plane.

     

     

  • cylinder

    A cylinder is a three-dimensional figure with two parallel, congruent, circular bases, formed by translating one base to the other. Each pair of corresponding points on the bases is connected by a line segment.

  • face

    Any flat surface on a three-dimensional figure is a face.

    A cube has 6 faces.

  • prism

    A prism is a solid figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a “pentagonal prism.”

    rectangular prism

    triangular prism

    pentagonal prism

  • pyramid

    A pyramid is a solid figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a “hexagonal pyramid.”

    square pyramid

    pentagonal pyramid

  • solid of rotation

    A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.

    The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.
  • sphere

    A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.

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