Lesson 5

Scaling and Unscaling

Lesson Narrative

In this lesson, students practice working with areas of scaled figures, connecting them to cross sections and encountering a common misconception.

Then, they create a graph representing \(y=\sqrt{x}\) and use it to answer questions about a situation. They analyze the shape of the graph to better understand the relationship between an area and the scale factor needed to achieve it.

When students consider two arguments (one of which includes a common misconception) and decide which is correct, they are critiquing the reasoning of others (MP3).

Learning Goals

Teacher Facing

  • Describe (using words and other representations) the relationships between scale factors and areas using square root graphs and calculations.

Student Facing

  • Let’s examine the relationships between areas of dilated figures and scale factors.

Required Materials

Required Preparation

Devices are required for the digital version of the activity Graphing Areas and Scale Factors.

Learning Targets

Student Facing

  • I can use square root graphs and do calculations to interpret the relationships between scale factors and areas.

CCSS Standards

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.