Lesson 4

Scaling and Area

Lesson Narrative

In previous units, students learned that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. In the last lesson, students dilated a rectangle by a factor of $$k$$ and observed that the area was not multiplied by $$k$$. In this lesson, students analyze the result of scaling on area. This concept will be essential to creating a volume formula for pyramids later in the unit. Additionally, students will encounter the graph representing the equation $$y=\sqrt{x}$$ in a geometric context when, in upcoming lessons, they analyze the relationship between scaled areas and factors of dilation. This will build on work students did in grade 8 evaluating square roots of small perfect squares and determining that $$\sqrt{2}$$ is irrational.

Knowing the relationship between scale factor and area is a basis for understanding how measurement and dimension are related in geometry. As students observe a pattern in the area of a figure after several different dilations, they are looking for regularity in repeated reasoning (MP8).

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Learning Goals

Teacher Facing

• Comprehend that when figures are dilated by a scale factor of $k$, their areas are multiplied by $k^2$.

Student Facing

• Let’s see how the area of shapes changes when we dilate them.

Student Facing

• I know that when figures are dilated by a scale factor of $k$, their areas are multiplied by $k^2$.

Building On

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.

• 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.”

• 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.”

• 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.

• sphere

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

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