Lesson 6

Scaling and Area

Lesson Narrative

This lesson is optional. In this lesson, students are introduced to how the area of a scaled copy relates to the area of the original shape. Students build on their grade 6 work with exponents to recognize that the area increases by the square of the scale factor by which the sides increased. Students will continue to work with the area of scaled shapes later in this unit and in later units in this course. Although the lesson is optional, it will be particularly helpful for students to have already had this introduction when they study the area of circles in a later unit.

In two of the activities in this lesson, students build scaled copies using pattern blocks as units of area. This work with manipulatives helps accustom students to a pattern that many find counterintuitive at first (MP8). (It is a common but false assumption that the area of scaled copies increases by the same scale factor as the sides.) After that, students calculate the area of scaled copies of parallelograms and triangles to apply the patterns they discovered in the hands-on activities (MP7).

Learning Goals

Teacher Facing

  • Calculate and compare (orally and in writing) the areas of multiple scaled copies of the same shape.
  • Generalize (orally) that the area of a scaled copy is the product of the area of the original figure and the “square” of the scale factor.
  • Recognize that a two-dimensional attribute, like area, scales at a different rate than one-dimensional attributes, like length and distance.

Student Facing

Let's build scaled shapes and investigate their areas.

Required Preparation

Prepare to distribute the pattern blocks, at least 16 blue rhombuses, 16 green triangles, 10 red trapezoids, and 7 yellow hexagons per group of 3–4 students.

Copy and cut up the Area of Scaled Parallelograms and Triangles blackline master so each group of 2 students can get 1 of the 2 shapes.

Learning Targets

Student Facing

  • I can describe how the area of a scaled copy is related to the area of the original figure and the scale factor that was used.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • area

    Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.

    For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac12\) square unit.

    Figure A on the left composed of 8 identical shaded squares arranged in 3 rows. Figure B on the right consists of one square with a diagonal segment from corner to corner. Half of the square is shaded.
  • corresponding

    When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.

    For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).

    2 triangles with corresponding parts
  • reciprocal

    Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac25\) is \(\frac52\).

  • scale factor

    To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.

    In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).

    2 triangles
  • scaled copy

    A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.

    For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).

    2 triangles