Lesson 6

Lesson 6: Scale Factors and Areas of 2-D Shapes

Focus

Have you ever noticed how some cartoon characters or figurines have heads that seem much bigger than the rest of their bodies? Bratz dolls, bobblehead dolls, and some cartoon characters, such as Betty Boop, are all examples of characters with heads that are not in proportion to the size of their bodies. Can you think of other examples of a character, doll, or figurine with a head that is oversized (or undersized) compared to the rest of its body?

In cartoons, large heads do not always mean the character has a large ego! The disproportionately large heads take up a larger area in the cartoon and make the characters stand out. Unlike cartoons, scale diagrams require dimensions and areas that are proportional. This ensures that the enlarged or reduced shapes are similar to the original ones.

This lesson will help you answer the following critical questions:

• When a shape is enlarged or reduced, what is the relationship between scale factor and a shape’s dimensions?

• When a shape is enlarged or reduced, what is the relationship between scale factor and a shape’s area?

Assessment

• Math Lab: Scale Factors and Areas of 2-D Shapes

All assessment items you encounter need to be placed in your course folder.

Materials and Equipment

• calculator
• ruler

Launch

Are You Ready?

1. What does area mean? Answer
   
   
2. Show the formula for the area of a
   
   
   1. rectangle
      
      
   2. square
      
      
   3. triangle
      
      
   4. circle
      
      
   5. trapezoid
      
      
   6. parallelogram
   
   Answers
   
   
3. Determine, to the nearest square unit, the area of the following shapes:
   
   
   1. 
      
      
   2. 
      
      
   3. 
      
      
   4. 
      
      
   5. 
   Answers

Refresher

How did you do? Did you remember how to find the area of basic shapes? Great work, if you remembered! If you did not, carefully read this Refresher section.

Discover

Kunal is not impressed with his new room in the basement of their new smaller house. He tells his sister Akuti that his old room is twice as big as his new one. Go to Kunal and Akuti’s Conversation to hear their conversation.

Let’s help Kunal figure out what the difference is.The following Math Lab will let you and Kunal figure out what the real size difference is between his old and new rooms.

Math Lab: Scale Factors and Areas of 2-D Shapes

Akuti completed the Math Lab at school today and is trying to help Kunal understand the difference between twice as much area and twice the dimensions. Go to Kunal and Akuti’s Conversation Continues to hear the rest of their conversation.

Explore

Does doubling the dimensions of an object double its area? Many people think it does. But as Kunal learned, this is not true! As you discovered in the Math Lab, when a scale factor (k) is applied to a 2-D shape, it creates a similar shape. The area of the similar shape (A2) is the product of the square of the scale factor (k2) and the area of the original shape (A1). We could write this as:

area of similar 2-D shape = k2 × area of the original shape

or

A2 = k2 × A1

Rearranging this formula allows you to determine the square of the scale factor that relates two similar 2-D shapes.

By taking the square root of the area of a shape divided by the area of the original shape, you can determine the scale factor.

Connect

If you haven’t done so already, submit your certificate of completion from Math Lab: Scale Factors and Areas of 2-D Shapes to your teacher for the assessment in this lesson.

Going Beyond

The activity “Fractured Pictures” allows you to generate interesting, irregular fractals. You start with a polygon and you input the number of sides. You then enter in a fractal scale factor. The program draws the figure by recreating a scaled copy of the original figure at each vertex. Give it a try! You can click on the link. Choose the “Learner” tab to see some example fractals that were created using the program.

Lesson 6 Summary

Unlike some cartoon characters, scale diagrams require dimensions and areas that are proportional. This ensures that the enlarged or reduced shapes are similar to the original ones. The dimensions of similar 2-D shapes are related by a scale factor.

A common misconception is that the area of two similar shapes is also related by the scale factor. But as you discovered in your Math Lab, the area of two similar shapes are related by the square of the scale factor (k2). The area of the similar shape is the product of the square of the scale factor (k2) and the area of the original shape.

area of similar 2-D shape = k2 (area of original shape)

You can rearrange this formula to isolate the square of the scale factor that relates the two similar 2-D shapes.

The scale factor can be determined by taking the square root of the area of a shape divided by the area of the original shape.

In Lesson 7 you will investigate the relationship between the scale factor and the surface area of two similar 3-D objects and the relationship between the scale factor and the volumes of two similar objects.