Illuminations: Linking Length, Perimeter, Area, and Volume

Linking Length, Perimeter, Area, and Volume

Go With Green Rectangles

This Internet Mathematics Excursion is based on E-example 6.3 from the NCTM Principles and Standards for School Mathematics. This is the second in a sequence of four lessons designed for students to understand ratio, proportion, scale factor, and similarity using perimeter and area of various rectangular shapes. Students manipulate 2-dimensional rectangles to focus on the relationship between the scale factor and ratio of perimeters of similar rectangles, and the relationship between scale factor and ratio of areas of similar rectangles.

Learning Objectives
 
Students will:
  • compare the perimeter and area ratios of similar rectangles with various scale factors
  • discover and articulate the relationship between scale factor perimeter and area through the use of tables and graphs
  • recognize the area relationship of similar rectangles and the square of the scale factor
  • create pattern units of squares, predict how patterns with different numbers of squares will appear when repeated in a grid, and check their predictions
  • analyze how repeating patterns are generated

Materials
 
Side Length and Area of Similar Figures Applet
Green Rectangles Activity Sheet
Grid paper
Graphing calculators

Instructional Plan

As students experiment with similar rectangles and different scale factors relative to side lengths, they have the opportunity to observe and interpret the changes in the perimeter and area. Students should consider the relationships between perimeters and areas of similar figures as they relate to the scale factor. Graphing will be used as a method to visualize the linear and non-linear relationships.

Each student turns to a neighbor and states two or three facts they know about perimeter and area. Several pairs of students share one fact about perimeter and area with the rest of the class.

Students should open the Side Length and Area of Similar Figures Applet. When the applet is open, there are two similar rectangles, one blue and one green. Along the top of the screen is a sliding dot for scale factor. Students should move the dot and watch the results to become familiar with this applet.

Students should set a scale factor and observe what happens to the blue rectangle. Also notice the ratios on the right side of the screen which are perimeter of B: perimeter of A, and area of B: area of A.

Next, students should change the green rectangle size by pulling the red dot at the bottom to change the width or at the top to change the length.

(If the green rectangle is too large, the words on the right side of the screen do not show. Students may want to adjust the size of the green rectangle.)

Guiding Questions to Ask Students:

  • As the size of the green rectangle changes, what do you notice about the scale factor and the ratios?
  • Why do some numbers change and other numbers do not?

Students should record scale factor, length, width, perimeter and area for Rectangle A and Rectangle B in the appropriate spaces in the table on the Green Rectangles activity sheet and repeat Steps 3 and 4 at least 10 times.

Green Rectangles Activity Sheet Green Rectangles Activity Sheet

Guiding Questions to Ask Students:

  • What is the relationship between the scale factor and the perimeter ratio?
  • Why is the scale factor the same as the perimeter ratio?
  • What is the relationship between the scale factor and the area ratio?
  • Why is the scale factor the square root of the area ratio? (It is easiest for students to see this relationship when the scale factor is 3.)

On the applet, students should click on the Graphs button. Using the Scale Factor, length and width from their table, they should create graphs for each of their 10 trials.

Guiding Questions to Ask Students:

  • What do both graphs have that are the same?
  • What do both graphs have that are different?
  • When the Scale Factor is changed, how do the points on the Perimeter vs Scale Factor graph change?
  • What is the ratio of Perimeter to Scale Factor for the point shown?
  • How does the Perimeter vs Scale Factor graph illustrate relationship between perimeter and scale factor?
  • When the Scale Factor is changed, how do the points on the Area vs Scale Factor graph change?
  • What is the ratio of Area to Scale Factor for the point shown?
  • What is the relationship between the area of a rectangle and the scale factor?
  • How does the Area vs Scale Factor graph illustrate the relationship between area and scale factor?

Questions for Students
 

If the scale factor is 1.377, what will the perimeter ratio be?

If the scale factor is 1.377, what will the area ratio be? (Students should verify answers by returning to the applet and setting scale factor at 1.377, noting ratios.)

Repeat the above questions with the following scale factors:

2.213
1.648
0.922
3.00

Explain how you determine the perimeter ratio if you know the scale factor.

Explain how you find the area ratio if you know the scale factor.

NCTM Standards and Expectations
 
Geometry 6-8
  1. Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
  2. Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
  3. Use geometric models to represent and explain numerical and algebraic relationships.
  
2 periods   

NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
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