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Blue Squares and Beyond

  • Lesson
6-8
2
Geometry
Unknown
Location: Unknown

This Internet Mathematics Excursion is a pre-activity for E-example 6.3 from the NCTM Principles and Standards for School Mathematics. This is the first in a sequence of four lessons designed for students to understand ratio, proportion, scale factor, and similarity. This lesson invites students to manipulate two rectangles to create examples of similarity and to study the effects on area ratios. Students sketch similar figures, verify proportionality, and apply these concepts to structures in their world.

To maximize student learning, certain prerequisites are necessary to use this activity. Thus, it would be appropriate to include this activity as part of a more fully developed Standards-based lesson, but it should not be used as a complete stand-alone lesson.

Discuss with the whole class the concepts of ratio, proportions, proportional, proportional shapes. Ask students to briefly answer the following questions:

  1. Why is it important for models that represent real structures to be in proportion to the actual structures?
  2. If two or more things are in proportion, which dimensions are affected?

Discuss student responses as a class.

Distribute the Blue Squares activity sheet to each student.

pdficon  Blue Squares Activity Sheet 

Students should open the Side Length and Area of Similar Figures Applet. When the applet opens two blue rectangles appear on both grids A and B. Beneath the grids there are ratios for width A : width B, height A : height B, and area A : area B.

Guiding Questions to Ask Students:

  • What do you notice about the ratios?
  • Why are the ratios 1:1?

Using the slide bars, students should create a new Rectangle A. They should record the dimensions and the ratios of Rectangle A and Rectangle B in Chart 1 on the Blue Squares activity sheet.

Below is a chart that will be used for the next several steps.

Rectangle A  

Rectangle B 

Ratios 

Width:     

Width:     

Width A ÷ Width B:     

Height:

Height:

Height A ÷ Height B:

Area:

Area:

Area A ÷ Area B:


Students should now change Square B into a rectangle similar to Rectangle A. Once again, they should record the dimensions and the ratios of Rectangle A and Rectangle B in Chart 2 on the Blue Squares activity sheet.

Next, students will change Rectangle B so it is not similar to Rectangle A. They will record the dimensions and the ratios of Rectangle A and Rectangle B in Chart 3 on the Blue Squares activity sheet.

Students should change Rectangle A to a new figure of any size. Change Rectangle B to a figure out of proportion to Figure A. As previously, students should record the dimensions and the ratios of Rectangle A and Rectangle B in Chart 4 on the Blue Squares activity sheet.

Once the students have created similar figures, use the ratios on the Blue Squares activity sheet charts to verify similarity. Repeat the procedure from above using various dimensions and figures.

Students should now change Figure A to a 3 x 4 rectangle and multiply both width and height by 2 to determine dimensions to create rectangle B.

Guiding Questions to Ask Students:

  • Are these two shapes proportional? How can you tell by looking? How can you tell mathematically?
  • What factor was used to determine the dimensions for Rectangle B?
  • What is the scale factor of these two rectangles?
  • If Rectangle A is 2 x 3 and Rectangle B is 16 x 24, what is the scale factor? (Students may need to act out more examples using the applet before answering this question.)

Students should now choose a scale factor to create rectangles too large to fit on the screen. They should record the dimensions of these rectangles in Chart 5 on the Blue Squares activity sheet.

Next, students should create a new Rectangle A and record the dimensions and the number of squares it covers in Chart 6 on the Blue Squares Activity Sheet. Repeat for Rectangle B.

Students should now create a set of two similar rectangles and record the dimensions and area of each, labeling each correctly. They should record the ratios in Chart 7 on the Blue Squares activity sheet.

Finally, students should reate four more sets of similar rectangles, recording the dimensions and area of each set in Chart 7 on the Blue Squares activity sheet.

Assessments 

Students should complete the following in pairs to assess their understanding of this lesson.

  1. Students should choose and sketch a structure (e.g. car, bridge, building). Estimate the dimensions and area. Label.

    Next, students should sketch a similar figure. Label the estimated dimensions and record area of the similar figure.

    Have students verify mathematically that their two sketches are proportional.

  2. Ask students to explain why it is necessary to have models in proportion to the real structure.
 
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Geometry

Go With Green Rectangles

6-8
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.
1033icon

Fill'r Up

6-8
This Internet Mathematics Excursion is based on E-example 6.3.2 from the NCTM Principles and Standards for School Mathematics. This is the third in a sequence of four lessons designed for students to understand scale factor and volume of various rectangular prisms. In this lesson, the student can manipulate the scale factor that links two three-dimensional rectangular prisms and learn about the relationships between edge lengths and volumes.
Geometry

Purple Prisms

6-8
This Internet Mathematics Excursion is based on E-example 6.3.2 from the NCTM Principles and Standards for School Mathematics. This is the last activity in a sequence of four lessons designed for students to understand scale factor and surface area of various rectangular prisms. Students manipulate the scale factor that links two three-dimensional rectangular prisms to learn about edge lengths and surface area relationships.

Learning Objectives

Students will:

  • Apply the concept of similarity
  • Construct and interpret figures by using scale factor
  • Apply the scale factor of two rectangles to the ratio of area
  • 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

Common Core State Standards – Mathematics

Grade 6, Ratio & Proportion

  • CCSS.Math.Content.6.RP.A.1
    Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

Grade 8, Geometry

  • CCSS.Math.Content.8.G.A.4
    Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.