Section 2

Section 2 Introduction

Library and Archives Canada; Copyright: Canada Post Corporation, 1927.. Reproduced with permission.

This commemorative stamp was issued in 1927 at the 60th anniversary of Canada's Confederation. It's amazing to be able to fit Canada, which stretches more than 9300 km from the Atlantic to the Pacific, on one small 12-cent stamp! The map, of course, is rather small in scale.

In Lesson 1 you will explore one-point perspective drawings of three-dimensional objects. The concept of scale in drawings, maps, and models will be discussed in Lessons 2 through 4. In the Module 5 Project you will create a scale model of a three-dimensional object.

Lesson 1: One-Point Perspective

Focus

Have you ever driven along a prairie highway and watched the road narrow, and even vanish, at a distant point on the horizon? This classic prairie landscape in the photo captures the way your eyes see depth and your mind perceives distance.

In this lesson you will explore a method of drawing objects the way your eyes would see them in three dimensions.

Lesson Question

In this lesson you will investigate the following question:

• How are views drawn in one-point perspective?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to the Mathematics 20-3: Glossary Terms
  
  
• Lesson 1 Assignment (Save a copy of your lesson assignment to your course folder now.)

Materials and Equipment

• blank sheets of paper
• pencils
• pencil crayons
• eraser
• straight edge
• square from a geometry set

Time

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 1.

Are You Ready?

In this lesson you will apply your knowledge of some of the terminology associated with one-point perspectives.

1. What is a horizontal line? Answer
   
   

2. What is a vertical line? Answer
   

If you answered the Are You Ready? questions without problems, move on to Discover.

If you found the Are You Ready? questions to be difficult, complete Refresher to review these topics.

Refresher

If you don’t know the answers in Are You Ready?, or require more information, click on the buttons to study the Horizontal Line and Vertical Line applets.

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty with the questions.

Discover

In this section you will investigate several principles behind one-point perspective drawings. A perspective drawing represents three-dimensional objects in such a way that the objects appear three dimensional rather than being flat on a two-dimensional surface. There are one-point, two-point, and three-point perspective drawings. In this course you will focus only on one-point perspective drawings.

First, you must understand an important concept. A horizon line is a horizontal line that represents the viewer’s eye level. If the horizon is above an object, the viewer must be looking down on top of the object.The vanishing point is an area on the horizon where the object moves out of sight. As you approach the vanishing point, objects get smaller and smaller until they disappear. The vanishing point is the farthest point in the perspective drawing.

Use One-Point Perspective Interactive to investigate the effects on a perspective drawing when you move the vanishing point.

Try This 1

Use One-Point Perspective Interactive to answer the following questions.

1. What side of the object do you see when the vanishing point is moved to the left side?
   
   
2. What side do you see when the vanishing point is on the right side?
   
   
3. What side of the object do you see when the vanishing point is above the object? Why?
   
   
4. What side do you see when the vanishing point is below the object? Why?

Share 1

Share your answers to Try This 1 with a partner or with a group of people. Then discuss the following question:

• How does the view of an object (the side you see) change as the vanishing point moves?

If required, save a copy of your discussion to your course folder.

Lesson 1 Summary

In 1973, W. O. Mitchell’s book The Vanishing Point was published. The setting for this novel was patterned after the Eden Valley Reserve (part of Stoney Nakoda Nation) near Calgary. The novel deals with relationships between the First Nations people on the reserve and people in the wider community. Why do you think Mitchell chose The Vanishing Point as his title?

road: © JMB/9491797/Fotolia; foot bridge: © micahbowerbank/3773650/Fotolia; grass field: © Dreamframer/14438802/Fotolia

If you were designing a cover for Mitchell's book, which image would you choose? Where might the vanishing point be?

In this lesson you explored one-point perspective—the art of drawing a three-dimensional object so the object appears as your eyes see it.

Lesson 2: Drawing Scale Diagrams

Focus

Architects, drafters, salespeople, home buyers, and contractors all make use of blueprints from a home's conception to its completion. Blueprints outline all the home’s features from the exterior to basement construction, framing, wiring, heating, and details for the interior design. Blueprints are drawn to scale so required materials can be ordered in the correct sizes and to ensure that overall dimensions and interior partitions can be easily understood.

In this lesson you will explore how to draw scale diagrams, of which blueprints are just one example. The scale diagrams will include representations of both two-dimensional and three-dimensional objects.

Lesson Question

In this lesson you will investigate the following question:

• How is a scale diagram of a two- or three-dimensional object drawn?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your additions to the Mathematics 20-3: Glossary Terms
  
  
• Lesson 2 Assignment (Save a copy of your lesson assignment document to your course folder now.)
  
  
• the Project Connection

Materials and Equipment

• isometric dot paper
• ruler
• geometry set
• calculator

Time

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 2.

Complete the following Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your teacher.

Are You Ready?

In Module 3 and in previous mathematics courses, you solved equations involving ratios and proportions. Solving these types of equations is an essential skill when working with scale diagrams.

1. What is a ratio? Answer
   
   
2. Solve for x.
   
   
    
   
   
   Answer
   
   
3. Solve for x.
   
   
    
   
   
   Answer
   
   
4. Solve for x.
   
   
    
   
   
   Answer

If you answered the Are You Ready? questions without problems, move on to Discover.

Refresher

If you don't know the answers in Are You Ready?, or require more information, go to the Rates and Ratios applet to study equations involving ratio and proportion.

Go back to Are You Ready?, and try the questions again. Contact your teacher if you continue to have difficulty with the questions.

Discover

Have you used the zoom feature on a computer monitor? You may have used this device to enlarge small print. In Try This 1, you will identify patterns between the percentage of zoom and the change in an object's size.

Try This 1

1. Set the zoom feature to 100%. Measure the width and height of the eagle image with a ruler—use millimetres (mm). Record the measurement in a chart similar to the one shown.
   
   
2. Use “zoom” to enlarge and reduce the image to match the zoom setting in the chart. For each zoom setting, measure the height and width of the eagle image. Record all your observations.
   
   
   iStockphoto/Thinkstock
   
   

   Zoom Setting	Width	Height	Ratio   Width Height	Reduced Ratio   Width Height
   100% (original)
   50%
   25%
   150%
   200%

   
   
3. Does the zoom feature change the drawing's overall shape? Why or why not? How do you know?
   
   
4. Does the ratio between the width and the height change in the table? How do you know? Add another column to the chart so you can confirm your answer.
   
   
5. How does “zoom” affect the dimensions of the original drawing?

Save your responses to your course folder.

Share 1

Share your responses to the questions in Try This 1 with a classmate or with a group of people.

• How did the patterns observed in each table compare to yours? What patterns did you discuss that were new to you?

If required, save a copy of your discussion in your course folder.

Lesson 2 Summary

In this lesson you were introduced to scale used in maps, blueprints, technical diagrams, and photographs. You discovered that scale can be expressed with units to reference, or scale may be outlined without units as a ratio or as a fraction.

As well as studying scale, you practised drawing scale diagrams. This lesson was just an introduction. In Lesson 3 you will spend more time exploring scale.

Lesson 3: Constructing a Model to Scale in Three Dimensions

Focus

Models of future aircraft are tested in wind tunnels, such as the pictured one at NASA’s Langley Research Center. Constructing and placing scale models in wind tunnels at each stage of development helps aeronautical engineers determine the forces applied and the performance of their designs.

In this lesson you, too, will construct three-dimensional models to a given scale.

Lesson Question

In this lesson you will investigate the following question:

• How are three-dimensional models built according to a given scale?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to Mathematics 20-3: Glossary Terms
  
  
• Lesson 3 Assignment (Save a copy of your lesson assignment document to your course folder now.)
  

Materials and Equipment

• ruler or geometry set
• scissors
• glue stick or tape

Time

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 3.

Are You Ready?

When working with scale, you must be familiar with the units of linear measure in both SI and the imperial system. The following questions include both measures.

1. How many inches are there in 1 ft? Answer
   
   

2. How many centimetres are there in 1 m? Answer
   
   
3. Name three common items that are about 1 in wide. Answer

If you answered the Are You Ready? questions without problems, move on to Discover.

If you found the Are You Ready? questions difficult, complete Refresher to review these topics.

Refresher

If you don’t know the answers to the questions in Are You Ready?, or if you require more information, go to Lesson 1 of Mathematics 10-3: Module 5 to review linear measure in the imperial system.

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty with the questions.

Discover

Try This 1

Architects use a scaling ruler to directly calculate scale measurements. Click on Scale Rulers to download and print a file that contains two scale rulers.

For this activity you will need

• blank paper
• scale ruler page (from the file you just printed)
• straight edge or ruler
• pencil
• scissors
• tape

1. Cut out the scale ruler pieces and assemble them to make one big ruler. Attach the paper ruler to the straight edge or ruler using tape.
   
   
2. Using the ruler and the net of a rectangular prism, construct a model of the rectangular prism that is either half or double the original's size.
   
   Hint: If you are stuck, watch Using a Scale Ruler, a short animation on how to use a scale ruler to scale up or scale down.
   
   
   

Share 1

With a partner, or with a group of people, share your answers from Try This 1. Then discuss the following question:

• How does the scaling ruler help you scale an object up or down quickly?

If required, save a summary of your discussion in your course folder.

Example

It is not likely that a full-scale model of the Pyramid of Khafre at Giza will ever be built. This pyramid was built with a square base of approximately 215 m on each side and the slant height of each triangular face is approximately 180 m.

1. If a 1:2000 scale model were constructed, how long, to the nearest millimetre, would each side of the square base be?

2. To the nearest millimetre, what would the slant height be for each triangular face in the model?

Solution

Description of Step	Find the side length of the scale model.	Find the slant height of the scale model.
To use the scale of 1:2000, you need to have all measurements in the same units—millimetres will work well.   Convert the side length to millimetres. Remember that 1 m = 1000 mm.
Identify the ratios that relate the scale to the actual model lengths. Let s represent the scale model square base length. Let x represent the scale model slant height.	s: 215 000 = 1:2000	x: 180 000 = 1:2000
Set up a proportion and solve for s and x.   Each side of the model’s square base would be about 108 mm and the slant height would be 90 mm.

Try This 3

You saw how the dimensions of a 1:2000 square model of Khafre’s pyramid are calculated. Use the scale model dimensions from the example to build a scale model of the pyramid. You can modify the pyramid net provided or create the model from scratch.

You will need

• ruler
• geometry set
• tape or a glue stick

Save your work to your course folder or somewhere safe. Your teacher may ask to view your model.

Self-Check 1

iStockphoto/Thinkstock

This photo is of Kukulkan Pyramid at Chichen Itza in Mexico. The height of the temple is 24 m and the square base is 55.3 m on a side. What scale would you use to construct a three-dimensional model of this Mayan temple so that the base of the model is 11.06 cm on each side? Answer

Throughout the history of people, Three-Dimensional Models have been made from a variety of materials.

images: PhotoObjects.net/Thinkstock

Lesson 3 Summary

Flying radio-controlled model aircraft is a popular pastime for people from all age groups. From the photo, can you identify the type of aircraft? What do you think is the scale of the model?

In this lesson you explored constructing a three-dimensional scale model.

One of your tasks was modifying a geometric net to accurately represent a given object.

Lesson 4: Interpreting Scale Diagrams

Focus

Credit: Atlas of Canada, (Ottawa: Natural Resources Canada, 1990), <http://atlas.nrcan.gc.ca/ > (24 November 2011).

If you look at early maps of Canada, you will notice that political divisions have changed. In 1901, the provinces of Alberta and Saskatchewan did not exist. Alberta and Saskatchewan were just two districts of the Northwest Territories, which stretched from Canada’s Arctic to the American border. As you can see from the map, the District of Saskatchewan spanned over 800 km from parts of present-day Alberta and Manitoba. Without the scale shown on the map, it would be impossible to judge distances or appreciate how large these districts were.

In this lesson you will explore the concept of scale in more detail. You will examine scale on maps and on other scale diagrams.

Lesson Question

In this lesson you will investigate the following question:

• How is proportional reasoning used to solve problems involving scale diagrams?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to the Mathematics 20-3: Glossary Terms
  
  
• Lesson 4 Assignment (Save a copy of your lesson assignment to your course folder now.)
  
  
• the Project Connection

Materials and Equipment

• ruler
• calculator

Time

Launch

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 4.

Complete the following Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your teacher.

Are You Ready?

In Mathematics 10-3 and in Module 3 of this course, you explored changing from one unit of measurement to another. For example you changed pounds to kilograms, metres to millimetres, and yards to inches. Converting from one unit to another is an essential skill to master so you can solve problems involving scale.

1. Convert 4 yd to feet. Answer
   
   
2. Convert 12 km to centimetres. Answer
   
   
3. Convert 0.1 mi to inches. There are 5280 ft in a mile. Answer

If you answered the Are You Ready? questions without problems, move on to Discover.

If you found the Are You Ready? questions difficult, complete Refresher to review these topics.

Refresher

If you don’t know the answers in Are You Ready, or if you require more information, go back and review Module 3: Section 2, Lesson 2.

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty with the questions.

Discover

Have you ever played with Google Maps? If so, you’ve probably already looked at a satellite image of your home. How much detail were you able to see? In Try This 1 you will look up your home or a famous landmark using Google Maps to examine the effects of scale.

Try This 1

1. Open up http://maps.google.ca/ and locate your home or another landmark by typing the address in the search engine. What is the scale of the first map that appears? If you were to write the scale as a ratio, what would it look like?
   
   
2. Zoom in by pressing the + button once. Describe what happens to the scale. By how much does the scale increase or decrease? Why do you think this happens?
   
   
3. As you zoom in, do you see more or less area of the neighbourhood?
   
   
4. Predict what will happen if you press the + button one more time. Discuss your prediction in terms of scale and area displayed on the map. Check your prediction.
   
   
5. Predict what will happen if you press the − button four times. Discuss your prediction in terms of scale and area displayed on the map. Check your prediction.

Save your responses to your course folder.

Share

Share your responses to the questions in Try This 1 with a classmate or with a group of people. Work together to

• create a statement that describes how the zoom feature in Google Maps affects the scale of the map and the area in view
  
  
• test your statement using Google Maps

If required, save a copy of your discussion in your course folder.

Lesson 4 Summary

David Thompson is one of the world’s greatest explorers and cartographers (map makers). He arrived at Hudson Bay at the age of 14 and spent 28 years of his life exploring and mapping almost one-sixth of North America. His maps have been used for more than 200 years!

In this lesson you explored problems involving scale. Maps are just one of the many contexts in which scale is an essential component. In addition to scale, you reviewed converting between units and using proportions.

Section 2 Summary

This photograph shows a bleak winter landscape and captures an image of a long, cold walk across the bridge. The sense of distance is created though one-point perspective. The bridge narrows as it approaches the far shore, just as elements farther away appear smaller.

In Lesson 1 you explored the elements of one-point perspective in photography and drawing. In Lessons 2 and 4, you examined the concept of scale. Using proportional reasoning, you solved a variety of problems involving scale in maps, diagrams, art, and three-dimensional models. In Lesson 3 you created three-dimensional scale models. The creation of a three-dimensional scale model was the central theme for the Module 5 Project: Produce a Set of Drawings and a Scale Model.