Section 1

Lesson 1: Slope in the Real World

 

Focus

 

Have you heard of “Roof Over Your Head Day?” This day occurs every December 3 and was organized to let everyone appreciate how fortunate people are who have a home.

 

Most roofs in Canada are not flat. How would you describe the shape of most roofs in Canada? Some words you might use are angled, steep, or sloped. The most common reason for roofs not being flat is to help snow slide off. This helps keep roofs from collapsing under the snow’s weight. In many parts of Canada, where snowfalls of two or three feet are common, this is an important safety issue.

 

If you build a roof, some questions you have to consider are these:

• What are the building code requirements?

• Are there any height restrictions set by the community?

• Which type of roof requires less building materials?

In this lesson you will explore ways to describe slopes, identify slopes, and think about the safety issues that surround slopes.

 

Lesson Questions

 

In this lesson you will investigate the following questions:

• How can slopes be identified and described?
  
  
• When dealing with slopes, what safety features are necessary?

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 1 Assignment (Save a copy of your lesson assignment document to your course folder now.)
  
  
• the Project Connection

In this course you may come across Self-Check questions, Try This questions, and other activities that may or may not be assessed.

 

Remember that these questions and activities provide you with the practice and feedback that you need to successfully complete this course. You should respond to all the questions and place those answers in your course folder. Your teacher may wish to view the work that you have stored in your course folder to check on your progress and to see if you require assistance.

 

Materials and Equipment

• ruler(s)
• pencil
• measuring tape
• stairs

Time

 

This lesson has been designed to take 150 minutes; however, it may take more or less time depending on how well you are able to understand the lesson concepts. It is important that you progress at your own pace based on your own learning needs.

 

Are You Ready?

1. Where does a vertical line point? Answer
   
   
2. Where does a horizontal line point? Answer
   
   
3. What angle is formed when a vertical line meets a horizontal line? Answer
   
   
4. What is the longest side of a right triangle called? 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

 

The media pieces Right Triangle and Legs of a Right Triangle will help you review right angles, right triangles, and parts of right triangles.

 

 

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

 

Look at these three photos and consider the questions that follow.

 

wheelchair ramp: Hemera/Thinkstock; bike rider: Hemera/Thinkstock; stairs: iStockphoto/Thinkstock

• Could you easily go up and down in each case?
  
  
• Which photo seems to show the steepest slope?
  
  
• What activity would be the most dangerous?

1. Fill in the following chart. It will help you describe similarities and differences in the photos.
   
   

   Similarities	Differences

   
   
2. Think about and record some safety issues related to the activities. Use the following chart to record your ideas.
   
   

   Safety Issues

Save both charts in your course folder to use in the following Share 1 activity.

 

Share 1

 

Share your charts from Try This 1 with a partner or with a group of people.

• Compare your charts with the other charts.
  
  
• Explain reasons for any similarities and differences between the charts.

As you work on this Share section, refer to the Share Rubric. The rubric will help you understand what your teacher expects of you.

 

Explore

 

In Discover, you saw a variety of photos with one thing in common: slope. The images show a wheelchair ramp, a bike jump, and stairs up the outside of a temple. What problems could result if the slope of the wheelchair ramp were three times steeper than what is shown in the photo? At what point does a slope become too steep?

 

Try this experiment. Place a pencil about 1 in from the end of a 12-in ruler, as shown in the photo below. How high can you lift that end of the ruler before the pencil slides or rolls down?

 

 

How can you describe the steepness right before the pencil begins to move?

 

Lesson 1 Summary

 

In this lesson you saw slope in roofs, stairs, ramps, pieces of equipment, and in nature. You shared and discussed ways of describing slope and the safety issues surrounding slopes. You also learned that the formula used to calculate slope is made up of vertical distance, called the rise, and horizontal distance, called the run.

 

In Lesson 2 you will use your knowledge of slope and safety to calculate slope to ensure safety standards are met.

 

Lesson 2: Calculating and Understanding Slope

 

Focus

 

 

Trucks, especially those with big loads, have to slow down at some points along roads. This is especially true when the vehicles are being driven downhill. To help the truck drivers, there are signs telling them

• the road’s steepness

• the speed they should be driving

• how long the stretch of road is that they need to slow down for

In Lesson 1 you learned that slopes can be found all around. You also learned that slope is defined as In this lesson you will make calculations to find slope. You will also decide what slope values tell you about the steepness of a slanted object.

 

Lesson Questions

 

In this lesson you will investigate the following questions:

• How is slope calculated?

• Under what conditions will a slope be constant, zero, or undefined?

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 Math 20-3: Glossary Terms and the Formula Sheet
  
  
• Lesson 2 Assignment (Save a copy of your lesson assignment document to your course folder now.)
  
  
• the Project Connection

Materials and Equipment

• calculator
• graph paper (1 cm × 1 cm)

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?

1. Reduce the following fractions to their simplest (lowest) forms. Also give them as a decimal to two places.
   
   
   1. Answer
      
      
   2. Answer
      
      
   3. Answer
      
      
2.  
   1. What is a numerator? Answer
      
      
   2. What is the numerator of the fraction ? Answer
      
      
3.  
   1. What is a denominator? Answer
      
      
   2. What is the denominator of the fraction ? Answer
      
      
4.  
   1. What are equivalent fractions? Answer
      
      
   2. What are two fractions that are equivalent to ? 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

 

Review equivalent fractions using the Equivalent Fractions applet.

 

 

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 the following Try This activity you will explore slope at various points along different graphs of lines and curves.

 

 

Try This 1

 

Complete the table for Slopes 1, 2, and 3 by finding the slope for multiple points on each graph.

 

Slopes of Lines
Slope 1			Slope 2			Slope 3
Point 1	Point 2	Slope	Point 1	Point 2	Slope	Point 1	Point 2	Slope
(−6, −8)	(2, 8)	2

1. What did you notice about the slopes of lines 1, 2, and 3? Did you observe any patterns?

Share 1

 

Discuss your observations and answer to Try This 1 with a partner or with a group of people.

• What similar patterns were observed?
  
  
• Did anyone identify a pattern different from yours? Explain.

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

 

In Explore you will calculate slopes in different contexts.

Self-Check 1

 

 

1. Calculate the slope between two different places on the diagram above.
   
   
   1. Calculate the slope of the line between the black points. Use the black triangle to find the rise and run. Answer
      
      
   2. Calculate the slope of the line between the yellow points. Use the yellow triangle to find the rise and run. Answer
      
      
2. How do the two slopes on the same line compare? Answer
   

---

Refer back to Discover. What patterns did you discover for the slope of lines? You may have noticed that the slope of a curved line changed. The slope on the curved line depended on where you measured the rise and run on the curve. This slope was very different than the slope of a straight line. The slope of a straight line is the same no matter where you measure the rise and run. This slope is called a constant slope.

 

A straight line has a constant slope. A curved line does not.

 

Self-Check 2

 

Answer true or false to questions 1 and 2.

1. Line segment PS would have a constant slope. Answer

2. Line segments PQ, QR, and PR would have different slopes. Answer

3. Calculate the slope of PS. How did you calculate the slope? Answer

Share 2

 

Ninety-five percent of avalanches occur on mountains with slopes between and .

 

With a partner or with a group of people, discuss the following questions:

1. Is an avalanche likely to occur in the right graphic? Explain.

2. Is an avalanche likely to occur in the left graphic? Explain.

Save a summary of your conclusions to your course folder.

 

 

 

Look at the partial diagram of pipes to the right. The water could flow

• vertically (red arrow)

• in a loop where it would flow horizontally and vertically (blue arrow)

• horizontally (green arrow)

Calculate the slope of the blue, red, and green lines. Are any of the lines more challenging to calculate than others?

 

You may have found it challenging to calculate the slope of the green horizontal line since there was no rise to measure. The run may have been hard to measure for the red line.

 

Try This 3

 

Explore the Roller Coaster activity, which describes the slope of a car on a roller coaster. Pay attention to where the slope is equal to zero or undefined.

 

Lesson 2 Summary

 

In this lesson you discovered how to calculate slope. You also described situations where the slope is constant, equal to zero, or undefined. You viewed each of these situations in relation to a roller coaster.

 

Horizontal lines, like the red line segment, have a slope of zero because there is no rise.

 

 

Vertical lines, like the blue line segment, have an undefined slope because there is no run.

 

 

Oblique lines, like the green line segment, have slope found by dividing rise by run.

 

You saw that a straight line has a constant slope. You also determined that curved lines have slopes that change depending on what points you use to calculate their slopes.

 

Lesson 3: Implications of Slope—Safety and Use

 

Focus

 

Water parks are great fun for families and people of any age. How do slide designers make sure everyone has a fun time? Slides that are fun for a young child will be quite boring for most people aged 18. Meanwhile, slides intended for adults are very dangerous for some Grade 1 children. What factors influence slide design?

 

In this lesson you will explore some of these design factors as you compare slopes and use these comparisons to discuss safety and functionality.

 

Lesson Questions

 

In this lesson you will answer the following questions:

• How can the difference between two slopes be described?

• What safety and functionality issues are created by a slope?

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 Math 20-3 Glossary Terms and the Formula Sheet
  
  
• Lesson 3 Assignment (Save a copy of your lesson assignment document to your course folder now.)
  
  
• the Project Connection

Are You Ready?

 

Look at the two photos of signs that indicate road grades. Note that the road’s slope is given as a percentage. Review your skills by rewriting the percentages in the following forms:

• fraction

• fraction in simplest form

• fraction as a ratio

• decimal to the nearest hundredth (two places)

Grade (in percentage)	Fraction	Fraction in Simplest Form	Ratio Form	Decimal Form
8%
7%

 

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 reviewthese topics.

 

Refresher

 

If you don’t know the answers in Are You Ready?, or require more information, then use the following information to review concepts.

 

To review percentages, go to Percent.

For help with decimals, go to Decimals.

 

In addition to being written in fraction or decimal form, slope can also be written as a ratio. Look at the following example of ratio notation to see a number of ways ratio can be expressed. (This material comes from an Individual Learning Module for carpentry. You can check out the whole document at Calculating Ratio and Proportion, Mechanical Advantage and Percentage if you are interested.)

Ratio Notation

 

Suppose you are comparing the ratio of hammers (2) to screwdrivers (6) in your toolbox.

 

 

The ratio can be stated in one of three common ways:

• 2 to 6,
  
  
• 2:6 and
  
  
• 

All of these ratios are read the same way: the ratio is 2 to 6.

 

Because a ratio can be expressed as a fraction, it can also be reduced to its simplest terms. Reducing the ratio of to its simplest terms gives you . This means that there is 1 hammer for every 3 screwdrivers in your toolbox.

 

Note

The order in a ratio matters: a ratio of 1:3 is not the same as 3:1.

 

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

 

Discover

 

Water slides come in all heights and lengths. These measurements will determine how fast you can go and how long your ride will be. In Try This 1 you will investigate slopes of water slides.

 

Try This 1

1. Calculate the slope of the green water slide in the first photo and the slope of the blue water slide in the other photo. Write your answer in fraction form .

2. Describe the similarities and differences between the two slopes.

3. Which slide do you think would be the fastest? What is the fastest slide's slope?

Share 1

 

Once you have completed the Try This 1 activities, discuss the following questions with a partner or with a group of people.

• What is the relationship between the value of the slope and the steepness of the slide?

• What do you notice about the steepness of the water slides when the rise and run of the slope are switched?

Place a summary of your discussion and responses to the questions in your course folder.