Math 21 Home Book 10: Slope & Elevation Name: Start Date: Completion Date:
Year Overview: Earning and Spending Money Home Travel and Transportation Recreation and Wellness 1. Budget 2. Personal Banking 3. Interest 4. Consumer Credit 5. Major Purchases 6. Scale Drawings & Ratios 7. Area & Volume 8. Angles 9. Triangles 10. Slope & Elevation 11. Travel Project 12. Puzzles & Games 13. Understanding Statistics 14. Budgeting Recreation Topic Overview There is a lot of mathematics that can help you understand, design, and create things at home. Scale drawings help you to design decks and buildings, or read architectural drawings. Ratios not only help you to interpret scale drawings, you also see them in TVs and computer monitors. This section of the home unit is designed to help you understand and create scale drawing and understand the ratios around you. Suggested Timeframe: 8 Hours Outcomes Overlapping Outcomes in Scale Drawings and Ratios M21.1 Extend and apply understanding of the preservation of equality by solving problems that involve the manipulation and application of formulae within home, money, recreation, and travel themes. Theme Specific Outcomes M21.4 Demonstrate an understanding of slope 2
Contents Topic Overview... 2 Outcomes... 2 Overlapping Outcomes in Scale Drawings and Ratios... 2 Theme Specific Outcomes... 2 Slope... 4 Check Your Skills... 5 10.1 Calculating Slope... 7 10.1 Practice Your Skills Calculating Slope... 10 10.1 Practice Your Skills - Slope... 11 10.2 Angle of Elevation... 16 10.2 Practice Your Skills Calculator Practice... 18 10.2 Practice Your Skills Angle of Elevation and Slope... 19 Student Evaluation... 20 Learning Log... 22 3
Slope What is slope? If you have ever walked up or down a hill, then you have already experienced a real life example of slope. As you go up hill, you may feel like you are spending lots of energy to get yourself to move. The steeper the hill, the harder it is for you to keep yourself moving What is slope? (Explain in your own words) Look at the picture below. Label at least 4 more locations where slope is important in the construction of a home. 1 1. Gutter - Directs water off the roof in a desired direction. 2. - 3. - 4. - 5. - 4
Check Your Skills 1. Reduce the following fractions to simplest form. Remember to divide the top and bottom number (numerator and denominator) by the largest common number. a) 20 100 = 5 b) 2 100 = 1? c) 10 4 = d) 21 81 = e) 12 15 = 2. Convert Decimal Numbers to Fractions a) 0. 1 = 10 4. Multiply or Divide: a) -6 1 = b) (-5) X -6 = c) -4 X 7 = b) 0. 24 = 100 c) 0.68 = d) 1.68 = e) -0.8 = 3. Simplify each ratio: a) 2:6 = 1: b) 28:21 = : d) 8 = 8 5. Subtract: a) -4 - -6 = b) 3 (-4) = c) -10 5 = d) -8 - (-8) = c) 12:18 = d) -6:4 = 5
6. Find the length of the missing side for each of the following triangles. a) b) c) d) 6
10.1 Calculating Slope 7
Example: Duncan and Casey are using hand trucks to move small boxes from a house to a garage. They lay a loading ramp against the house steps, which are 18 high. The slope of the ramp is 0.2. What is the horizontal distance (run) from the base of the ramp to the point where the ramp rests on the top stair? Solution: The known values are: Slope = 0.2 Run =? Rise = 18 in Using the formula for slope: ssssssssss = rrrrrrrr rrrrrr Recognizing that we can use a formula triangle to isolate the unknown: So, calculating the run, or distance from the bottom of the ramp to the point where it touches the top of the step: 18 iiii rrrrrr = 0.2 rrrrrr = 90 iiii 8
In this investigation, you will be comparing the slope of the line segments AB, BC, and AC. How do you think these slopes will compare? The roof truss diagram has been copied to the graph paper below. C B A 9
10.1 Practice Your Skills Calculating Slope Using the roof truss diagram on graph paper, 1. Construct a right triangle with line AB as the hypotenuse. Find the rise and run of this triangle. Calculate its slope. 2. Construct a right triangle with line BC as the hypotenuse. Find the rise and run of this triangle. Calculate its slope. 3. Construct a right triangle with line AC as the hypotenuse. Find the rise and run of this triangle. Calculate its slope. 4. Compare the slope of the three lines AB, BC, and AC. What do you notice? 10
10.1 Practice Your Skills - Slope 1. Driving through the Rocky Mountains in British Columbia is a beautiful experience. Many of the passes that roads go through are very steep. Which is steeper, the Kootenay Pass or the Coquihalla Pass? a) A section of the Kootenay Pass in British Columbia gains 610 m of elevation over a horizontal distance of 12 100 m. Sketch the pass and calculate its slope. b) A section of the Coquihalla Pass in British Columbia gains 845m of elevation over a horizontal distance of 19 880 m. Sketch the pass and calculate its slope. c) Which section of highway is steeper? 11
2. The slope of a roof is called its pitch. Pitch is reported as the number of inches that a roof rises for every foot (12 inches) of run. For example, a roof that rises 6 inches in a foot is called a 6:12 roof. The pitch of a roof can be measured by placing a level on the roof, marking off 12 on the level, and then measuring the vertical distance between the level and the roof. The diagrams below show the profiles for 3 different roofs. Each square in the grid is 1 ft wide and 1 ft high. Calculate the pitch of each of the triangles and report it in the form :12 a) b) c) 12
3. Sandy is building a three-step staircase along the pathway to her garden. The total rise is 21 and the total run is 63. Assume that each step is the same height. Sketch the staircase. What is its slope? What is the rise and run of each step? 13
4. An extension ladder is built so that it can be made longer or shorter, depending on how high you need to go. To set a ladder safely, you need to follow the 1 in 4 rule. The distance from the bottom of the ladder to the wall should be ¼ of the distance from the top of the ladder to the ground. a) What is the slope of a ladder leaning safely against a wall? b) When an extension ladder is 6 m long, how far should its base be from the wall it is leaning against? c) When an extension ladder is 16 m long, how far should its base be from the wall it is leaning against? 14
d) How far do you need to move the base of the ladder when you change its length from 6 m to 16 m? e) Why do you think the 1 in 4 rule exists? 15
10.2 Angle of Elevation The angle of elevation is often mentioned when you are looking at an object at a distance, for instance: What this means is: The angle of elevation to the top of the tree 50m away is 35. The angle of elevation is the angle from the ground up to the line of sight to the object. Angle of elevation is an idea that combines two things that we already know about, slope and the Tangent Ratio: ssssssssss = rrrrrrrr rrrrrr TTTTTT AA = OOOOOO AAAAAA Opposite/Rise If Rise = the Opposite Side, and If Run = the Adjacent Side, then Slope = Tan A, where the angle A is the Angle of Elevation, Adjacent/Run 16
There is a button on your calculator that looks like TAN -1. This is called the Inverse TAN. You can use it to find out the size of Angle A, or the Angle of Elevation. If the slope = 3.5, then what is the Angle of Elevation? Slope = Tan A 3.5 = Tan A Tan -1 3.5 = A 75.0 = A Therefore, the angle of elevation for a slope of 3.5 is 75. The tan -1 button is usually found when you SHIFT tan. 17
10.2 Practice Your Skills Calculator Practice Ensure your calculator is set to degrees. 1. tan 45 = Is your calculator set to degrees? Here is a way to check: Input Tan 45, if you get an answer of 1, your calculator is on the right setting! 2. tan 0 = 3. tan 70 = 4. tan -1.577 = Enter.577 then press the tan -1 button on your calculator you will get an answer of 30 if you are in degree mode. 5. tan -1 0 = 6. tan -1 1 = 18
10.2 Practice Your Skills Angle of Elevation and Slope 1. Al wants to make a small copper picture frame similar to the one in the picture. He needs to make sure the angle is correct to balance the picture. a) What is the angle at the base of the picture frame? b) What is the total length of copper that Al needs to create this picture frame? (Hint: Use the Pythagorean Theorem to calculate the length of the hypotenuse) 2. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33º. How tall is the tree? Be sure to sketch the problem! 19
Student Evaluation Insufficient Evidence (IE) Student has not demonstrated the criteria below. Developing (D) Growing (G) Proficient (P) Exceptional (E) Student has rarely demonstrated the criteria below. Student has inconsistently demonstrated the criteria below. Student has consistently demonstrated the criteria below. Student has consistently demonstrated the criteria below. In addition they have shown their understanding in novel situations or at a higher level of thinking than what is expected by the criteria. Proficient Level Criteria IE D G P E M21.1 Extend and apply understanding of the preservation of equality by solving problems that involve the manipulation and application of formulae within home, money, recreation, and travel themes. a. I can prove whether given forms of the same formula are equivalent and justify the conclusion. b. I can describe, using examples, how a given formula is used in a home. c. I can create, solve, and verify the reasonableness of solutions to questions that involve a formula. d. I can find errors in an example of a student s work in a question that involves a formula. I can correct this mistake and explain what the student did wrong. e. I can solve questions that involve the application of a formula that: Does not require manipulation Does require manipulation 20
Proficient Level Criteria M21.4 [WA 20.9] Demonstrate an understanding of slope. a. I can research and present contexts that involve slope including the mathematics involved (e.g. ramps, roofs, road grade, skateboard parks, ski hills, treadmill). b. I can explain why I agree or disagree with the statement: It requires less effort to independently use a wheelchair to climb a ramp of a certain height that has a slop of 1:12 rather than a slope f 1:18. c. I can explain using examples and illustrations why slope can be described as rise over run. d. I can analyze slopes of objects, such as ramps or roofs, to determine if the slope is constant and explain the reasoning. e. I can analyze and explain the relationship between slope and angle of elevation (e.g. for a ramp [or pitch of a roof, grade of a road, slope in pipes for plumbing, azimuth in the sky] has a slope of 7:100, the angle of elevation is approximately 4 degrees). IE D G P E 21
Learning Log Date Starting Point Ending Point 22