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PBS Mathline Activity 1: How Many Shingles?

PBS Mathline Activity 1: How Many Shingles?. Real life application for area. Problem Solving. It’s about applying the math we know to help us in our life.

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PBS Mathline Activity 1: How Many Shingles?

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  1. PBS Mathline Activity 1: How Many Shingles? Real life application for area

  2. Problem Solving • It’s about applying the math we know to help us in our life. • There are some math problems where we need to know the math to survive, but there are many other situations where knowing math can save us TIME, MONEY, & FRUSTRATION.

  3. How can using math save us TIME, MONEY, & FRUSTRATION? • Ms. Dalton’s bathroom project - involved knowledge of measurement, geometry (area, perimeter), and algebra (calculating costs). Picture courtesy of Loftus Land on Flickr

  4. Could we do the project without this math? Yes, but it definitely added • TIME, • MONEY, • & FRUSTRATION.

  5. What are some other scenarios where problem solving can save us TIME, MONEY, & FRUSTRATION? • Making purchases - ex. Picking out a cell phone plan, cable plan, etc. • Building anything / home improvement - ex. Landscaping • Calculating your average grade • Creating a budget • And the list goes on and on….

  6. So… how do we approach problem solving?

  7. A Plan for Problem Solving • Explore • Plan • Solve • Examine

  8. 1. Explore - • Read (or identify) the problem carefully. • Pick out the important information • Determine what it is you need to find. Picture courtesy of IceSabre from Flickr

  9. 2. Plan • Pick out a strategy to use. • Some strategies: • Look for a pattern • Draw a diagram • Make a table • Work backward • Use an equation or formula • Make a graph • Guess and Check • Estimate the answer Picture courtesy of dSeneste.dk on flickr

  10. 3. Solve • Use your strategy • Be sure to answer the question. Picture courtesy of Johan Koolwaalj on Flickr

  11. 4. Examine • Check your answer. Does it make sense? Is it close to your estimate? Picture courtesy of CIMMYT

  12. Review of Skills • Pythagorean Theorem – When given a right triangle, leg2 + the other leg2 = the hypotenuse2 OR a2 + b2 = c2 • Proportions – Set two ratios equal to each other. Be sure to label the parts. Cross multiply and solve for the variable. 1 x n = 5 x 12 1n = 60 n = 60 60 eggs

  13. The surface area of a roof is measured in squares of shingles. Each square covers 100 square feet. However, when you buy them, shingles are usually priced per bundle. In most cases, 3 bundles = 1 square.

  14. To calculate the number of bundles needed for a roof, measure the roof’s square footage (length x width). Divide that number by 100 to get the number of squares needed. Use the equivalency above (3 bundles per square) to determine the number of bundles needed to shingle the roof. ** Hint – When it comes to calculating bundles, think proportions

  15. 1. You are planning to roof a house that has exterior measurements of 60 feet by 24 feet. The house has a gable roof. Use the information provided in Figures 1 and 2 to determine the number of bundles of shingles you must order to completely re-roof this house.

  16. Think – what shape is the roof?

  17. Form a Plan • Find the area that needs to be shingled • Determine how many squares are needed • Determine how many bundles are needed

  18. Form a Plan to find the Area to be Shingled • Identify the shape. • Identify the formula(s) • Identify the values needed for the formula • “Plug and Chug” • Check to see if your answer makes sense

  19. What shape is the roof? It can appear to be a parallelogram; however, if we look at it straight on, it is a rectangle. Two rectangles – one on each side.

  20. Form a Plan to find the Area to be Shingled • Identify the shape. Rectangle • Identify the formula(s) • Identify the values needed for the formula • “Plug and Chug” • Check to see if your answer makes sense

  21. Form a Plan to find the Area to be Shingled • Identify the shape. Rectangle • Identify the formula(s) A = lw • Identify the values needed for the formula • “Plug and Chug” • Check to see if your answer makes sense

  22. What is the length and width of the rectangle? The length is 60 ft. But what is the width? We know the width of the house, but we do not know the width of the rectangle. We can figure it out though.

  23. How do we find the length of the blue side? Do you remember Pythagorean theorem? When given a right triangle, the measure of the leg2 + the other leg2 = the hypotenuse2 OR a2 + b2 = c2

  24. How do we find the length of the blue side? a2 + b2 = c2 What is the length of the red side? 122 + 92= c2 144 + 81 = c2 225 = c2 √225 = √c2 15 = c OR in this case – the width to our rectangle

  25. What is the length and width of the rectangle? The length is 60 ft. The width is 15 ft. A = lw A = 60 15 A = 900 ft2 For the one side. There are two sides, so how much in all?

  26. Go back to the plan • Identify the shape. • Identify the formula(s) • Identify the values needed for the formula • “Plug and Chug” • Check to see if your answer makes sense Does my answer make sense?

  27. Go back to the master Plan • Find the area that needs to be shingled - done 1800 square feet • Determine how many squares are needed • Determine how many bundles are needed

  28. Go back to the master Plan • Find the area that needs to be shingled - 1800 square feet • Determine how many squares are needed 1800 square feet / 100 square feet in a square = 180 squares 3. Determine how many bundles are needed

  29. Go back to the master Plan • Find the area that needs to be shingled - 1800 square feet • Determine how many squares are needed 1800 square feet / 100 square feet in a square = 180 squares • Determine how many bundles are needed

  30. 2. You are again planning to roof a house that has exterior measurements of 60 feet by 24 feet. This house has what is called a hip roof.

  31. a. Describe the geometric figures (shapes) used to determine the areas of the two roof styles. The first house’s roof was made of two rectangles. This house’s roof is made of two congruent trianglesand two congruent trapezoids.

  32. b. Explain how you must change your method of calculating the are of the roof for the hip roof style. Identify the shape. Because the shape is different, we will use different formulas Identify the formula(s) Identify the values needed for the formula “Plug and Chug” Check to see if your answer makes sense

  33. b. Explain how you must change your method of calculating the are of the roof for the hip roof style. Identify the shape. Because the shape is different, we will use different formulas Identify the formula(s) Triangle – A = ½ bh, Trapezoid – A = ½h(b1 +b2) Identify the values needed for the formula “Plug and Chug” Check to see if your answer makes sense

  34. Identify the values needed for the formula c. The ridge line of the hip roof is 9 feet above the first floor, and it’s length is 2/3 the length of the house. Determine the number of bundles necessary to re-roof this house.

  35. Identify the values needed for the formula c. The ridge line of the hip roof is 9 feet above the first floor, and it’s length is 2/3 the length of the house. Determine the number of bundles necessary to re-roof this house. Triangle – A = ½ bh base = 24 ft. h b height = ?? What do we know that could help us figure out the height of the triangle?

  36. Let’s look at the house differently. Pretend you have sliced the house in half and are looking at it head on. Ridge line Roof 9 feet First floor 60 feet The ridge line of the hip roof is 9 feet above the first floor, and it’s length is 2/3 the length of the house.

  37. Let’s look at the house differently. Pretend you have sliced the house in half and are looking at it head on. 2/3 of 60 ft. Roof 9 feet 60 feet First floor The ridge line of the hip roof is 9 feet above the first floor, and it’s length is 2/3 the length of the house.

  38. Let’s look at the house differently. Pretend you have sliced the house in half and are looking at it head on. 40 ft. 2n + 40 = 60 -40 -40 2n = 20 n = 10 10 ft. Roof 9 feet ? ? 60 feet First floor The ridge line of the hip roof is 9 feet above the first floor, and it’s length is 2/3 the length of the house.

  39. Let’s look at the house differently. Pretend you have sliced the house in half and are looking at it head on. 40 ft. Roof 9 feet 10 ft. 60 feet First floor The ridge line of the hip roof is 9 feet above the first floor, and it’s length is 2/3 the length of the house.

  40. How do we figure out the slanted side which is the “height” of the triangle? a2 + b2 = c2 102 + 92 = c2 100 + 81 = c2 181 = c2 13.45 = c

  41. Triangle – A = ½ bh b= 24 h=13.45 Multiply that by 2 because there are two triangles in the roof. h

  42. Let’s go through a similar process with the trapezoid sides A = ½h(b1 +b2) What values do we know? b1= b2= h = 60 ft. 40 ft. ??

  43. In order to find the height of the trapezoid, we’ll go through a similar process of that of the triangle. Pretend you have sliced the house in half the other way and are looking from the side a2 + b2 = c2 9 ft. 12 ft. 24 ft.

  44. Let’s go through a similar process with the trapezoid sides A = ½h(b1 +b2) What values do we know? b1= 60 b2= 40 h = 15

  45. What is the total area for the hip roof? • How many squares would that be? • How many bundles would that be?

  46. Pictures courtesy of: • Bradfort Timeline on flickr- https://www.flickr.com/photos/bradford_timeline/6876098717 • Douglas Sprott on flickr - https://www.flickr.com/photos/dugspr/6949786993 • Alan Bernau Jr. on flickr - https://www.flickr.com/photos/alansfactoryoutlet/4295913752 • Elliott Brown on flickr - https://www.flickr.com/photos/ell-r-brown/5610415467

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