HEADING 1 - TW Cen MT Condensed (18 Pt) s2

Math-in-CTE Lesson Plan Template

Lesson Title: 1_ Carpenter on the Roof Lesson # 1 Author(s): Phone Number(s): E-mail Address(es): Steve Ratzlaff [email protected] Megan McCarthy [email protected] Occupational Area: Construction Technology

CTE Concept(s): Work Keys Level 6

Math Concepts: Rise/Run (slope), Pythagorean Theorem, Area, Unit conversion, Computation, Estimation, Rounding, Ratio, Proportions

Performance Standard: [9] GLE: M2.4.1, 2.4.3, 2.4.4, 3.4.1, 5.4.6, 10.4.2

Content Standards: N-3, N-4, MEA-4, E&C-1, F&R-3, F&R-7, G-6, PS-5

Lesson Objective: Students will calculate the roof area based on wall dimensions and slope angle and will estimate the amount of roofing materials required for the job.

Supplies Needed:

THE "7 ELEMENTS" TEACHER NOTES (and answer key) 1. Introduce the CTE lesson. Identify, and define and calculate the following: Show “roof problems” slides slope, pitch, rise/run.

Roof slope is a very important aspect and it is considered the primary factor in roof design. The slope of a roof has an effect on the interior volume of a building, the drainage, the style, and the material you use for your covering. Also it determines the amount of materials used. The slope of your roof is determined by the vertical rise in inches for every horizontal twelve inch (12") length (called the "run"). It is expressed with the rise mentioned first and the run mentioned second. For instance, if your roof has a four inch (4") rise for every horizontal foot, then it has a "4 in 12" slope (or pitch). Show “roof pitch terminology” slide

A fairly easy way to determine the slope is to take a 12" level and set one end on the roof surface and make it level. Then take a tape measure or ruler and measure from the other end down to the roof surface. This will give you the slope of the roof. The slope of the roof in the following picture would be expressed as "4 in 12" or "4 on 12" and written as 4/12, 4:12, or 4":12"

The area of a roof is determined by the amount of “square”, which means how many 10’ x 10’ square (100ft2).

Explain the different types of basic roof design.

Show “Basic Roof Style” slide 2. Assess students’ math awareness as it relates to the CTE lesson. What dimensions do you need to calculate the area of a rectangle? A = Length x Width How do you determine the area of a rectangle? How do you determine surface area? How do you determine the surface area of a roof? Figure out your linear distances, or dimensions. The difficult part is What are the variables needed to figure the amount of materials calclulating the width, which is determined by the rise/run and slope. needed for the roof? You can use the Pythagorean theorem If the roof gets steeper, what happens to the surface area of the roof? Show slide of a steep roof vs. a shallow pitch roof If we have more surface area, how will that affect the amount of materials required?

3. Work through the math example embedded in the CTE lesson. Show picture of a building with 20’ x 20’ walls. If we have a gable roofed building with the wall dimensions of 20’ x 20’, and a slope, or rise and run of 4/12, what is the area of the roof surface?(do not include any overhang) First we need to figure the length of the roof slope. How do we do this? Step 1 4 Calculate this by setting up a proportion. Rise over run is 4/12. or 12 4 x 4(120) 480 Since 10’ = 120”, = ; x = = = 40” 12 120 12 12 This works because we are going up 4” for every 12” in run (4/12), we will multiply 4” x 10, because the mid point of the roof is 10 feet from the edge of the wall. The resulting rise is 40”. The mid point of the roof is 40” higher than the top of the wall.

Step 2 Now, we can use the Pythagorean theorem to calculate the slope length of the roof. Be careful to use the same units of measurement. A2+B2 = C2 (40”) 2 + (120”) 2= 1600” + 14400” = 16000” C = 16000" = 126.49”  126-1/2” The length of the slope (hypotenuse) is approximately 126-1/2”

Step 3 Since we are working in inches, we will convert the 40 ft. wall length dimension into inches. To do this, we will take 40 ft. and multiply by 12. 20 x 12 = 240” The length of the roof is 240”

Step 4 We now have the information required to find the surface area of the existing roof multiply them and convert into square footage. The length of the roof is 240 inches and the width is 126-1/2 inches. A = L x W A = 240 x 126.5 = 30360 inches2

Step 5 Now we will convert this figure into ft2 by dividing by 144, this number is 12” x 12”. ft2 = 30360 / 144 = ft2 The area of ONE SIDE of the roof is 210.8 ft2. We will round to 211.

Step 6 We can figure the area of both sides by multiplying by 2. Total roof area = 211 x 2 = 422 ft2

Step 7 To order shingles for the roof we will need to calculate how many “square” there are on the roof. Each “square” is a 10’ x 10’, or 100 ft2. Total square = area / 100 = 422 / 100 = 4.22 We will need to order at least 4.22 square of roofing material.

Step 8 How many bundles of shingles are in a square? The answer is 3. So we will multiply 4.22 x 3 = 12.66 bundles, or 13 bundles (plus estimated waste of 15%.)

Step 9 How many sheets of plywood will we need to order? Each sheet of plywood is 4’ x 8’, which covers 32 ft2. We can estimate the required amount of roof sheathing by taking the roof area and divide by 32. Sheathing amount = 422 / 32 = 13.18 sheets of sheathing (round up to 14 sheets and add 15% for waste).

So, if things work perfectly we will need the following: 13 bundles of shingles 14 sheets of roof sheathing material

Further steps Add the dimensions of the overhangs. Add 15% for waste by multiplying these figures by 0.15. 4. Work through related, contextual math-in-CTE examples. 6 a. Rise over run is 6/12. or 1. You are building a gabled roof cabin with the dimensions of 30’ x 22’, 12 and a slope, or rise and run of 6/12, what is the area of the roof surface? 6 x 6(132) 792 Since 11’ = 132”, = ; x = = = 66” (rise) 12 132 12 12 b. Use Pythagorean theorem to calculate the slope length of the roof. Be careful to use the same units of measurement. A2+B2 = C2 (66”) 2 + (132”) 2= 4356” + 17424” = 21780” C = 21780" = 147.58”  147-1/2” The length of the slope (hypotenuse) is approximately 147-1/2”

c. Convert the 30 ft. wall length dimension into inches. 30 x 12 = 360” The length of the roof is 360”

d. A = 360 x 147.5 = 52920 inches2

e. 52920 / 144 = 367.5 ft2

f. 367.5 x 2 = 735 total ft2

g. 735 / 100 = 7.35 total square

h. 7.35 x 3 = 22.05 bundles of shingles i. 735 / 32 = 22.875 sheets of sheathing

2. You have a 5' x 5 outhouse with a shed roof that has a steep 8/12 slope. How much plywood do you need for the roof? 8 a. Rise over run is 8/12. or 12 8 x 8(60) 480 Since 5’ = 60”, = ; x = = = 40” (rise) 12 60 12 12 b. Use Pythagorean theorem to calculate the slope length of the roof. Be careful to use the same units of measurement. A2+B2 = C2 (60”) 2 + (40”) 2= 3600” + 1600” = 5200” C = 5200 = 72.11”  72-1/8” (slope length)

c. Convert the 5 ft. wall length dimension into inches. 5 x 12 = 60” The length of the roof is 60”

d. A = 60 x 72.125 = 4327.5 inches2 (roof area in inches)

e. 4327.5 / 144 = 30.05 ft2

f. There is only one slope since it is a shed roof, so no need to double the surface area.

g. 30.5 / 100 = .30 total square

h. .3 x 3 = .9 bundles of shingles i. 30.5 / 32 = .95 sheets of sheathing 5. Work through traditional math examples. Math Practice Questions

1. Find the pitch for the following roofs:

Total Rise Span Pitch

8’ 24’ 8’ 24’ 8/24 =1/3

6’ 36’ 6’ 36’ 6/36 =1/6

9’ 24’ 9’ 24’ 9/24 =3/8

12’ 36’ 12’ 36’ 12/36 = 1/3

8’ 32’ 8’ 32’ 8/32 =1/4

Math Practice Problems Solutions

Area (A) =Length (L) x Width (W)

2 2. Ima Genius is trying to convince her dad to buy new 2. A = 10 ft. x 12 ft = 120 square feet = 120 ft carpet for her bedroom. The dimensions of her room are 10’ by 12’. How many square feet of carpet would her dad need to buy? Draw and label a diagram and solve showing your work. 3. First find the area of the room: A = L x W

3. Heeszo Smart (Ima’s cousin) decided to carpet his A = 10 ft x 14 ft bedroom also. He found 3 dozen carpet samples at a = 140 square feet garage sale. Each one measures 2’ by 2’. His room is 10’ by 14’. Does he have enough carpet samples? Explain. = 140 ft2

Then find the area of each carpet sample:

A = L x W

A = 2 ft x 2 ft

= 4 square feet

= 4 ft2

Divide the room area by the carpet sample area to get the

number of samples needed:

140 ft2 / 4 ft2 = 35; He does have enough because he bought 3

dozen (3 x 12= 36)

4. Ima’s room area from #1 in inches:

Area (A) =Length (L) x Width (W) A = 120 in. x 144 in

= 17,280 square inches

2 4. Ima’s dad finds bundles of floor tiles at a yard sale. = 17,280 in He bought 6 bundles of 10 tiles. Each tile is 16” by 16”. How many tiles will be needed to cover the floor of her Each tile is 16” x 16” or 256 square inches. room? Does he have enough tiles? Divide the room area by the tile area to get the number of tiles

needed.

17,280 in2 / 256 in2 = 67.5 tiles needed; Since he bought 6 bundles

of 10 tile (6 x 10=60) he needs 8 more tiles. Maybe she will get her

carpet after all!

Use the Pythagorean Theorem:

A2 + B2 = C2 ; or

The diagram shows the ladder length is 25 feet = C and 24 feet is one of the legs, say A. The unknown length (distance from the house) is B.

(24 ft)2 + B2 = (25 ft)2 B2 = 625 ft2 – 576 ft2 = 49 ft2

B = 49 = 7 ft

Using the Pythagorean Theorem again:

A2 + B2 = C2

We are looking for the hypotenuse or C

A2 + B2 = C2

62 + 42 = C2

36 + 16 = C2

52 = C2

C = 52 4. Show your work! 5. Show your work! 6. Students demonstrate their understanding. 1. Determine slope Ask: to estimate the roof materials based on wall dimensions, what are 2. Calculate rise of roof the steps required? 3. Calculate length of slope 4. Convert all measurements into inches (or feet) 5. Calculate area of roof 6. Calculate number of square 7. Multiply square by roofing material needs per square 8. Calculate sheathing needed 7. Formal assessment. You are ordering materials for a one story house with wall dimensions Step 1 of 30’ x 30’ and a roof slope of 6/12. Find the following information for the Bill of Matarials: 6 Calculate this by setting up a proportion. Rise over run is 6/12. or Area in Square footage of the roof: 12 6 x 6(180) 1080 Number of bundles of shingles: Since 15’ = 180”, = ; x = = = 90” 12 180 12 12 Number of sheets of OSB: The resulting rise is 90”.

Step 2 Now, we can use the Pythagorean theorem to calculate the slope length of the roof. Be careful to use the same units of measurement. A2+B2 = C2 (90”) 2 + (180”) 2= 8100” + 32400” = 40500” C = 40500" = 201.24”  201-1/4” The length of the slope (hypotenuse) is approximately 201-1/4”

Step 3 Since we are working in inches, we will convert the 30 ft. wall length dimension into inches. To do this, we will take 30 ft. and multiply by 12. 30 x 12 = 360” The length of the roof is 360”

Step 4 We now have the information required to find the surface area of the existing roof multiply them and convert into square footage. The length of the roof is 360“ and the width is 201-1/4” A = L x W A = 360 x 201.25 = 72446.4 inches2

Step 5 Now we will convert this figure into ft2 by dividing by 144, this number is 12” x 12”. ft2 = 72446.4 / 144 = ft2 The area of ONE SIDE of the roof is 503.1 ft2. We will round to 504 ft2.

Step 6 We can figure the area of both sides by multiplying by 2. Total roof area = 504 x 2 = 1008 ft2

Step 7 To order shingles for the roof we will need to calculate how many “square” there are on the roof. Each “square” is a 10’ x 10’, or 100 ft2. Total square = area / 100 = 1008 / 100 = 10.08 We will need to order at least 10.08 squares of roofing material.

Step 8 How many bundles of shingles are in a square? The answer is 3. So we will multiply 10.08 x 3 = 30.24 bundles, or 31 bundles (plus estimated waste of 15%.) Step 9 How many sheets of plywood will we need to order? Each sheet of plywood is 4’ x 8’, which covers 32 ft2. We can estimate the required amount of roof sheathing by taking the roof area and divide by 32. Sheathing amount = 1008 / 32 = 31.5 sheets of sheathing (round up to 32 sheets and add 15% for waste).

So, if things work perfectly we will need the following: 31 bundles of shingles 32 sheets of roof sheathing material

Further steps Add the dimensions of the overhangs. Add 15% for waste by multiplying these figures by 0.15.

NOTES: