Math-in-CTE Lesson Plan Template
Lesson Title: Cutting a Common Rafter / Lesson #6Author(s): / Phone Number(s): / E-mail Address(es):
Schiewe / (907) 352-0431 /
Taylor / (907) 352-0456 /
Occupational Area: Building Trades/Math
CTE Concept(s): Constructing a common rafter
Math Concepts: right triangles, slope, decimal conversions, fractions, Pythagorean Theorem, square roots
[9] MEA-2, [8] MEA-2, [8] MEA-3, [8] MEA-1
Lesson Objective: / Students will be able to accurately layout and cut a common rafter with provided span and slope within a
certain tolerance.
Supplies Needed: / Tape measure, carpenter pencils, angle squares, framing square, Skilsaw, rafter, ridge board, temporary
supports, 16 penny nails, student tool belts.
The "7 Elements" / Teacher Notes
(and answer key)
Prior Knowledge Required / Students should be able to use the Pythagorean Theorem to find a hypotenuse and be able to subtract in feet and inches with borrowing to complete this lesson.
1.Introduce the CTE lesson.
While most people don’t build their own rafters, there are certain situations where this skill can be cost effective and appropriate.
Successfully completing this task requires attention to detail. Imperfections in building materials to the slightest degree can cause inaccurate results.
Few carpenters can complete this task without error and having this skill can make you more marketable as a carpenter. / Span – the total width of the structure
Total Run – half the span
Total Rise – the height or altitude of the roof triangle
Unit Run – the unit of measurement given in inches
Unit Rise – the amount of rise per foot of run
Slope – the incline of a roof expressed as a ratio of rise to run
Pitch – the slope of a roof expressed as a ration of rise to span
Line length – the hypotenuse of the roof triangle
Ridge board – the horizontal framing member that rafters are aligned against to resist their downward force
Centerline – half the thickness of the ridge board (typically 3/4”)
Rafter Tail – the line length of the rafter doesn’t not include the tail. The length of the tail is a separate calculation, based on the roof overhang.
Overhang – The horizontal measurement from the wall to the end of the rafter tail. If the plan shows the overhang at 24” , that means 24” horizontally from the vertical plane of the wall to the vertical plane of the rafter tail.
Bird’s Mouth – This is a notch cut into a rafter to provide a bearing surface where the rafter intersects the top plate of the wall. The bird’s mouth notch is comprised of 2 cuts in the rafter. A seat cut, which is the horizontal cut where the rafter bears on the top wall plate, and the plumb cut, which is the vertical cut of the bird’s mouth.
Height at Plate (HAP) – the distance measured vertically from the intersections of the seat cut and plumb cut, to the top edge of the rafter.
Ridge drop – the amount the ridge board is dropped to meet the top edge of the rafters.
2. Assess students’ math awareness as it relates to the CTE lesson.
Image (Figure 10-15 pg 109)
What makes it a “common” rafter?
Go over terminology.
What is the difference between rise and run?
How is slope defined?
What is the Pythagorean Theorem?
Give a triangle and have them find the hypotenuse.
What tools do you use to mark and cut a slope on a board? / A “common” rafter is a rafter used for a roof with 2 sloping sides.
Rise is horizontal change. Run is vertical change.
Slope is rise/run.
Pythagorean Theorem – a2 + b2 = c2 in a right triangle with hypotenuse c.
Mark and cut slope – use angle square
3. Work through the math example embedded in the CTE lesson.
Given a structure with a span of 32’ and a slope of 4/12 and an 1’ 6” overhang. The plan calls for 2x8 rafters and a 2x10 ridge board and a 4” HAP.
Step 1
Total run: If the span is 32’ and the run is ½ of the span, what is the run? (answer 16’)
Step 2
Rise constant: The slope is 4/12 so the rise constant is 4÷12 = .33
Step 3
Total rise: is the rise constant times total run. (.3333 * 16’ = 5.3328’) What is .3328’? It is closest to 1/3 of a foot, 1/3 of 12” = 4”. So the Total rise is 5’ 4”.
Step 4
Line length constant: The slope is 4/12, so we do Pythagorean Theorem ’, then divide by 12 to convert to a unit rate = 1.054
Step 5
Line length: Take the line length constant times total run. (1.054 * 16’ = 16.865’) So .865 * 12 = 10.38” then convert the decimal to a fraction (.38 is close to 3/8) So the total line length is 16’ 10 3/8”
Step 6
Length of rafter tail: Take the line length constant times the overhang (in decimal 1’6” is 1.5’). 1.054 * 1.5’ = 1.581’ then convert to inches (.581 times by 12) = 1’ 7”
Step 7
Total Rafter length: Line length plus rafter tail length = 16’ 10 3/8” + 1’ 7” = 18’ 5 3/8”
Step 8
Ridge drop: Use image (10-16 Pg 113) Rise constant times half of the ridge board thickness (usually ¾” = .75). .3333 * .75 = .2499 so it is ¼”
Step 9
Length of ridge support: Add total rise and HAP and then subtract the height of the ridge board and the ridge drop. 5’ 4” + 4” = 5’ 8” then – ¼” – 9 ¼” = 4’ 10 ½” / Step 1
Total Run: 32÷2 = run of 16’
Step 2
Rise Constant: The rise constant is the “unit rate” at which the roof rises. .33’ rise per 1 foot run.
Step 3
To convert from ft to in multiply by 12: .3328’ * 12 = 4”
Step 4
Line length constant: the dividing by 12 is to convert to a unit rate, not to convert into inches. The unit rate is still in feet.
Step 5
Use table to look up decimal .865 and find nearest useful fraction.
Step 6
Convert 1.581’ to inches. Multiply .581 times 12 = 7” and add the 1’
Step 7
Add fractions. Some problems may require carrying.
Step 8
If necessary, look up the decimal in table to find nearest useful fraction.
Step 9
Calculation: The additions
The numbers we are going to subtract:
= To do this subtraction we need to borrow from the 5’. In the math world this would look like:
But it can be much easier… you know you can’t subtract 9 ½” from 8” so how much short are you? 9 ½ - 8 = 1 ½” so we are now at 5’ and need to go down another 1 ½”. 5’ – 1 ½” = 4’ 10 ½”.
4. Work through traditional math examples.
#1 Pythagorean Theorem
Find the hypotenuse of a triangle with legs of length 5’ and 12’.
#2 Convert decimal to fraction
What is the closest common fraction to .3127?
#3 Subtract feet and inches from each other with borrowing
1’ 10” – 5 13/16” = / These exact problems will be the ones used in Element 5, so students who struggle with the math can still be successful in practice phase.
#1 = 13
#2 .3125 is 5/16
#3 1’ 10” – 5 13/16” = 1’ 4 3/16”
5. Work through related, contextual math-in-CTE examples.
Given a structure with a span of 8’ and a slope of 5/12 and an 1’ overhang. The plan calls for 2x4 rafters and a 2x6 ridge board and a 2” HAP.
Find the total rise, line length, tail line length, total rafter length, ridge drop, and support length. / Students draw and make calculations on paper.
Step 1
Run = ½ of span
Span 8’ so run is 4’
Step 2
Slope is 5/12
Rise constant 5÷12 = .4166
Step 3
Total rise = run times rise constant
4’ * .4166 = 1.66’ convert to inches 1’ 8”
Step 4
Line length constant= Pythagorean Theorem with slope numbers and then divide by 12.
= 13. 13÷12 = 1.0833
Step 5
Line length = line length constant times run
1.0833 * 4’ = 4.33’ times 12 = 4’ 4”
Step 6
Length of rafter tail = line length constant times overhang
1.0833 * 1 = 1.0833’ convert to inches (* 12) = 1’ 1”
Step 7
Total rafter length = line length + rafter tail
4’ 4” + 1’ 1” = 5’ 5”
Step 8
Ridge drop = rise constant * half ridge board width
.4166 * .75” = .3124” (round to .3125”) = 5/16”
Step 9
Length of ridge support = total rise + HAP – ridge board length – ridge drop
1’ 8” + 2” – 5 ½” – 5/16” =
1’ 10” – 5 13/16” =
1’ 4 3/16”
6. Students demonstrate their understanding.
Students are going to build a rafter. They will calculate the lengths and set the ridge board support at the proper height to complete this task.
7. Formal assessment.
Book (Pg 115)
NOTES: