Lesson Development
Math-in-CTE Lesson Plan Template
Lesson Title: Roof pitch and design
Author(s):
Daren Harmon
Lesson #
Phone Number(s):
M6
E-mail Address(es):
David Hagel
Occupational Area: Architectural Drafting
CTE Concept(s): Roof styles and pitch design
Math Concepts: Slope of line, Pythagorean theorem, ratios and proportions, conversion
Lesson Objective:
Students design and draw elevation view for various roof styles
Supplies Needed:
Worksheets, pencil, Architectural scale, house plans project
THE "7 ELEMENTS"
TEACHER NOTES
(and answer key)
1. Introduce the CTE lesson.
“We have previously discussed various Roof
styles and terms and you have developed your
plan view and have started the wall elevations
for your residential plan project.”
“We now need to add the roof elevation to our
plans.”
Prior Knowledge:
Beginning CAD class
Blueprint reading
Roof Styles, walking tour of local
neighborhood.
Developed Roof Plan view
To do:
Ask students what styles they have selected.
Hip, Gable, Shed, Gambrel
Worksheet #1 - Introduce and Identify roof
framing components
(See Key for W.S. #1)

Review different roof styles
models, pictures, blueprints, ?

Vocabulary of roof pitch terms and
abbreviations

Labeling roof pitch diagram
W.S. #1 will include
Vocabulary/abbreviations, truss/rafter picture,
example(s)
“You will need to determine these values for
your project”
National Research Center for Career and Technical Education
Lesson Development
2. Assess students’ math awareness as it
relates to the CTE lesson.
“Your assignment will require you to find missing
values to several different roof truss styles. You
may use any past experiences or knowledge to
complete it. Please show all work.”

Given the rise and run, the W.S. will include
calculating rafter length, pitch, unit rise(Uri),
unit run (URu is always 12), and unit line
length (ULL).
Worksheet #2
3. Work through the math example
embedded in the CTE lesson.
“Out of the Vocabulary terms from worksheet
#1, which have we NOT yet used?”
“Get out W.S. #2, and sketch extension and
dimension lines showing Rafter Length.”
Bridge question:
Rafter Length(RL), Unit Line Length(ULL)
Refer to W.S. #1 for extension/dimension
lines.
“How could we find these two lengths?” RL, ULL
Pythagorean Theorem. a 2  b 2  c2
Instruction – Work example
Split truss samples down middle, add right
angle symbol on examples.
Worksheet #3 (back of W.S. #2)

Find the Rafter length(RL) and Unit line
length (ULL) - Pythagorean Theorem
“(1) Turn your paper over (W.S. #2) and with
straight edge, draw the span and the roof line to
form a large triangle.
(2)Now draw a vertical line from the roof peak to
the span line to split this triangle in two.”
“(3)What is the angle that this line makes with
the span line?” ( 90 or right angles)
“We now have two right triangles”
(4) Place a, b, c according to the Pythagorean
thm. with a and b as legs of right triangle and c
is the hypotenuse.
National Research Center for Career and Technical Education
a, b are legs of right Triangle, c is the
hypotenuse.
Lesson Development
From the Pythagorean theorem we know that
a 2  b2  c2
a 2  b2  c2 .
c2  a 2  b2
If we solve for c, we get c  a 2  b 2
“But we have different names for a, b, and c.
What are they?
c2  a 2  b2
c  a b
2
2
example: pick a number, 7
so what is the square root of 7 squared?
7 2  49  7
a is Total Rise(TRi), b is Run, and c is Rafter
length(RL)
“If we substitute our terms for a, b, and c, then
we get RL  TRi 2  run 2 ”
Example #1- Room-in-attic
“Substitute 20 for TRi, 24 for run and calculate
the Rafter length(RL).
We want the answers in feet and inches.
31.24 feet equals 31’ 2.88”.
We should round to the nearest 1/8th in.
Say 31’ 2 7/8”
Example #2 –Scissors
RL  82  162
RL  64  256
3
RL  19.70 feet = 19' 8 "
8
Example #3 – Vault
RL  19.5 feet = 19' 6"
National Research Center for Career and Technical Education
RL  320
5
RL  17.89 feet = 17' 10 "
8
RL  7.52  182
RL  56.25  324
RL  380.25
RL  19.5 feet = 19' 6"
Lesson Development
Example #4 – Dual Pitch
(Work to come later)
4. Work through related, contextual math-inCTE examples.
Review question:
Total Rise(TRi) divided by the Run.
“What is the definition of pitch?”
Bridge question:
Slope (of a line)
“What is pitch called in a Math class?”
“In math class, the slope is defined as the rise
divided by the run, which is exactly what we
have just been doing.”
“What other real life applications do we come
across the concepts of slope/pitch?”
Related contextual CTE example(s)
Drainage, ramps, landscape, stairs, bracing,
driveways, patios, streets, etc.
Worksheet #4
#1 Ramp Problem
Rise of the Ramp – use proportions
What are we tying to find, and how are we going
to do it?
Ramp length – Use Pythagorean thm.
#2 Drainage Problem
Length of drainage pipe needed from
cleanout to main sewer line. – Students will
need to calculate the total drop(proportions)
and then use it to find pipe length using Pyth.
Thm.
What are we tying to find, and how are we going
to do it?
(work to come later)
(work to come)
National Research Center for Career and Technical Education
Lesson Development
#3 Stair Problem
What are we tying to find, and how are we going
to do it?
Find unit rise (length of tread) and the unit
run(height of riser). - proportions
Stringer length – Pythagorean thm.
5. Work through traditional math examples.
Worksheet #5

Proportion Problems

Conversion Problems

Right Triangle Problems (Pythagorean
thm.)
6. Students demonstrate their
understanding.
Students complete roof elevation on own house
plans.
7. Formal assessment.
Quiz on figuring pitch and rafter length.
(To come later)
National Research Center for Career and Technical Education
(answer key to come later)
All work will be completed on student
residential house plans.
(To come later)