3, 4, 5 TRIANGLE LESSON PLAN
Lesson title: How would you layout a 90 degree angle
for a landscape structure such as a shed, deck, or
rectangular structure?
Teacher used training aids: 6,” 8” and 10” plywood or card stock squares
Marked with 2” grids
Additional 8” square cut into 4 pieces
16’ tape
Framing square with rafter table (optional)
12’ rope loop with knots every foot (optional)
Materials needed per student: 11” x 17” paper
12” ruler
Pencil
Calculator with √ key & memory +/- functions
Principles of Square Root Applied to Right Triangles
Right triangles puzzle
Bell work/Do Now:
What does the little square in the inside corner of a triangle indicate?
Lesson Objectives:
 Illustrate the layout of a three different triangles based upon 3, 4, 5 triangles on
11” x 17” paper, accurate to within 1/16”.
 Demonstrate comprehension of the concept of Pythagorean Theorem through the
use of proper terms and symbols.
 Apply formula to solve variety of construction problems using a calculator to
compute accurately.
PA Academic Standards:
2.10.11.A: Identify, create, and solve practical problems involving right triangles using
the trigonometric functions and the Pythagorean Theorem.
2.5.11.A: Develop a plan to analyze a problem, identify the information needed to solve
the problem, carry out the plan, check whether an answer makes
sense, and explain how the problem was solved in grade
appropriate contexts.
2.5.11. B: Use symbols, mathematical terminology, standard notation, mathematical
rules, graphing and other types of mathematical representations to
communicate observations, predictions, concepts, procedures,
generalizations, ideas, and results.
PA Agriculture Benchmarks Addressed:
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19.2 Measure site and develop scaled drawings.
9.4 Layout a structure foundation, erect batter boards frames or forms.
9.9 Demonstrate skills in the construction of an agricultural structure.
Summary of concept/principle:
The right triangle is used extensively for layout in landscape design construction.
The simplest way to lay out a 90° angle is to use the 3-4-5 method. Understanding is
demonstrated by applying the formula (a² +b²=c²) to solve additional construction related
problems using a calculator is highly desired in many areas of industry and agriculture.
Key Terms/Vocabulary:

Altitude (rise)

Angle

Base (run)

Diagonal

Hypotenuse

Right angle

Square (²)

Square root (√)

Off-set

Framing square
Interest Approach:
Prior to closing the door after all students have entered the classroom wet your hands if
possible. When you lean out to close the door try to have your hair out of view of all
students. Use your wet hands or dry to mess your hair up as much as possible, standing
on end. Shut the door and enter the room, allow students time to notice your appearance!
Once noticed run your hands through your hair and bug your eyes out to look stressed!
Explain to the students you have gotten an urgent request from “a customer” to build a
square gazebo for their daughter’s wedding and future enjoyment. Ask students how they
think you should do a quality job and ensure its square. Once you guide them to a 90
degree or square angle begin the lesson as follows.
Introduction/Narrative:
In building layout and floor framing, buildings are checked for square. The 3-4-5
method is commonly used. This is a very old method developed by the Greeks. It’s called
the Pythagorean Theorem. Here's the deal; there was this Greek guy named Pythagoras,
who lived over 2,000 years ago during the sixth century B.C. Pythagoras spent a lot of
time thinking about math, astronomy, and music. One idea he came up with was a
mathematical equation that's used all the time in architecture, construction, and
measurement. Don’t worry about remembering the name, he’s dead now. What is
important is how to use the theory to layout right triangles. I’ll use this rope to
demonstrate. Today you will work in your teams to layout triangles on paper. Then use
calculators to solve construction related problems.
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Body:
1. Draw on white board and explain:
Angle,
90° angle and
right triangle
2. Right triangles are special:
a. Used extensively in construction.
b. 45°-45°-90° and 30°-60°-90° have unique qualities.
3. Construction terms and uses:
a. Base is run. Used in roof and stair design & construction
b. Altitude is rise. Also applied in the construction of roofs and stairs.
c. Electricians, plumbers, carpenters, architects, and engineers also use right
triangle. Off-set is another term.
d. Hypotenuse is diagonal when squaring rectangles; used in framing& forms
4. Demonstration for 3-4-5
a. Measure the up 3” on end of transparency & mark.
b. Measure allow long edge 4” & mark
c. Draw a line from mark to mark.
d. Measure the length of the line (5”).
e. Note: this can be done on a white board using a 16” tape. Measurements
become 3’, 4’ and 5.’ Layout is done by swinging an arch to establish pts.
5. Student practice layout and measure 90° triangles
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17”
9”
15”
6”
3”
11”
10”
5”
4”
8”
12”
6. Why does this work? Show cutouts. Also demonstrate how the cut 4² pieces fit
around 3² to form 5².
Cuts of spare 4” square
7. Demonstrate the division of the above squares based upon the Pythagorean
Theorem drawing a grid for a 3-4-5 triangle where the smaller red square is
divided into 9 blocks, the medium red box is divided into 4 squared or 16 blocks,
and the big blue box is 5 squared or 25 blocks. May simply use and overlay like
this below:
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8. Hand out calculators and Principles of Square Root Applied to Right Triangles
a. Explain keys (memory +/-, √ etc)
b. Practice drill with calculators
9. Review a² + b² = c² on board
a. Insert 3-4-5
b. Have students practice keys.
10. Explain relationship of formula substitute 3-4-5
a. a² + b² = c²
b. a² = c² - b²
c. b² = c² - a²
11. Solve problems using √ key
a. Work in teams
b. Encourage correct responses
12. Explain landscape design uses for the right triangle
a. Wall braces, stair stringers, roof rafters, floors, and foundations.
b. Direct responses to landscape design and excavation questions.
c. Assist student teams solve 3-4-5 triangle questions
Review:
1. What are some of the ways right triangle is used in construction?
2. Explain 3-4-5. What does c² equal? Who can explain “Square root”?
3. The framing square has a table printed (all runs are 12”). Show table.
4. What is the hypotenuse of a triangle with a 12” run and 5” rise? (use calculator)
5. Questions?
Laboratory Section: (Students should be in teams of 3-4)
The preferred setting for this lab exercise is outside. The required tools and
equipment or acceptable substitutes for an outside exercise are:
10 Tape measures
1-2 Rolls or one bale of thin twine of string
1” Wooden dowel rod, or ½” rebar, or ¾” square stake, etc.
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10 hammers to pound stakes into the ground
2-3 Scissors to cut twine or string
Inside lab exercise plan (inclement weather):
The required tools and equipment or acceptable substitutes for an inside exercise
are:
4-5 Chalk lines or 10 pieces of chalk
10 tape measures
10 yard stick or acceptable straight edge
5 roles of masking or duct tape
8 large point sharpie markers
Laboratory exercise/narrative:
Escort students to the lab/shop or outside depending upon your situation. Ask
students to carry the supplies to designated areas that are spread out enough for
them to layout 3-4-5 and larger triangles.
Demonstrate the process of laying out a 3-4-5 triangle
Instruct students to layout the triangles the customer is considering
Notify students that you as the teacher is the “code inspector” and their triangles
must be measured and checked for accuracy within 1” at most.
Upon approval INSPECTOR MUST INITIAL THEIR JOB SHEET after
each triangle is “APPROVED!!”
Be sure to ask students which side is the, b, c, hypotenuse or/and diagonal and
where the right angle is on each triangle.
Be sure to ask the expected length of each side and the diagonal/hypotenuse.
Let the students figure out how to take the triangle to a rectangle gazebo and hint
at simply “flipping” the triangle measures along the diagonal/hypotenuses.
Corners should be staked
Lines should be illustrated with the twine wrapped once around the state from
stake to stake for both the triangle and the “gazebo” to allow students to visualize
the shapes, angles, etc.
Note: inside use the tape on the floor with and X marked in sharpie (or
chalk). Snap the lines with chalk lines or draw them with chalk.
Additional resources: word search Right triangles puzzle
http://www.ams.org/samplings/feature-column/fcarc-surveying-one
http://www.rowan.edu/open/mcsiip/impact/impactlessonplans/math/looking_for_pythago
ras_investigation_3_fassett.html
http://home.avvanta.com/~math/y2l9p.html
Learning Assessment - Students will be evaluated for participation in the activities will
be assessed 10 points for each triangle construction exercise for 30 points IF INITIALLED. An
additional 10 points will be assessed as part of their Daily Participation Grade. Guided notes
are worth 10 points. 50 total points available in this lesson(s).
Cognitive Connect –
Yesterday – We finished our exploration of greenhouse capacity for Mrs. Greenjeans!
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Today – We determined what it will take to site a square gazebo for Mrs. Greenjeans!
Tomorrow – We will dig into the topography of landscape design.
Adaptations/Accommodations for Special Needs - Every effort will be made to
accommodate and adapt the lessons and material for exceptional learners in accordance
with their IEP. It is the intent of the agriculture education program to facilitate success for
these learners with a fair chance to show what they know in the medium that best suits
their abilities. Students with exceptional needs will receive guided worksheets with all
the correct answers verified upon completion for study and test preparation. Exceptional
students exceeding the pace will be provided additional open ended questions to develop
a deeper understanding of the material presented. Testing and assignment
accommodations may be arranged to allow the student to have the test presented audibly,
visually, or electronically as well as more time allotted for satisfactory completion.
Participation in sample collection can be simulated by having them scoop some snow for
the house of cards test, or they can search the internet for average snowfall events in your
locality.
Total Ag Program –
Academic Enhancement: This unit is math based and contains very technical terms and
applications that are cross disciplinary with math, science, and engineering classes the
students may be exposed to. It also presents more in-depth information on formulary
calculations that may be very helpful in the future when the students move on to other,
more advanced math and science courses.
In addition to plant science, this unit also helps the students’ writing and research skills
because the assignments and assessments allow the students to show their understanding
through observation, sampling, and worksheets with short answer and open ended
questions. The laboratory aspects of this unit will also help students begin to understand
the importance of research and aid in the development of citation and summarizing skills,
all of which are used extensively in other core subjects.
SAE Integration: The information in this unit would be helpful to students interested in
starting, continuing, or enhancing their entrepreneurship books. The terms and
information would apply to several types of placement SAE and would help students
understand some of the concepts used in agriculture, greenhouse management, and landscape
design. It also may expose students to some new concepts they may want to utilize in
their CDE’s.
FFA Integration: Surveying topics are part of a Land Judging CDE exams. Through this
knowledge acquisition and applying it in the classroom/field/lab, students will be able to
answer these questions. The lessons on any of the processes could also serve as topics in
the various public speaking events. This exposure could give students the basic
knowledge and interest prompting them to do further research and develop a relevant
topic based on their findings to use for a speech in their CDEs.
Career Path Options:
Agricultural engineering technician
Landscape design technician
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Greenhouse design technician
Land Survey Apprentice
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