Lesson Title: Calculating Diagonals using the Pythagorean Theorem. Lesson #?
... In carpentry, the word square is used for many different things. How.
Math-in-CTE Lesson Plan Template Lesson Title: Calculating Diagonals using the Pythagorean Theorem Author(s):Henry Begler and
E-mail Address(es):
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Phone Number(s): 406-324-2182
Clay Burkett
Lesson #??
406-324-2322
[email protected]
Occupational Area: Carpentry CTE Concept(s): Finding the diagonal of a building to insure that it’s square Math Concepts: Pythagorean Theorem, converting feet-inch-fraction of inch measurements to decimals, converting decimals back to feet-inch-fraction of inch measurements. Lesson Objective: This is the first in a series of lessons to be abl calculate diagonals. Some students need to go back to the basic converting the measurements between fractions and decimals. T lesson objective will be to take decimal answers and convert them in feet-inches. Supplies Needed: tape measure, basic calculator that d square/square root
TEACHER NOTES (and answer key)
THE "7 ELEMENTS" 1. Introduce the CTE lesson.
So today we are going to determine the diagonals for laying out a Discuss the difference between square in this context and the shape of a garage slab to insure your slab is square. So who can tell me what it square. means for a slab to be square? Why is it important for a slab to be square?
Discuss
the
issues
a
building
has
when
it
is
not
square.
Hopefully a student will mention the Pythagorean Theorem or a2 + b2 = c2. It’s important to point out that a and b are the legs of a right triangle and c The Pythagorean Theorem states that in a right triangle, the sum of is the hypotenuse. the squares of the legs is equal to the square of the hypotenuse. What formula do you know that we could use to find diagonals?
Discuss some contexts where square can mean different things in In carpentry, the word square is used for many different things. How carpentry and maybe even some context clues the students could use to is “square” used in the Pythagorean Theorem, different from the know which “square” is meant. square we talked about earlier? 2. Assess students’ math awareness as it relates to the CTE lesson.
Student’s ability levels may vary greatly. This is the first in a series of lessons written because of this very situation—students that did not have the basic math skills to convert between fractions and decimals and do a diagonal calculation.
Draw a rectangle on the board and label its length and width as 6” x 8” and ask if anyone can figure out (or knows) what the length of the diagonal (hypotenuse) would be for this to be “square”. Some students may know the 6-8-10 Pythagorean Triple. Many are familiar with a 3-4-5 right triangle and the following relationship: 32 + 42 = 52 (9 + 16 = 25) Draw a square on the board and label it 12’ x 12’ and ask students to calculate the diagonal for this to be square and to give the answer as a number of feet and inches (to the nearest 1/16”) The Pythagorean Theorem will yield a decimal answer here that the students will have to convert into feet-inches. It is likely that many students will not be able to do this successfully. (122 +122 = c2, c is approximately 16.97’) In feet-inches this is 16’ 11 5/8” The last assessment problem incorporates the carpentry method of writing feet-inches. A corner stake for the concrete forms of a 3 building is measured 35-6 ¾ in one direction, and then 40-9 ¼ The answer for the last problem, in carpentry notation, is 54-116. another direction (at a right angle to the first). You need to calculate Explanation to follow. the diagonal to place the stake in the right location. What is the length of the diagonal to the nearest 1/8”?
3. Work through the math example embedded in the CTE lesson. If the assessment turns out as expected, most students will not be able to convert the decimal answer from the 12’ x 12’ problem to a form they can read on the tape measure. So at this point, working through the steps to take that 16.97’ answer and get the answer 16’ 11 5/8”
0.97’ means a
97 100
proportion
and that needs to turn in to 12ths. This can be done using 97 𝑥 = . 100 12
Solve
this
proportion
using
cross
products
97×12 100
97×12=100× 𝑥. To solve for x, you do which equals 11.64 which is a number of inches—at this point it would be good to not mention that this is a number of inches and ask the class what the 11.64 means.
The next step is to take the 0.64” and set up another proportion to turn that in to a fraction of an inch. In this setting it is appropriate to calculate the After working through second problem, you might opt to work through answer to the nearest sixteenth of an inch so you would set up and solve the third problem (which involves converting the fraction to a decimal 64 𝑦 before doing the Pythagorean Theorem, then converting the answer the following proportion: 100=16 back to feet-inches-fraction of an inch). One point of emphasis here is 10 that out answer must be something that can measured on a tape The answer here is 10.24 which we have to know means 16 inches which 5 measure. would be ”. 8
Another way of setting up the proportion and solving would be to use .64 𝑥 = , which yields .64x16=𝑥. Many CTE teachers use this method, but it 1 16 is important for students to first learn proportional reasoning so that they know more than just a “shortcut method” of doing the work and not have a mathematical understanding that transfers to many different applications. In the last problem, 6 ¾” is 6.75”—divide by 12 to get about 0.56’. Next, 9 ¼” is 9.25”—divide by 12 to get about 0.77’. Using the Pythagorean Theorem, 35.562 + 40.772 = c2; c is about 54.10’. To convert the 0.10’ to 10 𝑥 inches, first set up a proportion = (or multiply 0.1 by 12). To solve the 100 12 proportion, multiply 10 by 12 and set equal to 100 times 𝑥. To solve for 𝑥, divide by 120 by 100 to get 1.2”. To change the 0.2” into the nearest 2 𝑦 sixteenth, set up a proportion = (or multiply 0.2 by 16). Solving this 10
16
3
proportion in a similar fashion (using cross-products) gives you 16” (rounded to the nearest sixteenth). The final answer in carpentry notation is 3 54-1 . It is appropriate to discuss how accurate a measurement is 16 calculated depending on the type of work for which it is used (foundations vs. framing vs. cabinetry for example)
4. Work through related, contextual math-in-CTE examples. The challenge for many students is not the Pythagorean Theorem itself, but the changing units and forms of the numbers, so another related example is to find the diagonal for a 35’x50’ slab. The main challenge here will be to turn the decimal answer to a “tape measure friendly” answer.
The decimal answer here is 61.03’. The 0.03’ must be turned to inches using proportions as demonstrated earlier. This gives 0.36”. Turning this in to a fraction (to the nearest sixteenth of an inch) gives 5.76—round this to 3 3 6, reducing gives an answer of ". Final answer in carpentry form: 60-0 . 8
8
5. Work through traditional math examples.
Convert from given measurement to fraction form of the measurement. Answers: 1. 10.25 in.
1. 10 ¼” 1
2. 3.33 in. ( to nearest 16 in.) 1 8
3. 20.18 ft (to nearest in.)
6. Students demonstrate their understanding. 2. 14.67 ft. (to nearest in.) 3. 72.111 ft. ( to nearest
7. Formal assessment.
5 “ 16
1 8
5 8
3. 20’ 2 ” or 20-2
Answers: 5
1. 2. 7.625 in. 1 16
2. 3
1. 7 8 "
2. 14’ 8” in.)
5 16
3. 72’ 1 "
In a future lesson, students will “stake out” a square foundation on a piece of wood using four nails, string, and the Pythagorean Theorem to insure Since this is a multi-step lesson/process, the formal assessment that the foundation is square. Students will also go outside and stake out piece occurs in a later lesson where students will have to lay out a the foundation for a 24’x24’ garage. foundation and measure the diagonal to make sure it is square,
NOTES:
