# Last Step in Using Pythagorean Formula = Square Root

```Carpentry T-Chart
Last Step in Using Pythagorean Formula
= Square Root of a Number
Duty: Building and Building Foundation
Layout
Task: Locate lines on batter boards and do a
diagonal check
CARPENTRY ASSOCIATED WORDS:
Diagonals, checking for square, 3-4-5
Same method is to be done here.
Square Rooting a Number for a Diagonal Check
is the last step when using the Pythagorean
Formula
Foundation for a Building
?
8 ft
12 ft
PSSA Eligible Content - M11.A.1.1.1
Find the square root of an integer to the
nearest tenth using either a calculator or
estimation
MATH ASSOCIATED WORDS:
Hypotenuse, diagonal, legs
Perfect squares, square root, “root”,
Pythagorean Triplet (3-4-5)
Calculator Method to find Square Root:
Two Lines Display Screen Calculator:
Ex.
7 - press
enter 7
press Enter or =
2.64575 ≈2.6
One line Display Screen:
enter 7
Ex. 7 press
press Enter or =
2.64575 ≈ 2.6
Nearest
Estimation Method to find Square Root:
Ex. 7
Pick two perfect
squares around number
to be square rooted, one
below and one above
Pythagorean Theorem:
a2+b2=c2
12 2 + 8 2 = c 2
208 = c 2
208 208 = c
4
Calculator:
208 = 14.422 ≈ 14.4 ft.
Estimation:
196
=
14
208
=
14.4
210.5
=
14.5
Now, 14.4 ft is about 14 ft 5 1/16 in
=
2
225
=
15
7
9
=
3
4 = 2 and
9 = 3, so
7 must be between 2
and 3.
The middle between 4
and 9 is 6.5
and the middle between
2 and 3 is 2.5., so the
7 must be a little
bigger than 2.5.
An estimate around 2.6 to 2.7 would be fine.
4
=
2
6.5 7 9
=
= =
≈2.5 ? 3
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
In Carpentry, you are taking square roots of measurements. Measurements are considered to
be positive numbers. In math, there are occasions that we try to square root negative whole
numbers (called negative integers). We cannot square root a negative number. Your calculator
will give an error message or domain error.
In carpentry, the decimal part of the answer may need to be converted to inches or even
fractions of an inch.
7 ≈ 2.6.
could mean 2.6 feet. (Convert the .6 ft to inches by X 12 for the 12 inches in a foot.)
.6 X 12 = 7.2.
So, 2.6 ft would be 2 ft 7.2 in OR 2’7”
7 ≈ 2.6.
could mean 2.6 inches. (Convert the .6 in to 16th inch by X 16.)
.6 X 16 = 9.6
So, 2.6” would be 2
9
10
in or 2
in. (The 10/16 would reduce to 5/8.)
16
16
When taking the square root of a number, the answer can actually by + or -. So, 4 is actually
±2. The reason is because Square Root answers the question “What number do I multiply by
itself to get the number under the root?” In this case “2 times 2 = 4 AND -2 times -2 = 4”.
Because in carpentry, the value needed is a measurement, we only concern ourselves with the
positive value.
Common Errors Made by Students:
o Unfamiliar with the calculator – students that borrow calculators or keep switching
between styles and models have to continually determine how to enter the square root of
a number. Suggestion: try doing 4 on the calculator. You know the answer is 2.
o Estimation – most errors from estimation with out a calculator will come from not know
perfect squares or not being able to find the middle between other values quickly and
easily.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. The diagonal for an 11’ X 15’ room is
346 . How many feet is this?
2. A concrete foundation is 6’x13’. You
measure a diagonal to be 14.75’. How far
off are you if the diagonal is supposed to
be 205 ?
3. When excavating a rectangular foundation,
you need to be with in . You .1 foot (1 3/16
inch) measure the diagonals to be
11.0’and 11.2’. The diagonal is supposed
to be 125 . Are you with in
specifications?
Related, Generic Math Concepts
4. Estimate the length of a rafter that has a
rise of 6’ and run 9’. Use the following
formula to help.
rise 2 + run 2
5. The length of a handicap ramp for a 30”
rise and a 600” run is 360900 . How long
is the ramp in inches?
6. After doing the Pythagorean Theorem for a
picture frame,
c = 57 . What is the c to the nearest tenth
inch?
PSSA Math Concept Look
7. Simplify:
361
A. 17
B. 18
C. 20
D. 19
PDE/BCTE Math Council
8.
If X = 2.5 cm and Y = 6 cm, what is the length
of Z?
A. 3.5 cm
B. 4.5 cm
C. 9.5 cm
D. 6.5 cm
9.
What is the best estimate for
5?
A. 2.0037...
B. 2.2360...
C. 4.0987...
D. 3.1209...
ANSWER KEY
Occupational (Contextual) Math Concepts
1. The diagonal for an 11’ X 15’ room is
346 . How many feet is this?
2. A concrete foundation is 6’x13’. You
measure a diagonal to be 14.75’. How far
off are you if the diagonal is supposed to
be 205 ?
Calculator:
346 = 18.601… ~ 18.6
Calculator:
205 = 14.317… ~ 14.3
PDE/BCTE Math Council
Calculator:
125 = 11.180… ~ 11.2
3. When excavating a rectangular foundation,
you need to be with in . You .1 foot (1 3/16 Estimate (because the number is low
enough to do easily.):
inch) measure the diagonals to be
11.0’and 11.2’. The diagonal is supposed
121
125
144
to be 125 . Are you with in
=
=
=
specifications?
11
?
12
Something reasonable would be 11.1 or
11.2.
Related, Generic Math Concepts
4. Estimate the length of a rafter that has a
rise of 6’ and run 9’. Use the following
formula to help.
62 + 92 = 117 = 10.81665’ ~ 10.8’
rise 2 + run 2
5. The length of a handicap ramp for a 30”
rise and a 600” run is 360900 . How long
is the ramp in inches?
6. After doing the Pythagorean Theorem for a
picture frame,
c = 57 . What is the c to the nearest tenth
inch?
360900 = 600.7495” ~ 600.7”
57 = 7.5498” ~ 7.5”
PSSA Math Concept Look
7. Simplify:
A. 17
B. 18
C. 20
361
D.
361 = 19
D. 19
PDE/BCTE Math Council
8.
D.
2.52 + 62 =
42.25 = 6.5 cm
If X = 2.5 cm and Y = 6 cm, what is the length
of Z?
A. 3.5 cm
B. 4.5 cm
C. 9.5 cm
D. 6.5 cm
B.
Calculator:
5 = 2.236 …
9.
What is the best estimate for
A. 2.0037...
B. 2.2360...
C. 4.0987...
5?
Estimate:
4
=
2
5
6.5
=
?
~
2.4
9
=
3
D. 3.1209...
Something right between 2 and 2.4 would
be good – 2.2.
PDE/BCTE Math Council
Carpentry T-Chart
Square root of a number on a number line
OR Tape measure
Duty: Building and Building Foundation
Layout
Task: Locating/Staking a building
PSSA Eligible Content - M11.A.131
Locate/identify irrational numbers at the
approximate location on a number line.
MATH ASSOCIATED WORDS:
CARPENTRY ASSOCIATED WORDS:
Hypotenuse, diagonal, legs
Perfect squares, square root, “root”,
Irrational, non-repeating & non-termination
decimals
Diagonals, checking for square,
Same method is to be done here.
Square Rooting a Number for a Diagonal Check
is the last step when using the Pythagorean
Formula
Foundation for a Building
?
8 ft
12 ft
Pythagorean Theorem:
a2+b2=c2
12 2 + 8 2 = c 2
208 = c 2
208 = c
Calculator Method to find Square Root:
Two Lines Display Screen Calculator:
Ex.
7 - press
enter 7
press Enter or =
2.64575
One line Display Screen:
enter 7
Ex. 7 press
press Enter or =
2.64575
Convert 2.64575 ft to inches and fractional
inches.
2’ and .64575 of a foot.
.64575 X 12 (inches per foot) = 7.749
inches. (7 inches and .749 of an
inch.)
.749 X 16 (for 16th of an inch) =
11.984. (This would round to
12.) So, it would be 12/16” or
¾”.
Calculator:
208 = 14.422 ft
Convert the .422 to inches.
.422 X 12 =
7 = 2.64575 = 2’ 7 ¾”.
Now, 14.422 ft is about 14 ft 5 1/16 in.
OR
Convert the entire measurement in feet to
inches.
OR
Convert the entire measurement in feet to
inches.
PDE/BCTE Math Council
14.422 X 12 = 173.064”
Then convert the .064” to fractional inch.
.064 x 16 = 1.024 = 1/16”
14.422 ft = 173 1/16”
2.64575 X 12 = 31.749”
Then convert the .749” to fractional
inch.
.749 X 16 = 11.984. (This
would round to 12.) So, it
would be 12/16” or ¾”.
7 = 2.64575 = 31 ¾“
TEACHER’S SCRIPT FOR BRIDGING THE GAP
In Carpentry, you are taking square roots of measurements. Measurements are considered to
be positive numbers. In math, there are occasions that we try to square root negative whole
numbers (called negative integers). We cannot square root a negative number. Your calculator
will give an error message or domain error.
Students should know how to square root a number using a calculator or by estimation. (See
description below for both methods.)
Calculator Method to find Square Root:
Two Lines Display Screen Calculator:
Ex.
7 - press
enter 7
press Enter or =
2.64575 ≈2.6
One line Display Screen:
enter 7
Ex. 7 press
press Enter or =
2.64575 ≈ 2.6
Nearest
Estimation Method to find Square Root:
Pick two perfect squares around number to be square rooted,
Ex. 7
one below and one above
4
=
2
4
=
2
7
9
=
3
6.5 7 9
=
= =
≈2.5 ? 3
4 = 2 and
9 = 3, so
7 must be between 2
and 3.
The middle between 4
and 9 is 6.5
and the middle between 2 and 3 is 2.5., so the
7 must be a
little bigger than 2.5.
An estimate around 2.6 to 2.7 would be fine.
PDE/BCTE Math Council
Convert Square Root Estimate to Feet – Inches - Fractional inches.
In carpentry, the decimal part of the answer may need to be converted to inches or even
fractions of an inch.
7 ≈ 2.6.
could mean 2.6 feet. (Convert the .6 ft to inches by X 12 for the 12 inches in a foot.)
.6 X 12 = 7.2.
So, 2.6 ft would be 2 ft 7.2 in OR 2’7”
7 ≈ 2.6.
could mean 2.6 inches. (Convert the .6 in to 16th inch by X 16.)
.6 X 16 = 9.6
So, 2.6” would be 2
9
10
in or 2
in. (The 10/16 would reduce to 5/8.)
16
16
When taking the square root of a number, the answer can actually by + or -. So, 4 is actually
±2. The reason is because Square Root answers the question “What number do I multiply by
itself to get the number under the root?” In this case “2 times 2 = 4 AND -2 times -2 = 4”.
Because in carpentry, the value needed is a measurement, we only concern ourselves with the
positive value.
In math, students may need to plot both the + square root value and – square root value on a
number line or coordinate plane (X-Y Axes).
Common Errors Made by Students:
o Unfamiliar with the calculator – students that borrow calculators or keep switching
between styles and models have to continually determine how to enter the square root of
a number. Suggestion: try doing 4 on the calculator. You know the answer is 2.
o Estimation – most errors from estimation with out a calculator will come from not know
perfect squares or not being able to find the middle between other values quickly and
easily.
o When plotting on the negative side of the number line, it is helpful to list the values out
and then place you estimate.
o Students may have the most difficulties with converting the decimal part of the irrational
number to Feet – Inches – Fractional Inch.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. The diagonal for an 11’ X 15’ garage is
346 . How many feet is this? Convert the
value to Feet – inches.
2. A concrete foundation is 6’x13’. You
calculate a diagonal to be 205 . On a
tape measure, what should you be looking
for in Feet-Inch-Fractional Inch?
3. When excavating a rectangular foundation,
one diagonal is 11’ 1 ½ “. You have a 1”
tolerance. The diagonal is supposed to be
125 . Are you with in specifications?
Related, Generic Math Concepts
4. What is the length of a rafter that has a
rise of 6’ and run 9’. Use the following
formula to help.
answer inches.
rise 2 + run 2 . Give the
5. The length of a handicap ramp for a 30’
rise and a 600’ run is 360900 . How long
is the ramp in feet-inches?
6. After doing the Pythagorean Theorem for a
picture frame,
c = 57 . What is the c to the nearest 16th
inch?
PDE/BCTE Math Council
PSSA Math Concept Look
Number Lines
7.
At what position on the number line is the red dot located?
A.
B.
4
C.
D.
8.
2
What is the best estimate for
A. 2
11/16”
B. 3
7/8”
C. 3
5/8”
D. 3
3/16”
15 in inches?
9.
0
1
2
What position on the number line is the red dot located?
A.
6
B.
5
C.
4
D.
2
3
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
346 = 18.601’
1. The diagonal for an 11’ X 15’ garage is
346 . How many feet is this? Convert the
value to Feet – inches.
2. A concrete foundation is 6’x13’. You
calculate a diagonal to be 205 . On a
tape measure, what should you be looking
for in Feet-Inch-Fractional Inch?
3. When excavating a rectangular foundation,
one diagonal is 11’ 1 ½ “. You have a 1”
tolerance. The diagonal is supposed to be
125 . Are you with in specifications?
.601 X 12 = 7.213” ~ 7”
So,
346 = 18.601’ = 18’ 7”
205 = 14.3178’
.3178 X 12 = 3.814”
.814 X 16 = 13.024 ~ 13
205 = 14.3178’ = 14’ 3 11/16”
125 = 11.180’
.180 X 12 = 2.16”
.16 X 16 = 2.56 ~ 3
125 = 11.180’ = 11’ 2 3/16”
The 11’ 2 /316” minus the 1” tolerance is 11’
1 3/16”. The one diagonal of 11’ 1 ½” is
between those two measurements so it is
within specifications.
Related, Generic Math Concepts
4. What is the length of a rafter that has a
rise of 6’ and run 9’. Use the following
formula to help.
answer inches.
rise + run . Give the
2
2
62 + 92 = 117 = 10.81665’
10.81665 X 12 = 129.7998”
.7998 X 16 = 12.7968 ~ 13
117 = 10.81665’ = 129 13/16”
PDE/BCTE Math Council
5. The length of a handicap ramp for a 30’
rise and a 600’ run is 360900 . How long
is the ramp in feet-inches?
360900 = 600.7495’
.7495 X 12 = 8.994” ~ 9
360900 = 600.7495’ = 600’ 9”
6. After doing the Pythagorean Theorem for a
picture frame,
c = 57 inches. What is the c to the
nearest 16th inch?
57 = 7.5498”
.5498 X 16 = 8.7968 ~ 9
57 = 7.5498” = 7 9/16”
PSSA Math Concept Look
Number Lines
7.
C.
The red dot is located between 3
and 4 on the number line. Now,
look at your answer choices:
At what position on the number line is the red dot located?
A.
2 = 1.4142...
4=2
= 3.1415...
B.
4
= 1.7724...
C.
D.
Of the answer choices, only
be the correct answer.
can
2
B.
8.
Therefore, the 15 must be just a
little bit less than 4.
What is the best estimate for
15 in inches?
15 = 3.8729...
A. 2
11/16”
B. 3
7/8”
C. 3
5/8”
D. 3
3/16”
.8729 X 16 = 13.92 ~ 14
15 = 3.8729… = 3 14/16”
reduces to 3 7/8”.
PDE/BCTE Math Council
9.
D.
0
1
2
What position on the number line is the red dot located?
The square root of 1 equals 1.
3
The square root of 4 equals 2.
So, you know the red dot is located
A.
6
between 1 and 4 because the
red dot is located between 1 and 2.
B.
5
C.
4
D.
2
The red dot is located at the
square root of 2, which is
1.41421356237.
PDE/BCTE Math Council
Carpentry T-Chart
Estimate
=
Estimation
PSSA Eligible Content – M11.A.2.1.1
Solve Problems using operations with
rational numbers including rates and
percents (single and multi-step and multiple
procedure operations) (e.g., distance, work
and mixture problems, etc.)
Duty: Floor Framing
Task: Estimating Flooring
CARPENTRY ASSOCIATED WORDS:
TERM - FLOOR JOISTS, SHIPLAP, TOUNGE
AND GROOVE, CROSS BRIDGING,
SHEATHING
Formula
Additional amount of material to order
1.15 x Area x 15%
MATH ASSOCIATED WORDS:
TERM -rate, percent, ratio, proportion
Formula
% Divided by 100 = decimal
Decimal times 100 = %
Area of the top trapezoid
A = ½height (base 1 + base 2)
A = ½(2) (8 + 4)
A = ½(2) (12)
A = 12 square feet
Area of the top rectangle
A = base x height
A = 10.5 x 13
A = 136.5 square feet
Area of the top rectangle
A = base x height
A = 5 x 13
A = 65 square units
Area of the top trapezoid
A = ½height (base 1 + base 2)
A = ½(7) (4 + 10)
A = ½(7) (14)
A = 49 square units
When you add these areas together you get a total of
262.5 square units. 1.15x262.5x.15 = 45.28125
262.5 + 45.28125 = 307.78125
You would need to order 308 square
feet of wood.
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PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
You use rates and percents in both math class and in Carpentry.
How are the concepts similar?
The concepts are very similar. We may use examples in math class exactly like the examples
used in carpentry.
How do the concepts differ?
The formulas for computing the additional materials needed to do a job may vary. These
formulas may even be different from one carpentry book to another and one type of material to
another type of material. It is important that you be able to understand and apply any formula
as well as understanding the idea of percent.
Common Mistakes Students Make
A common mistake is to compute the amount of increase or decrease, but not apply this to the
original amount.
Another common mistake is to put the decimal point in the wrong place.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. If your flooring area is 144 square feet, then how much flooring material should you buy?
2. If your flooring area is 168 square feet, then how much flooring material should you buy?
3. If the dimensions of a room are 3½” by 5” in a scaled drawing, and the dimensions of
the actual room are 7’ by 10’, what is the scale?
Related, Generic Math Concepts
1. If you are purchasing materials at a 30% discount and the original price of the
materials is \$535.00, what is the sale price (include 6% sales tax)?
2. If a building company marks items up 60% and the original price was \$230.00,
what is the price after the markup?
3. If Harold makes \$45.00 for 5 hours, what is his hourly rate?
PSSA Math Concept Look
1. It takes Josie 3 hours to clean a home. How many homes can she clean in a week if
she does not want to work more than 10 hours a day and she does not want to work
more that 6 days a week?
2. If Jessica makes \$55.00 for 5 hours, what will her paycheck be for 30 hour if 25% is
deducted for taxes?
3. You need one can of paint for 100 square feet of wall space. You are painting 2 walls
that are 12’ by 8’. Including 10% for waste, do you have enough paint if you have 2
cans?
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. Since your flooring area is 144 square feet ,
(1.15)(144)(15%)
Use the formula for amount of additional materials to purchase.
(1.15)(144)(.15)
Change the decimal to a percent.
24.84
Calculate the amount of additional materials to purchase.
144 + 24.84
Add the amount to the original area.
168.84
When dealing with materials always round up.
You would need to order 169 square feet of wood for the floor.
2. Since your flooring area is 144 square feet ,
(1.15)(168)(15%)
Use the formula for amount of additional materials to purchase.
(1.15)(168)(.15)
Change the decimal to a percent.
28.98
Calculate the amount of additional materials to purchase.
168 + 28.98
Add the amount to the original area.
196.98
When dealing with materials always round up.
You would need to order 197 square feet of wood for the floor.
3. 3½” per 7’ and 5” per 10’ Since these ratios reduce to 1” per 2’ that would be
the scale factor.
The scale factor is 1”= 2’-0”.
Related, Generic Math Concepts
1. If you are getting 30% off then you are paying 70% of the price.
\$535 x .70 = \$374.50
Calculate the cost at 30% off.
\$374.50 x 1.06 = \$396.97 Calculate the amount plus 6% tax. (100% + 6%)
You would pay \$396.97 for the materials.
2. Since the materials will be marked up 60% the new price would be 100% +60%
the original price. \$230 x 1.6 = \$368
The new price would be \$368.00.
3. \$45 divided by 5 hours = \$9. He would make \$9 per hour.
PSSA Math Concept Look
1. Since she doesn’t want to work more than 10 hours per day, she can clean 3 houses
per day. If she works 6 days a week, she can clean 18 houses per week.
2. \$55 per 5 hours means that she would be paid \$11 per hour. \$11 x 30 hours = \$330.00
\$330.00 x 75% = \$247.50 (Since 25% is deducted. you can calculate 75% to get the
amount she will be paid.) She would be paid \$247.50 for 30 hours of work.
3. 12’ x 8’ = 96 square feet
96 sq. ft. x 2
192 sq. ft.
192 x 1.10 = 211.20 sq. ft.
Calculate the area of one wall.
Multiply the area by two since there are two rooms.
This would be the area of the two walls.
This would be the amount of paint needed with the 10%
waste calculated. (100% +10%)
Since one can of paint covers 100 sq. ft. two cans of paint would cover 200 sq. ft..
You would need enough paint to cover 211.20 sq. ft. and you only have enough paint
to cover 200 sq. ft..
You would not have enough paint.
PDE/BCTE Math Council
Carpentry T-Chart
Blue Print Reading and Sketching
Task: Interpret scale on architectural scale rule
CARPENTRY ASSOCIATED WORDS:
Scale = Proportion
PSSA Eligible Content – M11.A.2.1.2
Solve Problems Using Direct and Inverse
Proportions
MATH ASSOCIATED WORDS:
SCALE - direct, indirect, proportion, ratio
PROPORTION - similar, ratio, cross
products
Formula to compare units that vary directly or
indirectly:
Formula to compare units that vary
directly:
small small
=
l arg e l arg e
part
part
=
whole whole
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
Proportion is used in Carpentry when dealing with blue prints, floor plans or architectural
drawings. All of these relate an actual physical measurement to a related measurement on a
drawing or plan. Correctly interpreting these measurements is essential to accurate
construction
In both Carpentry and Math, it is important to set up the proportion correctly, comparing the
correct corresponding measurements. In carpentry, that usually means relating the drawing
measurement to its actual physical measurement for both sides of the proportion.
In math, the proportion can be set up several ways, just as long as the relationship on both
sides of the proportion is the same.
Plan
Plan
=
Actual Actual
Carpentry
small − length small − width
l arg e − width l arg e − length
small − length l arg e − length
=
=
=
OR
OR
small − width l arg e − width
l arg e − length l arg e − width
small − width small − length
Math- all of these will give you the correct missing measurement!
Any proportion can be solved by finding the cross products. These MUST be equal if the the
proportion is correct.
Common Mistakes Students Make
Obviously, the most common mistake is to set up your proportion incorrectly. You need to be
careful that the same relationship exists on BOTH sides of the proportion. This is not a buffet;
there is no mixing; only matching!
l arg e − length l arg e − width
=
small − width small − length
NO, NO, NO!
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. A room is 16 feet by 24 feet. If the scale
3
is in = 1 ft , what would be the length
8
and width on the blueprint?
2. The scale on a map is 1 in. = 25 miles. If
the two towns are 3 inches apart on the
map, what their actual distance apart?
3. The scale on the drawing is 1 in. = 20
feet. What is the actual width of a house
that is 3½ in on the drawing?
Related, Generic Math Concepts
4. If 4 widgets cost \$12, what is the cost of
10 widgets?
5. A 4 ft boy has a 10 ft shadow, how tall is
the boy next to him with a 5 ft shadow?
6. If 7 cakes cost \$52.50, what is the cost of
4 cakes?
PSSA Math Concept Look
7. Solve for x :
x 20
=
5 10
8. Triangle ABC is similar to triangle DEF,
AB = 7, DE = 21, BC = 12, EF = ?
9. Solve for y:
7 14
=
6
y
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
3
8
1. A room is 16 feet by 24 feet. If the scale
3
is in = 1 ft , what would be the length
8
and width on the blueprint?
x
1 16
3
y
8
=
1 24
2. The scale on a map is 1 in. = 25 miles. If
the two towns are 3 inches apart on the
map, what their actual distance apart?
1in
3in
=
25mi Xmi
3. The scale on the drawing is 1 in. = 20
feet. What is the actual width of a house
that is 3½ in on the drawing?
3 1 in
1in
= 2
Xft
20 ft
=
16 times 3/8 = 6 in.
24 times 3/8 = 9 in.
3 times 25 = 75/1 =75 miles
20 times 3 ½ = 70 ft.
Related, Generic Math Concepts
4. If 4 widgets cost \$12, what is the cost of
10 widgets?
4 10
=
12 x
12 times 10 = 120, 120/ 4 = \$30
5. A 4 ft boy has a 10 ft shadow, how tall is
the boy next to him with a 5 ft shadow?
4 x
=
10 5
4 times 5 = 20, 20/10 = 2 ft.
7
4
=
52.50 times 4 = 210.00
52.50 X
6. If 7 cakes cost \$52.50, what is the cost of
4 cakes?
210/ 7 = \$30
PSSA Math Concept Look
7. Solve for x :
x 20
=
5 10
8. Triangle ABC is similar to triangle DEF,
AB = 7, DE = 21, BC = 12, EF = ?
9. Solve for y:
7 14
=
y
6
10x = 100, x = 10
7 12
=
21 times 12 = 252
21 X
252/ 7 = 36
7 y = 84 , y = 12
PDE/BCTE Math Council
Carpentry T-Chart
Scale = Proportion
Duty: Blue Print Reading and Sketching
Task: Interpret Floor Plans
PSSA Eligible Content – M.11.A.2.1.3
Identify and/or use proportional relationships
in problem solving settings
CARPENTRY ASSOCIATED WORDS:
MATH ASSOCIATED WORDS:
SCALE - ratio
PROPORTION - ratio, similar
Formula to relate floor plans to actual
measure:
Formula to relate real life proportions :
drawing drawing
=
actual
actual
part
part
=
whole whole
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
Proportion is used in Carpentry when dealing with blue prints, floor plans or architectural
drawings. All of these relate an actual physical measurement to a related measurement on a
drawing or plan. Correctly interpreting these measurements is essential to accurate
construction
In both Carpentry and Math, it is important to set up the proportion correctly, comparing the
correct corresponding measurements. In carpentry, that usually means relating the drawing
measurement to its actual physical measurement for both sides of the proportion.
In math, the proportion can be set up several ways, just as long as the relationship on both
sides of the proportion is the same.
Carpentry
Plan
Plan
=
Actual Actual
small − length small − width
l arg e − width l arg e − length
small − length l arg e − length
=
=
=
OR
OR
small − width l arg e − width
l arg e − length l arg e − width
small − width small − length
Math- all of these will give you the correct missing measurement!
Any proportion can be solved by finding the cross products. These MUST be equal if the the
proportion is correct.
Common Mistakes Students Make
Obviously, the most common mistake is to set up your proportion incorrectly. You need to be
careful that the same relationship exists on BOTH sides of the proportion. This is not a buffet;
there is no mixing; only matching!
l arg e − length l arg e − width
=
small − width small − length
NO, NO, NO!
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. The scale on the plans is ¼ in =1 ft. If a
window measures ¾ in on the plans,
what size window do you install?
2. The scale on the plans is ¼ in = 1ft.
What is the width of a door that is ¾ in
on the plans?
3. The scale on the plans is ¼ in = 1 ft. If a
wall is to be built 16 ft long, how long will
it be on the plans?
Related, Generic Math Concepts
4. 1 oil change takes ¼ hr. How many
changes can be done in an hour?
5. Luke can print 5 posters in 15 minutes.
How many can print in one hour?
6. Mark works 35 hours and makes
\$420.00. How much does he make if he
works 25 hours at the same rate?
PSSA Math Concept Look
7. Vincent buys 4 burgers for \$ 20.00.
What is the cost of 10 burgers?
8. There are 27 pairs of shoes in a case.
How many pairs are there in 12 cases?
9. Margie can make buy 7 shirts for \$ 94.50
What would it cost if she only bought 4?
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. The scale on the plans is ¼ in =1 ft. If a
window measures ¾ in on the plans,
what size window do you install?
2. The scale on the plans is ¼ in = 1ft.
What is the length of a wall that is 4 in.
on the plans?
3. The scale on the plans is ¼ in = 1 ft. If a
wall is to be built 24 ft long, how long will
it be on the plans?
in 34 in
=
1 ft
Xft
1
4
1
4
in 4in
=
Xft
1 ft
1
4
in
Xin
=
1 ft 24 ft
¾ = ¼ X, 3ft = X
¼X = 4, x = 16 ft
¼ times 24 = X, 6in. = X
Related, Generic Math Concepts
4. 1 oil change takes ¼ hr. How many
changes can be done in an hour?
5. Luke can print 5 posters in 15 minutes.
How many can print in one hour?
6. Mark works 35 hours and makes
\$420.00. How much does he make if he
works 25 hours at the same rate?
1
4
hr 1hr
=
X
1
¼X = 1, X = 4
5 posters
Xposters
=
15 min utes 60 min utes
15X = 300
X = 20
35hrs 25hrs
=
\$420
\$X
35X = 10500
X = \$300.00
PSSA Math Concept Look
7. Vincent buys 4 burgers for \$ 20.00.
What is the cost of 10 burgers?
4
10
=
\$20 \$ X
8. There are 27 pairs of shoes in a case.
How many pairs are there in 12 cases?
27 prs
Xprs
=
1case 12cases
324 = X
9. Margie can make buy 7 shirts for \$ 94.50
What would it cost if she only bought 4?
7 shirts 4 shirts
=
\$94.50
\$X
378 = 7X, X = \$54
200 = 4X, X = 50
PDE/BCTE Math Council
Carpentry T-Chart
Order of Operations = PEMDAS
Duty: Lumber
Task: Calculate Board Feet
CARPENTRY ASSOCIATED WORDS:
LUMBER - board feet, linear feet
Formula to find board feet or linear feet:
1 board foot =144 cubic inches
TxWxL
bF =
12
L=
PSSA Eligible Content – M.11.A.3.1.1
Simplify/evaluate expressions using the
order of operations to solve problems (any
rational # may be used)
MATH ASSOCIATED WORDS:
ORDER OF OPERATIONS - surface area,
volume
Formula to find area or volume with
order of operations:
V = LxWxH
SA = 2(lw) + 2(wh) +2(hl)
bFx12
TxW
bF – Board feet
T –thickness
W – Width
L - Length
A – area
V – volume
L – length
W – width
H – height
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
In Carpentry, it is necessary to calculate the amount of materials needed for a job using a given
formula. When making calculations with that formula it is very important to perform the
numerical computations correctly.
In Math, it is always important to follow the order of operations (PEMDAS) when you are
simplifying or evaluating an expression. Of course, in Math class you will work with many
different formulas that you may not necessarily use on a daily basis.
Parentheses: any kind of grouping symbol including a fraction line
Exponents: those little numbers that indicate repeated multiplication
Multiplication:
Division: equally as important as multiplication, do them left to right
Addition:
Subtraction: equally as important as addition, do these left to right, also
Common Mistakes Students Make
The biggest mistake most students make is to perform operations out of order.
It is very important to look at the entire expression and decide which operation to do first.
Example:
4+3 x 5 = 60?
NO, NO, NO!
4+3 x 5 = do 3x5, and then add 4, equals 19!
YES, YES, YES!
It is a good idea to get a good calculator that does PEMDAS as well.
Check it with the above example BEFORE you buy it.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. Mike orders fifteen 8-ft studs and twelve
12-ft studs. How many linear feet did he
order?
2. How many board feet are in a 2x12 that
is 20 feet long?
3. Find the number of board feet in 140
pieces of 2x6 each 8 feet long.
Related, Generic Math Concepts
4. If Robert buys four CDs at \$15 each and
two DVDs at \$20 each, how much does
he spend altogether?
5. If a box is three feet wide, five feet high
and eight feet long, what is its total
surface area?
6. What is the volume of that same box?
PSSA Math Concept Look
7. Evaluate: 12 ⋅ 3 ÷ 4 + 5 ⋅ 2
8. Evaluate: (3+5) · (7 – 4)
9. Evaluate: 2·6 + 3(4 – 1)
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. Mike orders fifteen 8-ft studs and twelve
12-ft studs. How many linear feet did he
order?
2. How many board feet are in a 2x12 that
is 20 feet long?
3. Find the number of board feet in 140
pieces of 2x6 each 8 feet long.
8 times 15 plus 12 times 12 = 120+ 144
= 264
bF =
TxWxL
12
2 ⋅12 ⋅ 20 480
=
= 40
12
12
TxWxL
12
2 ⋅ 6 ⋅8
96
= 140 ⋅ = 140·8 = 1120
140·
12
12
bF =
Related, Generic Math Concepts
4. If Robert buys four CD’s at \$15 each and
two DVDs at \$20 each, how much does
he spend altogether?
4·15 + 2·20 = 60 + 40 = 100
5. If a box is three feet wide, five feet high
and eight feet long, what is its total
surface area?
SA = 2(lw) + 2(wh) +2(hl)
SA = 2·8·3 + 2·3·5 + 2·5·8
SA = 48 + 30 + 80
SA = 158
6. What is the volume of that same box?
V = LxWxH
V = 8·3·5
V = 120
PSSA Math Concept Look
7. Evaluate: 12 ⋅ 3 ÷ 4 + 5 ⋅ 2
8. Evaluate: (3+5) · (7 – 4)
9. Evaluate: 2·6 + 3(4 – 1)
12 times 3 = 36, 36 divided by 4 = 9
and 2 times 5 = 10
then 9 plus 10 = 19
3 plus 5 = 8
7 minus 4 = 3
8 times 3 = 24
4 minus 1 = 3
2 times 6 = 12 and 3 times 3 = 9
12 plus 9 = 21
PDE/BCTE Math Council
Carpentry T-Chart
Estimating
Duty: Estimating
Task: Estimate the materials, labor and cost of a
building a deck.
= Estimation
PSSA Eligible Content – M11.A.3.2.1
Use estimation to check the reasonableness
of calculations in problem solving situations
involving rational numbers (e.g., significant
digits; rounding not to exceed 3 decimal
places).
MATH ASSOCIATED WORDS:
CARPENTRY ASSOCIATED WORDS:
TERM - Estimating, costing, quality, unit price
Qty.
x
Unit
=
Price
TERM - Estimation, place value, round,
place value, sum, difference, product,
quotient
Round up the digit in the desired place value
only if the digit directly to its right is 5 or
more.
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
Estimating the cost of a job is an important aspect of construction management. An estimate is
a calculation of the quantity of various materials and the expenses likely to be incurred. The
estimated cost of a job is a close approximation of its actual cost. The agreement of the
estimated cost with the actual cost will depend on the accurate use of estimating methods and
correct visualization of the job to be done.
The purpose of estimating is to give a reasonably accurate idea of the cost of a project. A
Carpenter will need to consider the following costs when estimating a job:
1. Estimating the materials to determine what materials are needed and the quantity of
those materials required for the job.
2. Estimating the labor to determine the number and type of workers to be employed to
complete the job in a specified timeframe.
3. Estimating the plan to determine the amount and type of equipment to complete the job.
4. Estimating the time to determine the length of time required to complete the job.
Common Mistakes Students Make
The common mistake students make when estimating a job is to underestimate the material,
labor or time needed to complete a job.
Carpenters also need to check math calculations closely to assure that the estimation for the
project is correct.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. Estimate the total number of 2" x10" x16' floor
joists required to build a 20' x16' deck at
16" O.C.
Formula: Width x .75 + 1 = Number of Joists
2. Estimate the total cost of pressure treated
framing lumber based on the given table.
Formula: Quantity x Unit Price = Cost
Material
Quantity
4" x 4" x8'
Post
2" x10" x16' Joist
10
2" x 4" x8'
Bridging
2" x 4" x12'
Bracing
24
42
12
Unit
Price
13.50
ea.
19.65
ea.
6.47
ea.
8.95
ea.
Subtotal
Cost
6% Tax
Cost
Cost
3. Estimate the total labor cost of a project bid
on the given table.
Formula: Hourly Rate x Labor Hours = Cost
Description
Foreman
Carpenter
Carpenter’s
Helper
Laborer
Hourly
Rate
\$27.50
\$18.00
\$11.00
\$8.75
Labor
Hours
45
80
80
Cost
105
Total
Cost
PDE/BCTE Math Council
Related, Generic Math Concepts
4. Susie went shopping on Saturday for all of
the things she needed for her camping trip.
Susie had \$200.00 dollars to spend. Use
estimating skills to determine if Susie had
enough money to purchase the following items:
Tent
Sleeping bag
Insect Spray
Flashlight
Batteries
Grill
Charcoal
\$79.00
\$22.50
\$3.79
\$2.30
\$2.50
\$52.50
\$5.49
Will Susie have enough money to purchase all
of these items?
5. Five students drive to school rather than ride
the bus. Student #1 drives 3.6 miles, student
#2 drives 7.8 miles, student #3 drives 12.2
miles, student #4 drives 2 miles and student #5
drives 14.3 miles. Determine the approximate
total miles driven by the group of students.
6. A blank CD holds 750MB of music, which
equals approximately 72 minutes. If you have a
CD with twelve songs each approximately 6
minutes already burned onto the CD, can you
add an additional song that is 3 minutes and 28
seconds long?
PSSA Math Concept Look
7. Round 3,496.543 to the nearest tenth.
8. Calculate the sum of 14.657 and 24.581 to
the nearest hundredth. Use estimation to check
the reasonableness of your answer.
PDE/BCTE Math Council
9. Find the product of 28.7 and 41.96 . Use
estimation to check the reasonableness of your
answer.
ANSWER KEY
Occupational (Contextual) Math Concepts
1. Estimate the total number of 2" x10" x16'
Width x .75 + 1 = Number of Joists
floor joists required to build a 20' x16' deck at
(20' x.75) + 1 = x
16" O.C.
15 + 1 = x
16 = x
Formula: Width x .75 + 1 = Number of
Joists
2. Estimate the total cost of pressure
treated framing lumber based on the given
table.
Formula: Quantity x Unit Price = Cost
Material
Quantity
4" x 4" x8'
Post
2" x10" x16' Joist
10
2" x 4" x8'
Bridging
2" x 4" x12'
Bracing
24
42
12
Unit
Cost
Price
13.50
ea.
19.65
ea.
6.47
ea.
8.95
ea.
Subtotal
Cost
6% Tax
Cost
Quantity x Unit Price = Cost
Material
Quantity
4" x 4" x8'
Post
2" x10" x16' Joist
10
2" x 4" x8'
Bridging
2" x 4" x12"
Bracing
42
24
12
Unit
Cost
Price
13.50
\$135.00
ea.
19.65
471.60
ea.
6.47
271.74
ea.
8.95
107.40
ea.
Subtotal
985.74
Cost
6% Tax
59.14
Cost \$1044.88
PDE/BCTE Math Council
3. Estimate the total labor cost of a project
bid on the given table.
Hourly Rate x Labor Hours = Cost
Formula: Hourly Rate x Man Hours = Cost
Description
Foreman
Carpenter
Carpenter’s
Helper
Laborer
Hourly
Rate
\$27.50
\$18.00
\$11.00
Man
Hours
45
80
80
\$8.75
Cost
105
Total
Cost
Description
Forman
Carpenter
Carpenter’s
Helper
Laborer
Hourly
Rate
\$27.50
\$18.00
\$11.00
\$8.75
Labor
Cost
45
80
80
\$1237.50
1440.00
880.00
105
918.75
Total
Cost \$4476.25
Related, Generic Math Concepts
4. Susie went shopping on Saturday for all
of the things she needed for her camping
trip. Susie had \$200.00 dollars to spend.
Use estimating skills to determine if Susie
had enough money to purchase the
following items:
Tent
Sleeping bag
Insect Spray
Flashlight
Batteries
Grill
Charcoal
\$79.00
\$22.50
\$3.79
\$2.30
\$2.50
\$52.50
\$5.49
80 + 23 + 4 + 2 + 3 + 53 + 5 = 170
Yes, Susie will have enough money.
Will Susie have enough money to purchase
all of these items?
5. Five students drive to school rather than
ride the bus. Student #1 drives 3.6 miles,
student #2 drives 7.8 miles, student #3
drives 12.2 miles, student #4 drives 2 miles
and student #5 drives 14.3 miles.
Determine the approximate total miles
driven by the group of students.
4 + 8 + 12 + 2 + 14 = 40 miles
PDE/BCTE Math Council
6. A blank CD holds 750MB of music, which
equals approximately 72 minutes. If you
have a CD with twelve songs each
approximately 6 minutes already burned
onto the CD, can you add an additional song
that is 3 minutes and 28 seconds long?
12 x 6 = 72 minutes
No, the CD will be full to capacity
PSSA Math Concept Look
7. Round 3,496.543 to the nearest tenth.
3,496.543 = 3,496.5
Answer with work
14.657
+24.581
39.238
8. Calculate the sum of 14.657 and 24.581
to the nearest hundredth. Use estimation to
check the reasonableness of your answer.
39.238 = 39.24
Estimation:
15
+25
40 9Answer is reasonable.
31.96
x 28.7
22372
25568
6392
917.252
9. Find the product of 28.7 and 41.96. Use
Estimation to check the reasonableness of
your answer.
Estimation:
30
x30
900
9Answer is reasonable.
PDE/BCTE Math Council
Carpentry T-Chart
Transit Angle = Exterior angle of a polygon
Duty: Foundations
Task: Set up a transit.
CARPENTRY ASSOCIATED WORDS:
PSSA Eligible Content – M11.B.2.1.1
Measure and/or compare angles in degrees
(up to 360) (protractor must be provided or
drawn)
MATH ASSOCIATED WORDS:
TERM - TRANSIT, BUILDER’S LEVEL, LINE
OF SIGHT, VERNIER’S SCALE, QUADRANT
TERM –angle, degrees, minutes, seconds,
interior angles, exterior angles, vertical
angles, corresponding angles, polygon
Formula (type purpose of formula here):
Formula (type purpose of formula here):
Angle to set the transit = 360º / n
Exterior angle of a polygon = 360º / n
n is the number of sides of the figure
n is the number of sides of the figure
Set the compass to the measure of the
exterior angle of the shape of the area.
Exterior Angles
Hexagon
Octagon
Square
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
When you are setting up a transit you need to be able to set it to the correct angle. You can
take the number of sides divided by 360º to find that angle. In geometry class we call this the
exterior angle. Hours, minutes and seconds can be used to measure angles to a greater
degree of accuracy.
How are the concepts similar?
We use the same formula in both classes.
How do the concepts differ?
Carpenters use different terminology than geometry instructors when talking about calculating
exterior angles. Geometry instructors also have formulas for interior angles, the sum of interior
angles, and supplementary angles.
Here are some other formulas for angles in a polygon
n is the number of sides of the polygon.
The sum of interior angles of any polygon is (n – 2) 180º
The sum of the interior angles of a quadrilateral is 360º = (4 – 2) 180º.
An exterior angle of any regular polygon is n divided by 360º
An exterior angle of a regular quadrilateral is 4 divided by 360º = 90º
Supplementary angles have a sum of 180º.
Complementary angles have a sum of 90º.
1
2
3
4
5
6
7
8
Angles 1&4, 2&3, 5&8, 7&4 are vertical angles.
Angles 1&5, 2&6, 3&7, 8&4 are corresponding angles.
If the lines are parallel then these angles are congruent.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. What units are used to represent fractions of a degree?
2. Evaluate 130º + 7º - 15º30’
3. Evaluate 120º - 17º10’23”
Related, Generic Math Concepts
1. What is the supplement of an angle with the measure 110º22’?
2. The following protractors show measures of 50º clockwise and 50º counterclockwise.
What is the difference in their measures?
3. If angle A had a measure of 5x + 2 and angle B has a measure of 3x – 7, and
they are supplementary angles, what is x?
PSSA Math Concept Look
For problems 1-3 use the figure below.
1. If the measure of angle AHE = 30º and the measure of angle FKD = (8x -10) º, what is x?
2. If the measure of angle BHE = 50º and the measure of angle HKD = (12x -10) º, what is x?
3. If the measure of angle HKD = 2xº and the measure of angle KHB = (8x -20) º, what is x?
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PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. Minutes and seconds are used to represent fractions of a degree.
2. 130º + 7º - 15º30’ = 129º60’ - 15º30’ = (129 - 15)º(60 – 30)’ = 114º30’
3. 120º - 17º10’23” = 119º59’60” - 17º10’23” = (119 – 17)º(59 – 10)’(60 – 23)” = 102º49’37”
Related, Generic Math Concepts
1. The supplement of an angle with the measure 110º22’ would be 180º - 110º22’.
180º - 110º22’ = 179º60’ - 110º22’ = 69º38’.
2. A protractor showing a measure of 50º counterclockwise would be the same as a measure
Of 130º clockwise, so the difference would be (130-50) º or 80º.
3.
( 5x + 2) + (3x – 7) = 180
8x – 5 = 108
The sum is 180º
Combine similar terms
PSSA Math Concept Look
1. These angles are congruent so they are equal in measure.
8x – 10 = 30
Set up the problem.
8x = 40
Add 10 to each side of the equation.
x=5
2. These angles are congruent so they are equal in measure.
12x – 10 = 50
Set up the problem.
12x = 60
Add 10 to each side of the equation.
x=5
3. These angles are supplementary so their sum is equal to 180º.
2x + 8x – 20 = 180
Set up the problem.
2x + 8x = 200
Add 20 to each side of the equation.
10x = 200
Combine like terms.
x = 20
Divide each side of the equation by 10.
PDE/BCTE Math Council
Carpentry T-Chart
Surface Area = Surface Area
Duty: Estimation
Task: Estimate Surface Area
PSSA Eligible Content – M11.B.2.2.1
Calculate the Surface Area of prisms, cylinders,
cones, pyramids and/or spheres. Formulas are
provided.
CARPENTRY ASSOCIATED WORDS:
MATH ASSOCIATED WORDS:
TERM - Dimension, Estimate, Inch, Foot,
Yard, Square Inch/Foot/yard, Coverage
TERM – Surface Area, radius, perimeter, base,
slant length
Formula to find Surface Area:
Formula to find Surface Area:
Cylinder:
Cone:
Cube:
Pyramid:
Sphere:
SA = 2πr2 + 2πrh
SA = πr(l + r)
SA = 6e2
SA = B + ½lp
SA = 4πr2
r=radius, h=height, l=slant length, e=edge length,
p=perimeter
Cylinder:
Cone:
Cube:
Pyramid:
Sphere:
SA = 2πr2 + 2πrh
SA = πr(l + r)
SA = 6e2
SA = B + ½lp
SA = 4πr2
r=radius, h=height, l=slant length, e=edge length,
p=perimeter
38”
60”
diameter = 38”
length = 60”
Find the surface area of the cylinder above
(figure not to scale).
What is the surface area of this cylindrical
fuel tank?
Radius is ½ of diameter = ½ x 38 = 19”
h(height) is same as length = 60”
2
Cylinder SA = 2πr + 2πrh
SA = 2×π×192 + 2×π×19×60
SA = 722π + 2280π
SA = 3002×π
SA = 7162.8 sq. in.
T-CHART CARPENTRY M11B221 SURFACE AREA
PRINTER ON: 6/19/2008
r = ½ · 38” = 19”
h = 60”
2
Cylinder SA = 2πr + 2πrh
SA = 2π(19)2 + 2π(19)(60)
SA = 722π + 2280π
SA = 3002π
SA = 7162.8 sq. in.
PAGE 1 OF 5
TEACHER’S SCRIPT FOR BRIDGING THE GAP
Surface Area is the total area of all surfaces of a solid object. Unlike lateral area, it includes the
area of the bases (s) of the figure.
How are the concepts similar?
The surface area formulas used in carpentry are the same as in mathematics:
The formulas correspond to the areas of the individual surfaces of the objects:
2πr2
+
2πrh
Area of top & bottom(2 circles) +
Area of sides
2
Cone SA = πr
+
πrl
Area of bottom(circle) + Area of side
6e2
Cube SA =
Area of 6 sides (6 squares)
Pyramid SA =
B
+
½lp
Area of base + Area of sides
Cylinder SA =
How do the concepts differ?
When using these surface area formulas for carpentry applications, the student must
identify which parts of the formulas to use, as many applications will not be concerned
with ALL surfaces of an object.
A ornamental cone top piece that must be painted will be resting on its base, so the
base should not included in the calculation of area to be painted:
Full Cone Surface Area =
πr2 + πrl
but if the base is not included, leave out the πr2:
Area to be painted (sides of cone only) = πrl
Common Mistakes Students Make
Type a description of common mistakes made when performing the mathematical task
Using Incorrect Formula: Correctly identifying the type of object you are dealing with
and use the appropriate formula (2 formulas may be needed for complex objects)
Not “Removing” Unnecessary Surface Areas from Calculations: Depending on the
question, not all surface areas included in formula may be needed. Identify the areas
that are required for the calculation and remove from formula as needed.
Using Consistent Units: If the problem wants square feet instead of square inches, be
sure to convert your given measurements into feet first (inches ÷ 12 = feet) OR
convert your square inch answer into square feet (sq. inches ÷ 144 = sq. feet)
T-CHART CARPENTRY M11B221 SURFACE AREA
PRINTER ON: 6/19/2008
PAGE 2 OF 5
Occupational (Contextual) Math Concepts
1. A customer has asked you to construct a grain
silo with r=15’ and h=50’. What is the total
Surface Area of the top and sides of the silo?
2. You need to order house wrap to cover the
cone at the top of a water tower with d=18’, l=14’.
How much wrap will you need to cover the sides
of the cone?
3. You need to paint two decorative spheres with
d=10’ at the entrance to the new mall. A gallon of
paint covers 175 sq.ft. How much paint will you
need?
Related, Generic Math Concepts
1. You need fabric to cover a 4-sided pyramid
with base sides of 12’ & slant length of 20’. How
much fabric will you need to cover all sides of the
pyramid?
2. One soup can has a r=3” and h=4”, another
soup can has a r=4” and a h=3”. Which can has a
greater total surface area?
3. A size 7 regulation basketball has a d=9.39”. A
size 6 regulation basketball has a d=9.07”. What
is the Surface Area of each basket ball?
PSSA Math Concept Look
1. Find the Surface Area
of this cylinder
d=12.75’
h=28.45’
2. Find the Surface Area of a sphere that is
27.75” across.
3. Find the total Surface Area of cone with base
diameter =15.50” and 22.25” from base to the top
T-CHART CARPENTRY M11B221 SURFACE AREA
PRINTER ON: 6/19/2008
PAGE 3 OF 5
ANSWER KEY
Occupational (Contextual) Math Concepts
1. A customer has asked you to construct a grain
silo with r=15’ and h=50’. What is the total Surface
Area of the top and sides of the silo?
2. You need to order house wrap to cover the
cone at the top of a water tower with d=18’, l=14’.
How much wrap will you need to cover the sides
of the cone (sq. ft.)?
3. You need to paint two decorative spheres with
d=10’ at the entrance to the new mall. A gallon of
paint covers 175 sq.ft. How many gallons of paint
will you need to purchase?
Cylinder SA = 2πr2 + 2πrh
But only the top is need, so: SA = πr2 + 2πrh
SA = π(15)2 + 2π(15)(50)
SA = 225π + 750π
SA = 975π
SA = 3063 sq. ft.
Cone SA = πr2 + πrl
Only the side is needed, so: SA = πrl
Radius = r = 18/2 = 9
SA = π(9)(14)
SA = 126 π
SA = 396 sq. ft.
One Sphere SA = 4πr2
Radius = r = 10/2 = 5’
SA = 4π(5) 2
SA = 100 π
SA = 314 sq. ft.
2 Spheres = 314 + 314 = 628 sq. ft.
628 sq. ft. ÷ 175 = 3.59 gallons paint needed,
so 4 gallons must be purchased.
Related, Generic Math Concepts
1. You need fabric to cover a 4-sided pyramid with
base sides of 12’ & slant length of 20’. How much
fabric will you need to cover all sides and the base
of the pyramid (sq. yd.)?
2. One soup can has a r=3” and h=4”, another
soup can has a r=4” and a h=3”. Which can has a
greater total surface area?
3. A size 7 regulation basketball has a d=9.39”. A
size 6 regulation basketball has a d=9.07”. What
is the Surface Area of each basket ball?
T-CHART CARPENTRY M11B221 SURFACE AREA
PRINTER ON: 6/19/2008
Pyramid SA = B + ½ lp
Base of a 4 sided pyramid is a square, so B = side2 = 122
p = Perimeter = 4s = 4·12 = 48’
SA = 122 + ½ · 20 · 48
SA = 144 + 480
SA = 624 sq. ft.
1 sq. yd. = 27 sq. ft.
SA = 624 sq. ft. ÷ 27 = 23.1 sq. yd.
Can 1:
Can 2 :
SA = 2π (32 ) + 2π (3 × 4)
SA = 57 + 75
SA = 2π (42 ) + 2π (4 × 3)
SA = 101 + 75
SA = 132in 2
Ball 1: r = 4.695
SA = 176in 2
Ball 2 : r = 4.535
SA = 4π (4.6952 )
SA = 4π × 22.04
SA = 4π (4.5352 )
SA = 4π × 20.57
SA = 277in 2
SA = 259in 2
PAGE 4 OF 5
PSSA Math Concept Look
1. Find the Surface Area
of this cylinder
2
Cylinder SA = 2πr + 2πrh
d=12.75’
h=28.45’
r = radius = ½ d = 6.875’
SA = 2π(6.875)2 + 2π(6.875)(28.45)
SA = 94.53125π + 391.1875π
SA = 485.71875π
SA = 1525.9 sq. ft.
2. Find the Surface Area of a sphere that is 27.75”
across.
One Sphere SA = 4πr2
Radius = r = 27.75/2 = 13.875”
SA = 4π(13.875) 2
SA = 770.0625 π
SA = 2419.2 sq. in.
3. Find the total Surface Area of cone with base
diameter =15.50” and 22.25” from base to the top
Cone SA = πr2 + πrl
Slant length is distance from base to top, s = 22.25”
Radius = r = 15.5 = 7.75”
SA = π(7.75) 2 + π(7.75)(22.25)
SA = 60.0625π + 172.4375 π
SA = 232.5 π
SA = 730.4 sq. in.
T-CHART CARPENTRY M11B221 SURFACE AREA
PRINTER ON: 6/19/2008
PAGE 5 OF 5
Carpentry T-Chart
Volume = Volume / Displacement
Duty: Estimation
Task: Estimate (Calculate?) Volume
CARPENTRY ASSOCIATED WORDS:
TERM - Dimension, Estimate, Cubic Volume,
Diameter, Board Feet (12”x12”x1”) a unit of
Volume
Formula to Find Volume of a Sonatube:
Cubic inches = 3.14 x radius (inches) x radius
(inches) x height (inches)
1 cubic foot contains 1728 cubic inches
1 cubic yard contains 27 cubic feet
PSSA Eligible Content – M11.B.2.2.2
Calculate the volume of prisms, cylinders,
cones, pyramids and/or spheres. Formulas
are provided.
MATH ASSOCIATED WORDS:
TERM –Length, Volume, Diameter, Radius,
Height
Formulas to Find Volume:
V = π r 2 h (cylinder)
4
V = π r 3 (sphere)
3
1
V = π r 2 (cone)
3
1
V = Bh (pyramid, B = base area)
3
h=10’
d=15”
Answer must be in yd3
5”
h
10”
d
r = d / 2 = 15 / 2 = 7.5”
V = π r 2h
V = π (7.52 ×120)
V = π (56.25 ×120)
V = 21206in3
V = 21206 × .0006 OR 21206 ÷ 1725
V = 12.7 ft 3 (Round to 13ft 3 )
V = 13 ÷ 27
Diameter = 5 inches, Height = 10 inches
Radius = 5 / 2 = 2.5 inches
V = π r2h
V = π ·(2.5)2·10
V = π ·6.25·10
V = π ·62.5
V = 196.3495 cu. in.
V = .48 yd 3 (Round to .5 yd 3 )
T-CHART CARPENTRY M11B222 VOLUME DISPLACEMENT
PRINTER ON: 6/19/2008
PAGE 1 OF 5
TEACHER’S SCRIPT FOR BRIDGING THE GAP
Volume is the amount of space occupied by a 3-dimensional solid or gas, measured in cubic
units (inches, feet, yards, centimeters, etc.)
How are the concepts similar?
Whether in calculating concrete volume or mathematical volume, the math concepts and the
formulas used are very similar:
Volume formula:
V = π r 2h
If the volume involves a circular or spherical shape (cylinder, sphere, cone), then π will be part
of the calculation. The best way to use π in your calculations is to use a π key on the
calculator, if available. Otherwise, using 3.14 as an approximation is fine.
How do the concepts differ?
The mathematical formulas for volume indicate a certain type of orientation that may not match
the application in question. For example, h will designate height of a cylinder, but if the cylinder
is horizontal, h will be the same as the length!
Common Mistakes Students Make
Type a description of common mistakes made when performing the mathematical task
Most volume formulas need radius (r), NOT DIAMETER (d): If you are given a diameter,
halve it to get the radius before using the formula:
Diameter is 10 inches, Radius = 10 / 2 = 5 inches
Pay Attention to Units: If you want volume in cubic inches, convert your measurements to
inches before using the formulas:
3 feet = 3x12 = 36 inches
2.5 yards = 2.5 x 36 = 90 inches
Converting Between Cubic Measurements:
1 cubic foot is a box 12 inches by 12 inches by 12 inches, so the calculation to convert
cubic inches to cubic feet (or vice versa) must use 12x12x12 = 1,728:
12,096 cubic inches = 12,096 / 1,728 = 7 cubic feet
AND
1 cubic yard is a box 3 feet by 3 feet by 3 feet, so the conversion
of cubic feet to cubic yards uses 27:
94.5 cubic feet = 94.5 / 27 = 3.5 cubic yards
T-CHART CARPENTRY M11B222 VOLUME DISPLACEMENT
PRINTER ON: 6/19/2008
PAGE 2 OF 5
Occupational (Contextual) Math Concepts
1. A customer has asked you to construct an
above ground, rain water holding tank with r=12’
and h=25’. What will the total Volume of the water
tank?
2. You need to set 3-concrete piers to support an
about ground deck. Each pier d=12 inches &
h=60”. Find the Volume of one pier in in3, ft3 &
yd3?
3. You need to build 3 4-sided pyramids to accent
a retaining wall. Each side of the base (b) = 18”,
height (h) = 15”. What is the Volume of each
pyramid? What is the Volume of all 3- pyramids in
1
yd3? V = ( Area of base) × h ( Area of base=b 2 )
3
Related, Generic Math Concepts
1. Your car’s engine is a “301.” 301 means the
engine displaces 301in.3. You find the bore=4”, &
stroke=3” What is the Displacement of one
cylinder? This engine has _____ cylinders.
2. One soup can has a d=3” and h=4”, another
soup can has a d=4 and a h=3. Which can holds
more soup?
3. A size 7 regulation basketball has a d=9.39”. A
size 6 regulation basketball has a d=9.07”. What
is the volume of each basket ball?
4
Use the formula: V = π r 3
3
PSSA Math Concept Look
1. Find the Volume of a cylinder d=12.75’ h=28.45’
2. Find the Volume of a sphere d=27.75”
3. Find the Volume of 4-side pyramid b=10, h=25
T-CHART CARPENTRY M11B222 VOLUME DISPLACEMENT
PRINTER ON: 6/19/2008
PAGE 3 OF 5
ANSWER KEY
Occupational (Contextual) Math Concepts
1. A customer has asked you to construct an
above ground, rain water holding tank with r=12’
and h=25’. What will the total Volume of the
water tank?
2. You need to set 3-concrete piers to support an
about ground deck. Each pier d=12 inches &
h=60”. Find the Volume of one pier in in3, ft3 &
yd3?
3. You need to build 3 4-sided pyramids to
accent a retaining wall. Each side of the base (b)
= 18”, height (h) = 15”. What is the Volume of
each pyramid in in3?
V = π 122 25
V = π × (144) × 25 or V = 3.14 × (122 ) × 25
V = 11310 ft 3 (rounded from 11309.73355)
V = π 122 60
V = π × 144 × 60 V = 27143in3
V = 27143 ÷ 1725 V = 15.7 ft 3 V = 16 ÷ 27
V = .6 yd 3 (Rounded from 0.59)
1
V = ( Area of base) × h ( Area of base=b 2 )
3
1
1
V = (182 ×15) V = × 4860 V = 1620in3
3
3
Related, Generic Math Concepts
1. Your car’s engine is a “301.” 301 means the
engine displaces 301in.3. You find the bore = 4”,
& stroke = 3” What is the Displacement of one
cylinder? This engine has _____ cylinders.
Divide 301 by 37.7 =7.98 =8 cylinders.
2. One soup can has a d=3” and h=4”, another
soup can has a d=4 and a h=3. Which can holds
more soup?
3. A #7 regulation basketball has a d=9.39”. A #6
regulation basketball has a d=9.07”. What is the
volume of each basket ball?
42 π 3
or d 2π h ÷ 4
4
Piston Displacement = π (4) × 3 or 16 × π × 3 ÷ 4
Piston Displacement = π 22 × 3 or
Piston Displacement = π × 4 = 12.6 × 3 = 37.73
or 16 × π = 50.27 × 3 = 150.8 ÷ 4 = 37.7in3
V = π r 2h
Can 1: V = π (1.5) 2 4
Can 2:V = π (2)2 3
V = 28.27in.3
V = 37.70in.3
4
V = ×π × r2
3
V = 1.333 × π ×192.5
V = 1.333 × π ×13.8752
V = 806.14in3
PSSA Math Concept Look
V = π r 2h
1. Find the Volume of a cylinder d=12.50’
h=28.75’
V = π × 6.252 × 28.75
2. Find the Volume of a sphere d=27.75”
4
V = ×π × r2
V = 1.333 × π × 13.8752
3
V = 1.333 × π ×192.5 V = 806.14in3
3. Find the Volume of 4-side pyramid b=10, h=25
1
V = (102 × 25)
3
T-CHART CARPENTRY M11B222 VOLUME DISPLACEMENT
PRINTER ON: 6/19/2008
V = 3528.155 ft 3
1
V = × V = 2500in3
3
PAGE 4 OF 5
Carpentry T-Chart 3
Area/Perimeter/Circumference = Area
Duty: Estimation
Task: Estimate area, perimeter or
circumference of an irregular figure.
PSSA Eligible Content – M11.B.2.2.3
Estimate area, perimeter or circumference of an
irregular figure.
CARPENTRY ASSOCIATED WORDS:
MATH ASSOCIATED WORDS:
TERM - Depth, Dimension, Estimate, Width, TERM – Length, height, base, width, diameter,
radius, hypotenuse, area, perimeter,
Rise, Run, Pythagorean Theorem, Span,
circumference
Formula to find Area and Perimeter:
Rectangle:
A = lw
P = 2l + 2w
2
C = 2πr
Circle:
A = πr
(Circumference = circle perimeter)
1
Triangle:
A = bh
P=a+b+c
2
Pythagorean Theorem:
c2 = a2 + b2
6”
40”
B
12”
Formula to find Area and Perimeter:
Rectangle:
A = lw
P = 2l + 2w
2
Circle:
A = πr
C = 2πr
(Circumference = circle perimeter)
1
Triangle:
A = bh
P=a+b+c
2
Pythagorean Theorem:
c2 = a2 + b2
Calculate the area and perimeter of this figure:
22”
6”
A
10”
60”
T1
12”
The plan designer forgot length dimension AB
on the counter plan above. You need to add a
trim around the entire edge. What length of trim
is required to complete the job? Illustration NOT
to scale.
The lengths of all edges are known except
for the length between A and B.
To estimate the distance between A and B,
we need to identify a triangle that has AB as
its long side:
6”
If the length of the other 2 sides
of the triangle (BC, AC) are
calculated, AB can be
calculated.
12”
A
B to C = 22” – 6” – 10” = 6” = 3×2
A to C = 60” – 12” – 40” = 8” = 4×2
A 3-4-5 triangle! So, A to B = 5×2 = 10”
Trim = 10”+60”+22”+40”+6”+10”+12” = 160”
T-CHART CARPENTRY M11B22 AREA CIRCUMFERENCE
PRINTER ON: 6/19/2008
C
B
10”
R1
A
40”
B
R2
C
22”
60”
Draw segments BC and AC to include triangle ABC.
BC = 22” – 6” – 10” = 6”
Pythagorean Theorem:
AC = 60” – 40” – 12” = 8”
AB2 = 62 + 82
AB2 = 100
AB = 100
AB = 10”
Perimeter = 6”+40”+22”+60”+10”+12”+(AB)
= 150” + 10”
= 160”
To calculate area, break the figure further into 1
triangle and 2 rectangular sections:
Area T1 = ½ bh = ½ (3)(4) = 6 sq. in.
Area R1 = lw = (16)(10) = 160 sq. in.
Area R2 = lw = (22)(40) = 880 sq. in.
Total Area = 6 + 160 + 880 = 1046 sq. in.
PAGE 1 OF 5
TEACHER’S SCRIPT FOR BRIDGING THE GAP
Area is the total number of square units in a region, perimeter is the distance around the
outside of a shape or figure (a circle’s perimeter is called a circumference).
How are the concepts similar?
Area, perimeter or circumference problems use a
toolbox of formulas for basic shapes, but the critical
step is to break down the irregular shape into these
basic shapes (circle, rectangle, triangle) and apply
the correct formulas.
Whether trying to solve a trade application or a math
problem, you should try to draw in new lines that create
simple shapes within the complex shape.
How do the concepts differ?
In carpentry, many right triangles are planned to follow a
3-4-5 pattern that are easier to solve.
However, in math, the Pythagorean Theorem is used to solve
any right triangle where you are given 2 sides and want the third:
10
3
x
A regular hexagon consists
of 6 similarly sized triangles
A = 6 × (area of 1 triangle)
= 6 × ( ½ sh)
32 + x2 = 102
9 + x2 = 100
x2 = 91
x = 9.54
Common Mistakes Students Make
Mixing Perimeter and Area Formulas or Calculations: Perimeter formulas calculate the
length of the outside edge of an object, while area formulas calculate the space taken up by the
shape. Area will often calculate to a larger number than perimeter, BUT NOT ALWAYS.
Perimeter Calculations should not include inner edges: The perimeter of an irregular
object should follow the outer edge of the figure. If you use perimeter formulas for basic
shapes constructed within the irregular object, be sure to eliminate inner edges that don’t follow
the outside edge.
Finding basic shapes within irregular objects can be frustrating: Some irregular objects
can be broken into basic shapes with only a couple of extra lines, while others seem to take a
lot more. Don’t feel locked in to your first attempt if it is too messy.
Empty shapes in the figure require subtracting the area of the “hole:” If your plan
includes areas that create holes in the object, you will be subtracting out that area to get a final
answer (e.g., a deck plan that has a spot for a hot tub).
Final answer may include multiple parts: Don’t forget to total up all the various areas or
perimeters to get your final answer.
T-CHART CARPENTRY M11B22 AREA CIRCUMFERENCE
PRINTER ON: 6/19/2008
PAGE 2 OF 5
Occupational (Contextual) Math Concepts
1. How many sq.ft of plywood will you need to
cover one roof gable end that has a Run of 15ft &
a Rise (h) of 18.5ft? (The base (b) is ½ the
span of the gable end)
18.5’
15’
2. How many sq.ft of plywood will you need to
cover the same roof gable (in #1) that has 48in
Hexagon shape ventilation louver installed?
(each side s = 27.7”)
B
3. What is the area of
this patio (in ft2)?
48”
27.7”
15’
12’
A
25’
Related, Generic Math Concepts
1. You have been asked to build a fence all the
way around the patio. What is the length of the
fence around the patio pictured above
B
15’
2. What is the Area of the patio pictured if you
install a 6’ (d) round hot tub in the center?
12’
25’
A
3. How much sealer will you need to cover the
floor of a hexagon shaped gazebo of width 18ft
and side of 10.5 ft?
One gallon of sealer covers 200ft2
18’
10.5
PSSA Math Concept Look
45’
1. Find the area of the figure pictured.
2. Find the area of the unshaded area if a=5,
b=18, d=3, and e=1.
3. Find a if c=37 and b=24 of the figure pictured.
18’
a d
c
e
b
c
a
b
T-CHART CARPENTRY M11B22 AREA CIRCUMFERENCE
PRINTER ON: 6/19/2008
PAGE 3 OF 5
ANSWER KEY
Occupational (Contextual) Math Concepts
1. How many sq.ft of plywood will you need to
cover one roof gable end that has a Run of 15ft
& a Rise (h) of 18.5ft?
(The run is ½ the span
of the gable end)
18.5’
Base of “triangle” = span = 2×15 = 30’
1
A = bh
A = (.5 × 30 ×18.5)
2
A = 277.5 ft 2
15’
2. How many sq.ft of plywood will you need to
cover the same roof gable (in #1) that has 48in
Hexagon shape ventilation louver installed?
(each side s = 27.7”)
B
3. What is the area of
this patio (in ft2)?
Split diagram into triangle and rectangle:
15’
Triangle base = 25 – 15 = 10 ft.
12’
A
Area to cover = Gable Area – Louver Area
h of Louver = ½ × 48 = 24”, s of Louver = 27.7”
Louver Area = 6(½ × s × h) (6 triangles with base s, height h)
= 6(½ × 27.7 × 24)
= 1994.4 in2
= 1994.4 ÷ 144 = 13.85 ft2
Area to cover = 277.5 ft2 – 13.85 ft2
= 263.65 ft2
25’
Area triangle = ½ bh = ½ × 10 × 12 = 60 ft2
Area rectangle = lw = 15 × 25 = 262.5 ft2
Total Area = 262.5 + 60 = 322.5 ft2
Related, Generic Math Concepts
1. You have been asked to build a fence all the
way around the patio. What is the length of the
fence around the patio pictured above?
2. What is the Area of the patio pictured if you
install a 6’ (d) round hot tub?
All edges except AB are known (use Pythagorean):
AB2 = 102 + 122
= 244
AB = 15.62 ft.
Length of fence = perimeter
= 15’ + 12’ + 25’ + 15.62’ = 67.62’
Area = Area entire patio – Area of hot tub
= 322.5 ft2 – π(3)2
= 322.5 ft2 – 28.3 ft2
= 294.2 ft2
Area hexagon = 6 triangles of base s and height h
3. How much sealer will you need to cover the
floor of a hexagon shaped gazebo of width 18ft
and side of 10.5 ft? One gallon of sealer covers
200ft2
Height h = 18 / 2 = 9 ft.
Area = 6( ½ × 10.5 × 9)
= 283.5 ft2
Sealer needed = 283.5 ÷ 200 = 1.4175 gallons
T-CHART CARPENTRY M11B22 AREA CIRCUMFERENCE
PRINTER ON: 6/19/2008
PAGE 4 OF 5
PSSA Math Concept Look
Figure can be broken down into:
2 semi-circles (1 full circle)
18’ 1 rectangle
45’
1. Find the area of the figure pictured.
2. Find the area of the unshaded area if a=5,
b=18, d=3, and e=1.
Area = Area Rectangle + Area 1 Full Circle
= lw + πr2
(l=45, w=18, r = radius = ½ × 18 = 9’)
= (45)(18) + π(9)2
= 810 + 254.5
= 1064.5 ft.2
a d
Figure is a triangle with
2 “empty” circles
c
e
b
Area = Area triangle – Area circle 1 – Area circle 2
= ½ bh – πr2 - πr2 (radius circle 1 = ½ × 3 = 1.5
radius circle 2 = ½ × 1 = 0.5)
= ½ (18)(5) – π(1.5)2 π(0.5)2
= 45 – 7.1 - .8
= 37.1
c
a
3. Find a if c=37 and b=24 of the figure pictured.
T-CHART CARPENTRY M11B22 AREA CIRCUMFERENCE
PRINTER ON: 6/19/2008
b
Pythagorean: c2 = a2 + b2
a2 + 242 = 372
a2 + 576 = 1369
a2 = 793
a = 793
a = 28.16
PAGE 5 OF 5
Carpentry T-Chart
Duty: Hand Tools
Task: Layout and cut patters
CARPENTRY ASSOCIATED WORDS:
Radius = Radius
PSSA Eligible Content – M11.C.1.1.1
Identify and/or use the properties of a radius,
diameter and/or tangent of a circle (given
numbers should be whole)
MATH ASSOCIATED WORDS:
TERM - BRACE, AUGER, BORING BIT,
RADIUS, COMPASS. RASP
TERM – RADIUS, DIAMETER, TANGENT,
PERPENDICULAR, CONCENTRIC
CIRCLES.
Formula (type purpose of formula here):
Formula (type purpose of formula here):
Diameter = 2 x Radius
Diameter = 2 x Radius
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
In both Carpentry and in Math Class you learn about the relationship between radius and
diameter.
How are the concepts similar?
These concepts are very similar in both classes.
How do the concepts differ?
In Carpentry there are many additional terms and tools you use associated with radius and
diameter.
In Math Class we study the concept of tangent as well as radius and diameter.
A tangent touches the circle in exactly one point and is perpendicular to the radius of the circle
at that point.
Common Mistakes Students Make
A common mistake that students make in interpreting drawings in both mathematics and
carpentry is confusing the radius of a circle to the radius of a corner in a diagram.
Another common error is to include the diameter of a circle as a part of the length the
rectangular portion of an irregular figure that contains a rectangle with a semi-circle on each
end.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. If a circle has a radius of 7 yards what is the diameter?
2. If a circle has a diameter of 18 feet what is the radius?
3. If the radius of a circle is 4 inches and the corner radius is 9 inches,
how far should the circle measure from the corner of the pattern?
Related, Generic Math Concepts
For the following problems use the figure below.
1. If AB = 15, then what is AC?
2. If BO = 3, and AO = 5, what is the length of AB?
3. If BO = 12, what is the length of CO?
<"http://etc.usf.edu/clipart/""Visit Clipart ETC for a great collection of clipart for students and teachers."
PSSA Math Concept Look
1. If the radius of one circle is 3 feet and the radius of another circle is 10 feet,
what is the difference in their diameters?
2. If concentric circles have diameter of 16 inches and 22 inches,
what would be the shortest distance connecting the two circles?
3. If a tangent segment to a circle with a radius of 3 feet measures 4 feet,
what is the distance from the endpoint of the tangent segment outside the circle?
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. If the radius of a circle is 7 yards, then the diameter is 14 yards.
D = 2r
D = 2 (7yards)
D = 14 yards
Write formula
Substitute known values
Evaluate answer including correct units
2. If the diameter radius of a circle is 18 feet, then the radius is 9 feet.
D = 2r
18 feet = 2r
9 feet = r
Write formula
Substitute known values
Divide each side of the equation by 2.
3. If the radius is 4 inches and the corner radius is 9 inches, then the distance from the
circle to the corner of the pattern would be the difference of these two radii, 5 inches.
Related, Generic Math Concepts
1. Since AC and AB are tangents to the same circle they are congruent so,
AC = 15.
2. Since BO is a radius of circle O and AB is a tangent to circle O. AO would be a leg
of a right triangle ABO. This would be a 3-4-5 right triangle. If you did not recognize it
as a 3-4-5 right triangle, you could use the Pythagorean Theorem to find the length.
Since AO is the hypotenuse of this right triangle its length would be 5.
3. CO = 12 since BO and CO are radii of the same circle.
PSSA Math Concept Look
1. First you need to find the diameters of the circle, then you need to subtract those
diameters to get the difference.
D = 2r
D = 2r
Write formula
D = 2(3 feet)
D = 2(10 feet) Substitute known values
D = 6 feet
D = 20 feet
Calculate Diameters
20 feet – 6 feet = 14 feet
Subtract
The difference would be 14 feet.
2. Since concentric circles have the same center, the shortest distance between the points
would be the difference in the radii.
D = 2r
D = 2r
Write formula
22 inches = 2r
16 inches = 2r
Substitute known values
11 inches = r
8 inches = r
Divide each side by 2.
11 inches – 8 inches = 3 inches
Subtract
The difference is 3 inches.
3. Since a tangent always meets a radius at a right angle this would be a 3-4-5 right
triangle and CA = 5. If you did not recognize it as a 3-4-5 right triangle, you could use
the Pythagorean Theorem to find the length.
PDE/BCTE Math Council
Carpentry T-Chart
Equal Diagonals = Congruent Diagonals
Duty: Foundations
Task: Set up batter boards
CARPENTRY ASSOCIATED WORDS:
TERMS - BATTER BOARD, DRY LINE,
BUILDING LINES
Carpentry Concept
When you are setting up batter boards it is
important that you not only measure the sides of
the area but also the diagonals to see if the
corners are square.
PSSA Eligible Content – M11.C.1.2.2
Identify and/or use the properties of
quadrilaterals (e.g., parallel sides, diagonals,
bisectors, congruent sides/angles and
supplementary angles).
MATH ASSOCIATED WORDS:
TERMS – CONGRUENT,
QUADRILATERAL, TRAPEZIOD,
ISOSCELES TRAPEZIOD, KITE,
PARALLELOGRAM, RHOMBUS, SQUARE,
RECTANGLE
Geometry Theorem
A parallelogram is has right angles only if the
diagonals are congruent.
Quadrilaterals
See Carpentry image on the following page.
Square
Trapezoid
Rectangle
Parallelogram
Kite
Rhombus
Isosceles
Trapezoid
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
One of the most commonly used properties of quadrilaterals in Carpentry class is that the
diagonals of an object must be congruent if the object is square. This is true for foundations as
well as picture frames. This is a concept you will also study in math class.
How are the concepts similar?
This is basically the same concept in both classes. The terms associated with the concept may
be different, but the idea is the same.
How do the concepts differ?
In Carpentry class you will measure the diagonals to see that they are equal in measure, and in
geometry class you will say that the diagonals of the parallelogram that is square are congruent
meaning they are the same in both shape and size.
Some Common Properties of Quadrilaterals
A kite is a quadrilateral with two pairs of consecutive congruent sides; exactly one pair of opposite
angles congruent and the diagonals are perpendicular.
A trapezoid has exactly one pair of parallel sides.
An isosceles trapezoid has congruent legs; congruent base angles and the diagonals are
perpendicular.
Properties of Parallelograms
Opposite sides are parallel.
Opposite angles are congruent.
Opposite sides are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
You can prove that a quadrilateral is a
parallelogram if you can prove any of the
properties at the left are true.
Squares, Rectangles and Rhombuses are
special parallelograms.
Rectangles have four congruent angles, and the diagonals are congruent.
Rhombuses have four congruent sides, perpendicular diagonals, and the diagonals also bisect the
opposite angles.
Squares are both Rectangles and Rhombuses and have the properties of both.
Common Mistakes Students Make
A common mistake that students make is thinking that all parallelograms have congruent
diagonals.
Another common mistake is thinking that all quadrilaterals with congruent diagonals are square.
An isosceles trapezoid is a quadrilateral that has congruent diagonals and it is not square.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. If you are measuring to set up a foundation and one diagonal the foundation is 26 feet,
what should the other diagonal be? You have already measured the sides and the
opposite sides are equal in measure.
2.
If you are measuring to set up a foundation and one diagonal the foundation is 20 feet,
what should the other diagonal be? You have already measured the sides and the
opposite sides are equal in measure.
3.
If the diagonals of a foundation are not congruent what does that mean?
Related, Generic Math Concepts
1. Are all quadrilaterals parallelograms?
2. Are all quadrilaterals with congruent diagonals either a square or a rectangle?
3. If one diagonal of a rectangle is 2x + 3 and the other is x + 8, what is the value of x?
PSSA Math Concept Look
1. If ABCD is a square and AC = 7x + 10 and DB = 3x + 42, what is the value of x?
2. Which of the following is NOT a way to prove a quadrilateral is a parallelogram?
A.
B.
C.
D.
Opposite sides are congruent
Opposite angles are congruent
Diagonals are congruent
Diagonals bisect each other
3. Which of the following quadrilaterals always have congruent diagonals?
A.
B.
C.
D.
Rectangles, Squares, and Rhombuses
Rectangles, Squares, and Isosceles Trapezoids
Parallelograms, Rhombuses and Kites
Kites, Squares and Rhombuses
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. Since the diagonals of a foundation should be equal in measure, the other diagonal
should be 26 feet.
2. Since the diagonals of a foundation should be equal in measure, the other diagonal
should be 20 feet.
3. If the diagonals of a foundation are not congruent then it means that the foundation is
not square.
Related, Generic Math Concepts
1. No, all quadrilaterals are not parallelograms, kites and trapezoids are two examples of
quadrilaterals that are not parallelograms.
2. No, isosceles trapezoids have congruent diagonals.
3. Since the diagonals are congruent you can set up this equation and solve for x.
2x + 3 = x + 8
Set up the equation.
x+3=8
Subtract x from both sides of the equation.
x=5
Subtract 3 from both sides of the equation.
PSSA Math Concept Look
1. Since the diagonals are congruent you can set up this equation and solve for x.
7x + 10 = 3x + 42
Set up the equation.
4x + 10 = 42
Subtract 3x from both sides of the equation.
4x = 32
Subtract 10 from both sides of the equation.
x=8
Divide both sides of the equation by 4.
2. The answer is C., since the diagonals of an isosceles trapezoid are congruent but it is
NOT a parallelogram.
3. The answer is B., since Rectangles, Squares, and Isosceles Trapezoids have
congruent diagonals.
PDE/BCTE Math Council
Carpentry T-Chart
3-4-5 Method
Duty: Building Layout
Task: Measure, layout and verify a rectangle
building
CARPENTRY ASSOCIATED WORDS:
= Pythagorean Theorem
PSSA Eligible Content – M11.C.1.4.1
Find the measure of the side of a right
triangle using the Pythagorean Theorem.
MATH ASSOCIATED WORDS:
TERM - Diagonals (hypotenuse), square,
plumb, level
TERM - Hypotenuse (diagonals), right
angle, right triangle, legs, Pythagorean
Theorem right triangle
Formula:
Formula:
32 + 42
= 52
A2 + b2
= c2
a2 + b2 = c2
a = Leg
b = Leg
c = Hypotenuse or Diagonals
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
A Greek mathematician named Pythagoras discovered that the relationship between the
hypotenuse and the legs is true for all right angles. His discovery is known as the Pythagorean
Theorem and it is one of the earliest theorems that can be traced back to over 2,500 years ago.
The Pythagorean theorem states:
In any right triangle, the square of the lengths of the hypotenuse is equal to the sum of the
squares of the lengths of the legs.
Carpenters refer to the Pythagorean Theorem as the 3-4-5 method or diagonals. They use this
theorem extensively to check the diagonal of a foundation or framed wall to determine if it is
square. The right triangle is the key to making stair stringers and roof rafters.
.
Common Mistakes Students Make
The common mistakes that students make when measuring the layout of a building is not
making sure the lines of the layout are parallel, which then means the building will not be
square.
Incorrectly measuring lines is another common mistake Carpenters make when laying out a
building.
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. Determine the diagonal for a rectangular
structure that is 40' x32' . Use the 3-4-5 Method.
2. Determine the height of a framed wall that is
22' long with a diagonal of 24' . Use the
3-4-5 Method.
3. Determine the length of a framed wall that is
12' in height with a diagonal of 36' . Use the
3-4-5 Method.
Related, Generic Math Concepts
4. The tent has two slanted sides that are both
5 ft. long and the bottom is 6 ft. across.
What is the height of the tent in feet at the
tallest point?
5. The measures of three sides of a triangle are
9 ft., 16 ft. and 20 ft. Determine whether the
triangle is a right triangle.
6. On a baseball diamond, the bases are 90 ft.
apart. What is the distance from home plate to
second base using a straight line?
PDE/BCTE Math Council
PSSA Math Concept Look
7. In a right triangle, the lengths of the legs are
12m and 15m. What is the length of the
hypotenuse to the nearest whole meter?
8. In a right triangle ABC, where angle C is the
right angle, side AB is 25 ft. and side BC is 17ft.
Find the length of side AC to the nearest tenth
of a foot.
9. In the given triangle, find the length of a.
B
26 in.
A
10 in.
a
C
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. Determine the diagonal for a
rectangular structure that is 40' x32' .
Use the 3-4-5 Method.
2. Determine the height of a framed
wall that is 22' long with a diagonal
of 24' . Use the 3-4-5 Method.
3. Determine the length of a framed
wall that is 12' in height with a
diagonal of 36' . Use the
3-4-5 Method.
40 2 + 32 2 = c 2
1600 + 1024 = c 2
1600 + 1024 = c
2624 = c
51.22' = c
a 2 + 22 2 = 24 2
a 2 + 484 = 576
a 2 = 576 − 484
a = 576 − 484
a = 92
a = 9.59'
12 2 + b 2 = 36 2
144 + b 2 = 1296
b 2 = 1296 − 144
b = 1296 − 144
b = 1152
b = 33.94'
Related, Generic Math Concepts
4. The tent has two slanted sides
that are both 5 ft. long and the
bottom is 6 ft. across. What is the
height of the tent in feet at the tallest
point?
5 2 = 32 + x
2
25 = 9 + x
2
16 = x
4= x
PDE/BCTE Math Council
a2 + b2 = c
2
16 2 + 9 2 = 20 2
5. The measure of three sides of a
triangle is 9 ft., 16 ft. and 20 ft.
Determine whether the triangle is a
right triangle.
256 + 81 ≠ 400
Therefore it is not a right triangle.
6. On a baseball diamond, the bases
are 90 ft. apart. What is the distance
from home plate to second base
using a straight line?
90 2 + 90 2 = c 2
16,200
8100 + 8100 =
C= 127.28 ft.
PSSA Math Concept Look
7. In a right triangle, the lengths of
the legs are 12m and 15m. What is
the length of the hypotenuse to the
nearest whole meter?
a2 + b2 = c2
12 2 + 15 2 = c 2
144 + 225 = c 2
369 = c 2
19m = c
B
25 ft.
8. In a right triangle ABC, where
angle C is the right angle, side AB is
25 ft. and side BC is 17ft. Find the
length of side AC to the nearest
tenth of a foot.
A
a2 + b2 = c2
17 2 + b 2 = 25 2
289 + b 2 = 625
b 2 = 336
b
17 ft.
C
b 2 = 336
b = 18.3 ft.
PDE/BCTE Math Council
9. In the given triangle, find the
length of a.
B
26 in.
A
10 in.
a
a2
a2
a2
a2
+ b2 = c2
+ 10 2 = 26 2
+ 100 = 676
= 576
a 2 = 576
a = 24in.
C
PDE/BCTE Math Council
Carpentry T-Chart
PITCH
= SLOPE
Duty: Roof Framing
Task: Measure, layout, cut, install rafters
PSSA Eligible Content - M11.D.3.2.1
Apply the formula for the slope of a line to
solve problems (formula given on reference
sheet).
CARPENTRY ASSOCIATED WORDS:
MATH ASSOCIATED WORDS:
PITCH - Ridge, Plumb, Rise, Run, Base, Span,
Rafter Length
SLOPE - Rise, Run, Line, Coordinate, Rate
of Change
Formula to find the pitch of a roof:
Formula to find the slope of a line:
Pitch = Rise (in inches) = Rise per foot
Run (in feet)
slope =
Y2 − Y1 Rise ΔY
=
=
X 2 − X1 Run ΔX
4, 6, or 8 inches of rise per foot of run would give
a pitch of 4-12, 6-12, 8-12
Rise = 87.5 inches
Run = 14 feet
Pitch = 87.5 = 6.25
14
Slope =
5−2 3
= =3
2 −1 1
Pitch = 6.25-12
PDE/BCTE Math Council
TEACHER’S SCRIPT FOR BRIDGING THE GAP
Pitch and slope are very similar. They both represent rise divided by run.
There are three major differences between pitch and slope:
1. Pitch is always positive whereas slope can be both negative and positive.
2. Slope requires you to find the rise and run by subtracting the y and x values while the
rise and run for pitch are given.
3. The answers are written differently. For pitch, you divide the rise in inches by the run in
feet, take your answer and put a -12 after it to get a 2-12, 4-12 pitch. For slope, you
reduce the fraction so 2-12 would really be 1/6, and 4-12 would really be 1/3.
Common Errors Made by Students:
1. Students will often not subtract consistently among y and x values. For instance, for the
slope of line passing through the points (3, 5) and (-1,7):
(3, 5) and (-1,7)
7−5
3 − (−1)
or
5−7
−1− 3
INCORRECT
(3, 5) and (-1,7)
(3, 5) and (-1,7)
instead of the correct answer:
7−5
5−7
or
− 1 − 3 3 − (−1)
CORRECT
(3, 5) and (-1,7)
PDE/BCTE Math Council
Occupational (Contextual) Math Concepts
1. Determine the pitch of a roof with a 60”
rise and a 6 foot run.
2. Determine the pitch of a roof with a 66”
rise and a 16’ 6” run.
3. Determine the pitch of a roof with a 64”
rise and an 8’ run.
Related, Generic Math Concepts
4. A ramp increases from ground level to a
height of 5 feet over a span of 20 feet.
What is the slope (rate of change) of the
ramp?
5. A pipe is installed at an angle across a
20 foot wall. It is 8 feet above the floor at
one end and 10 feet above the ground at
the other end. What is the slope (rate of
change) of the piping?
6. A sidewalk increases from ground level
to a height of 3 feet over a span of 40
feet. What is the slope (rate of change)
of the sidewalk?
PSSA Math Concept Look
7. Find the slope of a line passing through
the points (3, 5) and (2, 1).
8. Find the slope of a line passing through
the points (-2, 1) and (4, -5).
9. Find the slope of a line passing through
the points (4, 2) and (-5, 6)
PDE/BCTE Math Council
ANSWER KEY
Occupational (Contextual) Math Concepts
1. Determine the pitch of a roof with a 60”
rise and a 6 foot run.
60
= 10
6
10-12 pitch
2. Determine the pitch of a roof with a 66”
rise and a 16’ 6” run.
66
=4
16.5
4-12 pitch
3. Determine the pitch of a roof with a 64”
rise and an 8’ run.
64
=8
8
8-12 pitch
Related, Generic Math Concepts
4. A ramp increases from ground level to a
height of 5 feet over a span of 20 feet.
What is the slope (rate of change) of the
ramp?
5
1
=
20 4
5. A pipe is installed at an angle across a
20 foot wall. It is 8 feet above the floor at
one end and 10 feet above the ground at
the other end. What is the slope (rate of
change) of the piping?
8 − 10 − 2 − 1
=
=
20
20 10
6. A sidewalk increases from ground level
to a height of 3 feet over a span of 40
feet. What is the slope (rate of change)
of the sidewalk?
3
40
PSSA Math Concept Look
1− 5 − 4
=
=4
2 − 3 −1
7. Find the slope of a line passing through
the points (3, 5) and (2, 1).
5 −1 4
= =4
3−2 1
8. Find the slope of a line passing through
the points (-2, 1) and (4, -5).
1 − (−5)
6
− 5 −1
−6
= −1 or
= −1
=
=
4 − (−2)
6
−2−4 −6
9. Find the slope of a line passing through
the points (4, 2) and (-5, 6)
6−2
4
4
=−
=
or
−5−4 −9
9
or
2−6
4
−4
=−
=
4 − (−5)
9
9
PDE/BCTE Math Council
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