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. <"http://etc.usf.edu/clipart/""Visit Clipart ETC for a great collection of clipart for students and teachers." 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? <"http://etc.usf.edu/clipart/""Visit Clipart ETC for a great collection of clipart for students and teachers." 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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