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Contextualized Learning Activities (CLAs) For the “other required credits” in the bundle of credits, students in a Specialist High Skills Major program must complete learning activities that are contextualized to the knowledge and skills relevant to the economic sector of the SHSM. Contextualized learning activities (CLAs) address curriculum expectations in these courses. This CLA has been created by teachers for teachers. It has not undergone an approval process by the Ministry of Education. Contact Information Board Waterloo Catholic District School Board Development date August 2009 Contact person Christine Stockie Michael Sullivan (519) 621-4050 ext. 623 (519) 621-4050 ext. 676 (519) 621-4057 Phone Fax E-mail [email protected] [email protected] Specialist High Skills Major Manufacturing Course code and title MBF 3C – Foundations of College Mathematics Name of contextualized learning activity Measurement in Design Brief description of contextualized learning activity. Students will review and consolidate skills involving operations with fractions and decimals, and converting between Metric and Imperial measures. They will apply this knowledge to real-life examples from the manufacturing sector, while studying and learning about 2- and 3dimensional design in a practical environment. Duration Approximately 6 hours: Lesson 1 - Review - Operations with Fractions (60 min) Lesson 2 - Review –Fractions and Decimals; Conversions with Metric and Imperial Measure (60 min) Lesson 3 – Proportional Reasoning, Ratios and Scales (60 minutes) Lesson 4 – Trigonometry (60 minutes) Lesson 5 – Designing and Building a Sawhorse (120 minutes) Page 1 of 20 Overall expectations Geometry and Trigonometry The student will: 1. represent, in a variety of ways, two-dimensional shapes and threedimensional figures arising from real-world applications, and solve design problems; 2. solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications. Specific expectations Geometry and Trigonometry The student will: 1.1 recognize and describe real-world applications of geometric shapes and figures, through investigation, in a variety of contexts, and explain these applications 1.3 create plans, and patterns from physical models arising from a variety of real-world applications, by applying the metric and imperial systems and using design or drawing software 1.4 solve design problems that satisfy given constraints using physical models or drawings, and state any assumptions made 2.1 solve problems, including those that arise from real-world applications, by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios 2.4 solve problems that arise from real-world applications involving metric and imperial measurements and that require the use of the sine law or the cosine law in acute triangles Essential Skills: Reading Text Understanding text in the form of sentences or paragraphs Document Use Using information displays including blueprints, signs, drawings Numeracy Use of numbers and quantities Writing Completing solutions of multi-step problem-solving questions Continuous Learning Ongoing process of learning and acquiring skills Work Habits: Thinking Skills Cognitive ability, problem solving Teamwork Work willingly and cooperatively with others Initiative Starts work with little or no prompting Work Habits Punctual, time effective, and able to follow directions Organization Written work is well laid out and neat Working Independently Accomplishes tasks independently Page 2 of 20 Instructional/Assessment Strategies Teacher’s notes The Math teacher should communicate with the Manufacturing teacher on a regular basis. Both teachers should be kept up to date on developments that correspond to each other’s course. The teacher should become familiar with the use of mathematics in the Manufacturing sector. Providing applicable real life examples from the manufacturing sector can be beneficial for student learning. Constant diagnostic feedback is important for consistent learning and student development (ie. Through use of student worksheets #1-4). If the class is a split group (not all SHSM students) it may be advantageous to group the SHSM students together, however, this CLA has benefits for all MBF students, not just those enrolled in the SHSM program. Allow students to supplement their learning with applicable computer programs. Context This CLA is designed for students that plan on entering an apprenticeship or college program in the Manufacturing sector. Assessment and Evaluation of Student Achievement Strategies/Tasks 1. Operations with Fractions Purpose Assessment for Learning (diagnostic, formative) or Assessment of Learning (summative, evaluation) Diagnostic Assessment (give consistent feedback on student Progress) 2. Fractions & Decimals; Conversions with Metric and Imperial Measures Formative Assessment (give consistent feedback on student Progress) 3. Proportional Reasoning and Scale Diagrams Formative Assessment (give consistent feedback on student Progress) 4. Trigonometry Formative Assessment (give consistent feedback on student Progress) 5. Design and Build a Scale Model of a Summative Assessment (see rubric in Appendix B) Sawhorse Additional Notes/Comments/Explanations Both Mathematics Text books and Manufacturing Texts and Manuals are a great source for additional questions, practice, and examples. See Attachments for further details Page 3 of 20 Resources Smith, Robert D. Mathematics for Machine Technology, 4th Edition, Delmar Publishers,1999 Cooke, Gordon et al, Pearson Math 11, Pearson Education Canada, 2008 Accommodations Individual Education Plans (IEP) should be followed at all times. Be sure to consult the SERT for additional information and suggestions; additional time may be needed for quizzes, tests, assignments, and in-class work; the activities and lessons outlined in this CLA allow for flexibility in the delivery of the material. Alternating teaching strategies can help students who are not progressing at the appropriate level; font can be increased for those students that have a sight issue; class rules, behaviours, and due dates should be posted in the classroom and talked about so that all students are aware of the expectations; if possible, more individual instruction time can be allotted to students in need; can account for student work habits when considering assignments; provide additional work for advanced students; provide time for peer-to-peer teaching; use audio aids if needed; provide additional assessment opportunities that are geared towards students strengths; if available, many computer programs can be used to supplement student learning. List of Attachments Operations with Fractions lesson Student Worksheet #1 Imperial and Metric Measurement lesson Metric/Imperial Conversion Table and Student Worksheet #2 Proportional Reasoning and Scale Diagrams lesson Student Worksheet #3 Trigonometry lesson Student Worksheet #4 Designing and Building a Sawhorse Assignment Sawhorse Rubric Page 4 of 20 LESSON 1: Operations with Fractions In the manufacturing sector, workers are expected to have a solid working knowledge of both fractions and decimals, and to be able to convert between the two types of numbers. It is important that students review their basic skills of putting fractions in lowest terms, finding equivalent fractions and performing operations (+, -, x, ÷) with fractions to accomplish this! Lowest Terms: Fractions are reduced to “lowest terms” by dividing both the numerator (top) and denominator (bottom) of the fraction by the same number (make it the biggest value you can): eg. a) 6 2 3 8 2 4 b) 12 4 3 = 16 4 4 Equivalent Fractions: You can determine an “equivalent fraction” by multiplying the numerator and denominator by the same number: eg. 1 5 5 2 5 10 a) b) 3 4 12 8 4 32 Improper and Mixed Fractions: A fraction where the numerator is larger than the denominator is called an “improper fraction”. A “mixed fraction” is a combination of a whole number and a fraction. To convert between the two, simply follow these instructions: eg. a) 1 3 (to change this mixed fraction into an improper fraction, multiply the whole number by the 4 denominator, then add the numerator, then place over the original denominator) eg. b) 7 4 1X4+3 4 27 (to change this improper fraction into a mixed fraction, divide the numerator by the denomator. 16 The quotient is the whole number, and the remainder is the numerator of the fraction part.) 11 1 16 Page 5 of 20 27 ÷ 16 = 1 with a remainder of 11 Adding and Subtracting Fractions: When adding or subtracting two or more fractions, first find a common denominator, then add the numerators. Once completed, fractions must always be put into lowest form. eg. a) 1 3 3 4 4 9 = 12 12 13 12 1 1 12 3 1 2 1 8 16 19 17 8 16 38 17 16 16 21 16 5 1 16 b) Change to improper fractions first Final answer is a mixed fraction, in lowest terms. Multiplying Fractions: When multiplying fractions, multiply the numerators to get the numerator of the answer, and then do the same thing for the denominators. Remember that your final answer should be expressed as a mixed fraction in lowest terms. eg. a) 1 3 2 1 3 2 1 3 2 1 1 2 3 1 2 8 2 19 1 8 2 19 16 3 1 16 b) Change the first mixed fraction to an improper fraction first numerator X numerator denominator X denominator Dividing Fractions: When dividing fractions, you need to multiply the first fraction by the reciprocal of the second one. (Note: a “reciprocal” is a “flipped” fraction – one where the numerator becomes the denominator, and vice versa) eg. a) 15 2 16 15 1 16 2 15 32 Page 6 of 20 1 3 2 4 7 4 2 3 28 6 4 4 6 2 4 3 b) 3 Change this improper fraction to a mixed fraction, in lowest terms Student Worksheet #1 Complete the following exercises to demonstrate your mastery of operations with fractions: 1. 2. ? Page 7 of 20 3. 4. 5. 1 1 A metal rod with a diameter of 1 inches is 19 feet long. It needs to be divided into 8 equal pieces. 4 2 What is the length of each piece? You need 12 pieces of 3 3 4 Page 8 of 20 inch square bar. What is the total length of steel that needs to be ordered? LESSON 2: Decimals and Fractions; Imperial and Metric Measurements In the manufacturing sector, as well as in most other trades, workers use their knowledge of fractions AND decimals everyday. Fractions are used extensively with Imperial measurement, and decimals are used mainly when measuring in Metric, however, it is important for students to be able to convert from one type of number into another, thus using fractions and decimals interchangeably. Converting Fractions to Decimals: To change a fraction to a decimal, divide the numerator by the denominator. 27 30 eg. a) b) = 27 ÷ 30 = 0.9 21 6 = 21 ÷ 6 = 3.5 Converting Fractions to Percent: To convert a fraction to a percent, first change the fraction into a decimal, then multiply by 100. 12 20 eg. a) b) = (12 ÷ 20) x 100 = (0.6) x 100 = 60% 29 24 = (29 ÷ 24) x 100 = (1.21) x 100 = 121% Converting Between Metric and Imperial Units of Measure: Refer to the tables below to help you complete the following conversions: eg. a) 15 yds = _______ ft b) 15 yds = _________ in 15 × 3 15 × 3 × 12 = 45 ft d) 28 cm = _________ m e) 28 cm = _________ mm = 0.28 m g) 4 m = ________ in 4 × 100 ÷ 2.54 = 157.48 in h) 1 15 yds = __________ m 15 × 0.9144 = 540 in 28 ÷ 100 Page 9 of 20 c) = 13.716 m f) 28 cm = __________ in 28 × 100 28 ÷ 2.54 = 280 mm = 11.0236 in 3 in = ________ cm 8 1.375 × 2.54 = 3.4925 cm i) 12.5 ft = __________ mm 12.5 × 12 × 2.54 ×10 = 3810 mm Student Worksheet #2 Metric and Imperial Conversion Charts: Length: ÷ Metric Imperial Metric Imperial 10mm = 1cm 100cm = 1m 1000m = 1km 12 inches = 1 foot 3 feet = 1 yard 1760 yards = 1 mile 1 inch 2.54 cm 1 foot 30.48 cm 1 yard 0.9144m 1 mile 1.609 km × Mass: Metric Imperial Metric Imperial 1000 mg = 1 g 1000 g = 1 kg 1000 kg = 1 t 16 ounces = 1 lbs. 2000 lbs. = 1 ton 1 ounce 28.35 g 1 pound 0.454 kg 1 ton 0.907 t Metric Imperial Metric Imperial 1000mL = 1L 16 fl. ounces = 1 pint 2 pints = 1 quart 8 pints = 1 gallon 1 fl. ounce 29.574 mL 1 pint 0.473 L (473 mL) 1 gallon 3.785 L Volume: 6. Complete the following conversions, using the tables provided above: b) 3⅛ ft = ________ in c) 186 in = ___________ ft e) 96 in = ________ yd f) 78 ft = ___________ yd g) 6.377 cm = __________mm h) 3.47 m = ____________cm i) 300.3 mm = __________ m j) 0.93 mm = _________ cm k) 5.032 cm = ___________ m l) 0.45 m = ____________ mm m) 0.75 ft = __________ m n) 2.400 m = ___________ in o) 17.300 cm = __________ in p) 8.000 m = __________ ft q) 0.360 in = ____________ mm r) 84.00 cm = ____________ ft s) 780.00 mm = __________ ft t) 3.50 yd = ____________ m u) 3 a) 0.4 yd = _________ in d) 1 4 yd = _________ ft Page 10 of 20 1 4 in = ______________ cm 7. A screw measures 5 in long. How long is this in millimeters (use decimals)? 8 8. You have a piece of lumber that is 3 yards long, but you only need a length that measures 2.3 m. How much is left over (in m)? What percent of the entire piece of lumber is left over? 9. You need 36 steel rods that each measure 18 inches long. How much steel do you need to order in feet? In yards? In metres? Page 11 of 20 Lesson 3: Proportional Reasoning and Scale Diagrams In architecture, construction or any kind of design and manufacturing, a SCALE is used to make accurate drawings that show sizes of rooms, placement of furniture, or the dimensions of an item. The scale is presented as a RATIO – where the first number represents the length in the drawing and the second number represents the actual length of the object. For example, a scale of 1:10 means that 1 cm on the drawing represents a distance of 10 cm on the actual object (or that the drawing is 10 times smaller than the object). eg. a) Length on drawing = 3.3 cm Now, using the scale of 1:7, we multiply the drawing length by 7 to find the actual length of the hammer….. 3.3 × 7 = 23.1 cm Scale 1 : 7 eg. b) If the real bee measures 0.8 cm in length, find the scale for this scale diagram. Length on drawing = 3.7 cm Actual length = 0.8 cm Now, set up a ratio using the drawing length first, then the actual length: 3.7 3.7 : 0.8 = n n 0.8 1 n = 3.7 ÷ 0.8 n = 4.625 :1 The scale is 4.625 : 1. A PROPORTION is a comparison of two ratios. The order of the numbers in a proportion is important, since it is often used to calculate for missing values. For example, if a piece of metal has a length : width ratio of 3 : 2, and the length is actually 15ft, the matching width value can be calculated using equivalent fractions: eg. c) 3 2 15 This proportion can be solved by “cross-multiplying”… 3m = 30 m m = 10 eg. d) Given the dimensions in the diagram below, find the width and height of a cinder block with a length of 24 in. l = 4.2 cm h = 1.0cm 4.2 24 1.0 h 1.2 w Note that the first fraction in the proportion represents length values, the second fraction represents height, and the third is width. In all cases, the numerators are the dimensions from the diagram, and the denominators are the actual dimensions of the block (Imperial). w = 1.2 cm Working with 2 fractions at a time, and cross multiplying, you get: 4.2 h = 24 h = 5.714 in Page 12 of 20 4.2 w = 28.8 w = 6.857 in Student Worksheet #3 10. Use a ruler to determine the actual length of this bolt, if the scale is 1:3. 11. Determine the actual height of this height gauge, if the scale is 7.5:1. 12. Given the floor plan below, use a ruler and the dimensions shown to determine the scale. Page 13 of 20 13. B C D A Page 14 of 20 Lesson 4: Trigonometry Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles in triangles. Since pretty much any polygon can be divided up into a series of triangles, trigonometry has widereaching uses in professions such as construction, architecture, aerospace, engineering… to name just a few. Accuracy is key in the manufacturing field when you are designing and machining objects, and using trigonometry is the way to get there! Right-Angled Triangles: The sides of a right-angled triangle are related using the Pythagorean Theorem, whereby the square of the hypotenuse is equal to the sum of the squares of the other two sides. Referring to the diagram, then, the Pythagorean Theorem states that: c2 = a2 + b2. The Primary Trigonometry Ratios are used to relate the angles of a right-angled triangle. To review briefly, the ratios are: sin ө = cos ө = tan ө = length of the side opposite from length of the hypotenuse length of the side adjacent to length of the hypotenuse length of the side opposite from length of the adjacent to eg. a) Solve the following triangle for V, W and r : V 18.200 in 7.620 in W r c2 = a2 + b2 18.2 2 = 7.62 2 + r 2 331.24 = 58.0644 + r 2 331.24 – 58.0644 = r 2 273.1756 = r 2 sin W = r 273.1756 r = 16.528 in tan V = 7.620 18.200 W = 24.751 o tan V = r 7.620 16.528 7.620 V = 65.249 o Page 15 of 20 eg. b) A ramp needs to be built at the loading dock of the school. Find the length of the ramp (k) and it’s angle of inclination ( ө ). r 2 = 0.765 2 + 1.6065 2 r 2 = 3.1661 r = 3.1661 r = 1.7793 m h = 0.765 m ө tan ө = j = 1.6065 m 0.765 1.6065 ө = 25.463 o Oblique Triangles: An oblique triangle is one without any right angles. Since a triangle such as this has no right angle, it has no hypotenuse. Without a hypotenuse, the Pythagorean Theorem and Primary Trig Ratios are useless. The tools available to solve these types of triangles are the Sine Law or the Cosine Law…. Sine Law: The Sine Law can be used to solve for any missing information in an oblique triangle where “opposite information” is given (meaning you know the measure of at least one angle and the matching sinA sinB sinC opposite side). The Sine Law states that . a b c eg. c) sin64 P 3.125 cm 3.125 sinP 1.901 sinQ y solve for P first using the first 2 fractions in the proportion y 64o 1.901 cm Q 1.901 × sin64 o = sin P × 3.125 sin P = 1.901 × sin 64 o 3.125 Sin P = 0.54675 P = 33.145 o To solve for angle Q, 180 – 64 – 33.145 = 82.855 o Now solve for the final fraction in the proportion: 3.125 × sin Q = sin 64 o × y y = 3.125 × sin 64 o sin 82.855 o y = 2.831 cm Page 16 of 20 Cosine Law: The Cosine Law is used to solve for missing information in an oblique triangle where no opposite information is given. The Cosine Law states that c 2 = a 2 + b 2 – 2ab cosC, where C is the one angle that you know, or are trying to find. eg. d) A hiker enters a park (at P) and walks 3.28km North. She then turns (at N) and heads Southeast for 2.13km. How far is she from her original position? First, draw a triangle to organize the information from the question. Note: “Southeast” indicates a heading of exactly 45 o from south to east, as pictured on the diagram. N 45 o N 2 = S 2 + P 2 – 2SP cos N N 2 = 3.28 2 + 2.13 2 – 2(3.28)(2.13) cos 45 o N 2 = 15.2953 – 9.8803 2.13 km 3.28 km S P Page 17 of 20 N = 5.4150 N = 2.33 km Student Worksheet #4 14. x A bridge MN is to be built across a river. Point L is 121 ft from M, and at an angle of 83 o. Angle L measures 38 o. Calculate the length of the bridge. 15. M N L U 16. A steel support beam must be fabricated, according to the diagram, along length ST. Calculate the length of the beam. 12 ft 29 o S 11 ft T Page 18 of 20 17. You are working in a manufacturing shop, and the belt breaks on the lathe. To order the new belt, you need to fill in the missing information from the schematic diagram below. Belt p a Hints: The belt consists of a continuous loop around both circular drives. ø This symbol represents diameter. All measurements are in inches. Angle a = ___________ Slanted length p = ___________ in Page 19 of 20 Summative Design Assignment: Design and Build a Scale Model of a Sawhorse You are required to design a sawhorse to be used in manufacturing classes. The design specifics have been provided for you below (all measurements are in inches). a) Calculate all missing lengths and angles. b) Create a scale diagram of your sawhorse. c) Create a 3-dimensional drawing OR a 3-dimensional model of your piece of equipment. ? ? ? ? ? Supports are added at the midpoint of the horizontal supports (as indicated below), to offer additional stability. Determine the lengths of these pieces. 30 30 ? Page 20 of 20