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Lesson Plan Title: Touching the sky Description – Students design buildings with similar height but of varying depth of foundation and use their mathematical skills to predict the depth the foundation of a building of a specified height should have. Target grade: 9 and 10 Subjects covered: Math, Science Goals – (i) To be able to measure the angle “θ” by the help of sextant. (ii) To be able to use trigonometric ratios. (iii) To be able to measure the height of a building. Prerequisites – 1.The students are familiar with right angled triangle 2. The students know about the trigonometric ratios of a triangle. 3. The students are familiar with values of tan θ for different angles. Instructional Objective(s) At the end of the lesson, students should be able to: 1: Measure the height of a building using mathematical formulae 2. Predict the depth foundation of a building of a specified height should have 3. Understand the criteria for stability of a building/ tower/bridge etc. Material Needed For each student: KWLH chart (provided ) For each group: Sextant Thermocol sheet Chart papers Glue Measuring tape Sketch pens Pencils Sharpeners Cutter Marker pen Wooden ply board Any other that the students need to create the structure For the entire class: Designing skyscrapers books, project book etc. Instructional Resources Introduction – (i) Begin by asking few questions like: What is a skyscraper? (Expected ans- A tall building with continuous habitable floors that are more than 40-50 in number) Which is the highest sky-scraper in the world right now? (Expected ans- Burj Khalifa in Dubai, UAE) How do sky-scrapers stand so tall but still do not fall during winds and storms? What makes them remain stand erect and strong? (Expected ans- by having a strong core and base, a deeper foundation, stronger building material etc) (ii) Giving pamphlets in which models and different skyscrapers are given so that they can think about it. Ask them to go through this video showing the various skyscrapers. https://www.youtube.com/watch?v=I6QOXOGN8I4 Engineering Designing Introduction Part 1i. Perform a simple activity with the students. Take a thin stick a few inches high and take a jar of soil. Embed the stick lightly and superficially in the soil. Try to bend the stick, it’ll bend easily and might get uprooted too. Now push the stick very deep and try pushing it again. Ask the students to observe carefully. The stick doesn’t bend that easily. Explain that the deeper and stronger grip the foundation of a building has on the frame of a building the higher it can stand. ii. Divide the students into groups of 3-4. iii. Instruct the students to assume the role of architects and design a skyscraper using the materials provided (they’re free to use additional materials too). After designing, they must calculate the height of their skyscraper and the depth of the foundation they used for the skyscraper to remain stable on gentle pushing. Part 2Recapitulate with the students the method of determining the height of a tall structure using mathematical formulae. Take a simple example of a tree and tell them how its height can be measured. Explain thus: (Source of information: http://www.tiem.utk.edu/~gross/bioed/bealsmodules/triangle.html ) Variables: θ an acute angle of a right triangle x side of a triangle adjacent to the angle y side of a triangle opposite the angle z the hypotenuse of a triangle Method: Trigonometry is a branch of mathematics dealing with measurements of the angles and sides of triangles, and functions based on these measurements. The three basic trigonometric functions that we are concerned with here (sine, cosine, and tangent) are ratios of the lengths of two sides of a triangle. These ratios are the trigonometric functions of an angle, theta, such that where θ (theta) is the angle of interest, "opp" is the length of the side of the triangle opposite θ (y), "hyp" is the length of the hypotenuse (z), and "adj" is the length of the side adjacent to the angle θ (x), as illustrated on the triangle below. If we know the lengths of two sides of a right triangle (recall from geometry that a right triangle has one angle that is 90 degrees [a right angle]), we can calculate the length of the third side using the Pythagorean theorem (opp 2 + adj2 = hyp2, or y2 + x2 = z2). For the triangle below, the side opposite is three units in length, and the side adjacent to θ is 1.5 units in length. The length of the hypotenuse is calculated 32 + 1.52 = hyp2 =11.25. Taking the square root of 11.25, we find that the hypotenuse is approximately 3.35 units long. We now know the lengths of each of the sides of the triangle, and can use these to find values for the trigonometric functions: But what can we do with this information? Well, for one thing we can use it to find θ by taking the inverse of any of the functions (for our purposes here can do this using a calculator). Doing this we see that θ is an angle of approximately 63.4 degrees. The drawing above shows a forester measuring a tree's height using trigonometry. Assuming that the tree is at a right angle to the plane on which the forester is standing, the base of the tree, the top of the tree, and the forester form the vertices (or corners) of a right triangle. The forester measures his or her distance from the base of the tree, and then uses a clinometer (a small instrument that measures inclination, or angle of elevation) to look at the top of the tree and determine θ . Interpretation: In this situation, rather than knowing the lengths of all of the sides, we know θ and the length of the adjacent side (x), and are interested in determining the length of the opposite side (y, the height of the tree). Which of the three trigonometric functions deals with the adjacent and opposite sides? We know θ , and we know "adj" (or x); multiplying both sides of the equation by "adj" or x and substituting our known values, we get The tree is approximately 44 feet tall. We can measure the height of a building in a similar manner. Part – 3 – Ask the students to fill-in the KWL sheet and proceed with making the models of their skyscrapers. Extensions: Students can be asked to research on environment friendly skyscrapers. Which components are added/replaced to make them eco-friendly and which components are retained? Assessment and Rubrics Pre activity – (i) Discussion questions What is a skyscraper? Which is the highest sky-scraper in the world right now? How do sky-scrapers stand so tall but still do not fall during winds and storms? What makes them remain stand erect and strong? Activity Embedded – 1) You know from your geometry class that all the angles of a triangle must sum to 180 degrees. Therefore, the third angle from the tree and forester triangle must be (180 degrees - 90 degrees - 31.8 degrees) = 58.2 degrees. Verify this using the three trigonometric functions (note that for this angle, y is the adjacent side and x is the opposite side). 2) What would the angle of elevation (θ) be if the tree was 100 ft. tall? 3) Assessing students’ engineering design worksheets. Post activity - RUBRICS: Criteria Design of structures Excellent Good Average The design matches The design somewhat The design doesn’t the criteria matches the criteria match the criteria at completely. all. Research work while The nature of design reflects that extensive designing research and experimentation has The nature of design reflects that basic research and experimentation has The nature of design reflects that almost no research and experimentation has gone into it. gone into it. gone into it. Materials Used Low cost, easily available and appropriate materials are used and suggested for creating the structure Slightly expensive but easily available and appropriate materials are used and suggested for creating the structure Irrelevant materials are used and suggested for creating the structure Collaborative Approach Active Involvement of Involvement of few Involvement of one or all students students two students Knowledge skyscrapers Mathematical approach about Extensive Accurate Average Poor Up to mark Below mark Suggested Further Reading for Students: https://en.wikipedia.org/wiki/Skyscraper www.dezeen.com/architecture/skyscrapers/ http://science.howstuffworks.com/engineering/structural/skyscraper.htm http://www.explainthatstuff.com/howbuildingswork.html Credits (i) (ii) Mr. Bikas Chandra Jha, TGT (Maths) JNV, Dumka. Jharkhand. Patna Region Mr. Pawan Kumar, TGT (Maths) JNV, Ranchi. Jharkhand. Patna Region