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Math-in-CTE Lesson Plan Lesson Title: Octagon Hay Feeder Lesson #: AM07 Occupational Area: Agriculture CTE Concept(s): Use a tape measure, use a carpenters square, construct projects and equipment using an arc welder. Math Concepts: Computation in context, rate, scale drawing, map model, apply geometric properties, function, equation, model problem situation, use algebra to solve problems Lesson Objective: Translate word phrases and sentences into expressions and equations and vice versa. (Pass: Algebra I, Standard 1-1) Solve linear equations by graphing or using properties of equality. (Pass: Algebra I, Standard 1-6A.) Draw and analyze 2- and 3-dimenzional figures. (Pass: Geometry, Standard 2-2) Use geometric tools (e.g., protractor, compass, straight edge) to construct a variety figure. (Pass: Geometry, Standard 2-6) Supplies Needed: Link to Accompanying Materials: Agriculture Mechanics AM07 Downloads TEACHER NOTES (and answer key) THE "7 ELEMENTS" 1. Introduce the CTE lesson. Due to the high demand of round bale feeders in our community, we are going to construct octagon bale feeders, not round. Due to the nature of an octagon, we need to look at a new approach to building these feeders. Although we will utilize the pipe bender to construct the angles for the octagon, we could also use the cutoff saw to build the same angles, or even a miter saw to cut wood projects. First we need to determine the angle of an octagon and how to draw it on paper. We will draw the octagon and use a ratio to scale the picture to the optimum size. 1 Due to the nature of this project, a wood project may be more “user friendly” than using metal. This could be a picture frame… 2. Assess students’ math awareness as it relates to the CTE lesson. Question the class. Define: Octagon – eight sided polygon Complimentary angles – two angles whose sum is 90 Perpendicular – two lines intersecting in a 90° angle. Polygon – A union of segments connected end to end, such that each segment intersects exactly two others at its endpoints. Proportion – a statement that expresses ratios which are equal. Unit of measure Linear feet or inches – measures length of 2-D object Does anyone know the formula for finding the angle of an octagon? How can we find the optimum diameter of the bale feeder? What is the diameter of a round bale? From a drawing, how will we determine the actual dimensions of the bale feeder? In discussing the complimentary angle, be sure the students know that a triangle has 180° total. For example, start with a right triangle with a 20° angle. To find the third angle C, you can say 180°= 90° + 20° + C. Now subtract 90° from both sides gives 90° = 20° + C. Then subtract 20° from both sides gives 70° = C. Examples can be quickly drawn on chalkboard to give students a visual. For a 7 ft diameter feeder the sides will be 2.9 ft and for a 6 ft diameter feeder the sides will be 2.5 ft. angle = (N-2) · 180° N Will the angle for a small octagon be the same angle for a larger one? How will we “build” the angles of the octagon? What would we do to find the angle to be cut using the cutoff saw? 2 (Where N is the number of sides) Take them to shop and show on miter or cutoff saw with scrap material. In addition, transfer the angle with a sliding T bevel to the saw. 3. Work through the math example embedded in the CTE lesson. Now that we have our formula we will find the angle of an octagon: angle = (N-2) · 180° · is the same as multiply (x) N (N = # of sides of the polygon) Angle = (8 2)180 6(180) 1080 135 8 8 8 The use of a protractor and T bevel may be used to transfer the angle to the saw. Next we need to practice constructing a 135° angle of various sizes (e.g. 6 inch). Construct an octagon on paper. Using our drawing we can take the welding rods and bend them to an exact fit of the drawing and welding them together to form a scaled model of the feeder. Now we need to address the conversion from scale to actual. The students will determine the ratio of the scale to actual. To make this process easier we will have the students build their model on a 1:12 scale. For example: 1 7 gives us 1x = 7(12) so x = 72. 12 x Therefore, the measurement of their model will take the place of the 7 in the above example to find the actual measure of the feeder. When a pipe bender is not available or for extra instruction where a cutoff saw would be more applicable to a project, we will address building the angle with the cutoff 3 saw. Special care needs to be taken to allow ample time for this to “click” for the students. Have the students look from the perspective of the saw, the middle of the octagon. Using the picture and scale that was constructed earlier, the students may get a hands-on approach to learning. If this approach does not develop into the correct findings then one must use the Pythagorean theorem, which will be familiar to some students. Draw a line from the center of the octagon to one of the “corners” of the octagon. Then draw another line from the center of the octagon perpendicular to one of the sides adjacent to the “corner”. See the example: Mathematically, the line segment AB bisects the angle “B” (cuts the angle in half). So the 135° is cut into 67.5°. A B Looking at the triangle, we can see the different perspectives to view the angles. “A” being the saw perspective and “B” is our perspective. The saw perspective is looking at the compliment of B, thus 90° - 67.5°= 22.5°. For a 7 ft diameter feeder, the sides will be 2.9 ft. For a 6 ft diameter feeder, the sides will be 2.5 ft. Now the students may go to the shop and produce the feeder. 4 **** Optional formula that maybe used to challenge the more “advanced” students to find the length of the sides is 2r · tan(180°/N) where r is the radius and N is the number of sides. 4. Work through related, contextual math-in-CTE examples. angle = (N-2) · 180° What is the angle of an equilateral triangle? N Square? Rectangle? Pentagon? (N = the number of sides) Hexagon? 60°, 90°, 90°, 108°, 120° 5. Work examples. through traditional math Find the complimentary angles of 20º; 45º; 70º; 45º; 53º 37º; A classroom has 3 girls and 9 boys. If 21 boys another classroom has the same ratio, how many boys are in the class if there are 7 girls? 6. Students understanding. demonstrate their Students will construct one level of the bale Scrap material (wood or metal) feeder. would be useful to make these In addition, students may also construct the “smaller” projects. various shapes, pentagon, hexagon, heptagon (7), nonagon (9), or decagon(10), to construct a picture frame or similar projects. 7. Formal assessment. Problems will be on unit and/or semester tests as necessary. 5