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Unit 5 Area, the Pythagorean Theorem, and Volume Geometry

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Unit 5 Area, the Pythagorean Theorem, and Volume Geometry ACC Unit Goals – Stage 1 Number of Days: 34 days 2/27/17 – 4/13/17 Unit Description: Deriving new formulas from previously discovered ones, the students will leave Unit 5 with an understanding of area and volume formulas for polygons, circles and their three-dimensional counterparts. Unit 5 seeks to have the students develop and describe a process whereby they are able to solve for areas and volumes in both procedural and applied contexts. Units of measurement are emphasized as a method of checking one’s approach to a solution. Using their study of circles from Unit 3, the students will dissect the Pythagorean Theorem. Through this they will develop properties of the special right triangles: 45°, 45°, 90° and 30°, 60°, 90°. Algebra skills from previous classes lead students to understand the connection between the distance formula and the Pythagorean Theorem. In all instances, surface area will be generated using the smaller pieces from which a given shape is constructed. The student will, again, derive new understandings from old ones. Materials: Construction tools, construction paper, patty paper, calculators, Desmos, rulers, glue sticks (optional), sets of geometric solids, pennies and dimes (Explore p. 458), string, protractors, paper clips, square, isometric and graph paper, sand or water, plastic dishpan Standards for Mathematical Transfer Goals Practice Students will be able to independently use their learning to… SMP 1 Make sense of problems • Make sense of never-before-seen problems and persevere in solving them. and persevere in solving • Construct viable arguments and critique the reasoning of others. them. SMP 2 Reason abstractly and quantitatively. Making Meaning SMP 3 Construct viable UNDERSTANDINGS ESSENTIAL QUESTIONS arguments and critique Students will understand that… Students will keep considering… the reasoning of others. • By cutting apart and rearranging figures, new formulas can be • Do we need to memorize new SMP 4 Model with mathematics. derived from old ones. formulas or can we use old formulas in 2 SMP 5 Use appropriate tools • Units are a key to differentiating between area, u , and volume, a new way? 3 strategically. u . • If we have a shape with a given area, SMP 6 Attend to precision. • The Pythagorean Theorem leads to the distance formula and and we cut that shape apart and SMP 7 Look for and make use of the equation of a circle. rearrange its pieces, will the area structure. • In a right triangle, the area of the square on the hypotenuse is remain constant? SMP 8 Look for and express the sum of the areas of the squares on the legs. • How are the area formulas for a regularity in repeated • Pi is the constant of proportionality between the circumference square, a rectangle, a trapezoid, and a reasoning. and the diameter of a circle. triangle related? • If a shape is divided into subregions, the sum of the areas of • Why do the solution points for Standards for Mathematical those subregions, even if they are rearranged, is equal to the (x − h)2 + (y − k)2 = r 2 form a circle? area of the original shape. Content Clusters Addressed • How can you find the volume of a solid Define trigonometric • [m] G-SRT.C When a triangle or parallelogram is rotated so that a different for which no formula is available? ratios and solve side is its base, the area of that triangle or parallelogram holds • How can a concept map or a Venn problems involving constant. diagram be constructed to organize right triangles. • To find the surface area of a prism, pyramid, cylinder or cone, the solids based on their [a] G-C.B Find arc lengths and combine the shape’s lateral surface area with the area(s) of the commonalities? areas of sectors of shape’s base(s). • How is finding the volume of a prism circles. • Sub-shapes or near-shapes can be used to approximate similar to finding the area of a surface area and volume. rectangle? LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 1 Posted 2/28/17 Unit 5 Area, the Pythagorean Theorem, and Volume Geometry ACC Unit Goals – Stage 1 [a] G-GPE.A Translate between Acquisition the geometric KNOWLEDGE SKILLS description and the Students will know… Students will be skilled at and/or be able to… equation for a conic • Formulas and methods for finding the areas of • Derive formulas for the areas of rectangles, section. rectangles, parallelograms, triangles, trapezoids, parallelograms, triangles, trapezoids, kites, regular [m] G-GPE.B Use coordinates to kites, regular polygons, circles, sectors, polygons, circles, annuluses, sectors, segments of prove simple segments, and annuluses. circles, and surface areas of spheres. geometric theorems • The Pythagorean Theorem and that, given a2 + b2 • Derive the fact that the length of the arc intercepted algebraically. = c2, a and b are the lengths of the legs. The by an angle is proportional to the radius. [a] G-GMD.A Explain volume length of the hypotenuse is c. • Apply surface area formulas to solve problems. formulas and use • The Converse of the Pythagorean Theorem. • Solve area application problems using various them to solve • If two figures are similar with a scale factor s, then problem-solving strategies. problems. their areas have scale factor s2. • Solve for the missing side length of a triangle using [m] G-MG.A Apply geometric • The area of a sector is the same part of the area the Pythagorean Theorem. concepts in of the whole circle as the measure of the central • modeling situations. Use the Converse of the Pythagorean Theorem to angle of the sector is of 360◦. prove that a triangle is right. • In a 45°-45°-90° triangle, if the legs have length • Solve for the lengths of the sides of a 45°-45°-90° m, then the hypotenuse has length m 2 . In a triangle and a 30°-60°-90° triangle. • 30°-60°-90° triangle, if the leg opposite the 30° Derive the distance formula from the Pythagorean has length a, then the hypotenuse has length 2a Theorem. • Use the distance formula to solve problems. and the other leg as length a 3 . • Derive the equation of a circle of given center and • The equation (x − h)2 + (y − k)2 = r 2 describes a radius using the Pythagorean Theorem. circle with points on the circle (x, y), a radial • Apply the Pythagorean relationship to problems distance of r, and center (h, k). involving circles. • The volume formulas for prisms, cylinders, • Use coordinates to compute perimeters of polygons pyramids, cones, and spheres. and areas of triangles and rectangles. • Derive formulas for finding the volumes of prisms, • cylinders, pyramids, cones, and spheres using dissection arguments and Cavalieri’s Principle. • The surface area formula for a sphere. • Solve applied problems involving polyhedra, cones, cylinders, spheres, or hemispheres. • Use displacement to find the volumes of irregularly shaped solids. LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 2 Posted 2/28/17 Unit 5 Area, the Pythagorean Theorem, and Volume Geometry ACC Assessed Grade Level Standards Standards for Mathematical Practice SMP 1 Make sense of problems and persevere in solving them. SMP 2 Reason abstractly and quantitatively. SMP 3 Construct viable arguments and critique the reasoning of others. SMP 4 Model with mathematics. SMP 5 Use appropriate tools strategically. SMP 6 Attend to precision. SMP 7 Look for and make use of structure. SMP 8 Look for and express regularity in repeated reasoning. Standards for Mathematical Content [m] G-SRT.C Define trigonometric ratios and solve problems involving right triangles. G-SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. [a] G-C.B Find arc lengths and areas of sectors of circles. G-C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. [a] G-GPE.A Translate between the geometric description and the equation for a conic section. G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. [m] G-GPE.B Use coordinates to prove simple geometric theorems algebraically. G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. [a] G-GMD.A Explain volume formulas and use them to solve problems. G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s Principle, and informal limit arguments. G-GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. [m] G-MG.A Apply geometric concepts in modeling situations. G-MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G-MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Key: [m] = major clusters; [s] = supporting clusters, [a] = additional clusters Indicates a modeling standard linking mathematics to everyday life, work, and decision-making CA Indicates a California-only standard LONG BEACH UNIFIED SCHOOL DISTRICT 2016-2017 3 Posted 2/28/17 Unit 5 Area, the Pythagorean Theorem, and Volume Geometry ACC Evidence of Learning – Stage 2 Assessment Evidence Unit Assessment Claim 1: Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.
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