Geometry, Quarter 3, Unit 3.1 Trigonometric Ratios Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Make and defend conjectures to solve problems using trigonometric ratios (sine, cosine, tangent). (3 days) Use trigonometric ratios (sine, cosine, tangent) to find missing sides and angles of a right triangle. (3 days) Construct geometric arguments and use geometric properties to solve real-world problems involving trigonometric ratios (sine, cosine, tangent). (3 days) Mathematical practices to be integrated Use appropriate tools strategically. Use various tools, including technology, to help solve problems. Use estimation to know if tools have worked correctly. Reason abstractly and quantitatively. Represent mathematically what is read and apply to solve a problem. Rewrite problems involving trigonometric ratios in simpler terms. Attend to precision. Calculate and compute accurately. Essential questions How can trigonometric ratios be used to make and defend conjectures? How do you find a side length or angle measure in a right triangle? How do trigonometric ratios relate to similar right triangles? How is the process to solve for a side of a right triangle different from the process to solve for an angle? When can you use the Pythagorean Theorem instead of trigonometric ratios to solve problems involving right triangles? What are specific examples of trigonometric ratios being used in the real world? Cumberland, Lincoln, and Woonsocket Public Schools C-19
Geometry, Quarter 3, Unit 3.1 Final, July 2011 Trigonometric Ratios (10 days) Written Curriculum Grade Span Expectations M(G&M) 10 2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts (e.g., Pythagorean Theorem, Triangle Inequality Theorem). (State) Clarifying the Standards Prior Learning In grade 7, students calculated perfect squares and related square roots. In grade 8, students were introduced to and applied the Pythagorean Theorem. In units 2.1 and 2.2, students learned about triangle properties and congruency of triangles, and, in unit 1.1, students made and defended conjectures to solve problems using the Pythagorean theorem (Delete special right triangles). Current Learning In this unit, students make and defend conjectures to solve problems using trigonometric ratios. Students apply trigonometric ratios to solve real-world problems. Future Learning In precalculus, students will use special right triangles, Law of Sines, Law of Cosines, unit circle trigonometry, trigonometric identities, and proofs. In calculus, students will continue to use trigonometry. Additional Research Findings According to Principles and Standards for School Mathematics, students in grades 9 12 will use right triangle trigonometry to solve a range of practical problems (p. 313). Cumberland, Lincoln, and Woonsocket Public Schools C-20
Geometry, Quarter 3, Unit 3.2 Perimeter and Area of Figures Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Solve problems involving perimeter, circumference, and area of polygons and circles, using appropriate units. (4 days) Make conversions within or across systems, using an appropriate degree of accuracy. (1 day) Determine how changes in dimensions affect perimeter, circumference, and area. (2 days) Find the perimeter and area of composite shapes both algebraically and graphically. (2 days) Essential questions How are the circumference and area of a circle related? How are the formulas for the areas of polygons related? How is the area and perimeter of a polygon affected by changing its dimensions? Mathematical practices to be integrated Model with mathematics. Use a simpler problem to solve more complex problems. Interpret results in the context of a perimeter or area problem. Improve a model if it needs modification (for example, when solving the area or perimeter of composite figures). Attend to precision. Use units of measure correctly (i.e., square units). Calculate and compute area and perimeter accurately. How is the area of a figure different from its perimeter? How can you find the perimeter and area of composite shapes both algebraically and graphically? Cumberland, Lincoln, and Woonsocket Public Schools C-21
Geometry, Quarter 3, Unit 3.2 Final, July 2011 Perimeter and Area of Figures (10 days) Written Curriculum Grade Span Expectations M(G&M) 10 6 Solves problems involving perimeter, circumference, or area of two-dimensional figures (including composite figures) or surface area or volume of three-dimensional figures (including composite figures) within mathematics or across disciplines or contexts. (State) M(G&M) 10 7 Uses units of measure appropriately and consistently when solving problems across content strands; makes conversions within or across systems and makes decisions concerning an appropriate degree of accuracy in problem situations involving measurement in other GSEs. (State) Clarifying the Standards Prior Learning In grade 2, students used models to demonstrate conceptual understanding of perimeter and area. In grades 2 8, students measured and used units of measure appropriately and consistently; they also made conversions within systems. Current Learning Students solve problems using the perimeter of a figure, circumference of a circle, and area of a figure. They determine how the change of a dimension affects the perimeter, area, or circumference and they find perimeter and area of composite shapes. In units of study 3.3 and 3.4, students apply area to calculate surface area and volume. Future Learning Students will use the concept of area when they calculate the area of polygons in grade 12. Additional Research Findings According to Benchmarks for Science Literacy, by the end of grade 5, students should know that length can be thought of as unit lengths joined together; area can be thought of as a collection of unit squares; and volume can be thought of as a set of unit cubes (p. 223). According to Principles and Standards for School Mathematics, instructional programs from prekindergarten through grade 12 should enable all students to understand and use formulas for the area, surface area, and volume of geometric figures (p. 320). Cumberland, Lincoln, and Woonsocket Public Schools C-22
Geometry, Quarter 3, Unit 3.3 Surface Area Overview Number of instructional days: 12 (1 day = 45 minutes) Content to be learned Solve problems involving surface areas of pyramids, prisms, cylinders, cones, spheres, and composite figures using appropriate units and an appropriate degree of accuracy. (8 days) Sketch diagrams (nets) by hand or use dynamic geometric software to generate threedimensional objects from two-dimensional perspectives or two-dimensional perspectives from three-dimensional objects. (2 days) Determine how changes in dimensions affect the surface area of a figure. (1 day) Mathematical practices to be integrated Model with mathematics. Interpret results in the context of surface area problems. Attend to precision. Use units of measure correctly (i.e., square units). Calculate and compute surface area accurately. Use appropriate tools strategically. Use technology to visualize results. Use estimation to know if tools have worked correctly. Essential questions Where would you use surface area in the real world? How do changes in the dimensions of a solid affect its surface area? How is the area of a figure related to its surface area? Why are the formulas for the surface area of solids different? How is sketching or technology helpful when calculating surface area? What are applications of nets in relation to geometric solids? Cumberland, Lincoln, and Woonsocket Public Schools C-23
Geometry, Quarter 3, Unit 3.3 Final, July 2011 Surface Area (12 days) Written Curriculum Grade Span Expectations M(G&M) 10 6 Solves problems involving perimeter, circumference, or area of two-dimensional figures (including composite figures) or surface area or volume of three-dimensional figures (including composite figures) within mathematics or across disciplines or contexts. (State) M(G&M) 10 7 Uses units of measure appropriately and consistently when solving problems across content strands; makes conversions within or across systems and makes decisions concerning an appropriate degree of accuracy in problem situations involving measurement in other GSEs. (State) M(G&M) 10 10 Demonstrates conceptual understanding of spatial reasoning and visualization by sketching or using dynamic geometric software to generate three-dimensional objects from twodimensional perspectives, or to generate two-dimensional perspectives from three- dimensional objects, or by solving related problems. (Local) Clarifying the Standards Prior Learning In grades 1 2, students were introduced to the concept of area by using manipulatives. In grades 3 6, students were introduced to area by using grids, units, models, and formulas for a variety of shapes. In grade 7, surface area was introduced for rectangular prisms only. In grade 8, this concept was expanded to include triangular prisms, cylinders, pyramids, and cones. Current Learning In this unit of study, students solve real-world problems using surface area of pyramids, prisms, cylinders, cones, and composite shapes, both algebraically and graphically. When possible, students use dynamic geometric software or other technology to enhance learning. Students use concepts from this unit of study when they study geometric probability in unit of study 4.4 in this course. Future Learning Students will use surface area with Cavallieri s Principle and the process of integration to calculate surface areas of non-regular figures. Additional Research Findings According to Principles and Standards for School Mathematics, instructional programs should enable all students to analyze properties and determine attributes of two- and three-dimensional objects (p. 308). According to Benchmarks for Science Literacy, by the end of grade 12, students should know that there are formulas for calculating the surface areas of regular shapes. Students should know that when the linear size of a shape changes by some factor, its area changes in proportion to the square of the factor (p. 225). Cumberland, Lincoln, and Woonsocket Public Schools C-24
Geometry, Quarter 3, Unit 3.4 Volume Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Solve problems involving volume of pyramids, prisms, cylinders, cones, spheres, and composite figures, using appropriate units and an appropriate degree of accuracy. (5 days) Determine how changes in dimensions affect volume. (2 days) Essential questions Why is volume an important concept, and where do you use it in the real world? How do changes in the dimensions of a solid affect its volume? How is surface area related to volume? Mathematical practices to be integrated Model with mathematics. Interpret results in the context of volume. Attend to precision. Use units of measure correctly (i.e., cubic units). Calculate and compute volume accurately. Use appropriate tools strategically. Use technology to visualize results. Use estimation to know if tools have worked correctly. Why are the formulas for the volume of solids different? How is sketching or technology helpful when determining volume? What are applications of nets in relation to geometric solids? Cumberland, Lincoln, and Woonsocket Public Schools C-25
Geometry, Quarter 3, Unit 3.4 Final, July 2011 Volume (8 days) Written Curriculum Grade Span Expectations M(G&M) 10 6 Solves problems involving perimeter, circumference, or area of two-dimensional figures (including composite figures) or surface area or volume of three-dimensional figures (including composite figures) within mathematics or across disciplines or contexts. (State) M(G&M) 10 7 Uses units of measure appropriately and consistently when solving problems across content strands; makes conversions within or across systems and makes decisions concerning an appropriate degree of accuracy in problem situations involving measurement in other GSEs. (State) M(G&M) 10 10 Demonstrates conceptual understanding of spatial reasoning and visualization by sketching or using dynamic geometric software to generate three-dimensional objects from twodimensional perspectives, or to generate two-dimensional perspectives from three- dimensional objects, or by solving related problems. (Local) Clarifying the Standards Prior Learning In grade 1, students were introduced to three-dimensional shapes. In grades 4 and 5, students identified and described three-dimensional figures. Also in grade 5, students measured volume. By the end of grade 8, students calculated the volume of right prisms, pyramids, cones, and cylinders. Current Learning In this unit of study, students solve real-world problems using volume of pyramids, prisms, cylinders, cones, and composite shapes both algebraically and graphically. When possible, students use dynamic geometric software or other technology to enhance learning. Future Learning In grade 12, students will solve problems involving volume using Cavalieri s principle. Students will use the process of integration in calculus to compute volumes of non-regular figures. Additional Research Findings According to Principles and Standards for School Mathematics, instructional programs should enable all students to analyze properties and determine attributes of two- and three-dimensional objects (p. 308). According to Benchmarks for Science Literacy, by the end of grade 12, students should know that there are formulas for calculating the volume of regular shapes. Students should know that when the linear size of a shape changes by some factor, its volume changes in proportion to the square of the factor (p. 225). Cumberland, Lincoln, and Woonsocket Public Schools C-26