The Pythagorean Theorem

Transcription

1 Grade 8 Mathematics, Quarter 4, Unit 4.1 The Pythagorean Theorem Overview Number of instructional days: 4 (1 day = 45 minutes) Content to be learned Apply the Pythagorean Theorem to find the missing side of a right triangle. Apply the Pythagorean Theorem to problemsolving situations. Mathematical practices to be integrated Use appropriate tools strategically. Use tools to solve, explore, compare, and visualize problems and to deepen knowledge/understanding. Detect errors using estimation and other mathematical knowledge. Make sense of problems and persevere in solving them. Plan a solution pathway. Analyze givens, constraints, relationships, and goals. Model with mathematics. Identify important quantities in a practical situation. Apply mathematics known to solve problems. Analyze relationships mathematically to draw conclusions. Essential questions What is the relationship among the three sides of a right triangle? How do you find a missing side of a right triangle? How do you use the Pythagorean Theorem in a problem-solving situation? Cumberland, Lincoln, and Woonsocket Public Schools C-43

2 Grade 8 Mathematics, Quarter 4, Unit 4.1 Pythagorean Theorem (4 days) Written Curriculum Grade-Level Expectations M(G&M) 8 2 Applies the Pythagorean Theorem to find a missing side of a right triangle, or in problem solving situations. (Local) Clarifying the Standards Prior Learning This GLE is not addressed from kindergarten to grade 6. In grade 7, students applied theorems or relationships to solve problems (triangle inequality or sum of the measures of interior angles of regular polygons). Students were not introduced to the Pythagorean Theorem in prior grades. Current Learning In grade 8, students learn and apply the Pythagorean Theorem to find the missing side of a right triangle or in problem-solving situations. This concept is introduced, reinforced, and mastered at this level. Future Learning At the high school level, students will be asked to make and defend conjectures, construct geometric arguments, use geometric properties, or use theorems to solve problems (e.g., Pythagorean Theorem). Additional Research Findings Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics states: Students need to explain why the Pythagorean Theorem is valid by using a variety of methods for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedral. (p. 20) C-44 Cumberland, Lincoln, and Woonsocket Public Schools

3 Grade 8 Mathematics, Quarter 4, Unit 4.2 Surface Area Overview Number of instructional days: 13 (1 day = 45 minutes) Content to be learned Demonstrate conceptual understanding of surface area of rectangular prisms, triangular prisms, cylinders, pyramids, or cones. Use surface area to solve problems involving rectangular prisms, triangular prisms, cylinders, pyramids, or cones. Expresses all measures using appropriate units. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Explain the meaning of the problem. Plan a solution pathway. Consider similar problems to gain insight into its solution. Reason abstractly and quantitatively. Consider units involved. Be able to flow between contextual and noncontextual situations during problem solving and make meaning of numbers and symbols. Model with mathematics. Apply the mathematics known to solve problems arising in everyday life, society, and the workplace. Identify important quantities in a practical situation and map their relationship using such tools as diagrams and formulas. Analyze mathematical relationships to draw conclusions. Use appropriate tools strategically. Use tools when solving a mathematical problem and to deepen understanding of concepts (i.e., graphing calculators, graph paper, measurement devices). Attend to precision. Specify units of measure. Strive for accuracy. Cumberland, Lincoln, and Woonsocket Public Schools C-45

4 Grade 8 Mathematics, Quarter 4, Unit 4.2 Surface Area (13 days) Look for and make use of structure. Apply and discuss properties. Look for and express regularity in repeated reasoning. Look for mathematically sound shortcuts. Use repeated applications to generalize properties. Essential questions Without using a formula, how could you determine the surface area of a threedimensional figure? How do you calculate the surface area of threedimensional figures? C-46 Cumberland, Lincoln, and Woonsocket Public Schools

5 Surface Area (13 days) Grade 8 Mathematics, Quarter 4, Unit 4.2 Grade-Level Expectations Written Curriculum M(G&M) 8 6 Demonstrates conceptual understanding of surface area or volume by solving problems involving surface area and volume of rectangular prisms, triangular prisms, cylinders, pyramids, or cones. Expresses all measures using appropriate units. (Local) Clarifying the Standards Prior Learning In grade 1, students developed an understanding of the length/height of a two-dimensional object using nonstandard units. In grade 2, they moved into perimeter and area using models or manipulatives to cover or surround polygons. In grade 3, students began using grids to find perimeter of polygons and area of rectangles. The measures were expressed in appropriate units. Students in grade 4 found the perimeter of polygons and area of polygons or irregular shapes on grids and they began to use formulas in their calculations. Measures continued to be expressed in appropriate units. In grade 5, the area of a right triangle and the volume of a rectangular prism were introduced. In grade 6, students demonstrated conceptual understanding of the perimeter of polygons, the area of quadrilaterals or triangles, and the volume of rectangular prisms by using models, formulas, or by solving problems. They also demonstrated understanding of the relationship of circle measures by solving related problems. In grade 7, students demonstrated conceptual understanding of the area of a circle and the area or perimeter of composite figures. They also demonstrated conceptual understanding of the surface area of rectangular prisms or the volume of rectangular prisms, triangular prisms, or cylinders. Current Learning At the eighth-grade level, students demonstrate conceptual understanding of surface area by solving problems with rectangular prisms, triangular prisms, cylinders, pyramids, or cones. They must express all measures using appropriate units. These topics are introduced and reinforced at this level. Future Learning In high school, students will continue to work on solving problems involving perimeter, circumference, or area of two-dimensional figures or surface area or volume of three-dimensional figures. These concepts will be applied in mathematics and across disciplines or contexts. Additional Research Findings Curriculum Focal Points states: By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. They select appropriate two- and three-dimensional shapes to model real-world situations and solve a variety of problems involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders. (p. 37) Cumberland, Lincoln, and Woonsocket Public Schools C-47

6 Grade 8 Mathematics, Quarter 4, Unit 4.2 Surface Area (13 days) C-48 Cumberland, Lincoln, and Woonsocket Public Schools

7 Grade 8 Mathematics, Quarter 4, Unit 4.3 Volume Overview Number of instructional days: 13 (1 day = 45 minutes) Content to be learned Demonstrate conceptual understanding of volume of rectangular prisms, triangular prisms, cylinders, pyramids, or cones. Use volume to solve problems involving rectangular prisms, triangular prisms, cylinders, pyramids, or cones. Express all measures using appropriate units. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Explain to self the meaning of the problem. Plan a solution pathway. Consider similar problems to gain insight into its solution. Reason abstractly and quantitatively. Consider units involved. Be able to flow between contextual and noncontextual situations during problem solving and make meaning of numbers and symbols. Model with mathematics. Apply the mathematics known to solve problems arising in everyday life, society, and the workplace. Identify important quantities in a practical situation and map their relationship using such tools as diagrams and formulas. Analyze mathematical relationships to draw conclusions. Use appropriate tools strategically. Use tools when solving a mathematical problem and to deepen understanding of concepts (i.e., graphing calculators, graph paper, measurement devices). Attend to precision. Specify units of measure. Strive for accuracy. Cumberland, Lincoln, and Woonsocket Public Schools C-49

8 Grade 8 Mathematics, Quarter 4, Unit 4.3 Volume (13 days) Look for and make use of structure. Apply and discuss properties. Look for and express regularity in repeated reasoning. Look for mathematically sound shortcuts. Use repeated applications to generalize properties. Essential questions Without using a formula, how could you determine the volume of a three-dimensional figure? How do you calculate the volume of threedimensional figures? C-50 Cumberland, Lincoln, and Woonsocket Public Schools

9 Volume (13 days) Grade 8 Mathematics, Quarter 4, Unit 4.3 Grade-Level Expectations Written Curriculum M(G&M) 8 6 Demonstrates conceptual understanding of surface area or volume by solving problems involving surface area and volume of rectangular prisms, triangular prisms, cylinders, pyramids, or cones. Expresses all measures using appropriate units. (Local) Clarifying the Standards Prior Learning In grade 1, students developed an understanding of the length/height of a two-dimensional object using nonstandard units. In grade 2, they moved into perimeter and area using models or manipulatives to cover or surround polygons. In grade 3, students began using grids to find perimeter of polygons and area of rectangles. The measures were expressed in appropriate units. Students in grade 4 found the perimeter of polygons and area of polygons or irregular shapes on grids, and they began to use formulas in their calculations. Measures continued to be expressed in appropriate units. In grade 5, the area of a right triangle and the volume of a rectangular prism were introduced. In grade 6, students demonstrated conceptual understanding of the perimeter of polygons, the area of quadrilaterals or triangles, and the volume of rectangular prisms by using models, formulas, or by solving problems. They demonstrated understanding of the relationship of circle measures by solving related problems. In grade 7, students demonstrated conceptual understanding of the area of a circle and the area or perimeter of composite figures. They also demonstrated conceptual understanding of the surface area of rectangular prisms or the volume of rectangular prisms, triangular prisms, or cylinders. Current Learning At the eighth-grade level, students demonstrate conceptual understanding of volume by solving problems of rectangular prisms, triangular prisms, cylinders, pyramids, or cones. They must express all measures using appropriate units. These topics are introduced and reinforced at this level. Future Learning In high school, students will continue to work on solving problems involving the perimeter, circumference, or area of two-dimensional figures and the surface area or volume of three-dimensional figures. These concepts will be applied in mathematics and across disciplines or contexts. Additional Research Findings According to Curriculum Focal Points: By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. They select appropriate two and three dimensional shapes to model real-world situations and solve a variety of problems involving surface areas, areas, and circumferences of circles, and volumes of prisms and cylinders. (p. 37) Cumberland, Lincoln, and Woonsocket Public Schools C-51

10 Grade 8 Mathematics, Quarter 4, Unit 4.3 Volume (13 days) C-52 Cumberland, Lincoln, and Woonsocket Public Schools

11 Grade 8 Mathematics, Quarter 4, Unit 4.4 Similarity and Scaling Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Apply concepts of similarity to determine the impact of scaling on surface area and volume of three-dimensional figures when linear dimensions are multiplied by a constant factor. Apply concepts of similarity to determine the length of sides of similar triangles. Solve problems using constant growth. Solve problems using rates. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Look for an entry point. Plan a solution pathway. Consider similar problems to gain insight into the solution. Check answers using a different method, and continually ask, Does this make sense? Reason abstractly and quantitatively. Consider units involved. Make sense of quantities. Attend to the meaning of quantities. Construct viable arguments and critique the reasoning of others. Use inductive reasoning. Justify conclusions. Give examples and nonexamples. Model with mathematics. Apply the mathematics known to solve problems arising in everyday life, society, and the workplace. Identify important quantities in a practical situation and map their relationship using such tools as diagrams and formulas. Analyze mathematical relationships to draw conclusions. Interpret mathematical results. Cumberland, Lincoln, and Woonsocket Public Schools C-53

12 Grade 8 Mathematics, Quarter 4, Unit 4.4 Similarity and Scaling (10 days) Use appropriate tools strategically. Use tools when solving a mathematical problem and to deepen understanding of concepts (i.e., graphing calculators, graph paper, measurement devices). Attend to precision. Specify units of measure. Strive for accuracy. Look for and express regularity in repeated reasoning. Look for mathematically sound shortcuts. Use repeated applications to generalize properties. Evaluate the reasonableness of intermediate results. Essential questions How does scaling the side lengths of threedimensional figures affect its surface area? How can you determine if two triangles are similar? How does scaling affect the volume of threedimensional figures? How do you determine if a figure has grown at a constant rate? C-54 Cumberland, Lincoln, and Woonsocket Public Schools

13 Similarity and Scaling (10 days) Grade 8 Mathematics, Quarter 4, Unit 4.4 Grade-Level Expectations Written Curriculum M(G&M) 8 5 Applies concepts of similarity to determine the impact of scaling on the volume or surface area of three-dimensional figures when linear dimensions are multiplied by a constant factor; to determine the length of sides of similar triangles, or to solve problems involving growth and rate. (Local) Clarifying the Standards Prior Learning In grade 3, students demonstrated conceptual understanding of similarity by identifying similar shapes. In grade 4, students applied scales on maps and used characteristics of similar figures to identify and problem solve with similar figures. Grade 5 students demonstrated understanding of how scaling has a proportional effect on the dimensions of rectangles and triangles. Grade 6 students extended the concept of scaling to polygons and circles. In grade 7, they explored the concept of similarity and how scaling impacts the area of polygons and circles when the linear dimensions are multiplied by a constant factor. Current Learning Eighth-grade students apply concepts of similarity to determine the impact of scaling on surface area and volume of three-dimensional figures when linear dimensions are multiplied by a constant factor. They also apply concepts of similarity to determine the length of sides of similar triangles. These concepts are introduced, reinforced, and mastered at this grade level. Future Learning At the high school level, students will apply the concepts of similarity by solving problems within mathematics or across disciplines or contexts. Additional Research Findings Curriculum Focal Points states that students should use two-dimensional representations of threedimensional objects to visualize and solve problems such as those involving surface area and volume and understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects. (p. 37) Cumberland, Lincoln, and Woonsocket Public Schools C-55

14 Grade 8 Mathematics, Quarter 4, Unit 4.4 Similarity and Scaling (10 days) C-56 Cumberland, Lincoln, and Woonsocket Public Schools