Identifying and Classifying Angles and Shapes

Transcription

1 Grade 5 Mathematics, Quarter 2, Unit 2.1 Identifying and Classifying Angles and Shapes Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Describe, compare, and classify attributes of right, acute, obtuse, equiangular, and equilateral angles. Describe, compare, and classify the properties of quadrilaterals rectangles, squares, rhombi, trapezoids, and parallelograms. Identify and describe parallel, nonparallel, and perpendicular line segments. Identify and describe similar and congruent figures. Uses properties of angles and/or sides to identify, describe, classify, or distinguish among different types of triangles and quadrilaterals. Measure and use benchmark units of measures appropriately. Mathematical practices to be integrated Use appropriate tools strategically. Make sound decisions about when tools are appropriate and helpful. When selecting tools recognize strengths and limitations. Use tools to deepen understanding of mathematical concepts. Attend to precision. Label for clarification. Communicate precisely to others using clear definitions. Construct viable arguments and critique the reasoning of others. Understand and use prior learning in constructing arguments. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed. Essential questions How would you classify these quadrilaterals according to their lines, sides, and angles? (rectangles, squares, rhombi, trapezoids, parallelograms) What are the attributes of a given triangle? How can objects be represented and compared using geometric attributes? How do you decide which measurement to use? How would you determine if two lines are parallel, non-parallel, or perpendicular? What tools and units are used to measure angles? What is the difference between right, obtuse, and acute angles? What are the differences between congruent and similar figures? What strategies would you use to make a conversion within a measurement system? Cumberland, Lincoln, and Woonsocket Public Schools C-17

2 Grade 5 Mathematics, Quarter 2, Unit 2.1 Identifying and Classifying Angles and Shapes (10 days) Written Curriculum Grade-Level Expectations M(G&M) 5 1 Uses properties or attributes of angles (right, acute, or obtuse) or sides (number of congruent sides, parallelism, or perpendicularity) to identify, describe, classify, or distinguish among different types of triangles (right, acute, obtuse, equiangular, or equilateral) or quadrilaterals (rectangles, squares, rhombi, trapezoids, or parallelograms). (State) M(G&M) 5 7 Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands. (State) See Benchmarks in Appendix B. Appendix B: Measures Grade 5 Length Clarifying the Standards Prior Learning Units (accuracy): Inch (to 1/8 inch); Foot; Centimeter (to 0.5 centimeter); Meter (to 0.5 centimeter); Yard; Mile (use in scale questions); Kilometer (use in scale questions) Equivalencies: 12 inches in 1 foot; 100 centimeters in 1 meter; 3 feet in 1 yard; 36 inches in 1 yard; 10 millimeters in 1 centimeter Grade 4 students used number of angles or sides to identify a shape. They were further able to identify parallelism or perpendicularity. Fourth-grade students identified, described, or distinguished among triangles, squares, rectangles, rhombi, trapezoids, hexagons, and octagons, and classified angles relative to 90 degrees as greater than, less than, and equal to. Students also measured and used units of measure to 1/4 in. and 1/2 cm. They used mile and kilometer in scale questions. They were further able to understand equivalencies of 12 to 1, 100 cm to 1 m, 3 to 1 yd., and 36 to 1 yard. Current Learning Grade 5 students use properties or attributes of right, acute, or obtuse angles or sides to identify, describe, classify, or distinguish among different types of triangles or quadrilaterals. Their ability to measure is to 1/8, and they add the equivalency of 100 cm to 1 meter. Angles and rotations are accurate to 2 degrees. The content is taught at the developmental and reinforcement levels. These concepts are found within the knowledge, comprehension, and analysis level of Bloom s taxonomy. Cumberland, Lincoln, and Woonsocket Public Schools C-18

3 Grade 5 Mathematics, Quarter 2, Unit 2.1 Identifying and Classifying Angles and Shapes (10 days) Future Learning In sixth grade, students will be able to identify, describe, classify, and distinguish scalene and isosceles triangles millimeters in 1 meter will be new equivalencies, as well as 360 degrees equals one circle and 90 degrees in one right angle. Units of measurement and accuracy understanding will be expanded to include 1/16, 1/10 cm, and 1/100 m. Additional Research Findings According to Principles and Standards for School Mathematics, students should develop more precise ways to describe shapes, focusing on identifying and describing the shapes and properties, and learning specialized vocabulary associated with these shapes and properties. To consolidate their ideas, students should draw and construct shapes, classify them, compare and discuss their attributes, and develop and consider definitions on the basis of a shape s properties. The book also states that the reasoning skills that students develop in grades 3 5 allow them to investigate geometric problems of increasing complexity and to study geometric properties. As they move from grade 3 to grade 5, they should develop clarity and precision in describing the properties of geometric objects and classify them by these properties into categories. They develop knowledge about how geometric shapes are related to one another and begin to articulate geometric arguments about the properties of these shapes (p. 165). Notes About Resources and Materials Resources will vary by district (i.e., Everyday Mathematics, Investigations, Pearson Scott Foresman, etc.). Suggested materials: Games Hands-on manipulatives Released test items from NECAP Cumberland, Lincoln, and Woonsocket Public Schools C-19

4 Grade 5 Mathematics, Quarter 2, Unit 2.1 Identifying and Classifying Angles and Shapes (10 days) Cumberland, Lincoln, and Woonsocket Public Schools C-20

5 Grade 5 Mathematics, Quarter 2, Unit 2.2 Measuring and Area/Perimeter Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Find the perimeter and area of polygons (rectangles and right triangles) through models, manipulatives, or formulas. Find the perimeter and area of polygons and irregular figures on grids using a variety of models, manipulatives, or formulas. Express the perimeter and area of polygons using appropriate units. Measure and use benchmark units of measure appropriately and consistently (see Appendix B). Convert metric/customary units within systems when solving problems. Mathematical practices to be integrated Model with mathematics. Solve problems in everyday life through the use of geometry. Identify important quantities and their relationships. Use appropriate tools strategically. Make sound decisions about when tools are helpful when solving area and perimeter problems. Recognize strengths and limitations when selecting tools. Using estimations to judge reasonableness of solutions. Look for and express regularity in repeated reasoning. Look for patterns and repeated calculations that lead to shortcuts. Essential questions How would you measure the perimeter of a figure? How would you measure the area of a figure? What tools and units are used to measure the area and perimeter of a figure? How can you use a rectangle to find the area of a right triangle? How would you find the area and perimeter of an irregular figure on a grid? How do you determine equivalent metric and customary units of measure for the area and perimeter of a figure? How does the model relate to the formula? Cumberland, Lincoln, and Woonsocket Public Schools C-21

6 Grade 5 Mathematics, Quarter 2, Unit 2.2 Measuring and Area/Perimeter (10 days) Written Curriculum Grade-Level Expectations M(G&M) 5 6 Demonstrates conceptual understanding of perimeter of polygons, and the area of rectangles or right triangles through models, manipulatives, or formulas, the area of polygons or irregular figures on grids, and volume of rectangular prisms (cubes) using a variety of models, manipulatives, or formulas. Expresses all measures using appropriate units. (State) M(G&M) 5 7 Measures and uses units of measures appropriately and consistently, and makes conversions within systems when solving problems across the content strands. (State) See Benchmarks in Appendix B. Appendix B: Measures Grade 5 Length Units (accuracy): Inch (to 1/8 inch); Foot; Centimeter (to 0.5 centimeter); Meter (to 0.5 centimeter); Yard; Mile (use in scale questions); Kilometer (use in scale questions) Equivalencies: Clarifying the Standards Prior Learning 12 inches in 1 foot; 100 centimeters in 1 meter; 3 feet in 1 yard; 36 inches in 1 yard; 10 millimeters in 1 centimeter Fourth-graders gained an understanding of perimeter of polygons, and the area of rectangles, polygons, or irregular shapes on grids. They used a variety of models, manipulatives, or formulas. Fourth-graders expressed all measures in appropriate units. They measured and used units of measure to 1/4 and 1/2 cm. They used mile and kilometer in scale questions, and they understood equivalencies of 12 to 1 ; 100 cm to 1 m, 3 to 1 yd., and 36 to 1 yard. Current Learning Fifth-graders apply their prior knowledge of perimeter and area to right triangles. Later in the quarter, students demonstrate an understanding of the volume of rectangular prisms. The content is taught at the developmental and reinforcement levels. Students learn to connect models and manipulatives to formulas for perimeter and area. These concepts are found within the application and comprehension level of Bloom s taxonomy. Cumberland, Lincoln, and Woonsocket Public Schools C-22

7 Grade 5 Mathematics, Quarter 2, Unit 2.2 Measuring and Area/Perimeter (10 days) Future Learning Sixth-graders will demonstrate understanding of perimeter of quadrilaterals or triangles and the volume of rectangular prisms. They will demonstrate understanding of the relationships among circle measures. Units of measurement and accuracy understanding are expanded to include 1/16, 1/10 cm, and 1/100 m. Miles and kilometers are used in scale and rate questions. Equivalency understanding expands to include 1000 mm in 1m and 360 degrees in one circle as well as 90 degrees in one right triangle. Additional Research Findings According to Principles and Standards for School Mathematics: Many elementary and middle-grade students have difficulty understanding perimeter and area. Often, these children are using formulas such as P = 2L + 2W or A = L W without understanding how these formulas relate to the attribute being measured or the unit of measurement being used. Teachers must help students see the connections between the formula and the actual object (p. 46). The notion that shapes that look different can have equal areas is a powerful one that leads eventually to the development of general methods (formulas) for finding the area of a particular shape, such as a parallelogram. In this investigation, students are building their ideas about the properties of classes of shapes, formulating conjectures about geometric relationships, exploring how geometry and measurement are related, and investigating the shapes with equal area (p. 166). In grades 3 5, students should deepen and expand their understanding and use of measurement. For example, they should measure other attributes such as area and angle. They need to begin paying closer attention to the degree of accuracy when measuring and use a wider variety of measurement tools. They should also begin to develop and use formulas for the measurement of certain attributes, such as area (p. 171). In these grades, more emphasis should be placed on the standard units that are used to communicate in the United States (the customary units) and around the world (metric system). Students should become familiar with the common units in these systems and establish mental images or benchmarks for judging and comparing size. Students should gain facility in expressing measurements in equivalent forms (p. 172). Notes About Resources and Materials Resources will vary by district (i.e., Everyday Mathematics, Investigations, Pearson Scott Foresman, etc.). Suggested materials: Games Hands-on manipulatives Released test items from NECAP Cumberland, Lincoln, and Woonsocket Public Schools C-23

8 Grade 5 Mathematics, Quarter 2, Unit 2.2 Measuring and Area/Perimeter (10 days) Cumberland, Lincoln, and Woonsocket Public Schools C-24

9 Grade 5 Mathematics, Quarter 2, Unit 2.3 Ordered Pairs Overview Number of instructional days: 4 (1 day = 45 minutes) Content to be learned Identify and plot ordered pairs on a one- and four-quadrant coordinate grid. Use ordered number pairs to name any point in four quadrants. Determine the horizontal and vertical distances between points on a coordinate grid in the first quadrant. Essential questions How do you plot ordered pairs on a onequadrant coordinate grid? How would you know in which of the four quadrants a point should be plotted? Mathematical practices to be integrated Reason abstractly and quantitatively. Consider units involved. Make meaning of numbers and symbols. Create a coherent representation of the problem. How do you find the horizontal distance between two points on a coordinate grid? How do you find the vertical distance between tow points on a coordinate grid? Cumberland, Lincoln, and Woonsocket Public Schools C-25

10 Grade 5 Mathematics, Quarter 2, Unit 2.3 Ordered Pairs (4 days) Written Curriculum Grade-Level Expectations M(G&M) 5 9 Demonstrates understanding of spatial relationships using location and position by interpreting and giving directions between locations on a map or coordinate grid (all four quadrants); plotting points in four quadrants in context (e.g., games, mapping, identifying the vertices of polygons as they are reflected, rotated, and translated); and determining horizontal and vertical distances between points on a coordinate grid in the first quadrant. (Local) Clarifying the Standards Prior Learning Fourth-graders were able to interpret and give directions between locations on a map or coordinate grid in the first quadrant only. They plotted points in the first quadrant in context. Fourth-graders also found the horizontal and vertical distances between points on a coordinate grid in the first quadrant. Current Learning Fifth-graders utilize all four quadrants to plot points in context. They are able to identify the vertices of polygons being reflected, rotated, and translated on quadrant grids. The content is taught at the comprehension and application levels. These concepts are found within the reinforcement level of Bloom s taxonomy. Future Learning There is no new GLE beyond fifth grade. Everything from this point will be at a reinforcement level. Additional Research Findings According to Principles and Standards for School Mathematics: Students at this level should learn how to use two numbers to name points on a coordinate grid and should realize that a pair of numbers corresponds to a particular point on a grid. Using coordinates, they can specify paths between locations and examine the symmetry, congruence, and similarity of shapes drawn on a grid. They can also explore methods for measuring the distance between locations on the grid. As students ideas about the number system expand to include negative numbers, they can work in all fours quadrants on the Cartesian Plane (p. 167). Cumberland, Lincoln, and Woonsocket Public Schools C-26

11 Grade 5 Mathematics, Quarter 2, Unit 2.3 Ordered Pairs (4 days) Notes About Resources and Materials Resources will vary by district (i.e., Everyday Mathematics, Investigations, Pearson Scott Foresman, etc.). Suggested materials: Games (Battleship) Hands-on manipulatives Released test items from NECAP Website: Cumberland, Lincoln, and Woonsocket Public Schools C-27

12 Grade 5 Mathematics, Quarter 2, Unit 2.3 Ordered Pairs (4 days) Cumberland, Lincoln, and Woonsocket Public Schools C-28

13 Grade 5 Mathematics, Quarter 2, Unit 2.4 Properties, Attributes, and Visualizing 3-D Figures Overview Number of instructional days: 11 (1 day = 45 minutes) Content to be learned Identify, compare, and describe 3-dimensional shapes: rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones. Use the attributes, shapes or properties of bases, number of lateral faces, or number of bases to identify, compare, and describe 3-D shapes. Build models such as rectangular and triangular prisms, cones, cylinders, and pyramids from 2- D and 3-D representations. Essential questions How can you identify solid (3-D) figures by using attributes and properties? How can you distinguish among spheres, cylinders, and cones? What are the similarities and differences between pyramids and prisms? Mathematical practices to be integrated Model with mathematics. Analyze relationships to draw conclusions. Make assumptions and approximations to simplify a complicated situation, realizing these may need revision later. Construct viable arguments and critique the reasoning of others. Justify conclusions, communicate them to others, and respond to arguments of others. Develop questioning strategies to generate information. How can you construct a corresponding 3-D shape given a 2-D representation? How are the names and shapes of some 3-D figures related? (For example, a square pyramid has a square base.) Cumberland, Lincoln, and Woonsocket Public Schools C-29

14 Grade 5 Mathematics, Quarter 2, Unit 2.4 Properties, Attributes, and Visualizing 3-D Figures (11 days) Written Curriculum Grade-Level Expectations M(G&M) 5 3 Uses properties or attributes (shape of bases, number of lateral faces, or number of bases) to identify, compare, or describe three-dimensional shapes (rectangular prisms, triangular prisms, cylinders, spheres, pyramids, or cones). (State) M(G&M) 5 10 Demonstrates conceptual understanding of spatial reasoning and visualization by building models of rectangular and triangular prisms, cones, cylinders, and pyramids from two- or threedimensional representations. (Local) Clarifying the Standards Prior Learning Fourth-grade students used properties or attributes (bases or number of lateral faces) to identify, compare, or describe 3-D shapes, rectangular prisms, triangular prisms, cylinders, or spheres. They demonstrated conceptual understanding of spatial reasoning and visualization related to triangles, squares, rectangles, circles, rhombi, trapezoids, hexagons, and octagons. Fourth-graders also built models of rectangular prisms from 2-D or 3-D representations. Current Learning Fifth-grade students address the number of bases in 3-D figures and are able to identify and use properties or attributes related to pyramids and cones. The content is taught at the reinforcement and developmental levels. Content addressed in M(G&M) 5 10 are at the reinforcement level. These concepts are found within the knowledge, analysis, comprehension, and application level of Bloom s taxonomy. Future Learning Sixth-graders will use the number of edges and vertices to identify, describe, classify, or distinguish among different types of triangles and 3-D shapes. Additional Research Findings According to Principles and Standards for School Mathematics, Students in grades 3 5 should examine the properties of 2-D and 3-D shapes and the relationships among shapes. They should be encouraged to reason about these properties by using spatial relationships. In additions to studying physical models of these geometric shapes, they should also develop and use mental images. Students at this age are ready to mentally manipulate shapes. They can benefit from experiences that challenge them and that can also be verified physically. Much of the work students do with 3-D shapes involves visualization. By representing 3-D shapes in two dimensions and constructing 3-D shapes from 2-D representations, students learn about the characteristics of shapes. Students should become experienced in using a variety of representations for 3-D shapes (p. 168). Cumberland, Lincoln, and Woonsocket Public Schools C-30

15 Grade 5 Mathematics, Quarter 2, Unit 2.4 Properties, Attributes, and Visualizing 3-D Figures (11 days) Notes About Resources and Materials Resources will vary by district (i.e., Everyday Mathematics, Investigations, Pearson Scott Foresman, etc.). Suggested materials: Games Hands-on manipulatives Released test items from NECAP Cumberland, Lincoln, and Woonsocket Public Schools C-31

16 Grade 5 Mathematics, Quarter 2, Unit 2.4 Properties, Attributes, and Visualizing 3-D Figures (11 days) Cumberland, Lincoln, and Woonsocket Public Schools C-32

17 Grade 5 Mathematics, Quarter 2, Unit 2.5 Measuring Volume of Rectangular Prisms Overview Number of instructional days: 5 (1 day = 45 minutes) Content to be learned Determine the volume of a rectangular prism using pictures, words, manipulatives, or formulas. Express the volume of a rectangular prism using appropriate units. Mathematical practices to be integrated Model with mathematics. Identify important quantities and their relationships. Draw conclusions, interpret results, revise models if needed. Reason abstractly and quantitatively. Consider units involved when problem solving. Attend to the meaning of quantities. Essential questions How could you find the volume of a rectangular figure given a 3-D model? Using manipulatives, how would you construct a rectangular prism with a given volume? Why do we use cubic units for measuring volume? How would you use the length, width, and height of a rectangular prism to determine its volume? Cumberland, Lincoln, and Woonsocket Public Schools C-33

18 Grade 5 Mathematics, Quarter 2, Unit 2.5 Measuring Volume of Rectangular Prisms (5 days) Written Curriculum Grade-Level Expectations M(G&M) 5 6 Demonstrates conceptual understanding of perimeter of polygons, and the area of rectangles or right triangles through models, manipulatives, or formulas, the area of polygons or irregular figures on grids, and volume of rectangular prisms (cubes) using a variety of models, manipulatives, or formulas. Expresses all measures using appropriate units. (State) Clarifying the Standards Prior Learning Fourth-grade students understood perimeter of polygons and the area of rectangles, polygons, or irregular shapes on grids using a variety of models, manipulative, or formulas. They expressed all measures using appropriate units. Current Learning Fifth-graders understand volume of rectangular prisms (cubes). This is new learning, so it is taught at the developmental level. Students are tested with pictorial representations of volume, therefore it is essential that teachers help students connect 3-D models to pictorial representations. These concepts are found within the application and synthesis level of Bloom s taxonomy because students are gaining new knowledge and applying it to problems and prior experiences. Future Learning Sixth-graders will come to understand the volume of rectangular prisms by solving problems. They will understand relationships of circle measurement, radius to diameter, and diameter to circumference, by solving problems related to circle measurement. Additional Research Findings According to Principles and Standards for School Mathematics: As students have opportunities to look for patterns in the results of their measurements, they recognize that their methods for measuring the area and volume of particular objects can be generalized as formulas. Students in grades 3 5 should develop strategies for determining surface area and volume on the basis of concrete experiences. They should measure various rectangular solids (using objects such as solids and cubes), organize the information, look for patterns, and then make generalizations. These concrete experiences are essential in helping students understand the relationship between the measurement of an object and the succinct formula that produces the measurement (p. 174). Cumberland, Lincoln, and Woonsocket Public Schools C-34

19 Grade 5 Mathematics, Quarter 2, Unit 2.5 Measuring Volume of Rectangular Prisms (5 days) Notes About Resources and Materials Resources will vary by district (i.e., Everyday Mathematics, Investigations, Pearson Scott Foresman, etc.). Suggested materials: Games Hands-on manipulatives Released test items from NECAP Cumberland, Lincoln, and Woonsocket Public Schools C-35

20 Grade 5 Mathematics, Quarter 2, Unit 2.5 Measuring Volume of Rectangular Prisms (5 days) Cumberland, Lincoln, and Woonsocket Public Schools C-36