Grade 7 Mathematics, Quarter 4, Unit 4.1. Probability. Overview

Transcription

1 Grade 7 Mathematics, Quarter 4, Unit 4.1 Probability Overview Number of instructional days: 8 (1 day = 45 minutes) Content to be learned Understand how to use counting techniques to solve problems involving combinations and permutations. Predict the theoretical probability of an event in which the sample space may or may not contain equally likely outcomes. Test predictions through experiments and simulations. Compare and contrast experimental and theoretical probability. Determine the experimental or theoretical probability of an event in a problem-solving situation. Mathematical practices to be integrated Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Construct viable arguments and critique the reasoning of others. Question and problem-pose. Justify their conclusions and communicate them to others. Model with mathematics. Make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Analyze mathematical relationships to draw conclusions. Use appropriate tools strategically. Use tools when solving a problem and deepen understanding of concepts. Make sound decisions about when each of these tools might be helpful. Essential questions What is the Fundamental Counting Principle? What is sample space? How do you find the theoretical probability of an event? Look for and make use of structure. Look for, develop, generalize, and describe a pattern orally, symbolically, graphically, and in written form. Apply and discuss properties (related to the Fundamental Counting Principle). How do you determine sample space? How do you make a prediction about the probability of an event? How can you test your prediction? Cumberland, Lincoln, and Woonsocket Public Schools C-43

2 Grade 7 Mathematics, Quarter 4, Unit 4.1 Probability (8 days) How do you find the experimental probability of an event? What are the similarities and differences between experimental and theoretical probability? C-44 Cumberland, Lincoln, and Woonsocket Public Schools

3 Probability (8 days) Grade 7 Mathematics, Quarter 4, Unit 4.1 Grade-Level Expectations Written Curriculum M(DSP) 7 4 Uses counting techniques to solve problems in context involving combinations or permutations (e.g., How many different ways can eight students place first, second, and third in a race?) using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, orsc others). (Local) M(DSP) 7 5 For a probability event in which the sample space may or may not contain equally likely outcomes, predicts the theoretical probability of an event and tests the prediction through experiments and simulations; and compares and contrasts theoretical and experimental probabilities.(local) M(DSP) 7 5 For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation. (State) Clarifying the Standards Prior Learning In grades1 3, students described probability events using terms ranging from more likely, less likely, or equally likely to certain and impossible. Students began, in grade 2, to solve problems involving combinations using a variety of strategies. In grade 3, they tested predictions through experiments and determined fairness. In grades 3 4, students used counting techniques to solve problems involving combinations and simple permutations. Fourth- and fifth-graders described the likelihood of an event as a part to whole relationship first in words, then with fractions. They also determined theoretical probability. Grade 6 introduced probability in problem-solving situations. In this grade, they used counting techniques to solve problems involving combinations or simple permutations. The Fundamental Counting Principle was introduced. Current Learning In grade 7, students use counting techniques to solve problems involving combinations or permutations using organized lists, tables, tree diagrams, models, Fundamental Counting Principle, or others (student choice). In this grade, they no longer use only simple permutations. Students continue to determine the experimental or theoretical probability of an event in a problemsolving situation in which the sample space may or may not contain equally likely outcomes. New learning involves comparing and contrasting theoretical and experimental probabilities. Future Learning The GLE for counting techniques is the same for grades In grades 11 12, students will be introduced to calculator functions and will find unions, intersections, and complements of sets. In grade 8, they will predict the theoretical probability of an event and test predictions through experiments and simulations. They will also compare and contrast theoretical and experimental probabilities. Cumberland, Lincoln, and Woonsocket Public Schools C-45

4 Grade 7 Mathematics, Quarter 4, Unit 4.1 Probability (8 days) Additional Research Findings According to Principles and Standards for School Mathematics, teachers should give middle-grade students numerous opportunities to engage in probabilistic thinking about simple situations from which students can develop notions of chance. They should use appropriate terminology in their discussions of chance and use probability to make predictions and test conjectures (pp ). C-46 Cumberland, Lincoln, and Woonsocket Public Schools

5 Probability (8 days) Grade 7 Mathematics, Quarter 4, Unit 4.1 Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-47

6 Grade 7 Mathematics, Quarter 4, Unit 4.1 Probability (8 days) C-48 Cumberland, Lincoln, and Woonsocket Public Schools

7 Grade 7 Mathematics, Quarter 4, Unit 4.2 Geometric Relationships Overview Number of instructional days: 9 (1 day = minutes) Content to be learned Develop the angle concepts of adjacent, vertical, and straight resulting from two or more intersecting lines Develop relationships formed by two nonparallel lines cut by a transversal to solve problems Develop relationships formed by two parallel lines cut by a transversal to solve problems Apply theorems and relationships to solve problems, which involve triangle inequality and the sum of the measures of interior angles of regular polygons Mathematical practices to be integrated Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Look for and make use of structure. Look for, develop, and generalize and describe a pattern orally, symbolically, graphically, and in written form. Construct viable arguments and critique the reasoning of others. Understand and use prior learning in constructing arguments. Justify conclusions, communicate them to others, and respond to the arguments of others. Use appropriate tools strategically. Use tools when solving a mathematical problem and to deepen understanding of concepts. Essential questions Why is it important to understand properties of angles and figures to solve problems? Suppose that a transversal intersects two lines that are not parallel. Do you think that the angle pairs would still be congruent? Justify your answer. How could you use the Triangle Sum Theorem to find the sum of the angles in any polygon? What are vertical, adjacent, and straight angles? Look for and express regularity in repeated reasoning. Use repeated applications to generalize properties. How can you justify that the number of triangles in a polygon is two less than the number of sides? How can you justify the congruent relationships created when two parallel lines are cut by a transversal? Is it possible for vertical angles to be adjacent angles? Describe a situation to justify your answer. Cumberland, Lincoln, and Woonsocket Public Schools C-49

8 Grade 7 Mathematics, Quarter 4, Unit 4.2 Geometric Relationships (9 days) Written Curriculum Grade-Level Expectations/Grade-Span Expectations M(G&M) 7 1 Uses properties of angle relationships resulting from two or three intersecting lines (adjacent angles, vertical angles, straight angles, or angle relationships formed by two non-parallel lines cut by a transversal), or two parallel lines cut by a transversal to solve problems. (State) M(G&M) 7 2 Applies theorems or relationships (triangle inequality or sum of the measures of interior angles of regular polygons) to solve problems. (State) Clarifying the Standards Prior Learning Starting in kindergarten, students had to sort, classify, and find polygons in the real world; these polygons needed to be described with one attribute. In grades 1 and 2, students drew polygons and named polygons with a combination of two or more attributes. In grade 3, students began to recognize and distinguish among different polygons and circles based on their angles and sides. They began to use the concepts of parallel and perpendicular in grade 4 and classified angles relative to 90 degrees. The vocabulary of right, acute, and obtuse was used in grade 5, as well as the concepts of congruency. Differentiating between the different types of triangles and different types of quadrilaterals also began in grade 5. More formal triangle vocabulary was introduced in grade 6. Current Learning In grade 7, students develop the angle concepts of adjacent, vertical, and straight resulting from two or more intersecting lines. They also develop relationships formed by two nonparallel lines cut by a transversal and/or two parallel lines cut by a transversal to solve problems. Students apply theorems and relationships to solve problems that involve triangle inequality and the sum of the measures of interior angles of regular polygons. Future Learning In grade 8, students will learn the Pythagorean Theorem to find a missing side of a right triangle or in problem-solving situations. In high school geometry, students will create formal proofs of various geometric properties and theorems, including those that involve angles, lines, polygons, and circles, as well as sine, cosine, tangent, the Triangle Inequality Theorem, and the Pythagorean Theorem. Additional Research Findings From Curriculum Focal Points, students are expected to understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects (p. 37). Students should come to the study of geometry in the middle grades with informal knowledge about points, lines, planes, and a variety of two- and three-dimensional shapes; with experience in visualizing and drawing lines, angles, triangles, and other polygons; and with intuitive notions about shapes built from years of interacting with objects in their daily lives. (Principles and Standards for School Mathematics, p. 233) C-50 Cumberland, Lincoln, and Woonsocket Public Schools

9 Geometric Relationships (9 days) Grade 7 Mathematics, Quarter 4, Unit 4.2 Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-51

10 Grade 7 Mathematics, Quarter 4, Unit 4.2 Geometric Relationships (9 days) C-52 Cumberland, Lincoln, and Woonsocket Public Schools

11 Grade 7 Mathematics, Quarter 4, Unit 4.3 Similarity and Congruency Overview Number of instructional days: 5 (1 day = 45 minutes) Content to be learned Apply congruency by solving problems on a coordinate plane involving reflections, translations, or rotations. Understand the concept of congruency. Understand the concept of similarity involving scaling up or down. Understand scaling and its impact on angle measures, linear dimensions, and areas. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Identify and execute appropriate strategies to solve the problem. Construct viable arguments and critique the reasoning of others. Understand and use prior learning in constructing arguments. Model with mathematics. Analyze mathematical relationships to draw conclusions. Look for and express regularity in repeated reasoning. Use repeated applications to generalize properties. Essential questions How do you determine and identify which sides and angles of a figure correspond? What must be true about the corresponding angles and the corresponding sides for two polygons to be similar? How do you determine whether two polygons are congruent? What effect does scale factor have on area? What effect does scale factor have on linear measures and perimeter? What effect does scale factor have on angle measures? Cumberland, Lincoln, and Woonsocket Public Schools C-53

12 Grade 7 Mathematics, Quarter 4, Unit 4.3 Similarity and Congruency (5 days) Written Curriculum Grade-Level Expectations M(G&M) 7 4 Applies the concepts of congruency by solving problems on a coordinate plane involving reflections, translations, or rotations. (State) M(G&M) 7 5 Applies concepts of similarity by solving problems involving scaling up or down and their impact on angle measures, linear dimensions and areas of polygons, and circles when the linear dimensions are multiplied by a constant factor. Describes effects using models or sc explanations. (State) 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, 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 extended the concept of scaling to polygons and circles. Current Learning In grade 7, students understand and apply concepts of congruency by solving problems on a coordinate plane involving reflections, translations, and rotations. They understand and apply concepts of similarity by scaling up and down and discovering the impact on linear dimensions, angle measures, and areas of polygons and circles when dimensions are multiplied by a constant scale factor. These concepts are introduced at this grade level and will be state-tested. Future Learning Eighth graders will determine the impact of scaling on surface area and volume. They will continue to determine the length of sides of similar triangles. At the high school level, students will apply the concepts of similarity by solving problems in mathematics or across disciplines or contexts. Additional Research Findings In Principles and Standards for School Mathematics, investigations into properties of and relationships among similar shapes can afford students many opportunities to develop and evaluate conjectures inductively and deductively (p. 234). C-54 Cumberland, Lincoln, and Woonsocket Public Schools

13 Similarity and Congruency (5 days) Grade 7 Mathematics, Quarter 4, Unit 4.3 Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-55

14 Grade 7 Mathematics, Quarter 4, Unit 4.3 Similarity and Congruency (5 days) C-56 Cumberland, Lincoln, and Woonsocket Public Schools

15 Grade 7 Mathematics, Quarter 4, Unit 4.4 Measurement Overview Number of instructional days: 13 (1 day = minutes) Content to be learned Demonstrate conceptual understanding of the area of circles or the area or perimeter of composite figures (quadrilaterals, triangles, or parts of circles). Demonstrate conceptual understanding of the surface area of rectangular prisms. Demonstrate conceptual understanding of the volume of rectangular prisms, triangular prisms, or cylinders using models and formulas or by solving related problems. Mathematical practices to be integrated Model with mathematics. Identify important quantities in a practical stituation and map their relationships using diagrams, tables, and formulas. Use appropriate tools strategically. Use tools when solving a mathematical problem and to deepen their understanding of concepts (e.g., geometric construction, measurement devices, and graph paper). Look for and express regularity in repeated reasoning. Use repeated applications to generalize properties. Attend to precision. Specify units of measure. Strive for accuracy. Essential questions How can surface area and volume be used to find answers to real-world problems? Why is it important to be able to relate twodimensional drawings with three-dimensional figures? How can you demonstrate a situation where a composite figure must be broken down in order to find area or perimeter? Cumberland, Lincoln, and Woonsocket Public Schools C-57

16 Grade 7 Mathematics, Quarter 4, Unit 4.4 Measurement (13 days) Written Curriculum Grade-Level Expectations/Grade-Span Expectations M(G&M) 7 6 Demonstrates conceptual understanding of the area of circles or the area or perimeter of composite figures quadrilaterals, triangles,or parts of circles), and the surface area of rectangular prisms, or volume of rectangular prisms, triangular prisms, or cylinders using models, formulas, or by solving related problems. Expresses all measures using appropriate units. (State) Clarifying the Standards Prior Learning In grades 1 3, students demonstrated conceptual understanding of area and perimeter using a variety of models and manipulatives. By the end of grade 3, they expressed all measures in appropriate units. In grade 4, students found the area and perimeter of polygons and irregular shapes. Formulas were introduced. In grade 5, the concept of right triangles was introduced, along with the volume of rectangular prisms with the use of cubes. In grade 6, students used the developed geometry concepts to solve problems and were introduced to circle measures. Current Learning In grade 7, students learn how to find area of circles, the area or perimeter of composite figures (composed of quadrilaterals, triangles, or parts of circles), surface area of rectangular prisms, and volume of rectangular prisms, triangular prisms, or cylinders. Finding basic two-dimensional measures is mastered at this point. The three-dimensional topics are continued from earlier grades and reinforced and built upon in current and future problem-solving situations. Future Learning In grade 8, students will solve problems related to surface area and volume. They will continue to work with the surface area and volume of rectangular prisms and triangular prisms. The surface area of cylinders will be introduced, as well as surface area and volume of pyramids or cones. In high school, students will apply these geometric concepts within math courses and other disciplines. In grades 11 and 12, trigonometric concepts will be introduced. Additional Research Findings Students should come to the study of geometry in the middle grades with informal knowledge about points, lines, planes, and a variety of two- and three-dimensional shapes; with experience in visualizing and drawing lines, angles, triangles, and other polygons; and with intuitive notions about shapes built from years of interacting with objects in their daily lives. In middle grades geometry programs based on these recommendations, students investigate relationships by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects. Many topics treated in the Measurement Standard for the middle grades are closely connected to students study of geometry. (Principals and Standards for School Mathematics, p. 233) C-58 Cumberland, Lincoln, and Woonsocket Public Schools

17 Measurement (13 days) Grade 7 Mathematics, Quarter 4, Unit 4.4 Students skills in visualizing and reasoning about spatial relationships are fundamental in geometry. Some students may have difficulty finding the surface area of three-dimensional shapes using twodimensional representations because they cannot visualize the unseen faces of the shapes. (Principals and Standards for School Mathematics, p. 237) Cumberland, Lincoln, and Woonsocket Public Schools C-59

18 Grade 7 Mathematics, Quarter 4, Unit 4.4 Measurement (13 days) Notes About Resources and Materials C-60 Cumberland, Lincoln, and Woonsocket Public Schools

19 Grade 7 Mathematics, Quarter 4, Unit 4.5 Spatial Reasoning Overview Number of instructional days: 5 (1 day = 45 minutes) Content to be learned Demonstrate conceptual understanding of spatial reasoning and visualization by sketching three-dimensional solids. Draw nets of rectangular and triangular prisms, cylinders, and pyramids and use the nets as a technique for finding surface area. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Consider similar problems to gain insight into its solution. Check answers to problems using a different method. Monitor and evaluate progress. Reason abstractly and quantitatively. Make sense of quantities in problem situations. Attend to meaning of quantities. Consider units involved. Know and flexibly use different properties of operations and objects. Construct viable arguments and critique the reasoning of others. Give examples and nonexamples. Model with mathematics. Apply math they know to solve problems. Identify important quantities. Interpret math results. Use appropriate tools strategically. Consider/use appropriate/available tools and recognize limitations. Use tools to solve, explore, compare, and visualize problems and to deepen knowledge/understanding. Attend to precision. Specify units of measure and label figures appropriately. Strive for accuracy. Cumberland, Lincoln, and Woonsocket Public Schools C-61

20 Grade 7 Mathematics, Quarter 4, Unit 4.5 Spatial Reasoning (5 days) Essential questions How can you convert a three-dimensional figure to a two-dimensional figure? Draw an example to support your answer. How can you use the net of a figure to find the surface area? C-62 Cumberland, Lincoln, and Woonsocket Public Schools

21 Spatial Reasoning (5 days) Grade 7 Mathematics, Quarter 4, Unit 4.5 Grade-Level Expectations Written Curriculum M(G&M) 7 10 Demonstrates conceptual understanding of spatial reasoning and visualization by sketching three-dimensional solids; and draws nets of rectangular and triangular prisms, cylinders, and pyramids and uses the nets as a technique for finding surface area. (Local) Clarifying the Standards Prior Learning In grades 3 and 4, students copied and drew two-dimensional shapes and built rectangular prisms from three-dimensional models. In grade 5, students expanded three-dimensional representations to include triangular prisms, cones, cylinders, and pyramids. Current Learning In grade 7, students demonstrate understanding of spatial reasoning by sketching three-dimensional solids and by drawing nets of rectangular and triangular prisms, cylinders, and pyramids, which they use as a technique for finding surface area. This concept is new and will not be state-tested. Future Learning In grades 9 and 10, dynamic geometric software will be used to generate three-dimensional objects from two-dimensional perspectives or to generate two-dimensional perspectives from three-dimensional objects. In grades 11 and 12, students will perform and justify constructions with compass and straightedge or dynamic geometric software. Additional Research Findings According to Principles and Standards for School Mathematics, students in the middle grades should also develop formulas for the surface areas and volumes of three-dimensional objects. Teachers can help students develop formulas for the volumes of prisms, pyramids and cylinders, and for the surface areas of right prisms and cylinders by having them construct models, measure the dimensions, estimate the areas and volumes, and look for patterns and relationships related to lengths, areas, and volumes. In their work with three-dimensional objects, students can make use of what they know about two-dimensional shapes. For example, they can relate the surface area of a three-dimensional object to the area of its twodimensional net (p. 245). Cumberland, Lincoln, and Woonsocket Public Schools C-63

22 Grade 7 Mathematics, Quarter 4, Unit 4.5 Spatial Reasoning (5 days) Notes About Resources and Materials C-64 Cumberland, Lincoln, and Woonsocket Public Schools