# Polygons and Area. Overview. Grade 6 Mathematics, Quarter 4, Unit 4.1. Number of instructional days: 12 (1 day = minutes)

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1 Grade 6 Mathematics, Quarter 4, Unit 4.1 Polygons and Area Overview Number of instructional days: 12 (1 day = minutes) Content to be learned Calculate the area of polygons by composing into rectangles and decomposing into triangles. Develop and justify formulas for finding the area of triangles, special quadrilaterals, and other polygons. Draw polygons in the coordinate plane given the coordinates. Find side lengths of polygons on a coordinate plane using the corresponding (x, y) coordinates. Apply these techniques of composing and decomposing to solve real-world and mathematical problems involving area of polygons. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Compose and decompose figures using manipulatives and drawings. Recognize the relationship of triangles and special quadrilaterals to rectangles. Use knowledge of the area of rectangles to find the area of other figures. Model with mathematics. Apply mathematics to everyday situations. Analyze relationship to draw conclusions about area. Use appropriate tools strategically. Use the grid on a coordinate plane to find side lengths. Use rulers and other measuring tools to draw and measure figures on a coordinate plane. Use math knowledge to estimate area. Attend to precision. Use specific units of measure and label appropriately. Use math vocabulary to communicate mathematical thinking effectively. 37

2 Grade 6 Mathematics, Quarter 4, Unit 4.1 Polygons and Area (12 days) Essential questions What is area? Where in the real world would you use the concept of area? How can it be applied and solved? How can you find the area of triangles, special quadrilaterals, and polygons? What does it mean to compose and decompose a figure to create other polygons? How do you find the side length of a figure on a coordinate plane? What are the methods for finding the area of triangles and rectangles? How are they related? How can you use the area of triangles and rectangles to find the area of other polygons? Written Curriculum Common Core State Standards for Mathematical Content Geometry 6.G Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.1 6.G.3 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 38

4 Grade 6 Mathematics, Quarter 4, Unit 4.1 Polygons and Area (12 days) Current Learning Students compose and decompose figures to find and relate area to rectangles. They apply these techniques to solve real-world and mathematical problems. Students also use the coordinate plane to find the side lengths of figures using the vertices. This is a supporting cluster according to PARCC. (p. 30) This is a critical area according to the CCSS. Future Learning In grade 7, students will find the area, volume, and surface area of two- and three-dimensional figures composed of triangles, quadrilaterals and polygons, cubes, and right prisms to solve real-world and mathematical problems. Additional Findings According to the Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, problems involving area and volumes, calling on students to find areas or volumes from lengths or to find lengths from volumes or area and lengths, are especially appropriate. These problems extend the students work in grade 5 on area and volume and provide a context for applying new work with equations. (p. 36) According to Principles and Standards for School Mathematics, students should be able to precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties. (p. 232) 40

5 Grade 6 Mathematics, Quarter 4, Unit 4.2 Surface Area and Volume Overview Number of instructional days: 14 (1 day = minutes) Content to be learned Find the volume of a right rectangular prism with fractional lengths by packing it with unit cubes. Understand that equivalent volumes can be found either when you pack it with unit cubes or use the formula. Apply the formula V = lwh and V = Bh to find volumes of right rectangular prisms with fractional side lengths in real-world and mathematical problems. Use nets made up of rectangles and triangles to represent three-dimensional figures to find surface area. Find the surface area of prisms and pyramids by decomposing them into pieces whose area can be determined. (This is a critical area of the CCSS. [p. 40]) Essential questions What is the difference between volume and surface area? Where in the real world would you use the concept of volume? How can it be applied and solved? How can a three-dimensional figure be made into a net? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Analyze three-dimensional figures to find volume and surface area. Model with mathematics. Apply the volume formula to real-world and mathematical problems. Use and apply knowledge of area of twodimensional figures to surface area. Use manipulatives such as cubes and nets to calculate the volume and surface area. Attend to precision. Use specific units of measure and label appropriately. Use math vocabulary to communicate mathematical thinking effectively. Accurately compute volume and surface area. How can a net be used to find surface area? How does area relate to surface area? How can volume be found? 41

6 Grade 6 Mathematics, Quarter 4, Unit 4.2 Surface Area and Volume (14 days) Written Curriculum Common Core State Standards for Mathematical Content Geometry 6.G Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.2 6.G.4 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of 42

8 Grade 6 Mathematics, Quarter 4, Unit 4.2 Surface Area and Volume (14 days) 44

9 Grade 6 Mathematics, Quarter 4, Unit 4.3 Statistical Variability and Graphing Overview Number of instructional days: 10 (1 day = minutes) Content to be learned Understand that a statistical question anticipates variability. Understand that data collected can be described by its distribution at the center, spread, and overall shape. Understand measure of center (mean and median) and find the interquartile range of a set of data on a graph to describe a pattern for a deviation from the pattern. Organize, display, and interpret data on a number line, dotplot, histogram and boxplot. Analyze sets of data, describe the attributes, and summarize patterns and trends. Mathematical practices to be integrated Make sense of problems and persevere in solving them. Give examples and nonexamples of statistical situations Explain relationships between graphs and tables and search for trends. Construct viable arguments and critique the reasoning of others. Make inferences about data distributions using evidence from graphs. Determine relevance of the graph to the real world. Use appropriate tools strategically. Determine which graph should be used to display data in a given situation. Essential questions What is a statistical question? How can you use measures of center, spread, and overall shape to describe a set of collected numerical data? What are the measures of center? How are they significant to a set of data? What is statistical variability? How is it used to describe a set of data? Look for and make use of structure. Find patterns in graphs and what observations and conclusions can be made. Use data to highlight distributions of data, including outliers, clusters, and gaps. When is it appropriate to use dotplots, histograms, and boxplots? How is data displayed in various graphs? Given a set of data, how can the data be displayed and plotted appropriately? 45

10 Grade 6 Mathematics, Quarter 4, Unit 4.3 Statistical Variability and Graphing (10 days) Written Curriculum Common Core State Standards for Mathematical Content Statistics and Probability 6.SP Develop understanding of statistical variability. 6.SP.1 6.SP.2 6.SP.3 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, How old am I? is not a statistical question, but How old are the students in my school? is a statistical question because one anticipates variability in students ages. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Summarize and describe distributions. 6.SP.4 6.SP.5 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Summarize numerical data sets in relation to their context, such as by: a. Reporting the number of observations. b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and 46

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