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Unit Title: Filling and Wrapping Notes: Big Ideas Contents Skills Assessments Lessons Anchors Surface Area and Volume of Rectangular Prisms, Cylinders, INVESTIGATION 1: Cones, and Spheres Building Boxes Visualize a net as a representation of the Nets, Surface Area surface area of a Problem 1.1: Making (Cubes) cube. Cubic Boxes M.8.B.2.2;M.8.C.1.1.1 Connect the area of Nets, Surface Area the net to the surface (Cubes) area of a cube. M.8.B.2.2;M.8.C.1.1.1 Visualize a net as a representation of the Nets, Surface Area surface area of a Problem 1.2: Making (Rectangular Prisms) rectangular prism. Rectangular Boxes M.8.B.2.2;M.8.C.1.1.1 Connect the area of the net to the surface Nets, Surface Area area of a rectangular M.8.B.2.2.1;M8.B.2.2.3; (Rectangular Prisms) prism. M.8.C.1.1.1 Visualize a net as a representation of the Problem 1.3: Testing Nets, Surface Area surface area of a Nets (Rectangular M.8.B.2.2.1;M8.B.2.2.3; (Rectangular Prisms) rectangular prism. Prisms) M.8.C.1.1.1 Use a net for a rectangular prism to develop a strategy for finding the Nets, Surface Area surface area of the M.8.B.2.2.1;M8.B.2.2.3; (Rectangular Prisms) prism. M.8.C.1.1.1 Find the volume of a rectangular prism by counting the number Volume (Rectangular of unit cubes it takes Prisms) to fill the prism. M8.B.2.2.2 Understand the relationship between the dimensions of a Problem 1.4: Flattening Surface Area rectangular prism a Box (Surface Area of (Rectangular Prism) and its surface area. Rectangular Prisms) M.8.B.2.2;M.8.C.1.1.1 Calculate the srface Surface Area area of a rectangular (Rectangular Prism) prism. M.8.B.2.2.1 Continue to develop visualization skills Nets, Surface Area relating rectangular Mathematical Reflection, (Rectangular Prisms) prisms to their nets. p. 18 M.8.C.1.1.1 INVESTIGATION 2: Designing Rectangular Boxes Connect the dimensions of a Volume and Surface rectangular prism to Problem 2.1: Packaging Area Connections its volume and Blocks (Finding Surface (Rectangular Prisms) surface area. Area) M8.B.2.2 Understand that rectangular prisms may have the same Volume and Surface volume but quite Area Connections different surface M8.B.2.2.1;M8.B.2.2.2;M (Rectangular Prisms) areas. 8.B.2.2.3 Predict which rectangular prism of those with a Volume and Surface common volume will Problem 2.2: Saving Area Connections have the smallest Trees (Finding the Least (Rectangular Prisms) surface area. Surface Area) M8.B.2.2.1;M8.B.2.2.3 Refine a strategy for finding the surface Surface Area area of a rectangular (Rectangular Prism) prism. M8.B.2.2.1;M8.D.1.1.3 Understand that prisms can be filled systematically in identical layers, and Problem 2.3: Filling that this layering Rectangular Boxes leads to the formula (Finding Volume of a Volume (Prisms) for volume. Rectangular Prism) M8.B.2.2.2;M8.B.2.2.3 Develop a formula for finding the Volume (Rectangular volume of a Mathematical Reflection, M8.B.2.2.2;M8.B.2.2.3;M Prism) rectangular prism. p. 31; Check Up 1 8.D.1.1.3 INVESTIGATION 3: Prisms and Cylinders Develop Problem 3.1: Filling understanding of Fancy Boxes (Finding Volume, Surface volume and surface the Volumes of Other Area (Prisms) area of prisms. Prisms) M8.B.2.2.2 Develop a strategy for finding the Problem 3.2: Filling volume of a cylinder Cylinders (Finding the Volume (Cylinder) using its dimensions. Volume of Cylinders) M8.B.2.2 Connect this strategy to the idea of layers in rectangular and Volume (Cylinder) other prisms. M8.B.2.2 Begin to develop a strategy for finding Surface Area the surface area of a (Cylinder) cylinder. M8.B.2.2 Problem 3.3: Making Develop a strategy Cylinders and Prisms for finding the from Nets (Finding the Surface Area surface area of a Surface Area of (Cylinder) cylinder. Cylinders and Prisms) M8.B.2.2 Apply understanding Problem 3.4:Making a Problem-solving of volume to solve New Juice Container (Volume) problems. (Comparing Volumes) M8.B.2.2.2 Understand that a variety of different three-dimensional figures can have the same volume but Volume and Surface different surface Mathematical Reflection, Area Connections areas. p. 47; Partner Quiz M8.B.2.2 INVESTIGATION 4: Cones, Spheres, and Pyramids Explore a volume Volume Connections relationship between (Cylinders and cylinders and Problem 4.1: Comparing Spheres) spheres. Spheres and Cylinders Extend students' understanding of volume as layering to other ways of filling a three-dimensional Volume figure. Explore and use a Volume Connections volume relationship Problem 4.2: Cones and (Cylinders, Cones, between cylinders, Cylinders, Pyramids and and Spheres) cones and spheres. Cubes Explore and use a volume relationship Volume Connections between prisms and (Prisms and pyramids to find the Pyramids) volume of a pyramid. Strengthen understanding of the volume relationships Problem 4.3: Melting Ice Volume Connections among cones, Cream (Comparing (Cones, Spheres, spheres and Volumes of Spheres, and Cylinders) cylinders. Cylinders, and Cones) Use the relationships amond cylinders, cones, and sphers to develop a strategy for finding the Volume (Cone or volume of a cone or Mathematical Reflection, Sphere) sphere. p. 61; Check Up 2 INVESTIGATION 5: Scaling Boxes Understand how changes in one or more dimensions of Problem 5.1: Building a a rectangular prism Bigger Box (Doubling the Volume (Rectangular affect the prism's Volume of a Rectangular Prism) volume Prism) Design rectangular Volume (Rectangular prisms with a given Prism) volume. Extend students' Problem 5.2: Scaling Up understanding of the Compost Box Similarity and Three- similarity to three- (Applying Scale Factors Dimensional Figures dimensional figures. to Rectangular Prisms) Understand the effect on surface area of appling a Similarity and Three- scale factor to a Dimensional Figures rectangular prism. Understand the effect on volume of applying a scale Similarity and Three- factor to a Dimensional Figures rectangular prism. Apply students' understanding of scale factor and its relationship to changes in 1-, 2-, Mathematical Reflection, Problem 5.3: Building Similarity and Three- and 3-dimensional p. 75; Unit Test; Unit Model Ships: Similarity Dimensional Figures measures. Project and Scale Factors DELETE THIS INVESTIGATION? Unit Title: Thinking With Mathematical Models Notes: Big Ideas Contents Skills Assessments Make tables and graphs to represent data. Describe relationships between variables. Use data patterns to make predictions. Make tables and graphs to represent data. Describe relationships between variables. Use data patterns to make predictions. Make tables and graphs to represent data. Describe relationships between variables. Use data patterns to make predictions. Compare and contrast linear and nonlinear relationships. Fit a line to data that show a linear trend. Write an equation for a line based on a grpah of the line. Use mathematical models to answer questions about linear relationships. Practice effective strategies for writing linear equations from verbal, numerical, or graphical information. Develop skill in solving linear equations with approximation and exact reasoning methods. Write inequalities to represent "at most" situations. Use equations to represent questions about problem situations and interpret the solutions in the context of the problem. Explore situations that can be modeled by inverse variation Inverse Variation relationships. Explore situations that can be modeled by inverse variation Inverse Variation relationships. Investigate the nature of inverse variation in familiar Inverse Variation contexts. Compare inverse variations with linear Inverse Variation relationships. Explore situations that can be modeled by inverse variation Inverse Variation relationships. Investigate the nature of inverse variation in familiar Inverse Variation contexts. Compare inverse variations with linear Inverse Variation relationships. Lessons Anchors INVESTIGATION 1: Exploring Data Patterns Problem 1.1: Testing Bridge Thickness (Finding Patterns and Making Predictions) M8.D.4.1 M8.D.4 M8.D.1.1 Problem 1.2: Testing Bridge Lengths (Finding Patterns and Making Predictions) M8.D.4.1 M8.D.4 M8.D.1.1 Problem 1.3: Custom Construction Parts (Extending Patterns) M8.D.4.1 M8.D.4 M8.D.1.1 ????? INVESTIGATION 2: Linear Models and Equations Problem 2.1: Linear Models M8.E.4.1.1 M8.D.1.1.3 M8.D.2.2 Problem 2.2: Equations for Linear Relationships M8.D.2.2.2 Problem 2.3: Solving Linear Equations M8.D.2.1 M8.D.2.1 Problem 2.4: Intersecting Linear Models M8.D.2 INVESTIGATION 3: Inverse Variation Problem 3.1: Rectangles with Fixed Area (Relating Length and Width) Should we eliminate this investigation? Problem 3.2: Bridging the Distance (Inverse Variation Patterns) Problem 3.3: Average Cost (Inverse Variation Patterns) Unit Title: Looking for Pythagoras Notes: Big Ideas Contents Skills Assessments Number Systems; Pythagorean Theorem Review the Coordinate System coordinate system. Explore distances on Coordinate System a coordinate grid. Properties of Review properties of Polygons quadrilaterals. Connect properties Coordinate System of figures to and Properties of coordinate Polygons representations. Draw shapes on a Coordinate System coordinate grid. Develop strategies for finding areas of irregular figures on a Area grid. Draw squares on 5 dot-by-5 dot grids Area and find their areas. Introduce the Squares and Square concept of square Roots root. Understand square root geometrically, as the side length of Squares and Square a square with known Roots area. Use geometric understanding of square roots to find lengths of line Squares and Square segments on a dot Roots grid. Deduce the Pythagorean Pythagorean Theorem through Theorem explorations. Use the Pythagorean Theorem to find unknown side Pythagorean lengths of right Theorem triangles. Reason through a geometric proof of Pythagorean the Pythagorean Theorem Theorem. Use the Pythagorean Theorem to find the Pythagorean distance between Theorem two points on a grid. Relate areas of Pythagorean squares to the Theorem lengths of the sides. Determine whether a triangle is a right Pythagorean triangle based on its Theorem side lengths. Relate areas of Pythagorean squares to the Theorem lengths of the sides. Learn the meanings of rational number and irrational Number Systems number. Estimate the values Squares and Square of square roots that Roots; Number are irrational Systems; Estimation numbers. Squares and Square Estimate lengths of Roots; Number hypotenuses of right Systems; Estimation triangles. Estimation; Estimate lengths of Pythagorean hypotenuses of right Theorem triangles. Apply the Pythagorean Pythagorean Theorem to a Theorem problem situation. Investigate the Properties of Special special properties of Triangles a 30-60-90 triangle. Use the properties of Properties of Special special right triangles Triangles to solve problems. Lessons Anchors INVESTIGATION 1: Coordinate Grids Problem 1.1:Driving Around Euclid (Locating Points and Finding Distances) M8.C.3.1.1 Problem 1.2: Planning Parks (Shapes on a Coordinate Grid) M6.C.1.1.1 M6.C.1.1.1; M8.C.3.1.1 M8.C.3.1.1 Problem 1.3: Finding Areas M7.B.2.1; M8.B.2.2 INVESTIGATION 2: Squaring Off Problem 2.1: Looking for Squares M7.B.2.1; M8.B.2.2 Problem 2.2: Square Roots M8.A.1.1.2 M8.A.1.1.2 Problem 2.3: Using Squares to Find Lengths M8.B.2.2; M8.C.1 INVESTIGATION 3: The Pythagorean Theorem Problem 3.1: The Pythagorean Theorem M8.C.1 M8.C.1.2.1 Problem 3.2: A Proof of the Pythagorean Theorem M8.C.1 Problem 3.3: Finding Distances M8.C.1.2.1 M8.C.1 Problem 3.4: Measuring the Egyptian Way (Lengths that Form a Right Triangle) M8.C.1 M8.C.1 Investigation 4: Using the Pythagorean Theorem Problem 4.1: Analysing the Wheel of Theodorus M8.A.1 M8.A.1.1.2 M8.A.3.2 Problem 4.2: Stopping Sneaky Sally (Finding Unknown Side Lengths) M8.A.3.2 M8.C.1.2.1 Problem 4.3: Analyzing Triangles M8.C.1 Problem 4.4: Finding the Perimeter M8.C.a