© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Grade 6 – Module 5.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Session Objectives Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons. Introduction to mathematical models and instructional strategies to support implementation of A Story of Ratios. Consideration of scaffolding to provide multiple entry points for students at all levels.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios A Story of Ratios Scaffolding Mathematics Instruction Key Points Amplify Language Move from Concrete to Representation to Abstract Give Specific Guidelines for Speaking, Reading, Writing, or Listening

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Amplify Language Give clear mathematical definitions Explain multiple meanings Maintain consistency and point out interchangeable terminology

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Move from Concrete to Representation to Abstract Use familiar contexts Visually organize thinking Provide multiple representations

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Give Guidelines for Speaking, Reading, Writing, or Listening Provide structured opportunities to speak and write in English Give explicit instructions in student-friendly language Use visuals or examples in giving instructions.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Key Points Amplify Language Move from Concrete to Representation to Abstract Give Guidelines for Speaking, Reading, Writing, or Listening

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios What difficulties would you anticipate with student understanding of the mathematics in this section? What scaffolds would be effective for addressing those difficulties? Turn and Talk

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios How does this task align with the Universal Design for Learning (UDL) in providing multiple options for: Representation: the “what of learning” Action/Expression: the “how” of learning Engagement: the “why” of learning Turn and Talk

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module’s Foundation Standards: Pages 18-20 in the Progressions Document (K-6,Geometry,) serves as a foundation.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios G6-M5 Topic Overview Topic A: Area of Triangles, Quadrilaterals, and Polygons Topic B: Polygons on the Coordinate Plane Topic C: Volume of Right Rectangular Prisms Topic D: Nets and Surface Area

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios LESSON 1 – THE AREA OF PARALLELOGRAMS THROUGH RECTANGLE FACTS Students show the area formula for the region bounded by a parallelogram by composing it into rectangles. They understand that the area of a parallelogram is the area of the region bounded by the parallelogram. 19

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios LESSON 2 – THE AREA OF RIGHT TRIANGLES Students justify the area formula for a right triangle by viewing the right triangle as part of a rectangle composed of two right triangles. 21

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios LESSON 3 – THE AREA OF ACUTE TRIANGLES USING HEIGHT AND BASE Students show the area formula for a triangular region by decomposing a triangle into right triangles. For a given triangle, the height of the triangle is the length of the altitude. The length of the base is either called the length base or, more commonly, the base. Students understand that the height of the triangle is the perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. Students understand that any side of a triangle can be considered a base and that the choice of base determines the height. 22

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios LESSON 4 – THE AREA OF OBTUSE TRIANGLES USING HEIGHT AND BASE Students construct the altitude for three different cases: an altitude that is a side of a right triangle, an altitude that lies over the base, and an altitude that is outside the triangle. Students deconstruct triangles to justify that the area of a triangle is exactly one half the area of a parallelogram. 24

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios You’re Turn!! Use the templates provided to try and prove that the area of any triangle is half that of its corresponding rectangle/parallelogram.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios LESSON 5 – THE AREA OF POLYGONS THROUGH COMPOSITION AND DECOMPOSITION Students show the area formula for the region bounded by a polygon by decomposing the region into triangles and other polygons. They understand that the area of a polygon is actually the area of the region bound by the polygon. Students find the area of the region bounded by a trapezoid by decomposing the region into two triangles. They understand that the area of the trapezoid is actually the area of the region bound by the trapezoid. Students decompose rectangles to determine the area of other quadrilaterals. 29

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios LESSON 6 – AREA IN THE REAL WORLD Students determine the area of composite figures in real-life contextual situations using composition and decomposition of polygons. Students determine the area of a missing region using composition and decomposition of polygons. 34

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 7 – Distance on the Coordinate Plane Students use absolute value to determine distance between integers on the coordinate plane in order to find side lengths of polygons. 37

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 8 – Drawing Polygons on the Coordinate Plane Given coordinates for the vertices, students draw polygons in the coordinate plane. Students find the area enclosed by a polygon by composing or decomposing using polygons with known area formulas. Students name coordinates that define a polygon with specific properties. 38

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 9 – Determining Area and Perimeter of Polygons on the Coordinate Plane Students find the perimeter of irregular figures using coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Students find the area enclosed by a polygon on the coordinate plane by composing or decomposing using polygons with know area formulas. 42

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 10 – Distance, Perimeter, and Area in the Real World Students determine distance, perimeter, and area in real-world contexts. 46

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Mid-Module Assessment Complete Question 4 from the Mid-Module Assessment. Share your completed task with a table partner. With your partner, refer to the rubric and sample student work for this question and compare your own findings. Which standards relate to this question? Which Mathematical Practices are embodied in this task? What challenges might students face as they complete this task? How does students’ work in Topics A and B prepare them for this task?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 11 – Volume with Fractional Edge Lengths and Unit Cubes Students extend their understanding of the volume of a right rectangular prism with integer side lengths to right rectangular prisms with fractional side lengths. They apply the formula V= l w h to find the volume of a right rectangular prism and use the correct volume units when writing the answer. 50

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Scaffolding: Take 4 minutes to read the scaffolding portion at the end of Lesson 11. Discuss at your tables the benefit of including this scaffold prior to instruction. How will you determine if students need this scaffold?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 12 – From Unit Cubes to the Formulas for Volume Students extend the volume formula for a right rectangular prism to the formula V=Area of base x height. They understand that any face can be the base. 57

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 13 – The Formulas for Volume Students determine, understand, and apply formulas for finding the volume of right rectangular prisms and cubes. 59

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 14 – Volume in the Real World Students understand that volume is additive and apply volume formulas to determine the volume of composite solid figures in real-world contexts. Students apply volume formulas to find missing volumes and missing dimensions. 62

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 15 – Representing Three- Dimensional Shapes Using Nets Students construct three-dimensional figures through the use of nets. They determine which nets make specific solid figures and determine if nets can or cannot make a solid figure. 66

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 16 – Constructing Nets Students construct nets of three-dimensional objects using the measurement of the solids’ edges. 77

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 17 – From Nets to Surface Area Students use nets to determine the surface area of three-dimensional figures. 83

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 18 – Determining the Surface Area of Three- Dimensional Figures Students determine that a right rectangular prism has six faces: top and bottom, front and back, and two sides. They determine that surface area is obtained by adding the areas of all the faces and develop the formula SA=2lw + 2lh+ 2wh. Students develop and apply the formula for the surface area of a cube as SA=6s2. 87

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 19 – Surface Area and Volume in the Real World Students determine the surface area of three-dimensional figures in real-world contexts. Students choose appropriate formulas to solve real-life volume and surface area problems. 91

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 19a – Applying Surface Area and Volume to Aquariums Students apply the formulas for surface area and volume to determine missing dimensions of aquariums and water level. 94

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios End-of-Module Assessment Count off at your tables 1, 2, 1, 2…. so that half of you will answer question 1 and the other half will answer question 5. Complete Questions 1 or 5. When everyone at your table is finished, the 1’s will discuss their question and then the 5’s will discuss their question. Whole group share: What is the progression from Topic A through Topic D?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Biggest Takeaway Turn and Talk: What questions were answered for you? What new questions have surfaced?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Grade 6 – Module 5.

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© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Grade 6 – Module 5.

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