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NYS COMMON CORE MATHEMATICS CURRICULUM Geometry Grade 7 – Module 6 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Session Agenda • Module Overview • Focus on Topics and Lessons • Module in Review • Implementation Discussions © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Session Objectives • Gain an understanding of the content enveloped in Module 6. • Understand what was, what is, and why the change. • See how geometry concepts from here forward are starting to take shape. • Gain the knowledge necessary to implement with fidelity. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Curriculum Overview of A Story of Ratios © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Grade 7 Module 6 Overview Let’s look at: • Foundational Standards (p.5) • Focus Standards (p.4) • Mathematical Practice Standards (p.5-6) • Table of Contents and Narrative © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic A: Unknown Angles 7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. • Prerequisites: G4 M4 and G7 M3 • Goal is to build fluency in these difficult problems • Build recognition and use of structure MP.7 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Complementary and Supplementary Angles The measures of two supplementary angles are in the ratio of 2: 3. Find the two angles. Lesson 1 / Example 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Solving for Unknown Angles Using Equations Two lines meet as the common vertex of two rays. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of ∠𝐴𝑂𝐵 and ∠𝐵𝑂𝐶. Lesson 2 / Exercise 4 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Solving for Unknown Angles Using Equations Two lines meet at the common vertex of two rays. In complete sentences, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of angles ∠𝐵𝐴𝐶 and ∠𝐵𝐴𝐻. Lesson 3 / Example 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Solving for Unknown Angles Using Equations Two lines meet at a point. Set up and solve an equation to find the value of 𝑥. Find the measurement of one of the vertical angles. Lesson 3 / Example 4 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Solving for Unknown Angles Using Equations 2 The measurement of an angle is the measurement of its supplement. 3 Find the measurement of the angle. Lesson 4 / Example 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic B: Constructing Triangles 7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. New Terminology New Tools New Approach © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Correspondence Two identical triangles are shown below. Give a correspondence that matches equal sides and equal angles. Lesson 5 / Example 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Drawing Geometric Shapes Use a ruler, protractor, and compass to complete the following: 1. 3 segments: 4 𝑐𝑚, 7.2 𝑐𝑚, 12.8 𝑐𝑚 2. Complementary angles, one of 35° 3. Vertical angles, one of 125° 4. 3 circles: 𝑟 = 2 𝑐𝑚, 𝑟 = 4 𝑐𝑚, 𝑟 = 6 𝑐𝑚 5. 3 adjacent angles: 25°, 90°, 50° 6. Rectangle 𝐴𝐵𝐶𝐷, 𝐴𝐵 = 8 𝑐𝑚, 𝐵𝐶 = 3 𝑐𝑚 7. Segment: 5 𝑐𝑚, cut a segment 5 𝑐𝑚 using your compass 8. Segment AB; Circle with center A and radius AB, and circle with center B and radius BA. Lesson 6 / Exploratory Challenge © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Drawing Geometric Shapes Use a ruler, protractor, and compass to complete the following: Lesson 6 / Exploratory Challenge © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Drawing Geometric Shapes Use a ruler, protractor, and compass to complete the following: Lesson 6 / Exploratory Challenge © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Drawing Parallelograms Use a setsquare, ruler and protractor to construct parallelogram 𝐷𝐸𝐹𝐺 so the measurement of ∠𝐷 = 40°, 𝐷𝐸 = 3 𝑐𝑚, the measurement of ∠𝐸 = 140°, and the altitude to 𝐷𝐸 is 5 𝑐𝑚. Lesson 7 / Exercise 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Drawing Triangles A triangle is to be drawn given the following conditions: the measurements of two angles are 30° and 60°, and the length of a side is 10 𝑐𝑚. Note that where each of these measurements is positioned is not fixed. Lesson 8 / Exploratory Challenge © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Three Sides Condition Two Sides and the Included Angle Condition 4. △ 𝑋𝑌𝑍 has side lengths 𝑋𝑌 = 2.5 cm, 𝑋𝑍 = 4 cm, and ∠𝑋 = 120°. Draw △ 𝑋′𝑌′𝑍′ under the same conditions. Use your compass and protractor to draw the sides of △ 𝑋′𝑌′𝑍′. Use your ruler to measure side lengths. Leave all construction marks as evidence of your work, and label all side and angle measurements. Under what condition is △ 𝑋′𝑌′𝐶′ drawn? Compare the triangle you drew to two of your peers’ triangles. Are the triangles identical? Did the condition determine a unique triangle? Use your construction to explain why. Lesson 9 / Exploratory Challenge 4 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Two Angles and a Given Side 3. A triangle △ 𝐽𝐾𝐿 has angles ∠𝐽 = 60° and ∠𝐿 = 25° and side 𝐾𝐿 = 5 cm. Draw triangle △ 𝐽′𝐾′𝐿′ under the same condition. Leave all construction marks as evidence of your work, and label all side and angle measurements. Under what condition is △ 𝐽′𝐾′𝐿′ drawn? Compare the triangle you drew to two of your peers’ triangles. Are the triangles identical? Did the condition determine a unique triangle? Use your construction to explain why. Lesson 10 / Exploratory Challenge 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Conditions on Measurements that Determine a Triangle Are there conditions on the lengths of the sides of a triangle? Are there conditions on the measures of the angles in a triangle? Lesson 11 / Exploratory Challenge 1&2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Conditions on Measurements that Determine a Triangle Is there a “Three Angles” condition for unique triangle? Lesson 11 / Exploratory Challenge 2d © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Two Sides and a Non-Included Angle Use your tools to draw △ 𝐴𝐵𝐶 provide 𝐴𝐵 = 5, 𝐵𝐶 = 3 𝑐𝑚, and ∠𝐴 = 30°. • What makes this drawing challenging? • Use a ruler and compass to locate 𝐶. Lesson 12 / Exploratory Challenge 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Two Sides and a Non-Included Angle Use your tools to draw △ 𝐷𝐸𝐹 provide 𝐷𝐸 = 5, 𝐸𝐹 = 3 𝑐𝑚, and ∠𝐹 = 90°. Lesson 12 / Exploratory Challenge 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Two Sides and a Non-Included Angle Use your tools to draw △ 𝐽𝐾𝐿 provide 𝐾𝐿 = 8, 𝐾𝐽 = 4 𝑐𝑚, and ∠𝐽 = 120°. Conclusions based on the results of Exploratory Challenges 1-3? Lesson 12 / Exploratory Challenge 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Checking for Identical Triangles Lesson 14 / Examples 1-2 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Using Unique Triangles to Solve Real World and Mathematical Problems Quadrilateral 𝐴𝐵𝐶𝐷 is a model of a kite. Diagonals 𝐴𝐵 and 𝐶𝐷 represent the sticks that help keep the kite rigid. Lesson 15 / Exercise 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Mid-Module Assessment • Assessment • Progression Toward Mastery Rubric • Exemplary Student Responses © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic C: Slicing Solids 7.G.A.3 Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Slicing 3-Dimensional figures by a plane and the resulting 2-Dimensional slices. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Slicing a Right Rectangular Prism with a Plane Lesson 16 / Exercises 5-6 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Right Rectangular Pyramid Bla Lesson 17 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Right Rectangular Pyramid Lesson 17 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Slicing a Right Rectangular Pyramid with a Plane Lesson 17 / Example 5 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Slicing on an Angle Lesson 18 / Example 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM Slicing on an Angle Lesson 18 / Example 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Understanding 3-Dimensional Figures If slices parallel to the tabletop are taken of this figure, then what would each slice look like? Given the level slices that you determined, how many unit cubes are in the figure? Lesson 19 / Exercise 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic D: Problems Involving Area and Surface Area 7.G.B.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Real World Area Problems Ten targets are being painted as shown in the following figure. The radius of the smallest circle is 3 𝑖𝑛., and each successive, larger circle is 3 𝑖𝑛. more in radius than the circle before it. A “tester” can of red and of white paint is purchased to paint the target. Each 8 𝑜𝑧. can of paint covers 16 𝑓𝑡 2 . Is there enough paint of each color to create all ten targets? Lesson 20 / Example 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Mathematical Area Problems Examine the change in dimension and area of squares as the lengths and widths are increased by the given amount. Lesson 21 / Example 1 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Area Problems with Circular Regions The diameters of four half circles are sides of a square with side lengths of 7 𝑐𝑚. Find the exact area of the shaded region. Lesson 22 / Problem Set #4 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Surface Area Determine the surface area of the right prism. Lesson 23 / Exercise 4 © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Surface Area A right rectangular pyramid has a square base with side lengths of 10 inches. The surface area of the pyramid is 260 𝑖𝑛2 . Find the height of the four lateral triangular faces. Lesson 24 / Example 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic E: Problems Involving Volume 7.G.B.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Volume of Right Prisms Two containers are shaped like right triangular prisms, each with the same height. The base area of the larger container is 200% more than the base area of the smaller container. How many times must the smaller container be filled with water and poured into the larger container in order to fill the larger container? Lesson 25 / Example 3 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Volume of Composite 3-Dimensional Objects Find the volume of the right prism shown whose base is the region between two right triangles. Use two different strategies. Lesson 26 / Example 2 © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Real-World Volume Problems Jim wants to know how much his family spends on water for showers. Water costs $1.50 per 1000 gallons. His family averages 4 showers per day (total). The average length of a shower is 10 minutes. He places a bucket in his shower and turns on the water. After one-minute, the bucket has 2.5 gallons of water in it. About how much money does his family spend on water for showers in a month (30 days)? Lesson 27 / Exit Ticket © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM End-of-Module Assessment • Assessment • Progression Toward Mastery Rubric • Exemplary Student Responses © 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module 6 in Review Take a few moments to discuss, with those around you, the major highlights of the Module. • Likes • What was and what is • Concerns • Major Takeaways © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Key Points • • • • • • • Student transitioning to more complex levels of geometry Using algebra to solve unknown angle problems Explore the structure of triangles Determine the conditions that determine unique triangles Identify identical triangles and use their relationship to solve problems Describe the slices of three-dimensional figures Apply strategies for area, surface area, and volume in solving real world and mathematical problems © 2012 Common Core, Inc. All rights reserved. commoncore.org