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Math 7: Unit 5 – Geometry (4 weeks) Unit Overview: Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. For a detailed conceptual foundation, click here. Content Standards: Draw, construct, and describe geometrical figures and describe the relationships between them. MCC.7.G.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. (ITBS) MCC.7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. MCC.7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. (ITBS) MCC.7.G.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. MCC.7.G.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. (ITBS) TRANSITIONAL STANDARD-TEACH 2012-13 MCC.6.G.3 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 realworld and mathematical problems. Standards for Mathematical Practice: 1. Make sense of complex problems and persevere in solving them. 2. Reason abstractly and quantitatively Diagnostic: Prerequisite Assessment Standards for Mathematical Practice (1, 2) EQ: How can what is known help determine where to begin or what to do next when solving a problem? (MP1) How do mathematically proficient students use both concrete and abstract reasoning to solve problems? (MP2) Learning Targets: I can … explain the meaning of a problem (MP1) make conjectures about the form and meaning of the solution (MP1) plan a solution pathway rather than simply jumping into a solution attempt (MP1) consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution (MP1) monitor and evaluate my progress and change course if necessary (MP1) draw diagrams of important features and relationships, and search for regularity or trends (MP1) use concrete objects or pictures to help conceptualize and solve a problem (MP1) check answers to problems using a different method (MP1) understand the approaches of others to solving complex problems (MP1) make sense of the quantities and their relationships in problem situations (MP2) bring two complementary abilities to bear on problems involving quantitative relationships (MP2) decontextualize-to abstract a given situation (MP2) represent a given situation symbolically (MP2) manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents (MP2) contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved (MP2) create a coherent representation of the problem at hand; considering the units involved (MP2) attend to the meaning of quantities, not just how to compute them (MP2) know and flexibly use different properties of operations and objects (MP2) Concept Overview: SMP1 Make sense of complex 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. Teachers who are developing students’ capacity to “make sense of problems and persevere in solving them” develop ways of framing mathematical challenges that are clear and explicit, and then check in repeatedly with students to help them clarify their thinking and their process. A teacher might ask her students to work in pairs to evaluate their approach to a problem, telling a partner to describe their process, saying “what they did, and what they might do next time.” Another teacher might post a set of different approaches to a solution, asking students to identify “what this mathematician was thinking or trying out” and evaluating the success of the strategy. Students may articulate a specific way of laying out the terrain of a problem and evaluating different starting points for solving. A teacher of adolescents and young adults might frame the task as a real-world design conundrum, inviting students to engage in a “tinkering” process of working toward mathematical proof, changing course as necessary as they develop their thinking. SMP2 Reason abstractly and quantitatively Teachers who are developing students’ capacity to "reason abstractly and quantitatively" help their learners “make sense of quantities and their relationships in problem situations” and introduces the skills of “creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Beginning with the noticing and wondering stages, teachers can help students focus on quantities and their values, and relationships or calculations they could do. Stress naming quantities and providing units over identifying values (the numbers in the problem). Students then use those quantities and relationships to get good at guess and check. Being able to identify the appropriate quantity to guess for, set up appropriate calculations, check results against appropriate constraints, and repeat the process in an organized way are vital to being able to set up a mathematical model of a situation. If the student can’t use guess and check successfully, it’s unlikely that they could set up and solve the problem algebraically. Resources: MP1 Inside Mathematics Website MP2 Inside Mathematics Website Constructing Two-Dimensional Figures Draw, construct, and describe geometrical figures and describe the relationships between them. MCC.7.G.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. MCC.7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. EQ: How can I use geometry to draw, construct, and describe the relationships between two- and three-dimensional figures? Learning Targets: I can … draw a geometric shape with specific conditions. (G2) construct a triangle when given three measurements: 3 side lengths, 3 angle measurements, or a combination of side and angle measurements. (G2) determine when three specific measurements will result in one unique triangle, more than one possible triangle, or no possible triangle. (G2) name the two-dimensional figure that represents a particular slice of a three-dimensional figure. (G3) Concept Overview: This cluster focuses on the importance of developing geometric intuition through visualization. Being able to visualize and then represent geometric figures on paper is essential to solving geometric problems. At this stage of students' learning trajectory, they should try to develop this intuition by drawing many pictures. Now “geometric intuition" is not a measurable quantity, so it is difficult to say whether one has “enough" geometric intuition or not. However, the following strategies may be used to help students develop it. Constructions facilitate understanding of geometry. Provide opportunities for students to physically construct triangles with straws, sticks, or geometry apps prior to using rulers and protractors to discover and justify the side and angle conditions that will form triangles. Explorations should involve giving students: three side measures, three angle measures, two side measures and an included angle measure, and two angles and an included side measure to determine if a unique triangle, no triangle or an infinite set of triangles results. Through discussion of their exploration results, students should conclude that triangles cannot be formed by any three arbitrary side or angle measures. They may realize that for a triangle to result the sum of any two side lengths must be greater than the third side length, or the sum of the three angles must equal 180 degrees. Students should be able to transfer from these explorations to reviewing measures of three side lengths or three angle measures and determining if they are from a triangle justifying their conclusions with both sketches and reasoning. Further construction work can be replicated with quadrilaterals, determining the angle sum, noticing the variety of polygons that can be created with the same side lengths but different angle measures, and ultimately generalizing a method for finding the angle sums for regular polygons and the measures of individual angles. For example, subdividing a polygon into triangles using a vertex (N-2) 180° or subdividing a polygons into triangles using an interior point 180°N - 360° where N = the number of sides in the polygon. An extension would be to realize that the two equations are equal. Slicing three-dimensional figures helps develop three-dimensional visualization skills. Students should have the opportunity to physically create some of the three-dimensional figures, slice them in different ways, and describe in pictures and words what has been found. For example, use clay to form a cube, then pull string through it in different angles and record the shape of the slices found. Challenges can also be given: “See how many different two-dimensional figures can be found by slicing a cube” or “What three-dimensional figure can produce a hexagon slice?” This can be repeated with other three-dimensional figures using a chart to record and sketch the figure, slices and resulting two-dimensional figures. This cluster lends itself to using dynamic software. Students sometimes can manipulate the software more quickly than do the work manually. However, being able to use a protractor and a straight edge are desirable skills. Vocabulary: right rectangular prism: A prism which has bases aligned one directly above the other and has lateral faces that are rectangles. The angles that the bases form with the lateral faces are right angles. right rectangular pyramid: A pyramid that has its apex (vertex) aligned directly above the center of the base. Sample Problem(s): Solutions to Sample Problems Sample Problem 1: Given the two side lengths, 6 and 7, what is the shortest possible length of the third side? What is the longest possible length of the third side? (G2) Sample Problem 2: Draw a triangle where one angle is twice as large as another. Measure the third angle. What is the relationship between the three angles? (G2) Sample Problem 3: Is it possible to draw a triangle with a 90˚ angle and one leg that is 4 inches long and one leg that is 3 inches long? If so, draw one. Is there more than one such triangle? (G2) Sample Problem 4: Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not? (G2) Sample Problem 5: Draw an isosceles triangle with only one 80 degree angle. Is this the only possibility or can you draw another triangle that will also meet these conditions? (G2) Sample Problem 6: Can you draw a triangle with sides that are 13 cm, 5 cm and 6cm? (G2) Sample Problem 7: Draw a quadrilateral with one set of parallel sides and no right angles. (G2) Sample Problem 8: Given the following angles, 0°, 10°, 15°, 30°, 35°, 65°, 70°, 75°, 80°, and 100°, find all the possible angle combinations that will form a triangle. Precisely draw all possible triangles using a protractor and ruler. (G2) Sample Problem 9: Explain how a single slice through the right rectangular prism can create a triangle, a rectangle, a trapezoid, or a square? (G3) Sample Problem 10: What are all the possible 2-D figures you can create by slicing an ice-cream cone? (G3) Sample Problem 11: Using a clay model of a rectangular prism, describe the shapes that are created when planar cuts are made diagonally, perpendicularly, and parallel to the base. (G3) Standard MCC.7.G.2 Topic Constructing triangles Resources Teacher Notes Holt 3 Angles in Triangles Section 7 – 3 pgs. 336 – 338 Student Misconceptions: Students may have misconceptions about how to read a ruler. Pearson 2 Drawing a Picture Test Taking Strategies pg. 365 Math 7 Unit 5- Good Questions Sides and Angles of a Triangle Activity Lab 7 – 3a pg. 335 Literacy Strategy: List strategies for teaching vocabulary and/or reading word problems. (K) Construct Triangle – 3 Sides (S, K) Constructing a 30-60-90 Triangles (S) Constructing Triangles Activity (S) Patty Paper Geometry by Michael Serra provides investigations for constructing Math 7 Unit 5- Differentiation Strategies congruent triangles. Investigation Set 8 pg. 125-137 MCC.7.G.3 Slicing 3dimensional figures to create 2-dimensional figures Textbook Section #’s Holt 3 Symmetry in 3 Dimensions Ch. 8 Extension – Example 2 pg. 446 Model Lesson: Cross Sections Play-Doh Activity Pearson 2 Cross Sections Activity Lab 8 – 10c. pg. 432 433 Cooperative Learning Strategy: Cross Sections Activity Math 7 Unit 5- Good Questions Math 7 Unit 5- Differentiation Strategies Literacy Strategy: List strategies for teaching vocabulary and/or reading word problems. (K) Holt 2 Cross Sections Extension pgs. 612 – 613 Learning Task: Cross Section of a Cube (K) Learning Task: Similar Cross Sections (K) Learning Task: What’s My Solid? (U) Solving Problems Involving Two- and Three-Dimensional Objects Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. MCC.7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. MCC.7.G.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. MCC.7.G.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. TRANSITIONAL STANDARD-TEACH 2012-13 MCC.6.G.3 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 realworld and mathematical problems. EQ: How can we use geometric reasoning to solve problems involving angle measure, area, surface area, and volume? Learning Targets: I can … state the formulas for finding the area and circumference of a circle. (G4) use formulas to compute the area and circumference of a circle (G4) determine the diameter or radius of a circle when the circumference is given (G4) use a ratio and algebraic reasoning to compare the area and circumference of a circle. (G4) state the relationship between supplementary, complementary, and vertical angles. (G5) use angle relationships to write algebraic equations for unknown angles. (G5) use algebraic reasoning and angle relationships to solve multi-step problems. (G5) determine the area of two-dimensional figures. (G6) determine the surface area and volume of three-dimensional figures (G6) solve real-world problems involving area, surface area, and volume. (G6) plot vertices in the coordinate plane to draw specific polygons (6G3) use the coordinates of the vertices of a polygon to find the length of a specific side. (6G3) plot points, draw figures, and find lengths on the coordinate plane to solve real-world problems. (6G3) Concept Overview: This is the students’ initial work with circles. Knowing that a circle is created by connecting all the points equidistant from a point (center) is essential to understanding the relationships between radius, diameter, circumference, pi and area. Students can observe this by folding a paper plate several times, finding the center at the intersection, then measuring the lengths between the center and several points on the circle, the radius. Measuring the folds through the center, or diameters leads to the realization that a diameter is two times a radius. Given multiple-size circles, students should then explore the relationship between the radius and the length measure of the circle (circumference) finding an approximation of pi and ultimately deriving a formula for circumference. String or yarn placed over the circle and compared to a ruler is an adequate estimate of the circumference. This same process can be followed in finding the relationship between the diameter and the area of a circle by using grid paper to estimate the area. Another visual for understanding the area of a circle can be modeled by cutting up a paper plate into 16 pieces along diameters and reshaping the pieces into a parallelogram. In figuring area of a circle, the squaring of the radius can also be explained by showing a circle inside a square. Again, the formula is derived and then learned. After explorations, students should then solve problems, set in relevant contexts, using the formulas for area and circumference. In previous grades, students have studied angles by type according to size: acute, obtuse and right, and their role as an attribute in polygons. Now angles are considered based upon the special relationships that exist among them: supplementary, complementary, vertical and adjacent angles. Provide students the opportunities to explore these relationships first through measuring and finding the patterns among the angles of intersecting lines or within polygons, then utilize the relationships to write and solve equations for multi-step problems. Real-world and mathematical multi-step problems that require finding area, perimeter, volume, surface area of figures composed of triangles, quadrilaterals, polygons, cubes and right prisms should reflect situations relevant to seventh graders. The computations should make use of formulas and involve whole numbers, fractions, decimals, ratios and various units of measure with same system conversions. Vocabulary: radius: The distance from the center of a circle to any point on the circle. Also, the line segment that has the center of the circle as one endpoint and a point on the circle as the other endpoint. diameter: The distance across a circle through its center. The line segment that includes the center and whose endpoints lie on the circle. circumference: The distance around a circle. area of a circle: The measurement of the region enclosed by a circle. The area of a circle is always expressed in squared units. pi: The ratio of the circumference of a circle to its diameter. supplementary angles: Supplementary angles are two angles whose sum is 180 degrees. complementary angles: Complementary angles are two angles whose sum is 90 degrees. vertical angles: Two nonadjacent angles formed by intersecting lines or segments. Also called opposite angles. adjacent angles: Angles in the same plane that have a common vertex and a common side, but no common interior points. Sample Problem(s): Solutions to Sample Problems Sample Problem 12: The seventh grade class is building a mini golf game for the school carnival. The end of the putting green will be a circle. If the circle is 10 feet in diameter, how many square feet of grass carpet will they need to buy to cover the circle? How might you communicate this information to the salesperson to make sure you receive a piece of carpet that is the correct size? (G4) Sample Problem 13: Tennis balls are packaged in a cylindrical container containing three balls. Without measuring, determine which is longer, the height of a tennis ball container or the distance around it? (G4) Sample Problem 14: Write and solve an equation to find the measure of angle x. (G5) Sample Problem 15: Write and solve an equation to find the measure of angle x. (G5) Sample Problem 16: Solve for x and y. (G5) Sample Problem 17: If 𝑚∠𝐵 = 102° and 𝑚∠𝐿 = 120°, find every other angle measure, explaining how you found each. (G5) Sample Problem 18: A cereal box is a rectangular prism. What is the volume of the cereal box? What is the surface area of the cereal box? (Hint: Create a net of the cereal box and use the net to calculate the surface area.) Make a poster explaining your work to share with the class. (G6) Sample Problem 19: Find the total volume for the house if the base of the house is 20 ft. X 50 ft. with side walls that are 10 ft. high and the peak of the house is 15 ft. from the ground. (G6) Sample Problem 20: 3 2 Design a container that will hold at least 300 ft of water, but that has a lateral surface area of less than 310 ft . (G6) Given the coordinates A (2, 5), B (-4, 5), C (-4, 1), and D (2, 1) Jose says that the distance between A and D can be found by subtracting 2 from 5. Prove or disprove. Explain your answer with words, pictures, and equations. (6G3) On a map, the library is located at (-2, 2), the city hall building is located at (0, 2), and the high school is located at (0, 0). Represent the locations as points on a coordinate grid with a unit of 1 mile. (6G3) o What is the distance from the library to the city hall building? The distance from the city hall building to the high school? How do you know? o What shape is formed by connecting the three locations? The city council is planning to place a city park in this area. How large is the area of the planned park? Standard MCC.7.G.4 Topic Solving problems involving area and circumference of a circle Resources Holt 3 Circles Section 8 – 3 pgs. 400 – 401 Pearson 2 Circumference and Area of a Circle Section 8 – 5 pgs. 394 – 395 Teacher Notes Model Lesson: Relating Circumference and Diameter Student Misconceptions: Students sometimes believe r2 is the same as r × 2; Students will sometimes interchange the formulas for circumference and area; Holt 2 Area of Circles Section 9 – 5 pgs. 538 – 539 Perimeter & Circumference Section 9 – 2 Example 3 pg. 525 Pearson 1 Area of a Circle Section 9 – 6 pgs. 444 – 445 Circles and Circumference Section 9 – 5 pgs. 438 – 439 Exploring Circles Activity Lab 9 – 5a pgs. 437 Learning Task: It’s As Easy As Pi (K) Apple Pi (U) Exploring c/d = π (K) Many students are confused when dealing with circumference (linear measurement) and area. This confusion is about an attribute that is measured using linear units (surrounding) vs. an attribute that is measured using area units (covering). There are no diagrams given with formulas on the formula sheet. Be sure students not only know how to substitute values into the formulas but can also describe how where these formulas came from. Students mistakenly use the diameter in the area formula in place radius (r × 2 instead of r2) or square the diameter instead of the radius. Students may believe Pi is an exact number rather than understanding that 3.14 is just an approximation of pi. Students do not label their answers with appropriate units (for example: not using units squared when referring to the area of a circle and/or using units squared for circumference) Math 7 Unit 5- Good Questions Math 7 Unit 5- Differentiation Strategies Cooperative Learning Strategy: Smart Cookie Activity Literacy Strategy: List strategies for teaching vocabulary and/or reading word problems. (K) MCC.7.G.5 Solving problems finding missing angle measures Holt 3 Points, Lines, Planes and Angles Section 7 – 1, pgs. 324 – 326 Pearson 2 Identifying and Classifying angles Section 7 – 2 pgs. 330 – 332 Student Misconceptions: Students will need reminders and review of the different types of angles formed by two lines cut by a transversal. Students will sometimes mistake angles that are supplementary with angles that are congruent, when two parallel lines are cut by a transversal. Math 7 Unit 5- Good Questions Holt 2 Explore Complementary and Supplementary Angles Hands on Lab 8 – 2 pgs. 446 – 447 Literacy Strategy: List strategies for teaching vocabulary and/or reading word problems. (K) Classifying Angles Section 8 -2 pgs. 449 Pearson 1 Special Pairs of Angles Section 8 – 3 pgs. 374 – 375 I Have a Secret Angle (U) TI Graphing Calculator: Angles for a Solution (K, U) MCC.7.G.6 Solving problems involving twoand threedimensional objects Holt 3 Volume of Prisms and Cylinders Section 8 – 5 pgs. 413 – 415 *Prisms only Volume of Pyramids and Cones Section 8 - 6 pgs. 420 – 422 Surface Area of Prisms and Cylinders Section 8 – 7 pgs. 427 – 428 *Prisms only Surface Area of Pyramids and Cones Section 8 – 8 pgs. 432 - 433 Find Surface Area of Pyramids Hands on Lab 8 – 8 pg. 431 Pearson 2 Surface Areas of Prisms and Cylinders Section 8 – 9 pgs. 414 – 416 *Prisms only Student Misconceptions: Many students are confused when dealing with circumference (linear measurement) and area. This confusion is about an attribute that is measured using linear units (surrounding) vs. an attribute that is measured using area units (covering). Math 7 Unit 5- Good Questions Math 7 Unit 5- Differentiation Strategies Literacy Strategy: List strategies for teaching vocabulary and/or reading word problems. (K) Volumes of Prisms and Cylinders Section 8 – 10 pgs. 421 – 423 *Prisms only Stained Glass Designs (K) Staircases (U) MCC.6.G.3 Solving problems involving drawing polygons on the coordinate plane Holt 3 Coordinate Geometry Section 7 – 5 Examples 3 & 4 pgs. 348 – 349 Pearson 2 Graphing Points in Four Quadrants Section 10 – 1 Example 3 pg. 487 Geometry in the Coordinate Plane Activity Lab 10 – 1b. pg. 490 Polygons in Coordinate Plane (K) Calculator Activity: Drawing Polygons (S) Perimeter & Area in Coordinate Plane (U) Student Misconceptions: Students may think: Units of measurement for perimeter, area and volume are interchangeable. Volume can be found by determining the area of a single face on a prism and multiply that by the total number of faces. Area and volume are the result of plugging numbers into a formula. They do not understand the concepts of area and volume. Math 7 Unit 5- Good Questions Literacy Strategy: List strategies for teaching vocabulary and/or reading word problems. (K) Unit #5 Summative Assessment