7 th Grade CCGPS Math LFS Unit 5: Geometry

Transcription

1 7 th Grade CCGPS Math LFS Unit 5: Geometry Standards: Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them. MCC7.G.2 (DOK2) 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. MCC7.G.3 (DOK2) Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. MCC7.G.4 (DOK2) 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. MCC7.G.5 (DOK2) 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. MCC7.G.6 (DOK2) 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. Geometry Page 1

2 K-U-D Unit 5: Geometry UNDERSTAND By the end of this unit, I want my students to understand Relationships between 2-D and 3-D geometric figures are useful for building new knowledge and solving real world and mathematical problems. Know By the end of the unit, the student will know: Do By the end of the unit, the student will be able to: Triangles can be drawn given three measures of angles or sides (G.2) When conditions determine a unique triangle, more than one triangle, or no triangle (G.2) The interior angles of a triangle add up to 180 (G.2) Cuts made parallel will take the shape of the base; cuts made perpendicular will take the shape of the lateral (side face). Cuts made at an angle through the right rectangular prism will produce a parallelogram; cuts made at an angle through the right rectangular pyramid will also produce a parallelogram. (G.2) Different ways to slice a 3D figure (vertical, horizontal, and angled (G.3) What two-dimensional figures resulting from slicing a three-dimensional figure (cylinders, cones, pyramids, and prisms) (G.3) NOTE: STANDARDS SAY RIGHT RECTANGULAR PRISMS AND RIGHT RECTANGULAR PYRAMIDS Attributes of a right rectangular prism and right rectangular pyramid. (G.3) Pi and 22/7 (G.4) The formulas for area and circumference of a circle (G.4) The relationship between diameter, circumference, and pi (G.4) Draw precise geometric figures based on given conditions (G.2) (DOK 2) Discover the conditions necessary for a given set of angles or sides to make a triangle (G.2) (DOK 2) Explore conditions that determine a unique triangle, more than one triangle, or not triangle. (G.2) (DOK 2) Describe the two-dimensional figures resulting from slicing a three-dimensional figure (cylinders, cones, pyramids, and prisms) (G.3) (DOK 1) NOTE: STANDARDS SAY RIGHT RECTANGULAR PRISMS AND RIGHT RECTANGULAR PYRAMIDS Use formulas for the area and circumference to solve problems (G.4) (DOK 1) Show and explain how the circumference and area of a circle are related (G.4) (DOK 2) Use properties of supplementary, complementary, alternating Interior, alternating exterior, vertical, and adjacent angles to solve for unknown angles in a figure (G.5) (DOK 1) Write and solve equations based on a diagram of intersecting lines with some known angle measures as variables (G.5) (DOK 2) (See example on page 15) Solve mathematical problems involving area Geometry Page 2

3 Properties of supplementary, complementary, vertical, and adjacent angles (G.5) VOCABULARY: Protractor, construction, three-dimensional, right rectangular prisms, right rectangular pyramids, area, angle ( ), angle measure (m ), acute, obtuse, right, degrees, vertex, line segment, circumference, radius, diameter, supplementary angles, complementary angles, vertical angles, adjacent angles, intersecting lines, volume, surface area, parallelogram polygons, cubes, right prisms, cross-sections, center, slant height, base, altitude, height of triangles, quadrilaterals, polygons, and composite figures (G.6) (DOK1) Solve real-world problems involving area of triangles, quadrilaterals, polygons, and composite figures (G.6) (DOK2) Solve mathematical problems involving volume of cubes, right prisms, and composite polyhedra (G.6) (DOK1) Solve real-world problems involving volume of cubes, right prisms, and composite polyhedra (G.6) (DOK 2) Solve mathematical problems involving surface area of cubes, right prisms, right pyramids, and composite polyhedra. (G.6) (DOK1) Solve real-world problems involving surface area of cubes, right prisms, right pyramids, and composite polyhedra (G.6) (DOK 2) Geometry Page 3

4 SLM Unit 5: Geometry Key Learning Relationships between 2-D and 3-D geometric figures are useful for building new knowledge and solving real world and mathematical problems. Unit EQ How are relationships between 2-D and 3-D geometric figures useful for building new knowledge and solving real-world and mathematical problem? Concept Concept Concept Concept Angles (G.5, G.6) Triangles (G.2) Cross-sections (G.3, G.6) Area, Volume, Perimeter, Surface Area (G.4, G.6) Lesson EQ s Lesson EQ s Lesson EQ s Lesson EQ s 1. How are angles classified? (Chapter 8 Lesson 1 and 2) 2. How can angle measurement be determined in systems of intersecting lines? (Chapter 8 Lesson 1 and 2) 1. What characteristics are required to make a triangle? (Chapter 8 Inquiry Lab on page 625 investigation 1 and 2) 2. How do you find missing angles? (Chapter 8 Lesson 3) 1. What 2-D figure is formed when 3-D figures are sliced? (Chapter 8 Lesson 5 and Hand-on lab) 1. How are area and circumference of a circle related? (Chapter 9 Lesson 1 and 2) 2. How is area of composite figures determined? (Chapter 9 Lesson 3) 3. How is volume determined? (Chapter 9 Lesson 4 and 5) 4. How do you calculate surface area? (Chapter 9 Lesson 6,7 and 8) Vocabulary Vocabulary Vocabulary Vocabulary supplementary angles, complementary angles, vertical angles, adjacent angles, intersecting lines, Protractor, construction, angle ( ), angle measure (m ), acute, obtuse, right, degrees, vertex, line segment, three-dimensional, right rectangular prisms, right rectangular pyramids, parallelogram, polygons, cubes, right prisms, crosssections, base, circumference, radius, diameter, volume, surface area, slant height, altitude, height Geometry Page 4

5 Geometry Page 5

6 Geometry Page 6

7 Domain: Geometry MCC7.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. Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them. What does this standard mean? Students understand the characteristics of angles that create triangles. For example, can a triangle have more than one obtuse angle? Will three sides of any length create a triangle? Students recognize that the sum of the two smaller sides must be larger than the third side. Examples and Explanations Conditions may involve points, line segments, angles, parallelism, congruence, angles, and perpendicularity. Examples: 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? Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not? 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? Mathematical Practice Standards 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. Can you draw a triangle with sides that are 13 cm, 5 cm and 6cm? 7.MP.8. Look for and express regularity in Geometry Page 7

8 Draw a quadrilateral with one set of parallel sides and no right angles. repeated reasoning. Suggested Instructional Strategy Give students the same three angle measures, have them draw triangles on their own, compare their triangles, and discuss how the triangles are similar and different from one another. Do the same activity given three side lengths. Could use Patty Paper also. Have students create triangles and other geometric figures using dynamic geometry software to determine conditions necessary to create polygons. Skill Based Task 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? Problem Task Draw a triangle where one angle is twice as large as another. Measure the third angle. What is the relationship between the three angles? 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. Instructional Resources/Tools Dynamic geometry software Patty Paper Protractor Ruler Internet Resources: Geometry Page 8

9 Domain: Geometry MCC7.G.3 Describe the two-dimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them. What does this standard mean? Students need to describe the resulting face shape from cuts made parallel and perpendicular to the bases of right rectangular prisms and pyramids. Cuts made parallel will take the shape of the base; cuts made perpendicular will take the shape of the lateral (side) face. Cuts made at an angle through the right rectangular prism will produce a parallelogram; cuts made at an angle through the right rectangular pyramid will also produce a parallelogram. Examples and Explanations Example: 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. Mathematical Practice Standards 7.MP.2. Reason abstractly and quantitatively. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.7. Look for and make use of structure. Suggested Instructional Strategy Slice a variety of objects (e.g. Styrofoam shapes, molding clay, or potatoes), using fishing line, dental floss, or string, and describe the two-dimensional figures that result from the slices. Explore topographical maps as a two-dimensional representation of a three dimensional landscape. Take potatoes and cut into right rectangular prisms or any other type of right prism. Cut potato prisms at different angles, color the Geometry Page 9

10 cross-section using a marker or stamp pad, and stamp it onto a piece of paper to help students see that the cross-section is 2D. Skill Based Task Explain how a single slice through the right rectangular prism can create a triangle, a rectangle, a trapezoid, or a square? Problem Task Create three-dimensional models based on topographical maps. What are all the possible 2-D figures you can create by slicing an ice-cream cone? an umbrella? a 3-D object of your choice? Instructional Resources/Tools National Library of Virtual Manipulatives - Platonic Solids Slicing Google Sketchup Internet Resources: Geometry Page 10

11 Domain: Geometry MCC7.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. Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. What does this standard mean? Students understand the relationship between radius and diameter. Students also understand the ratio of circumference to diameter can be expressed as. Building on these understandings, students generate the formulas for circumference and area. Know the formula does not mean memorization of the formula. To know means to have an understanding of why the formula works and how the formula relates to the measure (area and circumference) and the figure. This understanding should be for all students. Examples and Explanations The illustration shows the relationship between the circumference and area. If a circle is cut into wedges and laid out as shown, a parallelogram results. Half of an end wedge can be moved to the other end a rectangle results. The height of the rectangle is the same as the radius of the circle. The base length is ½ the circumference (2r). The area of the rectangle (and therefore the circle) is found by the following calculations: Arect = Base Height Area = ½ (2r) r Area = r r Area = r 2 Students solve problems (mathematical and real-world) including finding the area of left-over materials when circles are cut from squares and triangles or from cutting squares and triangles from circles. Examples: The seventh grade class is building a mini golf game for the school carnival. The end of the putting Mathematical Practice Standards 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. Geometry Page 11

12 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? Students measure the circumference and diameter of several circular objects in the room (clock, trash can, door knob, wheel, etc.). Students organize their information and discover the relationship between circumference and diameter by noticing the pattern in the ratio of the measures. Students write an expression that could be used to find the circumference of a circle with any diameter and check their expression on other circles. Students will use a circle as a model to make several equal parts as you would in a pie model. The greater number the cuts, the better. The pie pieces are laid out to form a shape similar to a parallelogram. Students will then write an expression for the area of the parallelogram related to the radius (note: the length of the base of the parallelogram is half the circumference, or πr, and the height is r, resulting in an area of πr 2. Extension: If students are given the circumference of a circle, could they write a formula to determine the circle s area or given the area of a circle, could they write the formula for the circumference? 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. 7.MP.8. Look for and express regularity in repeated reasoning. Divide a circle into equal parts; rearrange pieces into a parallelogram to model the derivation of the area of a circle. Relate the base of the parallelogram to half the circumference of the circle and the height of the parallelogram to its radius. (See picture) Suggested Instructional Strategy Geometry Page 12

13 Cut two strings, one the length of the circumference, and the other the length of the diameter. Compare the lengths of the two strings to show that the circumference is approximately three diameters. Have students measure the circumference and diameter of several circular objects of different sizes and take the ratio of the circumference to the diameter to discover pi. Skill Based Task Find the area and circumference of a circle with a radius of 4 cm. Round to the nearest tenth. Instructional Resources/Tools Problem Task 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? Sir Cumference and the First Round Table and/or any Sir Cumference books. circular objects of several different sizes string or yarn tape measures, rulers grid paper paper plates Square Circles This lesson features two creative twists on the standard lesson of having students measure several circles to discover that the ratio of the circumference to the diameter seems always to be a little more than 3. This lesson starts with squares, so students can first identify a simpler constant ratio (4) of perimeter to length of a side before moving to the more difficult case of the circle. The second idea is to measure with a variety of units, so students can more readily see that the ratio of the Geometry Page 13

14 measurements remains constant, not only across different sizes of figures, but even for the same figure with different measurements. From these measurements, students will discover the constant ratio of 1:4 for all squares and the ratio of approximately 1:3.14 for all circles. Circles and Their Area Given that units of area are squares, how can we find the area of a circle or other curved region? Imagine a waffle-like grid inside a circle and a larger grid containing the circle. The area of the circle lies between the area of the inside grid and the area of the outside grid.. Exploring c/d = π Students measure circular objects to collect data to investigate the relationship between the circumference of a circle and its diameter. They find that, regardless of the size of the object or the size of the measuring unit, it always takes a little more than three times the length of the diameter to measure the circumference. Apple Pi Using estimation and measurement skills, students will determine the ratio of circumference to diameter and explore the meaning of π. Students will discover the circumference and area formulas based on their investigations. Circle Tool With this three-part online applet, students can explore with graphic and numeric displays how the circumference and area of a circle compare to its radius and diameter. Students can collect data points by dragging the radius to various lengths and clicking the "Add to Table" button to record the data in the table. Geometry of Circles Using a MIRA geometry tool, students determine the relationships between radius, diameter, circumference and area of a circle. Internet Resources: Geometry Page 14

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16 Domain: Geometry MCC7.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. Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. What does this standard mean? Students use understandings of angles to write and solve equations. Examples and Explanations Angle relationships that can be explored include but are not limited to: Same-side (consecutive) interior and same-side (consecutive) exterior angles are supplementary. Examples: Write and solve an equation to find the measure of angle x. Mathematical Practice Standards 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with mathematics. 7.MP.5. Use appropriate tools strategically. Write and solve an equation to find the measure of angle x. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure. Geometry Page 16

17 Suggested Instructional Strategy Use patty paper or a reflecting device (mira) to explore relationships among the measures of angles formed by intersecting lines. Draw two intersecting lines. Measure one angle and find the measures of the others without measuring. Provide explicit examples of supplementary, complementary, vertical, and adjacent angles and demonstrate their relationships, including adjacent angles that are neither supplementary nor complementary. Solve for x and y. Skill Based Task Problem Task If m B=102 and m L=120, find every other angle measure, explaining how you found each. Instructional Resources/Tools Exploring Geometry with The Geometer s Sketchpad - Angles Formed by Intersecting Lines activity Internet Resources: Geometry Page 17

18 Domain: Geometry MCC7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. What does this standard mean? Students continue work from 5 th and 6 th grade to work with area, volume and surface area of two-dimensional and three-dimensional objects (composite shapes). Students will not work with cylinders, as circles are not polygons. Know the formula does not mean memorization of the formula. To know means to have an understanding of why the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding should be for all students. Surface area formulas are not the expectation with this standard. Building on work with nets in the 6 th grade, students should recognize that finding the area of each face of a threedimensional figure and adding the areas will give the surface area. No nets will be given at this level. Examples and Explanations Students understanding of volume can be supported by focusing on the area of base times the height to calculate volume. Students understanding of surface area can be supported by focusing on the sum of the area of the faces. Nets can be used to evaluate surface area calculations. Examples: Choose one of the figures shown below and write a step by step procedure for determining the area. Find another person that chose the same figure as you did. How are your procedures the same and different? Do they yield the same result? Mathematical Practice Standards 7.MP.1. Make sense of problems and persevere in solving them. 7.MP.2. Reason abstractly and quantitatively. 7.MP.3. Construct viable arguments and critique the reasoning of others. 7.MP.4. Model with Geometry Page 18

19 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. Find the area of a triangle with a base length of three units and a height of four units. Find the area of the trapezoid shown below using the formulas for rectangles and triangles. 12 mathematics. 7.MP.5. Use appropriate tools strategically. 7.MP.6. Attend to precision. 7.MP.7. Look for and make use of structure MP.8. Look for and express regularity in repeated reasoning. Suggested Instructional Strategy Have students design a way to determine the volume of various containers. Students can use their method to determine the volume of several containers and compare volumes. Test their hypotheses regarding the relative size of the containers by filling them with rice or beans. Skill Based Task 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. Problem Task Design a container that will hold at least 300 ft 3 of water, but that has a lateral surface area of less than 310 ft 2. Instructional Internet Resources: Geometry Page 19

20 Resources/Tools Geometry Page 20