Grade 7 Mathematics. Unit 8. Geometry. Estimated Time: 18 Hours

Transcription

1 Grade 7 Mathematics Geometry Estimated Time: 18 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization Grade 7 Mathematics Curriculum Outcomes 233

2 Grade 7 Mathematics Curriculum Outcomes 234

3 : Geometry Overview Introduction Students will develop a vocabulary of geometric terms and learn to create some simple constructions. Several Explore investigations will be performed to enable the learning of concepts and techniques. Students will be introduced to the Cartesian plane (or coordinate plane), and will advance their knowledge of transformations. The big ideas in this unit are: There are a variety of methods used to create parallel and perpendicular line segments. There are a variety of methods used to create line segment bisectors and angle bisectors Locations on a Cartesian plane are indentified using ordered pairs. Under translations, rotations, and reflections the objects and the images are congruent to each other. Transformations can be applied in sequence, one after the other. Objects will be denoted with capital letters (like A) and images will use the prime notation ( A for a single transforamtion or A for a combination of two transformations). Context The students will begin by identifying parallel and perpendicular line segments that are found in the environment. They will explore a variety of methods to construct their own parallel and perpendicular segments, all the while learning the correct language and definitions. The concept of a bisector, both line and angle, will be introduced and the construction of bisectors will be learned. The Cartesian plane will be presented and the four quadrants will be explained. Naming of locations, or points on the plane, will be done with ordered pairs. The students will learn the mathematical terms for slide, flip, and turn. A discussion of congruence and orientation will be made with each transformation. The use of the prime notation will be introduced, and the students will explore the properties of combined transformations. Why are these concepts important? Developing a good understanding of geometry will permit students to: Be active participants in many of today s fields that require a strong knowledge of geometric concepts. Fields like engineering, carpentry, surveying, interior decorating, architecture and more. Be good logical thinkers. Geometry requires the ability to reason in a linear and coherent manner; the ability to make statements and support them until a logical conclusion is reached. Whatever future path a student may walk the ability to analyze and reason logically will serve them well. Be better prepared for the higher order geometry that they will study in high school and beyond. And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art. Albrecht Durer ( ) Grade 7 Math Curriculum Guide 235

4 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Specific Outcome It is expected that students will: 7SS3. Perform geometric constructions, including: perpendicular line segments parallel line segments perpendicular bisectors angle bisectors. [CN, R, V] Elaborations: Suggested Learning and Teaching Strategies What makes shapes alike and different can be determined by an array of geometric properties. For example, shapes have sides that are parallel, perpendicular, or neither. (Van de Walle and Lovin 2006, p. 179) Students have been introduced to the concept of: lines (or line segments) that are parallel (never intersect) or that are perpendicular (meet at right angles) in familiar shapes and in the real world. identifying the parallel sides of squares, rectangles, hexagons, trapezoids, and parallelograms. Achievement Indicators 7SS3.1 Identify line segments on given diagrams that are either parallel or perpendicular. 7SS3.2 Describe examples of parallel line segments in the environment. 7SS3.3 Draw a line segment parallel to another line segment, and explain why they are parallel. identifying pairs of adjacent sides that are perpendicular. This achievement indicator is discussed in Lesson 8.1 (Parallel Lines) and Lesson 8.2 (Perpendicular Lines). Examples of parallel lines in the environment. Opposite sides of picture frames Railroad/Roller Coaster tracks Lines on loose-leaf paper Rows of siding on a house Lines of latitude Guitar strings The Explore activity on page 300 in the textbook allows students to discover their own methods of creating parallel lines using a wide variety of tools. It is open ended and most likely will lead to many novel approaches. The textbook has a host of methods (pp ) that illustrate how to draw parallel line segments. They are all effective and offer their own perspectives on the concepts involved. Grade 7 Mathematics Curriculum Outcomes 236

5 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Suggested Assessment Strategies Resources/Notes Activity Ask the students to make a list of as many pairs of parallel lines as they can find in the classroom in two minutes. After the two minutes are up have the students pass their list to another student. Then ask students, one at a time, to read an entry from the list in front of them. Everybody who has that entry on their list will cross it off. At the end, the list with the most remaining entries will be the winner. (Students can be paired up for this activity.) Paper and Pencil 1. Have students draw a line that is neither vertical nor horizontal. Then, using a method of their choice, draw a second line that is parallel to the first. 2. Where might you see this pattern and what is the purpose of these parallel lines? Journal Have students think of 2-D shapes (excluding quadrilaterals) that have parallel sides. Ask the students to include diagrams to illustrate their thinking. Math Makes Sense 7 Lesson 8.1 : Data Analysis TR: ProGuide, pp. 4 6 Master 8.8, 8.24, 8.15 CD-ROM Masters ST: pp Practice and HW Book pp Grade 7 Mathematics Curriculum Outcomes 237

6 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Specific Outcome Elaborations: Suggested Learning and Teaching Strategies It is expected that students will: 7SS3. Perform geometric constructions, including: perpendicular line segments parallel line segments perpendicular bisectors angle bisectors. [CN, R, V] (Cont d) Achievement Indicators 7SS3.4 Describe examples of perpendicular line segments in the environment. Examples of perpendicular lines in the environment. Crosses Railway tracks and railway ties Fence posts and fence rails Four way stops Lines of latitude and longitude A wall and a shelf 7SS3.5 Draw a line segment perpendicular to another line segment, and explain why they are perpendicular. The Explore activity on page 303 in the textbook allows students to discover their own methods of creating perpendicular lines using a wide variety of tools. It is open ended and most likely will lead to many novel approaches. The textbook has a host of methods (pp ) that illustrate how to draw perpendicular line segments. They are all effective and offer their own perspectives on the concepts involved. Grade 7 Mathematics Curriculum Outcomes 238

7 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Suggested Assessment Strategies Resources/Notes Activity Ask the students to make a list of as many pairs of perpendicular lines as they can find in the classroom in two minutes. After the two minutes are up have the students pass their list to another student. Then ask students, one at a time, to read an entry from the list in front of them. Everybody who has that entry on their list will cross it off. At the end, the list with the most remaining entries will be the winner. (Students can be paired up for this activity) Paper and Pencil Have students draw a line that is neither vertical nor horizontal. Then, using a method of their choice, draw a second line that is perpendicular to the first. Journal 1. Can two lines be both parallel and perpendicular? 2. Can a line have more than one other line that is perpendicular to it? Explain your reasoning. Can you think of an example? Math Makes Sense 7 Lesson 8.2 : Data Analysis TR: ProGuide, pp. 7 9 Master 8.9, 8.25, 8.16 CD-ROM Masters ST: pp Practice and HW Book pp Grade 7 Mathematics Curriculum Outcomes 239

8 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Specific Outcome It is expected that students will: 7SS3. Perform geometric constructions, including: perpendicular line segments parallel line segments perpendicular bisectors angle bisectors. [CN, R, V] (Cont d) Achievement Indicators 7SS3.6 Describe examples of perpendicular bisectors in the environment. Elaborations: Suggested Learning and Teaching Strategies A perpendicular bisector is a line or segment that intersects another segment at a right angle and divides it into two equal parts. AB CD AM Examples of perpendicular bisectors segments in the environment. = MB 7SS3.7 Draw the perpendicular bisector of a line segment, using more than one method, and verify the construction. The Explore activity on page 306 in the textbook allows students to discover their own methods of creating perpendicular bisectors using a wide variety of tools. It is open ended and most likely will lead to many novel approaches. The textbook has three strategies (pp ) that illustrate how to draw perpendicular bisectors. They are all effective and offer their own perspectives on the concepts involved. Grade 7 Mathematics Curriculum Outcomes 240

9 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Suggested Assessment Strategies Resources/Notes Journal The Reflect activity on page 309 in the textbook is suggested as an exercise. Paper and Pencil For the Perpendicular Bisector worksheet, refer to Appendix 8 A. Math Makes Sense 7 Lesson 8.3 : Data Analysis TR: ProGuide, pp Master 8.10, 8.26 CD-ROM Masters ST: pp Practice and HW Book pp Grade 7 Mathematics Curriculum Outcomes 241

10 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Specific Outcome It is expected that students will: 7SS3. Perform geometric constructions, including: perpendicular line segments parallel line segments perpendicular bisectors angle bisectors. [CN, R, V] (Cont d) Achievement Indicators 7SS3.8 Describe examples of angle bisectors in the environment. Elaborations: Suggested Learning and Teaching Strategies For every angle, there exists a line that divides the angle into two equal parts. This line is known as the angle bisector. Example in the environment: 7SS3.9 Draw the bisector of a given angle, using more than one method, and verify that the resulting angles are equal. The Explore activity on page 310 in the textbook allows students to discover their own methods of creating angle bisectors using a wide variety of tools. It is open ended and most likely will lead to many novel approaches. The textbook has three strategies (pp ) that illustrate how to draw angle bisectors. They are all effective and offer their own perspectives on the concepts involved. Grade 7 Mathematics Curriculum Outcomes 242

11 Strand: Shape and Space (3-D Objects and 2-D Shapes) General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them. Suggested Assessment Strategies Resources/Notes Pencil & Paper For the Angle Bisector worksheet, refer to Appendix 8 B. Informal Observation Mix-up Match-up: Create a set of cards. On one half write the terms in the list below; on the second half write definitions that match the terms on the first half. Distribute these cards to the students and have them circulate the room trying to match up the correct cards. Instruct them to sit together when they find their match. parallel lines perpendicular lines angle bisector perpendicular bisector line segment obtuse angle acute angle right angle straight angle reflex angle vertex radius diameter circumference Math Makes Sense 7 Lesson 8.4 : Data Analysis TR: ProGuide, pp Master 8.11, 8.27, 8.17, 8.18a,b CD-ROM Masters ST: pp Practice and HW Book pp Unit Problem P 338, 339 Grade 7 Mathematics Curriculum Outcomes 243

12 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Specific Outcome It is expected that students will: 7SS4. Identify and plot points in the four quadrants of a Cartesian plane, using integral ordered pairs. [C, CN, V] Elaborations: Suggested Learning and Teaching Strategies Key Terms coordinate plane the plane containing the x- and y-axes coordinate system the grid formed by the intersection of two perpendicular number lines that meet at their zero points ordered pair a pair of numbers used to locate any point on a coordinate plane quadrant one of the four regions into which the x- and y-axes separate the coordinate plane x-axis the horizontal number line on a coordinate plane y-axis the vertical number line on a coordinate plane x-coordinate the first number in a coordinate pair y-coordinate the second number in a coordinate pair Achievement Indicators 7SS4.1 Label the axes of a four quadrant coordinate plane (or Cartesian plane), and identify the origin. Identify the ordered pair that names point A. Step 1 Move left on the x-axis to find the x-coordinate of point A, which is -3. Step 2 Move up the y-axis to find the y-coordinate, which is 4. Point A is named by (-3, 4). 7SS4.2 Identify the location of a given point in any quadrant of a Cartesian plane, using an integral ordered pair. 7SS4.3 Plot the point corresponding to a given integral ordered pair on a Cartesian plane with units of 1, 2, 5 or 10 on its axes. (5, 4) Origin (0, 0) Grade 7 Mathematics Curriculum Outcomes 244

13 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Suggested Assessment Strategies Resources/Notes Games Have students play the commercial board game Battleship or create a game similar to battleship using the coordinate plane. Creating a game would probably be better because you can utilize all four of the quadrants. The game can also be played online at the link indicated below: For the website, refer to: Paper and Pencil For the Coordinate Plane worksheet, refer to Appendix 8 C. Graphic Organizer (Foldable) A four-door booklet can be created to organize students notes on a Cartesian plane. (Refer to the Appendix 8 D for the Coordinate Plane foldable.) Journal Using the internet research why coordinate planes are often called Cartesian planes. Write a brief paragraph explaining your findings. Math Makes Sense 7 Lesson 8.5 : Data Analysis TR: ProGuide, pp Master 8.20, 8.12, 8.28 PM 22 CD-ROM Masters ST: pp Practice and HW Book pp Grade 7 Mathematics Curriculum Outcomes 245

14 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Specific Outcome It is expected that students will: 7SS4. Identify and plot points in the four quadrants of a Cartesian plane, using integral ordered pairs. [C, CN, V] (Cont d) Achievement Indicators 7SS4.4 Draw shapes and designs in a Cartesian plane, using given integral ordered pairs. Elaborations: Suggested Learning and Teaching Strategies The following achievement indicators are addressed together. Students would be expected to perform both of these tasks: A 1. Given sets of points as ordered pairs, plot them on a coordinate plane and join those points to create a shape. i.e.: given point A(1, 3), point B., plot the points and connect to form a shape. 7SS4.5 Create shapes and designs, and identify the points used to produce the shapes and designs, in any quadrant of a Cartesian plane. 2. Using shapes drawn on a coordinate plane, identify the locations of the vertices. i.e. Given the shape in the diagram identify the ordered pair that describes each vertex. Grade 7 Mathematics Curriculum Outcomes 246

15 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Suggested Assessment Strategies Resources/Notes Paper and Pencil 1. Natasha is creating an X-pattern for her needlepoint project in home economics. She has plotted the X on a coordinate plane using these ordered pairs. A(3, 0) B(2, -1) C(1, -2) D(-3, -2) E(-1, -4) F(-1, 0) G(0, -1) H(2, -3) I(3, -4) Will Natasha make an X? If not, what ordered pair will she need to change to fix it? 2. Refer to Appendix 8 E for the worksheet entitled Coordinate Plane Worksheet: Newfoundland Flag. Math Makes Sense 7 Lesson 8.5 (continued) Grade 7 Mathematics Curriculum Outcomes 247

16 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Specific Outcome It is expected that students will: 7SS5. Perform and describe transformations (translations, rotations or reflections) of a 2-D shape in all four quadrants of a Cartesian plane (limited to integral number vertices). [C, CN, PS, T, V] Achievement Indicators 7SS5.1 (It is intended that the original shape and its image have vertices with integral coordinates.) Identify the coordinates of the vertices of a given 2-D shape on a Cartesian plane. 7SS5.2 Describe the horizontal and vertical movement required to move from a given point to another point on a Cartesian plane. Elaborations: Suggested Learning and Teaching Strategies Students have been exposed to transformational geometry in previous grades. An emphasis at this level should be the use of the formal language of transformations, such as translation, reflection, and rotation, instead of slides, flips, and turns. Students will be working with transformations and combinations of transformations in all four quadrants of the Cartesian plane. With respect to describing transformations, students should be able to recognize a given transformation as one of the following: a reflection, a translation, a rotation, or some combination of these. In addition, when given an object and its image students should be able to describe: a translation, using words and notation describing the translation (E.g. A B C is the translation image of ABC, or D (5, 8) is the translation image of D (-5, 8). Students will be taught that when describing translations they have to describe the horizontal change first and the vertical change second. a reflection, by determining the location of the line of reflection a rotation, using degree or fraction-of-turn measures, both clockwise and counterclockwise, and identify the location of the centre of a rotation. A centre of rotation may be located on the shape (such as a vertex of the original image) or off the shape. Note: A is read as A prime. It is used to label the point that matches point A after the transformation has been applied. 7SS5.3 Determine the distance between points along horizontal and vertical lines in a Cartesian plane. When investigating properties of transformations, students should consider the concepts of congruence, which were developed informally in previous grades. In discussing the properties of transformations, students should consider if the transformation of the image: has side lengths and angle measures the same as the given image; is both similar to and congruent to the given image; has the same orientation as the given image; appears to have remained stationary with respect to the given image. Grade 7 Mathematics Curriculum Outcomes 248

17 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Suggested Assessment Strategies Resources/Notes Group Activity(Discussion) Ask students to identify identical objects in the classroom (desks, books, posters, etc.). Discuss how these objects could be regarded as objects and images, and describe the transformations that relate them to each other. i.e. Desks are arranged in 5 rows with 6 desks in each row. What transformation could be applied to relate the first desk in the first row to the fourth desk in the third row? Paper and Pencil The transformation is a translation of two desks to the right, and three desks back. O is the object. A, B, C, and D are images of O. 1) Identify the coordinate pairs of the vertices for the object O and its images. 2) Describe the movement required to move from any point on O to each of its image points. Math Makes Sense 7 Lesson 8.6 Lesson 8.7 : Data Analysis TR: ProGuide, pp & pp Master 8.20, 8.13, 8.29 Master 8.20, 8.14, 8.30 PM 22 CD-ROM Masters ST: pp ST: pp Practice and HW Book pp pp Grade 7 Mathematics Curriculum Outcomes 249

18 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Specific Outcome It is expected that students will: 7SS5. Perform and describe transformations (translations, rotations or reflections) of a 2-D shape in all four quadrants of a Cartesian plane (limited to integral number vertices). [C, CN, PS, T, V] (Cont d) Achievement Indicators 7SS5.4 Describe the positional change of the vertices of a given 2-D shape to the corresponding vertices of its image as a result of a transformation, or successive transformations, on a Cartesian plane. 7SS5.5 Perform a transformation or consecutive transformations on a given 2-D shape, and identify coordinates of the vertices of the image. 7SS5.6 Describe the image resulting from the transformation of a given 2-D shape on a Cartesian plane by identifying the coordinates of the vertices of the image. Elaborations: Suggested Learning and Teaching Strategies Transformational geometry is another way to investigate and interpret geometric figures by moving every point in a plane figure to a new location. To help students form images of shapes through different transformations, students can use concrete objects such as cardboard cut-outs or geometry sets, figures drawn on graph paper, mirrors or other reflective surfaces, or appropriate technology. Some transformations, like translations, reflections, and rotations, do not change the figure itself. Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson: Can someone tell me where you might see a reflection in everyday life? Can anyone tell me what it means to rotate an object? Can anyone guess what it might mean to translate an object? Successive transformations are defined as more than one transformation being applied to an object. For example, a translation is applied to A and then a second transformation is applied to A creating A. The textbook addresses successive transformations in Lessons 8.6 and 8.7. There is no elaboration in the text on this topic but there are exercises. However, students have had prior experience with successive transformations in Grade 6. Grade 7 Mathematics Curriculum Outcomes 250

19 Strand: Shape and Space (Transformations) General Outcome: Describe and analyze position and motion of objects and shapes. Suggested Assessment Strategies Resources/Notes Paper and Pencil For the worksheets entitled Describing Transformations Worksheet and Successive Transformations Worksheet, refer to the Appendix 8 F and 8 G. Journal If a shape undergoes two transformations, one after the other, does it matter in what order they are applied? Will you get the same final image either way? Technology/Web Resources 1. A neat game addressing successive transformations aimed at ages 9-13 is available at 2. For a golf game: rm/golftrans.html This website was found at: Math Makes Sense 7 Lesson 8.6 Lesson 8.7 (continued) Examples of Islamic art, easily found on the Internet, would be useful for illustrating successive transformations. Some examples are found below. Grade 7 Mathematics Curriculum Outcomes 251

20 Strand: Shape and Space (Transformations) Grade 7 Mathematics Curriculum Outcomes 252