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1 Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson Title Math Learning Goals Expectations 1 Measuring and Construct acute, obtuse, right, and reflex angles. 7m46, 7m48 Bisecting Angles Estimate angle sizes and measure with a protractor. CGE 2a, 2c, 3f, 5a Bisect angles using a variety of methods, e.g., protractor, compass, paper folding, Mira. 2 Investigating and Classifying Triangles 3 Building and Classifying Quadrilaterals 4 Investigating Polygon and Quadrilateral Properties 5 Construct related lines (lesson not included) 6 Investigating Perpendicular Bisectors and Angle Bisectors 7 Investigating Parallel Lines 8 Investigating Related Lines Using The Geometer s Sketchpad 4 (lesson not included) Classify triangles by their sides and angles (scalene, isosceles, equilateral, acute, obtuse, right) Investigate triangle properties, e.g., the largest angle in a triangle lies across from the longest side. Classify triangles by the number of lines of symmetry they possess. Classify and name quadrilaterals and illustrate their characteristics. Classify quadrilaterals based on geometric properties, e.g., symmetry, number of equal sides, number of equal angles... Investigate the relationship between the number of sides in a regular polygon and the number of lines of symmetry it possesses. Investigate the relationship in a quadrilateral between the number of lines of symmetry and if it has 180 o or 90 o rotational symmetry. Construct lines that intersect at 30, 40, and 60, using a variety of tools and strategies. Construct perpendiculars and the perpendicular bisector of a line using a variety of tools and strategies. Use appropriate symbols to mark 90 o angles and equal line segments. Investigate perpendicular bisectors and angle bisectors of triangles. Construct parallel lines using a variety of tools. Determine angle properties created by parallel lines. Use angle properties to construct parallel lines. Investigate angles in a parallelogram. Create basic constructions using The Geometer s Sketchpad 4 Review skills using dynamic geometry software (midpoints and the midpoint line, perpendicular lines and perpendicular bisectors, bisecting angles). 7m47 CGE 4c, 5b 7m47 CGE 2b, 3c 7m47, 7m56 CGE 2b, 3c 7m46 CGE 2c, 3c 7m46, 7m47, 7m48 CGE 2a, 2b, 2c, 3c, 3f 7m46, 7m47 CGE 2c, 3c, 4c 7m46, 7m47, 7m48 CGE 4f TIPS4RM: Grade 7: Unit 6 Geometry 1

2 Day Lesson Title Math Learning Goals Expectations 9 Investigating Students use The Geometer s Sketchpad 4 to classify 7m47 Quadrilaterals Using quadrilaterals based on their sides and angles. The Geometer s Students make hypotheses related to triangles and quadrilaterals, CGE 3c, 4f Sketchpad 4 investigate using tools and strategies, and support their findings with data or find a counter-example. (lesson not included) 10 Examining Geometry Properties Using the Coordinate System Plot points on the coordinate plane in the first quadrant. Draw a triangle using ordered pairs in the first quadrant. Distinguish between similar shapes and congruent shapes. 7m48, 7m53, 7m54 CGE 2c, 2d TIPS4RM: Grade 7: Unit 6 Geometry 2

3 Unit 6: Day 1: Measuring and Bisecting Angles Grade 7 Math Learning Goals Materials Construct acute, obtuse, right, and reflex angles. compasses Estimate angle sizes and measure with a protractor. protractors Bisect angles using a variety of methods, e.g., protractor, compass, paper folding, Miras BLM 6.1.1, 6.1.2, Mira Assessment Opportunities Minds On Whole Class Demonstration Develop four different ways to describe a straight angle using the headings: mathematical characteristics, everyday examples, diagram, and explanation. (See BLM for sample responses.) Groups of 4 Exploring Angles Post eight pieces of chart paper around the room. In groups of four, students focus on a specific angle, i.e., acute, right, obtuse, and reflex. Each angle is done twice. They define the angle and show examples, using available resources, books, Internet, etc. Facilitate a class discussion using prompts such as: How did each group classify the angle? (by its degree range) Which angle(s) seems most common in the everyday world? Reflect on and explain why. (responses will variable) Alternatively, use the Frayer model (BLM 5.1.1). It is important that all four angles are represented. Word Wall bisect acute angle obtuse angle right angle reflex angle estimate Action! Groups of 4 Practice Students complete Part A (BLM 6.1.2) and reflect after each measurement: Do we need to revise our estimates? Are our estimates within 10º? Whole Class Demonstration Demonstrate how to bisect using a Mira, a compass, paper folding, and a protractor and mark equal angles using proper notation. Students complete each bisection, marking equal angles on BLM 6.1.2, Part B. Individual Reflection Students reflect, using guiding questions: What happened to the original angle? (bisected) What does bisect mean? (divides angle into two equal parts) How does this method compare to the others, i.e., compass, Mira, paper folding, and protractor? (responses will variable) Lesson may vary depending on what protractors are available (360 or 180 ). Consolidate Debrief Individual Practice: Bisecting Angles Students complete BLM 6.1.3, Part C. Ask: What do you notice about the two new angles created after bisecting the original angle? (They are equal.) What conclusions can you draw? (Bisecting an angle divides it into two new equal angles.) Curriculum Expectations/Observation/Mental Note: Assess students ability to bisect angles using at least two methods. Demonstrate paper folding using a prepared angle on a piece of paper. Copy protractors on overhead acetates and cut up for Home Activity. Practice Home Activity or Further Classroom Consolidation Using a protractor, a compass, and paper folding, complete the worksheet TIPS4RM: Grade 7: Unit 6 Geometry 3

4 6.1.1: Straight Angles (Teacher) Angle Type: Straight Mathematical Characteristics Everyday Example Angle is composed of two rays that meet at a common end point (vertex). The straight angle always measures 180º. Cereal Box Diagram Explanation A straight angle is 180º. It lies on any point along a straight line segment. Therefore any straight line has a straight angle. TIPS4RM: Grade 7: Unit 6 Geometry 4

5 6.1.2: Estimating, Measuring, and Marking Angles Part A Part B Bisecting Angles Bisect all angles in Part A, marking all equal angles, using a Mira for questions 1 and 4, compasses for questions 2 and 5, paper folding for question 3, and a protractor for question 6. TIPS4RM: Grade 7: Unit 6 Geometry 5

6 6.1.3: Bisecting Angles Part A Use a compass to bisect and measure the following angles. Mark equal angles. Part B Use a protractor to bisect and measure the following angles. Mark equal angles Part C On the back of this sheet draw 3 different types of angles and bisect by folding paper. Mark equal angles. TIPS4RM: Grade 7: Unit 6 Geometry 6

7 Unit 6: Day 2: Investigating and Classifying Triangles Grade 7 Math Learning Goals Classify triangles by their sides and angles (scalene, isosceles, equilateral, acute, obtuse, right). Investigate triangle properties (e.g., the largest angle in a triangle lies across from the longest side). Classify triangles by the number of lines of symmetry they possess. Assessment Opportunities Minds On Pairs Word and Picture Sort Students sort the set of cards into categories and explain their reasoning and criteria for the sort. Students then sort the cards in a different way, using different criteria (BLM 6.2.1). Action! Whole Class Discussion Discuss lines of symmetry and how this relates to the type of triangle. Pairs Investigation Students create a variety of triangles to determine the relationship between length of sides and angle sizes. Students can use geoboards, straws, paper strips, or GSP 4 to help them investigate the relationship. Focus the students attention on looking for the relationship between side length and opposite angle sizes: What relationship can you find between the length of the sides in a triangle and the size of the opposite angle? Students sketch a triangle, measuring and recording the size of each angle and the length of each side. They create enough triangles to notice a relationship and record their observations, e.g., the largest angle lies across from the longest side; the smallest angle lies across from the smallest side; if two angles are equal, then the two opposite sides are equal. Circulate, taking note of which students to ask to share in the whole-class discussion. Reasoning and Proving/Oral Questions/Checklist: Assess students ability to determine triangle properties through investigation. Materials geoboards straws paper strips protractor BLM Focus the students attention on what proving, reasoning, reflecting, and communicating looks like and sounds like. Record types of triangles on Word Wall. Consolidate Debrief Whole Class Discussion Students choose a triangle to share with the class and cut it out for posting. One student shares a sample triangle, then another student shares a different type of triangle. Continue until there are enough samples to discuss the relationship. Post their samples in categories (scalene; equilateral; isosceles; obtuse; right; 1, 2, 3 lines of symmetry). Discuss and record the relationships students discovered. Connect to lines of symmetry as well. Reflection Home Activity or Further Classroom Consolidation Write a journal report about discoveries you made about the relationship between the length of the sides in a triangle and the size of the opposite angle. Illustrate your report with diagrams. Find pictures of quadrilaterals used in daily life and bring them to class. Collect and assess students journal entries. TIPS4RM: Grade 7: Unit 6 Geometry 7

8 6.2.1: Classifying Triangles Word Sort Cut apart and place the set of cards in an envelope. Make sufficient copies for students to work in pairs. 1 line of symmetry Equilateral triangle All three angles are equal Obtuse-angled triangle All three angles are acute Right-angled triangle 3 lines of symmetry Two sides are equal Isosceles triangle Three sides are equal Acute-angled triangle Two angles are acute 0 lines of symmetry Scalene triangle 1 angle is obtuse TIPS4RM: Grade 7: Unit 6 Geometry 8

10 6.3.1: Anticipation Guide Name: Date: Before Statement After 1. Agree/Disagree The sum of all interior angles in a quadrilateral is equal to 360 degrees. Agree/Disagree 2. Agree/Disagree In a quadrilateral there is a relationship between the number of lines of symmetry and the number of pairs of equal sides. Agree/Disagree 3. Agree/Disagree In a quadrilateral the number of equal sides is always related to the number of equal angles. Agree/Disagree Name: Date: Before Statement After 1. Agree/Disagree The sum of all interior angles in a quadrilateral is equal to 360 degrees. Agree/Disagree 2. Agree/Disagree In a quadrilateral there is a relationship between the number of lines of symmetry and the number of pairs of equal sides. Agree/Disagree 3. Agree/Disagree In a quadrilateral the number of equal sides is always related to the number of equal angles. Agree/Disagree TIPS4RM: Grade 7: Unit 6 Geometry 10

12 6.4.1: Lines of Symmetry and Rotational Symmetry B A C Number of Lines of Symmetry: How many times do the shapes match in one full rotation? Rotational Symmetry: Number of Lines of Symmetry: How many times do the shapes match in one full rotation? Rotational Symmetry: Number of Lines of Symmetry: How many times do the shapes match in one full rotation? Rotational Symmetry: B A C Number of Lines of Symmetry: How many times do the shapes match in one full rotation? Rotational Symmetry: Number of Lines of Symmetry: How many times do the shapes match in one full rotation? Rotational Symmetry: Number of Lines of Symmetry: How many times do the shapes match in one full rotation? Rotational Symmetry: TIPS4RM: Grade 7: Unit 6 Geometry 12

13 6.4.2: Who s On the Computer? Your task is to develop a system to make sure each student has equal opportunity and time to use the computers. Use three different quadrilaterals to develop a system based on rotational symmetry for 15-minute (square: 90 rotation), 30-minute (rectangle: 180 ), and 60-minute (360 ) activities. Type of Quadrilateral Lines of Symmetry Rotational Symmetry (Degrees) Questions for Reflection: 1. Is there a relationship between the number of lines of symmetry and degrees of rotational symmetry? Explain your reasoning. 2. Describe other quadrilaterals that you did not experiment with that might be used for the 15-minute, 30-minute, or 60-minute activities? Explain how they can be used. TIPS4RM: Grade 7: Unit 6 Geometry 13

14 Unit 6: Day 6: Investigating Perpendicular Bisectors and Angle Bisectors Grade 7 Math Learning Goals Construct perpendiculars and the perpendicular bisector of a line using a variety of tools and strategies. Use appropriate symbols to mark 90 angles and equal line segments. Investigate perpendicular bisectors and angle bisectors of triangles. Minds On Whole Class Guided Investigation Guide students through a series of investigations to help them understand that there are similarities and differences between a perpendicular line and perpendicular bisectors (BLM 6.6.1). Curriculum Expectation/Mental Note: Observe students understanding of perpendicular lines and bisectors. Assessment Opportunities Materials board compass protractor board protractor Mira BLM A perpendicular line is not necessarily a perpendicular bisector. Action! Groups of 3 Investigation Use a cut-out sample of a triangle to demonstrate how to find the perpendicular bisector of one of the sides of a triangle. Problem to Investigate What are the special properties of the perpendicular bisectors of the 3 sides of a triangle? Students investigate what happens when they draw the perpendicular bisector for all three sides of an isosceles triangle and individually record what they notice, including a description, a diagram, and characteristics. They repeat this investigation with scalene and equilateral triangles to determine if other triangles have the same properties. Individual Investigation Problem to Investigate What are the special properties of the 3 angle bisectors of a triangle? Students investigate what happens when the angles of an isosceles, an equilateral, and a scalene triangle are bisected, and record their findings. Problem Solving/Observation/Anecdotal Note: Assess how students apply the problem-solving process during the investigation. Word Wall perpendicular lines bisector perpendicular bisector Consolidate Debrief Whole Class Discussion Record students findings on an overhead under Description, Diagram, Characteristics. Students add to and edit what they have included. Exploration Practice Home Activity or Further Classroom Consolidation Write a journal entry to describe what you learned about perpendicular bisectors and line bisectors. TIPS4RM: Grade 7: Unit 6 Geometry 14

15 6.6.1: Investigating Perpendicular Bisector and Angle Bisectors (Teacher) Part A: Perpendicular to a Line 1. Review the meaning of perpendicular lines. 2. Ask what tools they could use to create a perpendicular to a line segment. (Mira, paper folding, protractor, etc.) 3. Students draw a line segment using a pencil and ruler and create a perpendicular line segment by folding paper. 4. Demonstrate how to check if the line segment is perpendicular? (Use the corner of a ruler, the corner of a book, a protractor.) 5. Ask: Is there another line perpendicular to this line segment? Demonstrate another perpendicular. 6. Students find other perpendiculars to their line segment by paper folding, using a Mira, or using a protractor. 7. Ask: What is the relationship between these perpendicular lines? (They are parallel.) What kind of angles are created when a perpendicular line segment is created? 8. Mark it on the sample on the board. 9. Students mark these on their own diagram. 10. Ask: Does the perpendicular always cut the line in half? (No, it just creates a 90 angle.) 11. Demonstrate using folding and using a board protractor. Part B: Perpendicular Bisector 1. Ask: What does it mean when we bisect a line segment or an angle? How can we create a perpendicular bisector of a line segment? (folding, using a ruler, using a protractor, etc.) 2. Students, draw another line segment and create a perpendicular bisector, using paper folding. 3. Ask: How many perpendicular bisectors can you create for any given line segment? How do you know? How is this different from creating perpendicular lines? 4. Demonstrate how to mark 90 and equal line segments. 5. Students mark the 90 and equal line segments on their line. Part C: Finding the Perpendicular Bisector Using a Compass Demonstrate how to find the perpendicular bisector with a compass as an alternative way to find a perpendicular bisector. TIPS4RM: Grade 7: Unit 6 Geometry 15

16 Unit 6: Day 7: Investigating Parallel Lines Grade 7 Math Learning Goals Construct parallel lines using a variety of tools. Determine angle properties created by parallel lines. Use angle properties to construct parallel lines. Investigate angles in a parallelogram. Minds On Whole Class Review Provide students with two different-coloured sticky notes. Post the following statements in columns: Perpendicular bisectors meet inside triangles: (1 st colour) all the time sometimes never Angle bisectors meet inside the triangle: (2 nd colour) all the time sometimes never Students place the sticky note on the columns they deem as correct. Facilitate a whole-class discussion around student responses. Whole Class Demonstration Using a board compass, demonstrate how to construct parallel lines using a compass and ruler. Discuss the properties of parallel lines used to do this construction. Construct a parallelogram. Students practise constructing parallel lines. Assessment Opportunities Materials sticky notes (two colours) geoboards compasses protractors blank paper Answer: Perpendicular bisectors: sometimes (obtuse triangles do not meet inside triangles). Angle bisector: all the time These answers can be demonstrated using GSP 4 and a data projector. Action! Consolidate Debrief Concept Practice Pairs Investigation Students fold a blank paper to create two parallel lines, and then trace the lines with a ruler and pencil (make parallel lines approximately 10 cm apart). They then draw a diagonal line, called a transversal, that crosses the parallel lines at an angle (not 90º). Students measure and compare all angles and compile a list of all angle patterns they discover. They verify their results by making two parallel lines on a geoboard with a transversal and repeating their angle measurements. Selecting Tools/Demonstration/Anecdotal Note: Assess students selection and use of tools to measure angles. Whole Class Discussion Use a diagram of two parallel lines intersected by a transversal at 45º to initiate discussion. Discuss and record student findings. Ask: What do you notice? How can this angle property (F pattern or Z pattern) be used to construct a third parallel line? (I know that the transversal cuts both parallel lines at 45, so I could draw a third parallel line that intersects with the diagonal at 45. This new line is now parallel to the original two parallel lines.) Home Activity or Further Classroom Consolidation Construct parallelograms using pairs of parallel lines created by paper folding or by using a compass. Measure the interior angles of the parallelogram to determine any angle relationships that exist. Verify findings by constructing another parallelogram with a different orientation and measuring its angles. Describe your findings in a journal entry. Include welllabelled diagrams. Ask students how to ensure that the paper folding has created parallel lines. (Measure the distance between the lines at each end, perpendicularly, with a ruler.) Students should discover angle relationships in F, Z and C patterns. The terminology corresponding, alternate and interior angles are Grade 8 expectations. Assess individual student reasoning and proving and communication from the written journal entries. TIPS4RM: Grade 7: Unit 6 Geometry 16

18 6.10.1: Triangle Investigation Using Grid Paper and Midpoints 1. Draw in the y-axis and the x-axis on your grid paper, and label the scale. 2. Draw a triangle. 3. Label the ordered pairs of the triangle s vertices. 4. Find the midpoints of the sides of the triangle using a ruler or by paper folding. Join the midpoints to create an interior triangle. 5. Label the midpoint ordered pairs to the nearest tenth. 6. Investigate the new characteristics: a) Mark parallel sides. b) Mark equal angles. c) Mark the lengths of the sides. d) What relationship exists between the newly created triangles? How do you know? 7. Record all your findings. TIPS4RM: Grade 7: Unit 6 Geometry 18

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