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FEATURE LESSON PLAN Project Sirius: Curriculum Development Angles (Grade 7/8) By Daniel Zhu 37 Par t I: Teach er M ater ial s Objectives This lesson aims to teach students: - Angle relationships for parallel lines and transversals Angle sum in a triangle is always 180° How to solve problems involving interior, complementary, supplementary, opposite, alternate, and/ or corresponding angles Preparation - Materials: paper, scissors, rulers, protractors; cardboard and markers are also helpful. Prerequisites: basic understanding of what an angle is. If students have never seen an angle before, you can begin this lesson by introducing what an angle is and then proceed with the activities. Overview One of the best ways to teach students geometric concepts is for them to visualize and explore the concepts themselves---students will be much more engaged if you give them the opportunity to draw, measure, and manipulate things, creating a more effective learning experience. These activities aim to go beyond that---to see why the angles are congruent (have the same size), students will be asked to cut out the angles and spatially rearrange them. W henever you instruct students to draw a diagram or explore, do it with them on the board, and at the same time, you can ask them engaging questions or to share their examples. To make it easy, we have provided a student handout to guide them through the lesson. They will need paper/ cardboard and scissors to cut up the angles. Depending on your students' knowledge and preferences, you may find it helpful to adjust or supplement the activities. Angles and Lines "One of the best ways to teach students geometric concepts is for them to visualize and explore the concepts themselves" Divide the students into small groups and, according to the student handout, explore with them what happens when you cut and rearrange the angles---refer to Table 1 at the end of Part I for detailed teaching instructions and/ or if students seem stuck on a concept. W hen the students begin to grasp the idea that opposite angles are congruent, ask students to share their ideas with the rest of the class and see if they can piece together some reasons as to why that is true. Remind them how they moved one angle on top of another if they are stuck. W hen most students understand why this is true, wrap it up with an example similar to the following diagram: label one of the angles, and ask them if they can figure out the rest: 38 "Too often we give childr en a nswer s to r emember r ather tha n pr oblems to solve." The same goes for transversals that intersect parallel lines: the goal is for students to visualize that corresponding angles are congruent (have the same size). Draw diagrams and cut them with the students, making sure that they see the top half can stack perfectly on top of the bottom half. -Roger Lewin Once students come to that realization, ask them, "what angles in the above diagram are equal?" Notice that we have in the above diagram: ? a = ? d = ? e= ? h ? b= ? c= ? f = ? g Angles and Triangles For triangles, we have prepared an interactive activity for the students, where you will play an important role of guidance. Ask the students to draw their favorite triangle and then measure its angles. You can play the following game with eager students a few times until they discover/ want to learn the pattern: 1. Claim that you are a magician and can guess their favorite triangle 2. Ask the students for the measure of two of the angles, then, after some acting, guess the third angle using the fact that the angles of a triangle sum to 180° 3. Repeat & vary as necessary to engage your students W hen some students appear to know what you are doing, ask them to play the game with their neighbors. For more advanced students, draw the following diagram on the board and ask them to use it to show why your "magic" works: 39 Table 1: Summary of Concepts & Ideas Concept Supplementary angles Example Diagram Potential Guiding Questions - Complementary angles - Opposite angles - Corresponding angles - Alternate angles - 40 Possible Student Conjectures W hat is ? a+? b? W hat if a, bare under the horizontal line? Do the orientations of the angles matter? W hat if ? a is less than ? b? ? a+? b=180° Or: ? a+? b?180° due to measurement error The orientations, locations, and relative size of the angles don't matter W hat is one necessary feature of this diagram? W hat is ? a+? b? Do the orientations of the angles matter? W hat if ? a is less than ? b? - There must be a right angle ? a+? b=90° Or: ? a+? b?90° due to measurement error The orientations, locations, and relative size of the angles don't matter W hich is bigger, ? a or ? c?W hat about ? band ? d? Do the orientations, sizes, and/ or locations of the angles matter? W hy is this true? Think of supplementary angles! - W hich is bigger, ? a or ? e?W hat about ? band ? f? W hich is bigger, ? cor ? g?W hat about ? d and ? h? Do the orientations, sizes, and/ or locations of the angles matter? - ? a=? e ? b=? f ? c=? g ? d=? h The orientations, locations, and sizes of the angles don't matter W hich is bigger, ? cor ? f?W hat about ? d and ? e? Do the orientations, sizes, and/ or locations of the angles matter? W hy is this true? Think of oppositeand corresponding angles! - ? c=? f, ? d=? e The orientations, locations, and sizes of the angles don't matter ? b=? c(opposite) and ? b=? f (corresponding), so ? c=? f Similar reasoning for ? d=? e - - - ? a=? c ? b=? d The orientations, locations, and sizes of the angles don't matter ? a+? b=180°=? a+? d, so ? b=? d. ? a+? b=180°=? b+? c, so ? a=? c. Table 1: Summary of Concepts & Ideas Concept Example Diagram (Same-side) Interior angles Potential Guiding Questions - - Sum of angles in a triangle - W hat is ? d+? f? W hat is ? c+? e? Do the orientations, sizes, and/ or locations of the angles matter? W hy is this true? Think of supplementary and corresponding (or alternate) angles! W hy do you think these angles are named same-side interior angles? W hat is ? a+? b+? c? W hat if the triangle has a right angle? W hat if the triangle has an obtuse angle? W hat if all three angles are acute? Do the orientations of the angles matter? Possible Student Conjectures ? d+? f=180° ? c+? e=180° The orientations, locations, and sizes of the angles don't matter ? d+? b=180° and ? b=? f (corresponding), so ? d+? f=180° Or: ? d+? c=180° and ? c=? f (alternate), so ? d+? f=180° Similar reasoning for ? c+? e=180° ? a+? b+? c=180°? If ? a=90°, ? b+? c=90° The orientations and sizes of the angles don't matter Or: ? a+? b+? c?180° due to measurement error Supplementary Materials: For More Advanced Students For more advanced students, draw the following diagram on the board to introduce them to the concept of exterior angles: triangle is 180°-100°-50°=30°, and that x+30°=180°, so x=150°. More algebraically inclined students can directly reason that x=50°+100°=150° instead. If students seem stuck, prompt them to find the third angle of the triangle. If they still seem stuck, prompt them to look for a diagram in their table that looks like the region around x in the diagram on the left. This should be enough to guide them to the correct answer. By now, these students should have a good understanding of the reasoning behind why the angle sum of a triangle is always 180°. For this concept, it's best for students to reason out the correct answer (x=150°) algebraically or by inspecting the angles in the diagram, rather than cutting-and-placing which has been the teaching tool for all of the previous concepts. Student that have gotten this far should be able to reason that the third, unknown angle of the above Finally, if there is still time, you can add a layer of abstraction to the situation by replacing the 50° and 100° with variables a and b, and then asking your students to express x in terms of a and b. Students who can grasp this level of abstraction would thus have (implicitly) proven the property that they have just observed. Try your best not to bore your more advanced students. If a student has mastered all the concepts up until here, give him/ her the Student Problem Set (homework) and instruct him/ her to get a head start on it. The rest of your students can receive Student Problem Set as homework at the end of the class. Students are welcome to collaborate with their parents on the problem set, but otherwise should work alone. 41 Par t II: Stu den t M ater ial s Supplementary & Opposite Angles Angles are cool---you see them everywhere: they form whenever you cross two or more lines, and 3 angles make a triangle. But what interesting facts can we discover about them? Draw two intersecting lines. Then, measure the angles formed by these lines using your protractor. Do you notice anything interesting? Draw another picture and do the same. Do you have a hypothesis of what's going on? Are there any angles that add up to 180°?Are there any angles that are equal to each other?W rite your thoughts in your table. Now, take one of your drawings and cut out the four angles. Can you rearrange some of them and stack them on top of each other? W hat do you notice?Does this support your hypothesis? Try doing the same for other sets of intersecting lines, then share your ideas with the class. Move on to the next section when you are comfortable with stating your hypothesis and explaining why you think you are right to others. Complementary Angles Draw two lines that intersect at a right angle (90°) and a third line that cuts this right angle into two parts (not necessarily the same), as follows: "Sorry, we are not seeing any angels today." Do you notice anything interesting about these two parts (? a and ? babove) that you just created?W rite your thoughts in your table. Transversals & Parallel Lines You may have discovered that you get equal angles when two lines intersect, but what happens when the lines are parallel? Well, parallel lines don't form angles, so we need to draw a transversal that cuts (or intersects) both of them! 42 Does this remind you of what you just saw? W hat does the above diagram look like if we only look at the angles a, b, c, and d? Measure the angles of the above diagram (a, b, c, d, e, f, g, and h) and write them down next to their labels. W hat do you notice? Now, draw a bigger picture on cardboard or paper and cut out the angles (you should have 8 pieces). Can you somehow rearrange the angles and stack some them on top of each other again? Do you see something similar between the top part (? a, ? b, ? c, ? d) and the bottom part (? e, ? f, ? g, ? h)? These angles are called corresponding angles, since they are in the same region with respect to the two parallel lines. To see that, draw another diagram similar to the one on the previous page (with two parallel lines and a transversal) and cut your diagram between the two parallel lines. Can you translate the top four angles to lie on top of the bottom ones? There are other sets of angles that are related to each other. These are called alternate angles. Think about what this word might mean geometrically and use the guiding questions in your table to guide your search for these angles. Finally, there's a third set of angles, known as (same-size) interior angles, that have a special property associated with them. Think about what this word might mean geometrically and use the guiding questions in your table to guide your search for these angles. W hat other type of angles that you've played around with before does (same-size) interior angles remind you of? W hich angles are equal? W hich add up to 180°? W rite your thoughts in your table and share your ideas with your classmates. Triangles Let's take a closer look at triangles. Triangles come in all shapes and sizes, so draw your favorite triangle! Let's play a game! Measure the 3 internal angles of your triangle and, without letting your teacher know which triangle you drew, tell him/ her two of its internal angles. Do you believe that he/ she can guess the third? Try it out a few times. Do you notice any patterns? If you think you figured it out, take turns playing it with your neighbor: can you figure out the measure of their 3rd internal angle?W rite your thoughts in your table. Can you draw a triangle whose 3 internal angles sum up to 200°? W hat happens when you try to draw such a triangle?W hat about a triangle whose 3 internal angles sum to 150°? It turns out, much like how angle ? a and ? b are supplementary angles in the above diagram and add up to ________°, the angles of a triangle always sum up to ________° (fill in the blanks!). Otherwise, the three sides simply don't come together the right way. Can you convince yourself why?Hint: usethefollowing diagram! 43 Table: Observations & Hypotheses Concept 44 Example W hat do you notice? Supplementary angles - W hat is ? a+? b? W hy do you think this is true? Complementary angles - W hat is ? a+? b? W hy do you think this is true? Opposite angles - W hich is bigger, ? a or ? c?W hat about ? band ? d? W hy do you think is this true?Think of supplementary angles! Corresponding angles - W hich is bigger, ? a or ? e?W hat about ? band ? f? W hich is bigger, ? cor ? g?W hat about ? d and ? h? W hy do you think this is true? Alternate angles - W hich is bigger, ? cor ? f?W hat about ? d and ? e? W hy do you think this is true?Think of oppositeand corresponding angles! (Same-side) Interior angles - W hat is ? d+? f?W hat about ? c+? e? W hy do you think this is true?Think of supplementary and corresponding (or alternate) angles! Sum of angles in a triangle - W hat is ? a+? b+? c? Does the type of the triangle matter? Challenge: W hy do you think this is true? Par t III: Stu den t Pr obl em Set This is your homework. It's due ______________________________ Name: ______________________________ 1. Using the table that you filled out in class, fill in the following blanks: Date: ______________________________ a. Opposite angles have the __________ measure. b. Supplementary angles add up to __________. c. ____________________ angles add up to 90°. d. W hen two parallel lines are cut by a transversal, ____________________ angles and ____________________ angles are formed, which have the same measure. e. Same-side interior angles add up to __________. f. The angles of a triangle sum to __________. 2. Figure 1 on the right shows a diagram with 9 unknown angles labeled. Determine each of their measures (in degrees). Hint: dothem in theorder given & usetheother hintsgiven below! a: __________ d: __________ g: __________ b: __________ e: __________ h: __________ c: __________ f: __________ i: __________ Hints: a: corresponding b: triangle c: opposite d: alternate e: complementary f: supplementary g: triangle h: alternate i: alternate Lorem ipsum dolor sit amet consetetur sadipscing elitr. Figure 1 3. Using your answers to the previous problem and the given hints, briefly explain how you got each of your answers. Make sure to use the right words to identify the angles! a: ____________________________________________________________________________________ b: ____________________________________________________________________________________ c: ____________________________________________________________________________________ d: ____________________________________________________________________________________ Lorem ipsum dolor sit amet consetetur sadipscing elitr. e: ____________________________________________________________________________________ f: ____________________________________________________________________________________ g: ____________________________________________________________________________________ h: ____________________________________________________________________________________ i: ____________________________________________________________________________________ 4. Figure 2 on the right shows an isosceles right-angled triangle on top of a parallelogram with 4 unknown angles labeled. Determine each of their measures (in degrees). Hint: do them in the order given and use the fact that thetriangleisisoscelesand right-angled! a: __________ c: __________ b: __________ d: __________ Lorem ipsum dolor sit amet consetetur sadipscing elitr. Figure 2 45 Par t III: Stu den t Pr obl em Set This is your homework. It's due ______________________________ Name: ______________________________ Date: ______________________________ 5. In the space below, draw your favorite diagram that contains each of the following features: ? a, ? b: A pair of supplementary angles ? g, ? h: A pair of corresponding angles ? c, ? d: A pair of complementary angles ? i, ? j: A pair of alternate angles ? e, ? f: A pair of opposite angles ? k, ? l: A pair of same-side interior angles Label one example of each of the features above in your diagram using the angle labels given (you can give more than one label to the same angle). Make sure to only draw one diagram and add numerical angle measures where appropriate! Lorem ipsum dolor sit amet consetetur sadipscing elitr. 6. Briefly explain how you know ? cand ? d in your diagram from problem 5 are complementary: __________________________________________________________________________________________ __________________________________________________________________________________________ 7. Briefly explain how you know ? eand ? f in your diagram from problem 5 are corresponding: __________________________________________________________________________________________ __________________________________________________________________________________________ 8. Briefly explain how you know ? k and ? l in your diagram from problem 5 are same-side interior: __________________________________________________________________________________________ __________________________________________________________________________________________ 9. Can a triangle have two parallel sides? If so, draw a triangle that has two parallel sides and label all of its angles and side lengths. If not, use the properties of parallel lines and triangles to explain why not. Lorem ipsum dolor sit amet consetetur sadipscing elitr. 46 Par t IV: Qu iz No calculators, rulers, or protractors. Time limit: 20 minutes. Name: ______________________________ Date: ______________________________ 1. Figure 1 on the right shows a diagram with 5 unknown angles labeled. Determine each of their measures (in degrees). Hint: noticetheequilateral triangleand theright anglein thegiven diagram! a: __________ b: __________ c: __________ d: __________ e: __________ Lorem ipsum dolor sit amet consetetur sadipscing elitr. Figure 1 2. Using your answers to the previous problem, briefly explain how you got each of your answers. Make sure to use the right words to identify the angles! a: ____________________________________________________________________________________ ____________________________________________________________________________________ b: ____________________________________________________________________________________ ____________________________________________________________________________________ Lorem ipsum dolor sit amet consetetur sadipscing elitr. c: ____________________________________________________________________________________ ____________________________________________________________________________________ d: ____________________________________________________________________________________ ____________________________________________________________________________________ e: ____________________________________________________________________________________ ____________________________________________________________________________________ Lorem ipsum dolor sit amet consetetur sadipscing elitr. 47