Analyzing Congruency Proofs

Transcription

1 Lesson #28 Mathematics Assessment Project Formative Assessment Lesson Materials Analyzing Congruency Proofs MARS Shell Center University of Nottingham & UC Berkeley Alpha Version Please Note: These materials are still at the alpha stage are not expected to be perfect. The revision process concentrated on addressing any major issues that came to light during the first round of school trials of these early attempts to introduce this style of lesson to US classrooms. In many cases, there have been very substantial changes from the first drafts new, untried, material has been added. We suggest that you check with the Nottingham team before releasing any of this material outside of the core project team. If you encounter errors or other issues in this version, please send details to the MAP team c/o 2012 MARS University of Nottingham

2 Analyzing Congruency Proofs Teacher Guide Alpha Version January Analyzing Congruency Proofs Mathematical goals This lesson unit is intended to help you assess how well students are able to: Work with concepts of congruency similarity, including identifying corresponding sides angles within between triangles. Identify underst the significance of a counter-example. Prove, evaluate proofs in a geometric context. Common Core State Stards This lesson relates to the following Mathematical Practices in the Common Core State Stards for Mathematics: 3. Construct viable arguments critique the reasoning of others. 5. Use appropriate tools strategically. 7. Look for make use of structure. This lesson gives students the opportunity to apply their knowledge of the following Stards for Mathematical Content in the CCSS: G-CO: Underst congruence in terms of rigid motions. Prove geometric theorems. G-SRT: Underst similarity in terms of similarity transformations. Prove theorems involving similarity. Introduction In this lesson, students work on the concept of congruency whilst developing their understing of proof in a geometric context. Before the lesson, students complete a task designed to help you underst their current levels of understing of congruency proof. You analyze their responses, write questions to help them to improve their work. The lesson begins with an interactive whole-class discussion about establishing conditions for congruency from triangle properties. Students work alone to decide on the truth of a conjecture about congruency conditions for triangles. They then work in pairs to share their ideas, produce justify a joint response to conjectures. Working in the same pairs, they analyze sample responses produced by other students. In a whole-class discussion, you work with students to develop their understing of proof in this context. In a follow-up lesson students then use what they have learned from the lesson to improve their responses to a task similar to the initial assessment task. Materials required Each student will need a copy of the pre-assessment task, Finding Congruent Triangles, the sheet Instructions: Must the two triangles be congruent?, the cut-up cards Cards: Must the two triangles be congruent?, the post-assessment task, Finding Congruent Triangles (again.), a glue stick. Each small group of students will need a copy of the Sample Student Proofs. Provide paper, scrap paper, rules, compasses, protractors as needed. A board compass, protractor rule would be useful. There are slides to support class discussion. Time needed Twenty minutes before the lesson, a ninety-minute lesson (or two forty five minute lessons), twenty minutes in a follow-up lesson (or for homework). All timings are approximate MARS University of Nottingham 1

3 Analyzing Congruency Proofs Teacher Guide Alpha Version January 2012 Before the lesson Pre-Assessment task: Finding Congruent Triangles (20 minutes) Give each student the assessment task Finding Congruent Triangles, a rule, a compass, a protractor. Supply students with extra paper as needed. Introduce the task briefly help the class to underst the problem its context. You are asked to think about properties of triangles. What does property mean here? (Side lengths, angles, right triangles, scalene, isosceles, equilateral.) Suppose I have a triangle with a side 3" long it is isoscles. Your triangle has the same properties. Must your triangle be congruent to mine? [No.] Think about each statement, decide what s true, make sure to give good reasons for your answer. You might want to sketch, write, or construct diagrams to help explain your reasoning. I would like you to work alone, for fifteen minutes, answering these questions. Reassure students who do not complete the work: Don t be too concerned if you did not complete all the questions. We will be working on a lesson [tomorrow] that should help you improve your work. Assessing students responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understing, any difficulties they encountered. We suggest that you do not score students work. The research shows that this will be counterproductive as it will encourage students to compare scores distract their attention from what they might do to improve their mathematics. Instead, help students to make progress by summarizing their difficulties as a series of questions. Some suggestions for these are given on the next page. These have been drawn from common difficulties observed in the trials of this unit. We recommend that you write a selection of these questions on each piece of student work. If you do not have time, select a few questions that will be of help to the majority of students. These can be written on the board at the start of the subsequent lesson. 1. Finding Congruent Triangles Is Adeline correct to say that all triangles with these properties must be congruent? Explain your answer. 2. Solomon says, Adeline says, Ernie says, Suzie says, Is Suzie correct to say all triangles with these properties must be congruent? Explain your answer. Adeline, try to draw a triangle congruent to my triangle. It has one angle of 30º; another angle is 40º. That s easy Solomon, because all triangles that have those properties must be congruent. Suzie, draw a triangle congruent to my triangle. My triangle is a right triangle. One side is 1 long. That s easy, Ernie, because all triangles that have those properties must be congruent MARS University of Nottingham 2

4 Analyzing Congruency Proofs Teacher Guide Alpha Version January 2012 Common issues: Student draws unhelpful diagrams For example: The student draws triangles that do not have the given properties. Or: The students range of diagrams is restricted a counter-example is not found. Student does not use a counter-example correctly For example: The student does not draw a diagram showing a counter-example, falsely concludes that a conjecture is true. Or: The student does not recognize a triangle he/she has drawn as a counter-example. Or: The student draws multiple counter-examples to show a general statement is false. Student uses inductive reasoning For example: The student draws several examples, concludes that if the side is contained between two angles, all the triangles with the set of properties will be congruent (Q3). Although this is true, the reasoning is mathematically weak. Student provides inadequate justification For example: The student draws diagrams but does not explain them. Or: The student writes some facts derivations, but does not join them together into a cogent proof. Suggested questions prompts: How can you be sure you have tried enough different triangles? What properties must your triangles all have? You have drawn two right triangles; each has a 1" side. They are not similar to each other. What does this show you? (Q1) In all your examples, the side between the angles is the same length. Is this one of the set of properties in the question? What would you need to do to show that the statement all triangles with these properties are congruent is false? How many counter-examples are needed to show that a statement is false? How can you be sure that there is no triangle for which Lucy s statement is false? Try to find a stronger reason for supposing Lucy is correct. Write down in words how your work shows that the claim is true/false. Imagine you are trying to get a student in another class to believe you are correct from what you write. How much detail will you need to write down? MARS University of Nottingham 3

5 Analyzing Congruency Proofs Teacher Guide Alpha Version January Suggested lesson outline Introduction (20 minutes) Remind students of their work on the assessment task. I read through your scripts on Finding Congruent Triangles. I have written some questions about your work. Next lesson, I ll ask you to use my questions, what you learn today, to try some more questions like them. Give each student the sheet Instructions: Must two cards be congruent?, the card set Cards: Must the two triangles be congruent?, a rule, a compass, a protractor, a sheet of paper on which they will write solutions. Usually we would say sides of equal length are congruent. Just for today, we will use congruent only when we re talking about triangles, not sides or angles. That should help us to keep our ideas clear! Explain the activity with two examples. Make one example simple the other more complex. Example 1. Ask students to select Card One side of Triangle A is the same length as one side of I would like you to think about side lengths angles in two triangles; Triangle A Suppose one side of Triangle A is the same length as one side of Draw a pair of triangles A B that satisfies this condition. Make up some side lengths angles on your example. After students have had sufficient time to work on the task individually, ask them to show their examples. Take a look around. You ve all drawn different pairs of triangles. Are your triangles congruent? Mylo, show the class your pair of triangles. Stacey, are Mylo s triangles congruent? Why do you think that? Mylo s example shows us you can make Triangle A congruent to Kirsty, show the class your pair of triangles. Ruth, are Kirsty s triangles congruent? Why not? Emphasize the key questions: Must Triangle A be congruent to Triangle B? Is it possible that they are not congruent? In this case, it is clear that they can be congruent but do not have to be congruent. Example 2. Now students are to select Card Two sides of Triangle A are the same lengths as two sides of Triangle B one angle in Triangle A is the same size as one angle in Now draw me a pair of triangles that satisfies these conditions. Many students will produce a pair of congruent triangles Lets think about the key question again: Must Triangle A be congruent to Triangle B? Is it possible that A B are not congruent? 2012 MARS University of Nottingham 4

6 Analyzing Congruency Proofs Teacher Guide Alpha Version January Challenge students to find a pair of triangles that satisfy the conditions, but are not congruent. Give them a few minutes to think about this individually, then continue. Show how such a pair of triangles may be constructed, using a rule compass. The diagrams below show one way of doing this Individual work: Proving Congruency Conditions (15 minutes) Explain the purpose of the lesson. In this lesson, we re going to try to test out some more properties. Choose a card, draw some pairs of triangles that have the properties for that card, see if you can reach a decision. In particular, I want all of you to be sure to work on triangles with the properties shown in Card 8. Slide 1 of the projector resource; Must two triangles be congruent? describes how students should approach the task. Supply students with extra paper as needed. Ask students to work alone on at least two sets of properties. If students have difficulty getting started, suggest that they begin by drawing a triangle, then try to draw another triangle with one of the angles the same size, two of the side lengths the same. Encourage students to work towards a conjecture by trying a range of different examples. Whilst students work, you have two tasks: to notice students approaches to proof, to support their reasoning. Notice students approaches to proof Notice how students go about drawing examples as they conjecture. Do they tend to draw triangles that are similar in shape, size, orientation? Do they systematically vary properties look for extreme cases? Observe how students use their examples to reach a conclusion. Do they notice counter-examples? Do they provide evidence that they grasp the significance of a single counter-example? Do they make inductive generalizations based on a few or several cases? How typical are those cases: do they all share further, unspecified properties? Do they use transformation geometry? What is the quality of students written conjectures justifications? Do they refer back to their prior knowledge of congruency conditions? Do they clarify added constraints? Do they explain why their examples are relevant? Do students write down enough of their reasoning to make the solution transparent for the reader? You can use the information to support students reasoning, also to focus the whole-class discussion towards the end of the lesson MARS University of Nottingham 5

7 Analyzing Congruency Proofs Teacher Guide Alpha Version January Support students reasoning Try to support students in their proving activities, rather than steering them towards any particular method of proof. Prompt students to consider a wide range of examples, including variation in the relative positions of sides angles: Does it make any difference where the sides go where the angle goes? How do you know? Have you drawn a wide range of possible pairs of triangles? How did you choose your examples? Encourage students to write down their explanations, giving clear references to diagrams, using labeling, including lots of small details. You ve drawn some examples of triangles with that set of properties. What conclusion did you reach? Can you explain that in writing? Imagine you are writing this explanation for someone who is in an earlier grade. How much do you need to write to make your reasoning clear for a reader? Have you considered adding a diagram? Explain to your reader how the diagram helps. Help students to clarify the meaning of the questions, to use clear language. Encourage students to clarify the logic of the conditions on which they work. You may also want to use questions like those in the Common issues table to help students address their errors misconceptions. Small group work: Producing a joint response (20 minutes) Once everyone has produced some work to share on at least two sets of properties, organize students into pairs or small groups of three. Give each group two or three sheets of paper a glue stick. Ask students to take turns to explain their work to other members of the small group. Select a card one of you has worked on. Glue it to the middle of a blank sheet. Take turns to explain your work to the others. Explain your conclusion, how you reached that conclusion. Make sure everyone in your group understs your diagrams. When everyone who has worked on the same card has had a turn, work together to reach a joint decision. On your sheet produce an explanation together that is better than your individual explanations. Make sure to discuss Card 8. Slide 2 of the projector resource; Working Together, summarizes these instructions. Observe, listen, support students in their proving activity as before. When talking, students may point, use terms such as this angle that one rather than producing labels. It may be helpful to ask questions to get them to say the name of the triangle, to use the labels for angles sides. If students are able to do this when they talk, it may help students to be clearer in their written work. Encourage students to share their thinking in their groups, support those listening. If one student provides some reasoning, ask the other to state it or write it in his/her own words. Then ask the first student if the second s representation was correct, to encourage joint sense-making MARS University of Nottingham 6

8 Analyzing Congruency Proofs Teacher Guide Alpha Version January Small group work: Analyzing congruency proofs (15 minutes) When students have had sufficient time to work together on at least two sets of properties give each small group of students a copy of the two Sample Congruency Proofs. This work shows two attempts to tackle Card 8. Here is some work on this question produced by students in another class. Work in your pairs on one piece of work at a time. Read the work carefully, try to figure out what the explanation is about. Answer the questions about the student s reasoning that are written on the sheet. Try to help each other underst what that student was doing. During this small group work, observe support students as before. Notice students analytical activities. Do students underst what Kieran s diagram shows? Do they recognize this as a counter-example to the conjecture that all the triangles with these properties are congruent? Do they identify the relevance power of a counter-example? What connections do they make between the two proofs? Do they identify that when a further constraint is added, that the angle is between the two sides (SAS), all the triangles will be congruent? Do they notice the assumptions that Jorge has made? Support student proving activities. Ask students to explain the reasoning in the Sample Congruency Proofs, in particular, the links between words diagrams. Why do you think Kieran drew two circles? What connection do circles have with the set of triangle properties he worked with? What do you know from other math lessons about translations? constructions? How does that information help you underst critique Jorge s response? Ask students questions to guide their evaluation of the Sample Congruency Proofs. Do you agree with [Jorge s/kieran s] conclusion? Why?/Why not? Can they both be correct? You have done a lot of thinking together to underst this proof. How much of that thinking was written in the Sample Proof? How much extra did you have to do? If you have time to discuss the work with the class, there are slides showing the two proofs. Jorge Jorge adds an unacknowledged constraint to the given conditions. Jorge s solution is a proof of the congruency condition SAS. Without further information, we cannot know whether Jorge has realized that knowing two sides one angle does not, in general, specify congruent triangles. (Jorge s proof is intended to approximate Proposition 4 in Euclid s Elements (Book 1). The technique of the proof is to show that by applying or imposing one triangle on another, the parts of one triangle can be shown to correspond to the parts in another. This is not an uncontroversial method, but is intuitively acceptable for students, may be seen, perhaps, as a use of the rigid transformation, translation.) 2012 MARS University of Nottingham 7

9 Analyzing Congruency Proofs Teacher Guide Alpha Version January 2012 Kieran Kieran does little to communicate his reasoning to the reader. This is a major weakness with his proof: he makes no effort to explain his reasoning. Kieran provides a clever counter-example, by producing two triangles that have the required properties but are not congruent. The first side, x, is fixed, a circle is drawn with radius y > x. This is the locus of the point for the vertex of the known side y not affixed to x. The fixed angle, z, can either be placed between the two known sides, or just attached to one of them Whole-class discussion (20 minutes) Tell students that you are going to finish the lesson by discussing the rest of the cards. Begin by asking students about Card 8. Project Slide 3 of the projector resource. Look at this statement If there is disagreement, ask questions to clarify the issue, organize a debate until students are in agreement. Then ask students to look for ways to reduce the amount of work in completing the rest of the cards. Which cases do you think will be simplest? [It may be easier to work with fewer properties. It s easier to disprove than prove.] Ask students to explain their reasoning when they make a claim. Check whether those listening underst these explanations. If someone agrees, ask him to explain in his own words. If he does not, he can explain why MARS University of Nottingham 8

10 Analyzing Congruency Proofs Teacher Guide Alpha Version January Project Slide 6 of the projector resource. Cards: Must the two triangles be congruent? One side of Triangle A is the same length as one side of Two sides of Triangle A are the same lengths as two sides of Three sides of Triangle A are the same lengths as three sides of One side of Triangle A is the same length as one side of Triangle B one angle in Triangle A is the same size as one angle in Two sides of Triangle A are the same lengths as two sides of Triangle B one angle in Triangle A is the same size as one angle in 6. Three sides of Triangle A are the same lengths as three sides of one angle in Triangle A is the same size as one angle in One side of Triangle A is the same length as one side of Triangle B two angles in Triangle A are the same sizes as two angles in Alpha Version January 2012 Choose one of the remaining cards. Ask students to work on this on their mini-whiteboards collect their ideas. Ask students to share their reasoning. Show me what you have written so far. Xi, you think triangles with these properties must be congruent. Tell us why. Alan, do you agree? Explain your answer. It is more important that students explore a few cases in detail than that the whole table is finished. To end the lesson, ask students to explain in general how to show that a general statement is false. Make sure they underst the notion of a counter-example, the effect of a counter-example on the truth of a generalization, that only a single counter-example is required. Follow-up lesson: Finding congruent triangles (again) (20 minutes) Give students your comments from the individual pre-assessment task. If you have chosen not to write questions on individual scripts, write your list of questions on the board now. Give a copy of the post-assessment task: Finding congruent triangles (again) to each student. Supply students with extra paper as needed. Look at your original responses think about what you have learned this lesson. Carefully read through the questions I have written. Spend a few minutes thinking about how you could improve your work. You may want to make notes. Two sides of Triangle A are the same lengths as two sides of Triangle B two angles in Triangle A are the same sizes as two angles in 2011 MARS University of Nottingham Three sides of Triangle A are the same lengths as three sides of two angles in Triangle A are the same sizes as two angles in Projector Resources: Using what you have learned, try to answer the questions on the new task, Finding congruent triangles (again). When you think your work is complete, explain in words, using examples diagrams if you wish, what a counter-example is MARS University of Nottingham 9 6

11 Analyzing Congruency Proofs Teacher Guide Alpha Version January 2012 Solutions Finding Congruent Triangles 1. Adeline is incorrect: it is not true that all triangles with angles of 30º 40º must be congruent. The student could show this with a pair of triangles, showing two similar triangles of unequal size. Students might notice that the triangles are similar but this is not part of the question. 2. Suzie is incorrect: it is not true that all right-triangles with a side 1" long must be congruent. The student could show this by drawing a pair of triangles, for example, by showing two similar noncongruent right triangles or with non-similar right triangles with a common hypotenuse Wally is incorrect: it is not true that all triangles with a side 2" long, angles of 50º 30º are congruent. A single counter-example showing two non-congruent triangles with these properties is enough to answer this question. Students may have developed understing of adding a constraint, consider the effect of containing the fixed-length side between the two angles. If the triangles are AAS they will always be similar, but not necessarily congruent. The student could explain this in terms of the fixed angle sum of triangles (so two angles entails similarity) then exploring rigid transformations of the images. If the triangles are ASA then they must always be congruent. This could be argued by considering rigid transformations (reflections, translations, rotations) of a diagram showing that all images are congruent. Just drawing a few diagrams without the transformational reasoning would be an alternative weaker method of justification. Finding Congruent Triangles (again) These solutions are analogous to those above, so we will not repeat all the reasoning. 1. Ben is incorrect: it is not true that all triangles with angles of 70º 50º must be congruent. 2. Sanjay is incorrect: it is not true that all triangles with an angle of 60 a side length of 2" must be congruent. 3. Max is incorrect: it is not true that all triangles with a side 1" long, angles of 90º 20º are congruent MARS University of Nottingham 10

12 Analyzing Congruency Proofs Teacher Guide Alpha Version January Main lesson activity: Must the two triangles be congruent? The table below shows that the only conditions that ensure congruency are given in the right h column, where the three sides of triangle A are the same lengths as the three sides of triangle B. In all other cases, it is not necessary for the triangles to be congruent a single counter example will prove this each time. Below the table we offer some notes on particular cards, referring to the numbers in the cards. 1. One side of triangle A is the same length as one side of triangle B. 2. Two sides of triangle A are the same lengths as two sides of triangle B. 3. All three sides of triangle A are the same lengths as all three sides of triangle B. Not necessarily congruent Not necessarily congruent Must be congruent 4. One side of triangle A is the same length as one side of triangle B one angle in triangle A is the same size as one angle in triangle B. 5. Two sides of triangle A are the same lengths as two sides of triangle B one angle in triangle A is the same size as one angle in triangle B. 6. All three sides of triangle A are the same lengths as all three sides of triangle B. one angle in triangle A is the same size as one angle in triangle B. Not necessarily congruent Not necessarily congruent Must be congruent 7. One side of triangle A is the same length as one side of triangle B two angles in triangle A are the same sizes as two angles in triangle B. Not necessarily congruent (the triangles are similar) 8. Two sides of triangle A are the same lengths as two sides of triangle B two angles in triangle A are the same sizes as two angles in triangle B. Not necessarily congruent (the triangles are similar) 9. All three sides of triangle A are the same lengths as all three sides of triangle B. two angles in triangle A are the same sizes as two angles in triangle B. Must be congruent Students may try to show this by drawing several pairs of examples. Reasoning inductively, they might conclude that every pair of triangles with these properties will be congruent. Inductive reasoning is not mathematically compelling, but there is research evidence that suggests many students gain conviction this way. A fairly compelling proof accessible to students in High School would be to start to construct a generic example a typical triangle that does not have extra special properties such as angles of equal size - with two fixed side lengths. The student can then consider how many different triangles it is possible to make given a third, fixed length. 5. Strictly speaking, a single counter-example will establish this. However, the relative positions of angle sides matter here: if the angle is contained by the two sides the triangles must be congruent. Counter-examples occur when the angle is not contained between the two sides. So adding an extra property the containment of the angle between the two sides produces the familiar SAS condition. 6. This follows from 3. This follows from the fact that a triangle is determined by three side lengths. Adding a further property, an angle, will not change that. 7. These triangles will be similar, because the angles are equal. A counter-example can be produced by scaling, to change the position of the side of fixed length relative to the angles. Containing the known 2012 MARS University of Nottingham 11

13 Analyzing Congruency Proofs Teacher Guide Alpha Version January side between the two known angles adding the extra property produces the familiar ASA congruency condition. 8. This is perhaps the most difficult example. First notice that knowing two angles, we know three, so any counter-example will involve similar triangles. A counter-example is a single pair of similar triangles in which two sides are the same lengths in each triangle, but they are non-equivalent sides rotated around the triangle. $%&" #" '%(&" $%&" !" We do not expect student to generalize this, but include the proof for your interest: #"!#!"#!# $! ! These side lengths only specify a triangle if: a + a > ak (Assuming a > 0 k > 0). k!# " k > (a familiar number?) 9. This follows from the fact that a triangle is determined by three side lengths. Adding a further property, an angle, will not change that MARS University of Nottingham 12

14 Analyzing Congruency Proofs Student Materials Alpha Version January 2012 Finding Congruent Triangles 1. Solomon says, Adeline says, Adeline, try to draw a triangle congruent to my triangle. It has one angle of 30º; another angle is 40º. That s easy Solomon, because all triangles that have those properties must be congruent. Is Adeline correct to say that all triangles with these properties must be congruent? Explain your answer. 2. Ernie says, Suzie says, Suzie, draw a triangle congruent to my triangle. My triangle is a right triangle. One side is 1" long. That s easy, Ernie, because all triangles that have those properties must be congruent. Is Suzie correct to say all triangles with these properties must be congruent? Explain your answer MARS, University of Nottingham S-1

15 Analyzing Congruency Proofs Student Materials Alpha Version January Burt says, Wally says, K Wally, draw a triangle congruent to my triangle. One side is 2" long, it has angles of 50º 30º. That s easy, Burt, because all triangles that have those properties must be congruent to your triangle. Is Wally correct to say that all triangles with these properties must be congruent? Explain your answer MARS, University of Nottingham S-2

16 Analyzing Congruency Proofs Student Materials Alpha Version January 2012 Cards: Must the two triangles be congruent? One side of Triangle A is the same length as one side of Two sides of Triangle A are the same lengths as two sides of Three sides of Triangle A are the same lengths as three sides of 4. One side of Triangle A is the same length as one side of Triangle B one angle in Triangle A is the same size as one angle in 5. Two sides of Triangle A are the same lengths as two sides of Triangle B one angle in Triangle A is the same size as one angle in 6. Three sides of Triangle A are the same lengths as three sides of one angle in Triangle A is the same size as one angle in 7. One side of Triangle A is the same length as one side of Triangle B two angles in Triangle A are the same sizes as two angles in 8. Two sides of Triangle A are the same lengths as two sides of Triangle B two angles in Triangle A are the same sizes as two angles in 9. Three sides of Triangle A are the same lengths as three sides of two angles in Triangle A are the same sizes as two angles in 2012 MARS, University of Nottingham S-3

17 Analyzing Congruency Proofs Student Materials Alpha Version January 2012 Instructions: Must the two triangles be congruent? For each card: Draw examples of pairs of triangles A B that have the properties stated in the cell. Decide whether the two triangles must be congruent, record your decision. If you decide that the triangles do not have to be congruent, draw examples explain why. If you decide that the triangles must be congruent, try to write a convincing proof. Finally, make sure to include Card 8 in your work, as the whole class will discuss this statement. 8. Two sides of Triangle A are the same lengths as two sides of Triangle B two angles in Triangle A are the same sizes as two angles in 2012 MARS, University of Nottingham S-4

18 Analyzing Congruency Proofs Student Materials Alpha Version January 2012 Kieran Jorge both investigated statement 8: Sample Student Proofs Two sides of Triangle A are the same lengths as two sides of Triangle B two angles in Triangle A are the same sizes as two angles in Jorge says the triangles must be congruent. Do you agree with Jorge? Explain Jorge s reasoning. Explain how Jorge could improve his proof MARS, University of Nottingham S-5

19 Analyzing Congruency Proofs Student Materials Alpha Version January 2012 Kieran says the triangles need not be congruent. Do you agree with Kieran? Explain what Kieran s diagrams show. Explain how Kieran could improve his reasoning MARS, University of Nottingham S-6

20 Analyzing Congruency Proofs Student Materials Alpha Version January 2012 Finding Congruent Triangles (again) 1. Amy says, Ben says, Ben, try to draw a triangle congruent to my triangle. It has one angle of 70º; another angle is 50º. That s easy, because all triangles that have those properties must be congruent. Is Ben correct to say that all triangles with these properties must be congruent? Explain your answer. 2. Sara says, Sanjay says, Sanjay, draw a triangle congruent to my triangle. My triangle has one angle of 60. One side is 2" long. That s easy, Sara, because all triangles that have those properties must be congruent. Is Sanjay correct to say all triangles with these properties must be congruent? Explain your answer MARS, University of Nottingham S-7

21 Analyzing Congruency Proofs Student Materials Alpha Version January Vinay says, Max, draw a triangle congruent to my triangle. One side is 1" long, it has angles of 90º 20º. K Max says, That s easy, Vinay, because all triangles that have those properties must be congruent to your triangle. Is Max correct to say that all triangles with these properties must be congruent? Explain your answer MARS, University of Nottingham S-8

22 Must the two triangles be congruent? For each card: 1. Draw examples of pairs of triangles A B that have the properties stated in the card. 2. Decide whether the two triangles must be congruent, record your decision at the bottom of the card. 3. If you decide that the triangles do not have to be congruent, draw examples explain why. 4. If you decide that the triangles must be congruent, try to write a convincing proof. Alpha Version January MARS University of Nottingham Projector Resources: 1

23 Working Together 1 Select a card one of you has worked on. Glue it to the middle of a blank sheet. 2 Take turns to explain your work to the others. Explain your conclusion, how you reached that conclusion. Make sure everyone in your group understs your diagrams. 3 When everyone who has worked on the same card has had a turn, work together to reach a joint decision for that card. 4 On your sheet produce an explanation together that is better than your individual explanations. 5 Make sure you discuss Card 8. Alpha Version January MARS University of Nottingham Projector Resources: 2

24 Look at this statement 8. Two sides of Triangle A are the same lengths as two sides of Triangle B two angles in Triangle A are the same sizes as two angles in Alpha Version January MARS University of Nottingham Projector Resources: 3

25 Jorge s Proof Alpha Version January MARS University of Nottingham Projector Resources: 4

26 Kieran s Proof Alpha Version January MARS University of Nottingham Projector Resources: 5

27 Cards: Must the two triangles be congruent? One side of Triangle A is the same length as one side of Two sides of Triangle A are the same lengths as two sides of Three sides of Triangle A are the same lengths as three sides of One side of Triangle A is the same length as one side of Triangle B one angle in Triangle A is the same size as one angle in Two sides of Triangle A are the same lengths as two sides of Triangle B one angle in Triangle A is the same size as one angle in Three sides of Triangle A are the same lengths as three sides of one angle in Triangle A is the same size as one angle in 7. One side of Triangle A is the same length as one side of Triangle B two angles in Triangle A are the same sizes as two angles in Alpha Version January Two sides of Triangle A are the same lengths as two sides of Triangle B two angles in Triangle A are the same sizes as two angles in 2012 MARS University of Nottingham 9. Three sides of Triangle A are the same lengths as three sides of two angles in Triangle A are the same sizes as two angles in Projector Resources: 6