Student Work for Triangle congruence ARC

Transcription

1 Student Work for Triangle congruence ARC Max Ray-Riek The Math Forum / NCTM Deidra Baker Mid-Prairie High School Wellman Iowa

2 The Vision of the CRC Create an online mathematics education professional learning community that is the go-to site for teachers.

3 ARCs Activities with Rigor and Coherence Sequence of 2 4 lessons that support Principles to Actions addresses a specific math topic scaffolds effective teaching supports the 8 SMPs demonstrates the 5 Practices for Orchestrating Productive Mathematics Discussions integrates the wide array of NCTM resources

4 Insert screenshot of ARC on the NCTM website once available

5 Student Work for the Triangle Congruence ARC Students work for journal and assessment questions follow. The first examples of student work show students who are able to articulate understanding of the material. The final example will be used to show a student who is not yet able to articulate a concept, opening opportunities for discussion and follow up ideas.

6 ARC Assessment 1. In the diagram below, B is the center of the circle and ED and GF are diameters of the circle. Prove that EG = FD. Can you prove this using more than one method?

7 Answer: Assessment Question 1 (understanding) GB and EB are radius of the circle and ½ the diameter. Same for DB and ED (FB). They are both radiuses meaning they are half, so they are equal. Also, Angle DBF congruent to angle GBF because they are vertical angles. So we have SAS (side angle side) post(ulate). And if we patty papered them on top of one another EG and FD (segments) would line up. Arrow- bottom triangle if you reflect over point B you will see segments EG and FD line up.

8 B is center of circle: given Answer: Assessment Question 1 (understanding) Segments BE, BG congruent, and BF and Bd by def of radius Angles EBG and GBE (BFD) congruent by vertical angles, so triangles EBF and FBD are cpngruent by SAS and segments FD and Eg are congruent by correspondi ng parts of congruent triangles. This can be proved with more than 1 method. Like we could ve done segments EB and DB because of def of midpoint. Radii leave mulitple options.

9 Answer: Assessment Question 1 (opens more questions for discussion) The definition of diameter is the width of a circle and diameters for one circle. They are both the same. B is the center - given Segment BG is a radius- def of radius Segments, EB, FB, & DB are radiuses - def of radius The segments EB, BG, FB, and DB are all congruent - all radiuses are the same for same circle Segments ED and GF are congruent- segment addition Some possible questions for discussion: Which segments did you add? You have marked sides of the triangles in the picture as congruent. How can use this information? When we talked about proving triangles congruent how many parts did we need? - If you don t have enough parts are there any more parts you can show are congruent- which ones? How could this information help you?

10 Spongebob and Patrick are given a construction challenge by Squidworth. Squidworth says, Draw a circle with center Q and then draw a point P anywhere you want. Draw a line segment from center Q to point P. That will be one side of both triangles you re going to draw. Now draw a radius from Q to a point R anywhere on the circle. Make triangle QPR. Reflect Ray PR over line QP. Finish drawing triangle QPS by making point S where Ray PR (the reflected ray) intersects the circle. Squidworth claims that Patrick and Spongebob should both have drawn two congruent triangles, QPR and QPS. Patrick s triangles are congruent, but Spongebob s aren t. Make your own drawing. Can you make it so your triangles are congruent? So they aren t? Did Spongebob make a mistake? Is Squidworth wrong?

11 The triangles that we made are congruent. There is no way to make them so they aren t congruent because the side lengths will always be the same. (SSS) Spongebob made a mistake while drawing because there is no counterexample for this. Answer: Assessment Question 2 (understanding)

12 Answer: Assessment Question 2 (understanding) SSS because they share a side. They reflect a side and reflections are congruent. The distance from the points would be the same because the lines are the same. This work demonstrates student understanding that reflections images are the congruent to the pre-image, and that connecting points that reflections would create congruent segments as well. This student did not answer the second part of the question but seems confident that congruent triangle are created.

13 Assessment Question 2 (opens more questions for discussion) Yes but only if all the angles in the center are 90 degrees. Spongebob didn t necessarily do it wrong he just needed to specify that the middle had to be 90 degress in all 4 angles to be congruent. Some possible questions for discussion: Why do the angles have to be right angles? Why all 4 angles or which 4 angles? Did you try other cases? Are there any other ways to show the triangles are congruent?

14 After each lesson, students were asked some reflection questions. After the first lesson students were asked to explain or reflect on how they found the congruent halves.

15 Congruence worksheet reflections - Day 1 questions 1) Explain what you and your partner discussed as you decided how to split the figures, and then how to get them to line up with each other, what ideas did you try on the different shapes, was the patty paper helpful- if so how? 1) You wrote directions to get from shape A to shape B, how would those be different if another pair went from B to A? 1) Your advice/ concern/ reflections on this lesson for the authors?

16 Some responses for the first question: What direction the shape was compared to the other one. We turned the paper to see if another shape was there. Turning the paper and using the patty paper, so it was very helpful - Kaylee & Alex We found area and how to split it, we just looked at the shapes and we also used patty paper. For the more simple shapes we just looked at them by for the more complicated shapes we brought in other aspects. Yes because we could see if we were correct or not. - Kessa and Sophia

17 Most student responses for the second question indicated that going from the yellow shape to the green shape would be the opposite or the reverse of going from the the green to the yellow shape. - The only difference is that the order of transformations would be in the opposite order. Jaiden For shapes like this one, we had great class discussions about point of rotation, and the order of transformations. Some students wanted to rotate about the tip only, other students argued that you would have to translate before or after the rotation.

18 Students have some advice for the authors of this lesson: I like how this sheet has focuses on different kinds of ways to get the congruent halves and doesn't include just one of the many types. -Keaton I like puzzles and all, but if this kind of stuff were to be homework I would give plenty of time to finish it, because it is pretty difficult. - Nolan

19 On day 2 students investigated whether triangles with given side lengths would have to be congruent. They could use patty paper, rulers, protractors or compasses to decide if the triangles were congruent. The next slide contains some reflection questions about this lesson.

20 Some Day 2 reflection questions about shortcuts What did you learn about triangle side lengths and congruence? If you get the same 3 lengths in different orders, can you ALWAYS find a way to map one triangle onto the other triangle? Why or why not? What other key ideas did you learn today? Do you think that there are other combinations of 3 parts (besides sides) that might result in triangle congruence using transformations? What combinations do you predict might work?

21 (If all sides are equal then the triangle is congruent) If two triangles have the same side lengths, they are congruent.- Cassidy If two triangles have congruence on each each side with the matching side of the other triangle then the triangles are congruent. - Abby

22 Yes; because no matter if the same 3 lengths are in different orders the triangles will still be congruent. You would have to use transformations to move one triangle on top of the other one. -Jaiden You can rotate them, flip them, or whatever they are just going to match -Nenad This depends on the shapes themselves because some triangles you can simply rotate one or flip one to match the other but I feel that some triangles are not always congruent to each other. -Adam

23 (that you can use more than just side length to see if triangles are congruent. That aaa does not work -Josh If two triangles have congruence on each each side with the matching side of the other triangle then the triangles are congruent. -Adam

24 Yes SAS, ASA - Matt Yes; Flipping would work, also rotating and sliding would also work. -Jaiden Answers like Jaiden s gave me an opening to talk with the class about triangle parts and theorems about those parts, and transformations. There were some vocabulary misunderstandings.

25 Students then had the opportunity to explore other ways to decide if triangles are congruent. Classes generated possibilities to try with measurements to try. We investigated with paper-pencil tools, and online tools. Students were able to test SAS, ASA, AAA, AAS, and SSA. Some students were easily convinced which options would always work, and which would not while other students remained unsure what would always work to prove triangles congruent.

26 Lesson 3 Other Triangle congruences reflection questions 1. Right now, I can show triangles are congruent by OR. For me, right now, is the easier method because. I can image that if a problem were like, it might be easier to do the other method. 2. Patrick says that if you have two triangles and you know any two pairs of corresponding sides are congruent and any pair of corresponding angles are congruent, then the triangles have to be congruent. Sponge Bob says he doesn t think that s true but he s having a hard time explaining just when it is true. Please help Sponge Bob, using what you explored today. 1. Agree or Disagree: Looking at examples in the app proves that these shortcut sets of conditions guarantee triangle congruence. Why do you agree or disagree?

27 _side side side OR _side angle side _. For me, right now, _side side side is the easier method because _it is a lot neater and easier to write out_. I can image that if a problem were like _side side side_, it might be easier to do the other method. - Abby Right now, I can show triangles are congruent by determining what the method is OR showing whether it's vertical or not For me, right now, reflecting is the easier method because i know it is just a line that is put on the other side. I can image that if a problem were reflecting, it might be easier to do the other method. - Jacob reflexive OR rotating. For me, right now, reflexive is the easier method because its basically flipped. I can image that if a problem were like a shape is turned, it might be easier to do the other method. -Jillian This lesson reflection enabled me to discuss many misconceptions that students revealed in their reflections.

28 You need to have three points of information for it to be even close to correct. -Kessa If your only proof is set up SSA then the triangles dont have to be congruent. -Matt The sides will be the same and the other angle is the same so one of the triangles may need to be turned after the triangles are made. -Ryan This lesson was a wonderful way for students to explore triangle congruence in a hands on way, and the reflections confirmed my opinion that students really struggle with this concept. Having them write gave me a way to address misconceptions in an anonymous way.

29 I agree because if the sides and angles are congruent and in these orders then the others are set in place. -Matthew

30 After this series of lessons students continued to study triangle congruence, and worked on more traditional proofs, although using a 3-column format- demonstrating which three parts were congruent, in order of part (SAS) and how they could prove those parts were congruent. This series of lessons was a terrific way to introduce the need for more formal proof- students did not want to continue testing cases forever, they wanted to know which cases worked. As students worked on proof, they incorporated some strategies from the lessons like color coding or drawing a few cases to make sure they knew which parts were involved. Students still struggled with formal proofs, but has a much more intuitive idea about using transformations to decide if shapes would be congruent.

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