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Jencen Smith Math Methods Unit on Geometry Unit Title: Same Shape Shapes Unit Topic: Triangle Congruence Grade Level: 9th Length: 10 days Class Length: 55 Minutes The requirements are labeled in RED pen throughout the unit plan Big Ideas: Knowing the different types of triangles Being able to see different relationships between triangles Angle Relationships and the Triangle Sum Theorem Congruent Triangles Proving Triangle Congruence Unit Essential Questions: What are the different types and triangles and how can we compare and contrast them? How can we use angles within triangles to solve for the other angles and side lengths within the triangles? How can we use our knowledge of angle relationships between triangles to show that any two triangles are congruent? How can we use the SSS, SAS, ASA, AAS, HI, or CPCTC congruence theorems to show that two triangles are congruent? Unit Essential Topics/Daily Schedule: Triangle Congruence: Students will learn about congruence and transformations, classifying triangles, the triangle sum theorem, angle relationships, and proving triangle congruence using a multiple theorems. Sequence: In this unit, the students will first have to understand what the different types of triangles. We will have a lesson on all the types of triangles, where we will learn to identify and compare the different types of triangles. Then we will have a lesson on talking about angle relationships within triangles, which will prepare them to create the triangle sum theorem. Because our next lesson will be on congruent triangles, getting a refresher on congruence and how it relates to transformations will jog their memory and help them connect the information. I will do a lesson on congruence and transformations, which will show students that even if we move a shape or object about the coordinate plane, it will still be congruent. Then, we will talk specifically about congruent triangles. Then, we will have a day where we have a short quiz over the first four lessons and then we will work with some origami, where we will then see how what we have learned correlates with origami. Then, we will have three lessons discussing the SSS, SAS, ASA, AAS, HL, and CPCTC triangle congruence theorems, which will allow us to learn how to prove that two triangles are congruent. After all of this, we will have a review lesson, where I will have a review packet that we will go through to help the students study for the next day’s test. Day 1 2 Topic Lesson Overview In the first lesson, we will start off with a warm up, where the students will identify different types of angles and solving for a variable from side lengths of a triangle. Then, the class will translate their knowledge about different types of angles and Classifying learn how each of these angles is Triangles directly related to a specific type of triangle. They will take notes on each of the definitions of an acute, obtuse, equiangular, and right triangle. Once they do this, the class will work on a classifying triangles based on angle measures as well as side lengths. To begin this lesson, the students will be given three congruent triangles. They will be told to prove that all three angles inside a triangle add up to 180 degrees. This will lead into the discussion about the triangle Angle Relationships sum theorem. From here, the of Triangles and the class will learn about two Triangle Sum corollaries to the triangle sum Theorem theorem. We will then talk (FULL LESSON) about the exterior angle theorem. After this, the students will work on a worksheet that will help the students practice using the theorems and corollaries to solve for angles in different triangles. Common Core Standard CC.9-12.G.CO.10: Prove theorems about triangles. CC.9-12.G.CO.10: Prove theorems about triangles. 3 Congruence and Transformation 4 Congruent Triangles 5 Quiz and Origami Lesson (FULL LESSON) In this lesson, we will be talking about congruence and transformations. Because the students have already learned about transformations, their warm up will involve them graphing a translation and a reflection to refresh their brains. Then, we will create a table, representing transformations in the coordinate plane with translations, reflections, rotations, and dilations. We will go through an example of each using the formulas that we had come up with. After this, we will talk about which transformations produce congruent images. The students will then be asked to create three different transformations, two of which produce congruent images, and one does not produce a congruent image. They will give them to a neighbor and they will have to determine which ones are congruent images and which ones are not. This lesson is all about bringing the past three lessons together and using the theorems that the class has learned to prove that two triangles are congruent. We will first talk about corresponding angles and corresponding sides. The class will then prove that two triangles are congruent using the theorems that they had learned in the previous three lessons. Right when the students walk into class, they will take a quiz over the first four lessons. This CC.9-12.G.CO.6: Use geometric descriptions of rigid motions to transform figures and predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. CC.9-12.G.CO.7: Use this definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. CC.9-12.G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. CC.9-12.G.SRT.5: Use congruence and similarity criteria for 6 Congruence Theorems (SSS, SAS) will be used as a summative assessment to see how well each student understands the content that is being taught. Once everyone is done with the quiz, we will grade them in class so that each student knows how they did, and knows what they need more practice with. Then, we will move into our “art project” where each student will be making some origami. They will be given a worksheet with the directions on how to make a crane with a 3-inch by 3-inch piece of paper. Once each student makes a crane, we will unfold them and study the triangles that were created in the paper. We will then prove which triangles are congruent to each other. Now that the students understand the idea how to prove that two triangles are congruent, we will discuss two theorems that are shortcuts to figure out if two triangles are congruent. To start, the teacher will make two triangles using string and three different size straws to prove the SSS and SAS theorems. The students will work through a worksheet that will guide them through both theorems given two congruent triangles. triangles to solve problems and prove relationships in geometric figures. CC.9-12.G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions. CC.9-12.G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. CC.9-12.G.CO.7: Use this definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of 7 8 Congruence Theorems (ASA, AAS, HL) Congruence Theorems (CPCTC) In this lesson, the we will go through a similar process as the previous lesson except we will be discussing the ASA, AAS, and HL theorems. We will go through examples on a worksheet that will guide through these theorems. At the end of this lesson, we will also discuss why we cannot use SSA to prove that two triangles are congruent. For our last lesson, we will bring all the concepts together to talk about how we can use corresponding parts of congruent triangles to show that two triangles are, in fact, congruent. To do this, we will use the SSS, SAS, ASA, AAS, and HL theorems. Using these, we will be able to prove the CTCPC theorem. Again, we will go through a worksheet that uses all five theorems to prove the sixth theorem. sides and corresponding pairs of angles are congruent. CC.9-12.G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions. CC.9-12.G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. CC.9-12.G.CO.7: Use this definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. CC.9-12.G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. CC.9-12.G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motions. 9 Review for Unit Test 10 Unit Test This day will be dedicated towards reviewing for our test the following day. The teacher will create a review packet that picks out important elements of each lesson that will also appear on the test. The students will first work on the packet by themselves. Then, the students will pair up into pairs that the teacher will create, based on the class. During the last 15 minutes of class, we will go through any problem that students have questions on. The students will take their final summative assessment for the unit. All Previous Common Core Standards All Previous Common Core Standards