November 2013 Network Team Institute Grade 8 – Module 3 Module Focus Session TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X A Story of Ratios Grade 8 – Module 3 (1 min) Welcome! In this module focus session, we will examine Grade 8 – Module 3.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Session Objectives Examine the development of mathematical understanding across the module using a focus on concept development within the lessons. Identify the big idea within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum. (1 min) Our objectives for this session are to: Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons. Introduction to mathematical models and instructional strategies to support implementation of A Story of Ratios.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Agenda Introduction to the Module Concept Development Module Review (1 min) We will begin by exploring the module overview to understand the purpose of this module. Then we will dig in to the math of the module. We’ll lead you through the teaching sequence, one concept at a time. Along the way, we’ll also examine the other lesson components and how they function in collaboration with the concept development. Finally, we’ll take a look back at the module, reflecting on all the parts as one cohesive whole. Let’s get started with the module overview.

Curriculum Overview of A Story of Ratios Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Curriculum Overview of A Story of Ratios (1 min) The third module in Grade 8 is called Similarity. The module is allotted 25 instructional days. It challenges students to build on understandings from previous modules by defining similarity as a dilation, followed by of a sequence of rigid motions.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Agenda Introduction to the Module Concept Development Module Review

L1: What lies behind “same shape”? Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L1: What lies behind “same shape”? What does is mean for figures to be considered similar? Is describing them as “same shape” good enough? Introduction to dilations Center Scale factor Dilations do not preserve the lengths of segments. Rather, the length of a dilated segment is equal to the length of the original segment multiplied by the scale factor of dilation. (1 min) “In this lesson students learn that dilations do not preserve the lengths of segments. Rather, the length of a dilated segment is equal to the length of the original segment multiplied by the scale factor of dilation.” Lesson 1, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercises 2-6 (2 min) Allow participants to work on the exercises. When most have finished, advance to the next slide.

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X If the scale factor is r=3, what is the length of segment OP' ? The length of the segment OP' is 9 cm. Use the definition of dilation to show that your answer to Exercise 2 is correct. |OP’|=r|OP|, therefore, |OP’|=3×3=9 and |OP’|=9. If the scale factor is r=3, what is the length of segment OQ'? The length of the segment OQ' is 12 cm. Use the definition of dilation to show that your answer to Exercise 4 is correct. |OQ’|=r|OQ|, therefore, |OQ’|=3×4=12 and |OQ’|=12. If you know that OP=3, OP'=9, how could you use that information to determine the scale factor? (2 min) Ask participants to provide answers or click to have them appear on the slide.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students know that dilation does not preserve segment lengths, rather, the length of a dilated segment is equal to the scale factor multiplied by the length of the original segment.” Lesson 1, Student Debrief

L2: Properties of Dilations Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L2: Properties of Dilations Dilations maps lines to lines, segments to segments, rays to rays, and angles to angles. Dilations preserve the measures of angles. Properties verified experimentally (1 min) “In this lesson, students verify the properties of dilations.” Lesson 2, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Problem Set 1 (2 min) Provide participants with 2 minutes to dilate the figure. Then advance to the next slide.

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Problem Set 1 (1 min) “What was your strategy for dilating the figure?” Answer: draw a ray from center O through each vertex of the figure. Measure the length of the segment from the center to the vertex, divide it by 2, then mark that length along the ray. “When students perform dilations that have scale factors that are greater than 1, they use a compass. For scale factors that are less than 1, they use a ruler.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to know how to use the tools (compass and ruler) strategically to dilate a figure.” Lesson 2, Student Debrief

L3: Examples of Dilations Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L3: Examples of Dilations In the past, topics of congruence and similarity only existed in the world of rectilinear figures. Knowledge of the basic rigid motions and dilation in general provide the opportunity to explore these concepts with curvilinear shapes. How do we return a dilated figure back to its original size? (1 min) “In this lesson, students perform dilations on figures that are not rectilinear (e.g., circle, ellipse, heart). Then students learn how to determine the scale factor that is required to bring a dilated figure back to its original size.” Lesson 3, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Example 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “When presented to students, we give a segment a length, like 6 for AB. Then ask students what the length of the dilated segment would be. Since 6 x 1/3 is 2, then we know A’B’ is 2 units in length. Then students are asked how we could make A’B’ the length of AB again. It must be dilated by a scale factor of 3.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to know how to determine the scale factor needed to bring a dilated figure back to its original size.” Lesson 3, Student Debrief

L4: Fundamental Theorem of Similarity (FTS) Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L4: Fundamental Theorem of Similarity (FTS) FTS is explored in terms of dilation. Theorem: Given a dilation with center O and scale factor r, then for any two points P, Q in the plane so the O, P, Q are not collinear, the lines PQ and P’Q’ are parallel, where P’=dilation(P) and Q’=dilation (Q), and furthermore, |P'Q’|=r|PQ|. Teacher led activity based lesson using lined paper and a ruler. (1 min) “In this lesson, students learn about the Fundamental Theorem of Similarity in terms of dilations. The activity the students complete in this lesson is what we will do next.” Lesson 4, Concept Development

Activity On a lined piece of paper: TIME ALLOTTED FOR THIS SLIDE: Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Activity On a lined piece of paper: Choose a point near the top of the page on a line to use as our center of dilation. Label the point O. Draw a ray, from point O, through a whole number of lines, e.g., through 2 lines, or 5 lines, etc. Label this point P. Draw another ray, from point O, through the same number of lines. Label this point Q. Along ray (you may need to extend it), find another point, P’, a whole number of lines away from O. Along ray (you may need to extend it), find another point, Q’, the same whole number of lines away from O. Connect points P and Q. Connect points P’ and Q’. (5 min) Read the instructions on the slide. Model if necessary.

Activity What do you notice about lines PQ and P’Q’? Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Activity What do you notice about lines PQ and P’Q’? Lines PQ and P’Q’ are parallel, i.e., they never intersect. Write the scale factor that represents the dilation from O to P and O to P’, e.g., if you went 2 lines for P and 5 lines for P’, then scale factor r = 5/2. Why does it make sense for the scale factor to be r = 5/2 and not r = 2/5? If triangle OPQ was the original figure and it has been dilated to triangle OP’Q’ (where the lengths of the sides are longer than the original), that means that the triangle has been magnified, therefore the scale factor should be greater than 1. Now if triangle OPQ was the original figure and it has been dilated to triangle OP’Q’ (where the lengths of the sides are shorter than the original), that means the triangle has been shrunk, therefore the scale factor should be less than one (but greater than zero). (5 min) Read the instructions on the slide. Model if necessary.

Header July 2013 Network Team Institute Activity (cont.) TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Measure the distance from O to P, label your diagram. Measure and label OP’, OQ, and OQ’. Compare the length of the dilated segment to the original: OP’ to OP, and OQ’ to OQ. (Use values rounded to nearest tenths place.) What do you notice? You should notice that the length of the dilated segment OP’ divided by the original segment OP is equal to OQ’ divided by OQ. (5 min) Read the instructions on the slide. Model if necessary.

Header July 2013 Network Team Institute Activity (cont.) TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Measure the lengths P’Q’ and PQ. Compare the lengths as before. What do you notice? It is also equal! Now compare the ratio of the lengths to the scale factor. What do you notice? In each case, the ratio of the segment lengths is equal to the scale factor. (5 min) Read the instructions on the slide. Model if necessary.

Why does it work? The Fundamental Theorem of Similarity: Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Why does it work? The Fundamental Theorem of Similarity: If D is a dilation with center O and scale factor r, then for any two points P, Q in the plane so that O, P, Q are not collinear, the lines PQ and P’Q’ are parallel, where P’ = D(P), and Q’ = D(Q), and furthermore, Mathematically speaking; Therefore, (5 min) Read the bullets on the slide. Say “The activity we just completed is the experimental verification of this theorem. Students will participate in a similar investigation and then be presented with this theorem. Since students will have had hands on experience with the activity, they can make better sense of what is being stated in the theorem.”

What does it all mean? Are triangles OPQ and OP’Q’ similar? Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X What does it all mean? Are triangles OPQ and OP’Q’ similar? Use your transparency to trace angle OPQ. Translate along the vector Does angle OPQ map onto angle OP’Q’? What does that mean? Dilations preserve the measures of angles. The triangles are similar by AA criterion. The lengths of the sides of the triangles are in “proportion” and equal to the scale factor of dilation. In general, two figures are said to be similar if you can map one onto another by a dilation followed by a congruence. (5 min) Read the bullets on the slide. Say, “The reason we want to learn about dilations is to develop an understanding of similarity. If we can dilate one figure to be the “same size” as another, and then use the basic rigid motions to map it entirely onto another, then we know we have similar figures and we would expect them to have the stated properties of parallel lines and side lengths in proportion.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to know the basic properties of the theorem for similarity.” Lesson 4, Student Debrief

L5: First Consequence of FTS Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L5: First Consequence of FTS Converse of FTS. Students first experience the effect dilations have on points in the coordinate plane as an application of FTS. (1 min) “In this lesson, students apply what the learned in the previous lesson to determine the location of dilated points in the coordinate plane.” Lesson 5, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) Provide participants time to complete the exercise. Ask them to provide you with instructions or demonstrate the following using the document camera. Draw a ray OB along the x axis. Mark point B at 12. To find the location of A’, we need to show a scale factor of 5/12, which means we must put A’ along ray OA where it intersects with the vertical line through 5 on the x axis. Mark B’ at (5,0). We know lines A’B’ and AB are parallel, and we know that the segments have the relationship that the dilated segment is equal to the scale factor multiplied by the original. Count to find the length of AB. Then multiply by 5/12. The answer 3.33 is the y coordinate of point A’.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to know how to describe the effect of dilation on points in the coordinate plane.” Lesson 5, Student Debrief

L6: Dilations on the Coordinate Plane Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L6: Dilations on the Coordinate Plane Students generalize what they observed about dilations of points using FTS, that is, students recognize the multiplicative effect that dilations have on the coordinates. Given a dilation with scale factor r and center at the origin, a point in the plane (x, y), after the dilation will be located at (rx, ry). (1 min) “In this lesson, students look for patterns in their work from Lesson 5 to determine a “shortcut” for finding the coordinates of the dilated point.” Lesson 6, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “Students are shown this example in a sequence of 3 from Lesson 5. Then asked to state what they observed about the coordinates of the points and the dilated points.”

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “The “shortcut” is explained mathematically. That is, the basis for the shortcut is what we know about dilations.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to be able to name the coordinates of any dilated point (with center at the origin).” Lesson 6, Student Debrief

L7: Informal Proofs of Dilations Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L7: Informal Proofs of Dilations Optional lesson. Properties observed in Lesson 2 are proved informally. (1 min) “In this lesson, students verify the properties about dilations they observed in Lesson 2. This lesson is optional as it goes beyond the standard, but provides some informal proof of why lines map to lines, rays map to rays, etc.” Lesson 7, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “One of the discussions in this lesson shows that dilations preserve angle measures. In your handouts you see that there are a series of questions that help students think through the proof. Ultimately, it relies on their knowledge of FTS and angles associated with parallel lines.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to have a deeper understanding about the properties of dilations.” Lesson 7, Student Debrief

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L8: Similarity Is dilation enough to prove that two figures are similar? Similarity is defined as a dilation followed by a congruence. Show figures are similar by describing the sequence of the dilation and congruence. (1 min) “In this lesson, students learn that dilation is not enough to show that figures are congruent. Consider two figures that are similar but in different orientations. A rotation must be performed. Hence the need to define similarity as a dilation followed by a congruence.” Lesson 8, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) Provide participants 3 minutes to work on the Exercise. Then advance to the next slide.

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “A dilation from the origin by a scale factor of 2 creates the image A”’B”’C”’. Then we translate the image 4 units up and 12 units to the left.” Dilate A”B”C” by a scale factor of 2 from the origin. Then translate.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to be able to describe the sequence of a dilation followed by a congruence to prove a similarity.” Lesson 8, Student Debrief

L9: Basic Properties of Similarity Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L9: Basic Properties of Similarity Similarity is symmetric. If triangle A is similar to triangle B, then triangle B is similar to triangle A. Similarity is transitive. If triangle A is similar to triangle B, and triangle B is similar to triangle C, then triangle A is similar to triangle C. (1 min) “In this lesson, students learn that like congruence, similarity is symmetric and transitive.” Lesson 9, Concept Development

Exploratory Challenge 2 Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exploratory Challenge 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “Prior to this challenge, students work on showing that similarity is symmetric. Here, students investigate whether or not similarity is transitive by mapping ABC to A’B’C’, then A’B’C’ to A”B”C”, and finally ABC to A”B”C”. They are asked if they really needed to investigate the last pair of triangles. In other words, would you know they are similar without having to determine the sequence. The goal is for students to recognize that the size of the angles remain the same for dilations and that using the idea of similarity being symmetric, it’s not necessary to actually check the last pair of triangles.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to know that similarity is symmetric and transitive.” Lesson 9, Student Debrief

L10: Informal Proof of the AA Criterion for Similarity Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L10: Informal Proof of the AA Criterion for Similarity Straightforward proof, then a direct application using the triangle sum theorem from Module 2. Students practice presenting informal arguments to show that two triangles are similar. (1 min) “In this lesson, students learn that two triangles can be considered similar if they have two corresponding angles that are equal.” Lesson 10, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 4 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) Ask participants to respond to the exercise. “No, these triangles are not similar because angle B is not equal to angle B’. Because of the triangle sum theorem, we know that angle C is not equal to angle C’, therefore these triangles are not similar.”

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 5 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) Ask participants to respond to the exercise. “Yes, these triangles are similar because angle A is equal to angle A’. Because of the triangle sum theorem, we know that angle B is equal to angle B’, therefore these triangles are similar.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to know how to determine if two triangles are similar by comparing their corresponding angles. If two pairs of corresponding angles are equal, then by the AA criterion, the triangles are similar.” Lesson 10, Student Debrief

L11: More About Similar Triangles Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L11: More About Similar Triangles Find the length of a segment of a triangle Verify that triangles are similar Write ratios of corresponding sides. They are equivalent fractions, why? Find the number that makes the fractions equal. (1 min) “In this lesson, students learn that triangles can be considered similar by comparing their corresponding side lengths.” Lesson 11, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) Ask participants to complete the exercise. Then advance to the next slide.

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Based on the information given, is △ABC~△AB'C'? Explain. There is not enough information provided to determine if the triangles are similar. We would need information about a pair of corresponding angles or more information about the side lengths of each of the triangles. Assume line BC is parallel to line B'C'. With this information, can you say that △ABC~△AB'C'? Explain. If line BC is parallel to line B'C', then △ABC~△AB'C'. Both triangles share ∠A. Another pair of equal angles is ∠AB'C'and ∠ABC. They are equal because they are corresponding angles of parallel lines. By the AA criterion, △ABC~△AB'C'. Given that △ABC~△AB'C', determine the length of AC'. Let x represent the length of AC'. x/6=2/8 We are looking for the value of x that makes the fractions equivalent. Therefore 8x=12, and x=1.5. The length of AC' is 1.5.   (1 min) Ask the participants to share their answers. If curious, the answer to part d (not shown here) is done the same was as the last problem on this slide. The answer is 10.8.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to be able to use what they know about corresponding angles and corresponding side lengths to determine if a pair of triangles are similar.” Lesson 11, Student Debrief

L12: Modeling Using Similarity Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L12: Modeling Using Similarity Knowledge of similarity leads to ability to take an indirect measurement of distance across a lake. length of wood needed for a skate ramp. height of a building. height of a tree. (1 min) “In this lesson, students work through a series of real world problems that require an understanding and application of similar triangles.” Lesson 12, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) Provide participants time to work through the exercise. Note that the angle measure has been changed from 40 to 36 because it is more mathematically accurate. The height of the building is approximately 1282.9 feet.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to know how to use similar triangles to make indirect measurements in the real world.” Lesson 12, Student Debrief

L13: Proof of the Pythagorean Theorem Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L13: Proof of the Pythagorean Theorem Proof of Pythagorean theorem using similar triangles. Traditional application of Pythagorean theorem to find lengths of right triangles. (1 min) “In this lesson, students see their second proof of the Pythagorean theorem.” Lesson 13, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Proof TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “Now we prove the Pythagorean theorem using similarity.”

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “Place a point D along side AB so that CD is perpendicular to AB. Now we have 3 right triangles, shown at the bottom. Are they similar?” Participants should describe which pairs of corresponding angles can be used with the AA criterion to prove similarity.

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “Since we have similar triangles, then we know that their corresponding sides are equal in length.”

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “With our work so far, we have a^2 +b^2. What is the length AD + BD?” Participants should say that those segments are the same as AB. “Therefore we have AB x AB, which is AB^2, which is c^2. Therefore we have proven the Pythagorean theorem using similarity.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to understand the proof of the Pythagorean theorem using similarity.” Lesson 13, Student Debrief

L14: The Converse of the Pythagorean Theorem Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X L14: The Converse of the Pythagorean Theorem Proof of the converse of Pythagorean theorem. Given three lengths that satisfy the Pythagorean theorem, the lengths form a right triangle. Students give informal proofs about right triangles using the Pythagorean theorem. (1 min) “In this lesson, students learn how to verify if a triangle is a right triangle using the converse of the Pythagorean theorem.” Lesson 14, Concept Development

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Proof TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “Now we prove the converse, or at least part of it, by assuming that we do not have a right triangle.”

Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “The result 2am = 0 cannot be true. 2am is twice the length of am, therefore it cannot be zero. Since our work resulted in an untrue statement, it means that our assumption that we did not have a right triangle is not accurate.”

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X (1 min) “By the end of the lesson, we expect students to apply the converse of the Pythagorean theorem to determine if a triangle is a right triangle.” Lesson 1, Student Debrief

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Agenda Introduction to the Module Concept Development Module Review

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: X Biggest Takeaway Turn and Talk: What questions were answered for you? What new questions have surfaced? Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have? Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Key Points Dilation leads to an understanding of similarity. Dilations begin off the coordinate plane, then become fixed to examine the effect dilations and rigid motions have on coordinates. Let’s review some key points of this session.