Grade 7 – Module 6 Module Focus Session

Grade 7 – Module 6 Module Focus Session
May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Patty Paper Chart Paper Geometry NOTE THAT THIS FOCUS SESSION IS DESIGNED TO BE 250 MINUTES IN LENGTH. [Provide participants with a number 1, 2, 3, or 4 to be used for jigsaw-ing at several points throughout the session.] Welcome! In this module focus session, we will examine Grade 7 – Module 6. There was a significant amount of geometry material that was studied in Module 3 to practice algebraic reasoning with expressions, equations, and formulas. The material from Module 3 is then extended as we move in to module 6, geometry takes on the most challenging form that students have seen through the Story of Units and Story of Ratios curriculums. In this module we exhibit practices that will be found in Grade 8 and Grade 10 Geometry curriculums. Let’s take a look at the agenda for today’s session. Grade 7 – Module 6

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Session Agenda Module Overview Focus on Topics and Lessons Module in Review Implementation Discussions We’ll begin by taking a look at the Module using a “Big Picture” to foresee where previous topics have been leading to and where the topics and lessons in the module are leading to. We’ll then zoom in on the Module Overview, taking a specific look at the focus standards addressed in this module, and note the foundational standards to gain an understanding of what knowledge students should be coming to Grade 7 with. Next we’ll zoom in even further by taking a look at each topic in the module, and the lessons that the topic is comprised of. We will spend a great deal of time today working on problems from the lessons that range from teacher led to independent practice. While we work today, please try to approach problems from a student’s perspective, using the knowledge previously seen in this curriculum. Once we complete the module focus, we’ll glance back at the module and pull out and discuss the key points. In the final hour of our session, we’ll spend some time in discussion with regard to implementation of the Module, giving attention to

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Session Objectives Gain an understanding of the content enveloped in Module 6. Understand what was, what is, and why the change. See how geometry concepts from here forward are starting to take shape. Gain the knowledge necessary to implement with fidelity.

Curriculum Overview of A Story of Ratios
Grade 7 – Module 4 Module Focus Session February, 2014 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Curriculum Overview of A Story of Ratios Module 6 is the final module in the Grade 7 curriculum and is planned as a 35 day module, which means by the calendar, should have already begun*. Notice its proximity in time to the geometry module in Grade 6. Depending on your circumstances you may be able to use this to your advantage in filling student gaps. *It is understandable that this is likely not a reality in this transition year as the curriculum is new and scheduling has been impacted for most teachers. Understand that this curriculum is scheduled for the year 2021 student where the level of knowledge gaps if far less, providing for more efficient use of time. We’ll now start to zoom in on the module by examining the module overview.

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME 20 minutes MATERIALS NEEDED: Module Overview Document Grade 7 Module 6 Overview Let’s look at: Foundational Standards (p.5) Focus Standards (p.4) Mathematical Practice Standards (p.5-6) Table of Contents and Narrative Let’s take a bit of a different approach to the Module Overview. First, turn to p.5 and read through the foundational standards to gain understanding of what students are expected to know coming into this module. Divide standards by number: [1: 4.MD.C.7 2: 6.G.A.1 3: 6.G.A.2 4: 6.G.A.4 Everyone 7.G.B.4] Take 1-minute to read and understand the standard, then one at a time, tell the people at your table about your foundational standard. Discuss “what was”/“what is”, and “what will be” (2021 student). Now take another minute to read the Focus Standards for the Module: Divide standards by number: [1: 7.G.A.2 2: 7.G.A.3 3: 7.G.B.5 4: 7.G.B.6] Take 1-minute to read and understand the standard, then one at a time, tell the people at your table about your focus standard. Discuss “what was”/”what is”, and “what will be” (2021 student). Let’s now do the same jigsaw with the mathematical practice standards (p.5-6). Divide standards by number: [1: MP1 2: MP3 3: MP5 4: MP7] Take 1-minute to read and understand how the mathematical practice will be seen throughout the module, then share with those at your table. Now that we have established this overarching knowledge, take a few minutes to read through the module overview. Notice how the topics are broken down into lessons and highlight any major concepts. We’ll quickly address these major concepts before we focus in on Topic A.

Topic A: Unknown Angles
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Topic A Opener Topic A: Unknown Angles 7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. Prerequisites: G4 M4 and G7 M3 Goal is to build fluency in these difficult problems Build recognition and use of structure MP.7 As you saw in the module overview, students are introduced to angles, angle measurement, classifications of angles, the additive property of angle measure, and angle relationships including adjacent, complementary, supplementary, angles on a line, and angles at a point. They solved simple addition and subtraction problems involving these angle relationships. As they now move into grade 7, angle relationships were reviewed as application of algebraic reasoning in Module 3 as equations can be used to model the relationships between angles. Diagrams were drawn to scale so confirmation of work was possible via direct measurement with a protractor. In this module, students are engaged with some of the most challenging (and interesting) diagrams involving angle relationships that they’ve ever experienced. Through this 4 lesson topic, students are expected to synthesize their knowledge of algebra and angle relationships to solve problems. In lesson 1, we reintroduce complementary and supplementary angles.

Complementary and Supplementary Angles
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Complementary and Supplementary Angles The measures of two supplementary angles are in the ratio of 2:3. Find the two angles. In lesson 1, students revisit key vocabulary such as adjacent, vertical angles, and perpendicular lines, angles on a line, angles at a point (p.9-10) Students begin seeing and using standardized symbols and abbreviations to represent the relationships of these terms from this point forward. GIVE SOME EXAMPLES LESSON 1 Pg. 10 At this point, there is no distinction made between “an angle” and “the measurement of an angle”. Stating that two angles are equal simply indicates that the angles have the exact same measure. The use of “m” to indicate measure has been witnessed in M3, however students should begin discerning the meaning of notation based on the context. (for example… The primary focus of Lesson 1 is problems focused around supplementary and complementary angles, both with and without the use of diagrams (strictly verbal descriptions is new). As students approach each problem, they practice describing the angle relationships that are present in given diagrams, then they use their descriptions to develop equations that model those relationships. Diagrams begin stepping up in complexity, and so do the problems without diagrams. Here is one example. (EXAMPLE 1 ON DOC CAM) Lesson 1 / Example 1

Solving for Unknown Angles Using Equations
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Solving for Unknown Angles Using Equations Two lines meet as the common vertex of two rays. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of ∠𝐴𝑂𝐵 and ∠𝐵𝑂𝐶. In lesson 2, other previously learned angle relationships of vertical and adjacent angles are reintroduced to provide for more complex multi-step problems. Students are asked to describe the relevant angle relationships present in a diagram before trying to model the relationships with equations. This step encourages MP1, making sense of a problem and looking for an entry point. Teacher should model good practices of stating geometric reasoning when it is used in preparation for grade 8 and grade 10. There are several places in this lesson to help students distinguish ‘supplementary angles’ from ‘angles on a line’, i.e. supplementary angles refers to a pair of angles (that may or may not lie on a line) versus angles on a line that can refer to two or more angles that lie on a line with a common vertex. Students solve the equations that they use to model the angle relationships, then consider the reasonableness of their answer and even check by direct measurement. Lesson 2 / Exercise 4

Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Solving for Unknown Angles Using Equations Two lines meet at the common vertex of two rays. In complete sentences, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of 𝑥. Find the measurements of angles ∠𝐵𝐴𝐶 and ∠𝐵𝐴𝐻. In lesson 3, students are engaged in problems involving any and/or all previously learned angle relationships, now including angles at a point. Students begin labeling their diagrams with relevant information that is known in an abundance angle relationships. Included in this lesson is the use of angle relationships to establish connection between an equation with variables on both sides that models an angle relationship. Let’s take a look at Example 4. Lesson 3 / Example 3

Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Protractor Straight edge Solving for Unknown Angles Using Equations Two lines meet at a point. Set up and solve an equation to find the value of 𝑥. Find the measurement of one of the vertical angles. This problem highlights MP7 (Look for and make use of structure) by helping students see the connection between an angle diagram and the equation used to model it. Solving equations with variable on both sides is a G8 topic and extensions to these problems are provided if the teacher so wishes to address this. First, measure a 30° angle from the angle (𝑥+30)°. The remaining angle outside of 30° must then be 𝑥°. Students measure the resulting angle and partition the angle 3𝑥° into three angles of 𝑥°. Students can then use the modified diagram to write an equation that they can solve: 2𝑥=30 EXTENSION: Students can modeled the algebraic processes for solving the equation 3𝑥=𝑥+30 by modifying the diagram. This is an excellent place to preview the grade 8 skill. Lesson 3 / Example 4

Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Solving for Unknown Angles Using Equations The measurement of an angle is the measurement of its supplement. Find the measurement of the angle. The diagrams in lesson 4 really step up in complexity involving multiple lines, rays, and angle relationships. It also involves more verbal description based questions. Let’s take a look at Example 3. [Read the problem] Let’s use a tape diagram. [If time allows, the Exit ticket for Lesson 4 is included in the Participant Packet and exhibits the type of diagram that students are expected to be able to make sense of] PS #8 is a good challenger problem too! Lesson 4 / Example 3

Topic B: Constructing Triangles
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Topic B Opener Topic B: Constructing Triangles 7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. New Terminology New Tools New Approach Topic B comprises 11 lessons in which students perform a great deal of hands-on experimentation with geometry tools to draw geometric figures. Students then use correspondences between triangles to compare the triangles and their parts. This comparison will be used to determine if two triangles are identical or not. The construction tool that students use include the already familiar ruler and protractor, but now also include the compass and the setsquare. Over the course of several lessons, students discover the sets of conditions that determine unique triangles. The question is if two triangles are drawn with the same set of given conditions, is it possible that the triangles are different, or must they be identical? Notice that the term “congruent” is not mentioned, as it is not mentioned in the CCSS until grade 8. Congruence in grade 8 is based on rigid motions in the plane. For our purposes, we are using the terms unique triangle, and identical triangles. The major focus of topic B is to develop students’ intuitive understanding of the structure of a triangle.

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Patty Paper Correspondence Two identical triangles are shown below. Give a correspondence that matches equal sides and equal angles. Lesson 5 sets a major foundation for students to succeed in the following lessons as it looks at the concept of correspondences that exist between two triangles. They also learn how to label sides and angles that are known to be equal in measure. Two triangles and their respective parts can be compared systematically once a correspondence has been assigned. This also allows corresponding sides and corresponding angles to be identified. Triangles are identical if there is a correspondence assigned so that corresponding sides and angles are equal in measure. Example 2 in Lesson 5 asks students to provide a correspondence between two identical triangles that matches up sides and angles of equal measure. In the diagram they are given the markings that indicate these corresponding parts, but the triangles sit differently on the plane. A scaffold is provided with this Example: at the end of the lesson is an enlarged printable diagram that students can cut out and physically match up. The problem with the printable is that the vertices are labeled outside the triangles, and thus they are cut off when the triangles are cut out. To avoid this snafu, either relabel the vertices in the interior of the triangle, or use patty paper and trace one of the triangles. The patty paper idea can provide a sneak peak into what students will experience in grade 8 where congruence is determined by a sequence of rigid motions. Lesson 5 / Example 2

Drawing Geometric Shapes
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Drawing Geometric Shapes Use a ruler, protractor, and compass to complete the following: 3 segments: 4 𝑐𝑚, 7.2 𝑐𝑚, 12.8 𝑐𝑚 Complementary angles, one of 35° Vertical angles, one of 125° 3 circles: 𝑟=2 𝑐𝑚, 𝑟=4 𝑐𝑚, 𝑟=6 𝑐𝑚 3 adjacent angles: 25°, 90°, 50° Rectangle 𝐴𝐵𝐶𝐷, 𝐴𝐵=8 𝑐𝑚, 𝐵𝐶=3 𝑐𝑚 Segment: 5 𝑐𝑚, cut a segment 5 𝑐𝑚 using your compass Segment AB; Circle with center A and radius AB, and circle with center B and radius BA. This lesson is an exploratory lesson. Students will use 3 tools – (ruler, protractor and compass) to construct geometric shapes. Students already know how to use a ruler and protractor but the compass is a relatively new tool, especially in the way it is used in constructing triangles. Lesson 6 provides instruction and time for demonstration of how to properly use a compass. Students will have to practice using the compass properly before being comfortable to complete the exploratory exercises. Once students complete the exploratory exercises, they are to display their work and the class do a gallery walk to look at all the constructions. They should see and conclude how the tools can be used in different roles that aid in the construction process. For example, the compass is used to not only draw circles, but also mark segments of equal length. The majority of lesson 6 is an Exploratory Challenge that we will complete today. The first problems are fairly simple, but each step grows in complexity. By the end of this sequence, students are better prepared for the triangle constructions that they will perform starting in Lesson 8. Reveal each task one at a time, and provide time to complete. Lesson 6 / Exploratory Challenge

Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Drawing Geometric Shapes Use a ruler, protractor, and compass to complete the following: This problem requires the use of all 3 tools, while also requiring close attention to the directions and their sequencing. Lesson 6 / Exploratory Challenge

Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Drawing Geometric Shapes Use a ruler, protractor, and compass to complete the following: Problem 10 gives an excellent look into how a compass can be used to construct a triangle, a task that might seem unusual to students since they’ve previously used the compass for circles. Complete the problem What kind of triangles are ABC and ABD? What makes them isosceles? How do you know the sides are the same length? You can see how the tasks are scaffolded from basic use of two or more tools, and using those tools in strategic ways. The final task is in your participant packet. It asks students to determine all possible measurements of a given triangle and then use their tools to recreate the triangle. Provide a few moments to complete the task. Lesson 6 / Exploratory Challenge

Drawing Parallelograms
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Drawing Parallelograms Use a setsquare, ruler and protractor to construct parallelogram 𝐷𝐸𝐹𝐺 so the measurement of ∠𝐷=40°, 𝐷𝐸=3 𝑐𝑚, the measurement of ∠𝐸=140°, and the altitude to 𝐷𝐸 is 5 𝑐𝑚. Drawing parallelograms requires the ability to draw parallel lines. Using traditional construction procedures with a compass and straight edge, this task can become daunting. Lesson 7 introduces a new tool called a setsquare. This tool can be used to quickly and accurately construct parallel lines, and will be used in future grades for similar tasks. We are providing you with a setsquare to use, and you may choose to do the same with students, however there are directions to make your own setsquare by simply folding a taping a piece of card stock. [Demonstrate the use of the set square using the document camera] Students will now add the setsquare to their tool kit for use in constructing geometric figures with precision. Exercise 1 is a nice example that uses all but the compass. [Provide time for participants to complete the construction, then have someone provide the verbal steps necessary to complete the task and demonstrate using the doc camera] Lesson 7 / Exercise 1

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Chart Paper Drawing Triangles A triangle is to be drawn given the following conditions: the measurements of two angles are 30° and 60°, and the length of a side is 10 𝑐𝑚. Note that where each of these measurements is positioned is not fixed. In lesson 8, students use their tools and different given criteria to draw triangles, and explore which criteria result in many, a few, or one triangle. By the end of this lesson, students will possess the driving question for lessons (8),9-11: What conditions (i.e. how many measurements and what arrangement of measurements) are needed to produce identical triangles? Or, in other words, produce a “unique” triangle. Students are introduced to the use of prime notation to mark two or more figures that are related in some way. The Exploratory Challenge in Lesson 8 takes students through a series of constructions in which they use the same three given measurements (two angles and a side) and use them in different arrangements to draw triangles. You will find the Exploratory Challenge steps in your participant packet. Please take the next 8 minutes to complete these questions. <Hang some chart paper to keep track of the given conditions and our conclusions> [Provide 8 minutes to complete the Exploratory Challenge, then ask the discussion questions: In parts b-d, you were given two angles and a side. How many non-identical triangles were produced under these conditions? (3) Is it possible that we could draw more non-identical triangles under these conditions? (no, see part (e)) What conditions do you think will yield identical triangles no matter how many arrangements are drawn? (3 sides, 3 angles) Do you think it is possible to have less than all six measurements and still produce identical triangles? (we know this is yes) This final question will drive the subsequent lessons 9-11. Lesson 8 / Exploratory Challenge

Three Sides Condition Two Sides and the Included Angle Condition
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Chart Paper Three Sides Condition Two Sides and the Included Angle Condition △𝑋𝑌𝑍 has side lengths 𝑋𝑌=2.5 cm, 𝑋𝑍=4 cm, and ∠𝑋=120°. Draw △𝑋′𝑌′𝑍′ under the same conditions. Use your compass and protractor to draw the sides of △𝑋′𝑌′𝑍′. Use your ruler to measure side lengths. Leave all construction marks as evidence of your work, and label all side and angle measurements. Under what condition is △𝑋′𝑌′𝐶′ drawn? Compare the triangle you drew to two of your peers’ triangles. Are the triangles identical? Did the condition determine a unique triangle? Use your construction to explain why. The overarching question posed with Lesson 9 is is it possible to draw several non-identical triangles under a given set of conditions? If we measure all of the angles and sides and give all the relationships between angles and sides, then any other triangle satisfying the same conditions will be identical to our given triangle. If we give too few conditions on a triangle (such as one angle and one side), then there will be many non-identical triangles that satisfy the conditions. Sometimes, just a few specific conditions on a triangle make it so that every triangle satisfying those conditions is identical to the given triangle, meaning all triangles drawn under those conditions will be identical. In Lesson 8 we saw that at most 3 non-identical triangles could be drawn given the measures of two angles and the length of one side. We saw in this problem that a unique triangle is determined under the conditions of the lengths of two sides and the measure of their included angle. What difference(s) exist between these two sets of conditions? Two sides and an angle versus two angles and a side Arrangement of the conditions is a factor in this problem where it was not in Lesson 8 Reveal Unique triangle conditions posters for Three Sides Condition, and Two Sides and the Included Angle Condition Lesson 9 / Exploratory Challenge 4

Two Angles and a Given Side
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass Patty Paper Chart Paper Two Angles and a Given Side A triangle △𝐽𝐾𝐿 has angles ∠𝐽=60° and ∠𝐿=25° and side 𝐾𝐿=5 cm. Draw triangle △𝐽′𝐾′𝐿′ under the same condition. Leave all construction marks as evidence of your work, and label all side and angle measurements. Under what condition is △𝐽′𝐾′𝐿′ drawn? Compare the triangle you drew to two of your peers’ triangles. Are the triangles identical? Did the condition determine a unique triangle? Use your construction to explain why. In lesson 10 we return to the conditions explored in Lesson 8 where students were given two angles and a given side. This condition can exist in two possible arrangements, so in consideration of arrangements, this condition is split into two conditions. Two angles and the included side [Reveal poster of Unique Triangle Condition] this one is not hard Two angles and the side opposite one angle In Exploratory Challenge 3, students will quickly realize that this drawing is different than the others that they have done in the past few lessons. Take a few moments to complete Exploratory Challenge 3. [Provide a few moments to attempt the drawing. Ask if anyone experienced success, and if so, if they would share their strategy with the room. If not, introduce the strategy of using patty paper and a “floating” angle.] Here is a strategy that will accomplish this drawing with ease: Draw the adjacent parts in the condition (the adjoined angle and side) Draw a 60° on a sheet of patty paper, and float the angle until the rays of the 60° angle coincide with point K’ and the other side of angle L’. Once the 60° angle has been “floated” into place, the remaining measurements can be determined (or the 3rd vertex located). What students should conclude at the end of this lesson is that if we are given two angles and any side of a triangle, we can draw an identical triangle as long as we know the arrangement. There are two possible arrangements, so we consider this condition to be two conditions. Lesson 10 / Exploratory Challenge 3

Conditions on Measurements that Determine a Triangle
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Conditions on Measurements that Determine a Triangle Are there conditions on the lengths of the sides of a triangle? Are there conditions on the measures of the angles in a triangle? Now that we have looked at conditions that determine unique triangles, in Lesson 11 we look at the conditions on measurements that determine a triangle at all. First, draw segment AB on a piece of paper with a length of 10 𝑐𝑚. Next draw segment AC with a length of 3𝑐𝑚 on a piece of patty paper, and segment BC with a length of 5 𝑐𝑚 on a second piece of patty paper. Align the endpoints labeled A and the endpoints labeled B, and rotate the patty paper to form triangle ABC. (It can’t happen!) What must be true of the lengths of AC and BC for the two segments to meet? (must be equal to 10, or greater) What must be true of the lengths of AC and BC for the two segments to complete a triangle? (Must sum to greater than 10) This is a very fundamental understanding for the structure of triangles. This can be reiterated by considering two paths between points A and B. The shortest path is a straight line from A to B, which means that any path A to C to B must be long in distance than AB. One thing to consider is that this condition says that the sum of the lengths of any two sides of a triangle must be greater than the third side. Simply increasing the length of a particular side only determines a triangle to the point that its length is still less than the sum of the other two sides. This is what leads to the conclusion that the longest side length of a triangle must be less than the sum of the other two side lengths, and greater than the difference of the other two side lengths. Next, we look at conditions for a triangle regarding the measures of the angles. First, choose two angles for a triangle such that the sum of their measures is greater than 180°. (Doesn’t work because the sides of the angles that are not shared will not intersect to form a vertex.) Next, choose a pair of supplementary angles as two angles of your triangle. (Again, doesn’t work because the sides of the angles that are not shared will never intersect because they are parallel.) Finally, choose two angle measures whose sum is less than 180°. (This forms a triangle because the sides of the angles that are not shared must intersect at some point). This leads to the conclusion that any two angles of a triangle must have a sum of less than 180°. Lesson 11 / Exploratory Challenge 1&2

Conditions on Measurements that Determine a Triangle
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Conditions on Measurements that Determine a Triangle Is there a “Three Angles” condition for unique triangle? To finish out the Exploratory challenge in Lesson 11, students consider the possibility of a Three Angles condition for unique triangles. The task in the lesson directs students to draw a triangle using the same angle measurements as a given triangle, but with sides that are half as long. Students will quickly see that the triangles are not unique, however they are in fact scale drawings of one another. [Reveal diagram] If measured (taking human error into consideration), the corresponding angles are known to be equal in measure, and the lengths of the corresponding sides are proportional. Lesson 11 / Exploratory Challenge 2d

Two Sides and a Non-Included Angle
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass X Two Sides and a Non-Included Angle Use your tools to draw △𝐴𝐵𝐶 provide 𝐴𝐵=5, 𝐵𝐶=3 𝑐𝑚, and ∠𝐴=30°. What makes this drawing challenging? Use a ruler and compass to locate 𝐶. By looking at the title of this lesson, you may be thinking about that forbidden “congruence” criteria A-S-S, or S-S-A. Lesson 12 focuses in on this condition and students discover that even though the condition of two sides and a non-included angle will not always determine unique triangles, there are some cases where it will, and in fact, in those cases where it does not determine a unique triangle, it produces exactly two possible triangles. [Read and draw the given conditions on the screen] What makes this drawing challenging? (The challenge of this drawing is that the location of point 𝐶 is unknown. How many triangles are determined by the given conditions? (exactly 2) Extend the ray AC in anticipation of the location(s) of point C. Set your compass to a radius of 3cm to represent the remaining side BC and mark the location(s) at which BC intersects AC. As you can see, the given conditions determine 2 possible triangles, one with an acute angle at B, and the other with an obtuse angle at B. Lesson 12 / Exploratory Challenge 1

Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass X Two Sides and a Non-Included Angle Use your tools to draw △𝐷𝐸𝐹 provide 𝐷𝐸=5, 𝐸𝐹=3 𝑐𝑚, and ∠𝐹=90°. Exploratory Challenge 2 changes the given conditions slightly. Take a few moments to construct a triangle (or triangles) using the given conditions on the screen. [Provide 2 minutes to complete the triangle] How many triangles are determined by the given conditions? (1) What difference in Exploratory Challenge 1 and Exploratory Challenge 2 do you think limits the number of triangles determined to 1? [Compare the diagrams of the 1st set of conditions to the diagram of the second set of conditions. It may be helpful to prepare a construction involving two sides and an acute non-included angle near 90 degrees for comparison.] Lesson 12 / Exploratory Challenge 2

Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Clean Paper Ruler Protractor Compass X Two Sides and a Non-Included Angle Use your tools to draw △𝐽𝐾𝐿 provide 𝐾𝐿=8, 𝐾𝐽=4 𝑐𝑚, and ∠𝐽=120°. Conclusions based on the results of Exploratory Challenges 1-3? Finally we take look at a third set of conditions in which the non-included angle is obtuse. Take a few moments to complete the triangle(s) determined by the given conditions in Exploratory Challenge 3. How many triangles are determined by the given conditions? (1) What conclusions can we draw based on the triangle that were determined by the conditions given in Exploratory Challenges 1-3? (All three conditions contained two sides and a non-included angle, however the latter two conditions determined unique triangles whereas the first determined two different triangles.) In fact, if there are two triangles that have a correspondence with two sides and a non-included angle that are equal in measure, yet the triangles are not identical, one triangle can be made identical to the other by “swinging” the appropriate side along a circle with radius equal to the length of that side until the side again intersects the adjacent side. Lesson 12 / Exploratory Challenge 3

Checking for Identical Triangles
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet Checking for Identical Triangles In Lesson 13, students begin applying the learned conditions for unique triangles and determining if a given pair of triangles are identical based on those conditions. In Lesson 14, the application is extended into more complex diagrams where students must recognize and use angle relationships and structural relationships to determine if two triangles are identical based on known sets of conditions for unique triangles. These problems are more like the triangle congruence diagrams that you remember from high school geometry. Remember however that we are not using the term “congruent” since congruence by definition requires knowledge of rigid motions (8th grade). In Example 1, you can see that students must recognize that there is a shared side between the two triangles. Some students will find this difficult to recognize so it may be wise to consider each triangle as an individual diagram. The triangles are in fact identical by the three sides condition, having the triangle correspondence ∆𝑊𝑌𝑍↔∆𝑌𝑊𝑋. In Example 2 students must recognize the presence of vertical angles (or vertically opposite angles) at vertex 𝑂, and use the relationship of vertical angles to conclude that the triangles are in fact similar using the Two Sides and Included Angle Condition for identical triangles. Through the examples and exercises in these lessons, students are confronted with diagrams in which triangles are, or are not identical, and must justify their reasoning based on the conditions learned in previous lessons. (MP2) Lesson 14 / Examples 1-2

Using Unique Triangles to Solve Real World and Mathematical Problems
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Using Unique Triangles to Solve Real World and Mathematical Problems Quadrilateral 𝐴𝐵𝐶𝐷 is a model of a kite. Diagonals 𝐴𝐵 and 𝐶𝐷 represent the sticks that help keep the kite rigid. In lesson 15, students are introduced to a new challenge where they must determine if triangles are identical, and then show how their conclusions can be used to draw other conclusions about the diagram. Jon says that ∠𝐴𝐶𝐷=∠𝐵𝐶𝐷. Can we use identical triangles to show that Jon is correct? (Yes: ∆𝐴𝐶𝐷↔∆𝐵𝐶𝐷 by the Three Sides condition, and angles ACD and BCD are corresponding angles of identical triangles, therefore having equal measures) Jill says that the two sticks are perpendicular to each other. Use the fact that ∠𝐴𝐶𝐷=∠𝐵𝐶𝐷 and what you know about identical triangles to show that ∠𝐴𝐸𝐶=90°. (Triangles ACE and BCE are identical by the Two Sides and Included Angle Condition, and therefore angles AEC and BEC are equal in measure because they are corresponding angled in identical triangles. The angles are also angles on a line which means they have a sum of 180 degrees, so each angle must be 90 degrees, meaning the sticks must be perpendicular.) Lesson 15 / Exercise 2

Mid-Module Assessment
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Mid-Module Assessment Mid-Module Assessment Assessment Progression Toward Mastery Rubric Exemplary Student Responses The Mid-Module Assessment follows topics A and B and cover the standards 7.G.A.2 and 7.G.B.5. You will find in your binder materials the assessment, the Progression Toward Mastery Rubric, and exemplary student responses. In the interest of addressing the concepts and material in the lessons, we will not be directly addressing the assessment in this session. Please feel free to ask any questions that you may have with regard to the Mid- or End of Module Assessments during a break period.

Topic C: Slicing Solids
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Topic C Opener Topic C: Slicing Solids 7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Slicing 3-Dimensional figures by a plane and the resulting 2-Dimensional slices. In Topic C, students explore the cross sections (or slices) of three-dimensional figures and the resulting two-dimensional surfaces that result from different slices. Students start out with slices that are parallel to the base(s) of right rectangular prisms and pyramids, then begin slicing these solids such that the slice intersects a base in a line segment, first using slices that are perpendicular to the base, the those that are not perpendicular to the base. Students use comparisons of slices made to predict how to slice a given solid in order to yield a slice of a given shape (for example a hexagon). Finally students examine solids that are constructed of unit cubes (similar to those explored in Topic C of Module 3) and the whole-unit slices of the solid in order to determine the number of unit cubes in the solid.

Slicing a Right Rectangular Prism with a Plane
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Slicing a Right Rectangular Prism with a Plane Students first must come to a uniform understanding of the term “slice”. Students may consider several ideas of a slice, such as a slice of pizza, a slice of cake, etc. We will consider a slice as the result of a single-motion cut through a solid. To make this clear, consider the result of a single slice of cured deli meat. Further consideration needs to be made as to whether the resulting piece of deli meat that goes on your sandwich is then the slice, or whether the resulting surface of the cut is the slice. This leads into a discussion of the plane (a building block of geometry and an undefined term) and what it means for two planes to be parallel, and two planes to be perpendicular. We use this discussion to conclude that a slice is the intersection of a plane with a solid figure, meaning it is only the surface of the cut, 2-dimensions, without depth. Students first draw the representation of a slice that is parallel to a face of a right rectangular prism. Through examination, students conclude that the slice is also parallel to the opposite face, and is furthermore perpendicular to the other four faces of the prism. They then look at slices that are perpendicular to faces of a prism. Consider Exercises 5 and 6. The slice of the prism that is parallel to the base (bottom as shown) is a rectangle that is identical to the base, having dimensions of 20 𝑚𝑚 x 13 𝑚𝑚. In Exercise 6, the slice is perpendicular to the base (bottom as shown) and therefore will have the same height as the prism of 12 𝑚𝑚, however the width of the slice is at least 8 𝑚𝑚 (as long as the slice intersects the front and back faces as shown). If the slice happens to be perpendicular to the front and back as shown, then the slice has a width of 8 𝑚𝑚 and is parallel to the right and left faces (as shown). Have participants complete the Exit Ticket for Lesson 16 independently. Lesson 16 / Exercises 5-6

Right Rectangular Pyramid
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Right Rectangular Pyramid Bla Students were introduced to the right rectangular pyramid in Module 3, however it was not formally defined until now in Lesson 17. Students first discuss what they know about a pyramid, then narrow the scope of the pyramid to a rectangular pyramid, and finally a right rectangular pyramid. Lesson 17

Right Rectangular Pyramid
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Right Rectangular Pyramid Visuals are provided in the lessons to assist students in understanding this definition as it is very technical language. [Go over the images and captions on the screen] This leads into the definition of the “right” rectangular pyramid. In a right rectangular pyramid, the line segment that joins the pyramid’s vertex with the center of the base is perpendicular to the base plane. To assist students in understanding the structure of a right rectangular pyramid and the figures that will be obtained via slicing with a plane, Examples 1-3 have students draw the views of a pyramid from a variety of perspectives. Lesson 17

Slicing a Right Rectangular Pyramid with a Plane
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Slicing a Right Rectangular Pyramid with a Plane After slicing a pyramid parallel to its rectangular base, students now consider slices that are perpendicular to the base, and quickly realize that the slices become much more interesting that those of the rectangular prism. Have participants complete Example 5. Ask for a participant to share their answer and justification. (use document camera if necessary) Lesson 17 / Example 5

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Slicing on an Angle Things get really interesting in Lesson 18 as the slices of solids are no longer necessarily parallel or perpendicular to any of the faces or bases. Is it possible to get a triangular slice from a right rectangular prism? How? Lesson 18 / Example 1

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Slicing on an Angle Students see in Example 3 that a correspondence exists between the number of sides on the polygonal face resulting from the slice and the number of faces on the 3-dimensional solid. A slice cannot have more sides than the number of faces on the solid. [Provide a few minutes for participants to sketch slices of a right rectangular prism to produce slices in the forms of figures Show the results of participants using the document camera.] Lesson 18 / Example 3

Understanding 3-Dimensional Figures
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Understanding 3-Dimensional Figures If slices parallel to the tabletop are taken of this figure, then what would each slice look like? Given the level slices that you determined, how many unit cubes are in the figure? Students were introduced to solids formed of unit cubes in module 3 and found the surface areas of those solids, but volume was left for Module 6. Students now examine the link between the volume of a given figure and its slices as a precursor to similar skills that will be studied in Grade 10 Geometry. Figures in this lesson can be imagined as though a 1-unit x 1-unit grid is placed on the plane, and unit cubes are placed into the squares on the grid, then layers are stacked atop each other to form a solid. Students are able to use the slicing approach to determine the volume of such a figure, and further are able to reverse the process and build a three-dimensional figure based on a given set of layers. Start Exercise 2 off with participants, then have them complete the remaining layers of the figure. When finished, ask participants to determine the number of units cubes in the figure based on their layers. Lesson 19 / Exercise 2

Topic D: Problems Involving Area and Surface Area
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Topic D Opener Topic D: Problems Involving Area and Surface Area 7.G.B.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Students had some practice with area and surface area in Module 3, and they return to using strategies for area in Topic D to solve real world problems, make sense of algebraic reasoning using the distributive property, and solve problems involving composite figures involving polygonal regions, circular regions, and holes in the shapes of polygons and circles. These composite figures are more complex than those experienced in Module 3. In the last two lessons, the topic transitions from area to surface area once again, involving composite solids that may or may not include holes. The included holes are three-dimensional and this needs to be considered in calculations.

Real World Area Problems
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Real World Area Problems Ten targets are being painted as shown in the following figure. The radius of the smallest circle is 3 𝑖𝑛., and each successive, larger circle is 3 𝑖𝑛. more in radius than the circle before it. A “tester” can of red and of white paint is purchased to paint the target. Each 8 𝑜𝑧. can of paint covers 16 𝑓 𝑡 2 . Is there enough paint of each color to create all ten targets? Example 2 is a particularly tasty example of using area in a real world context to solve a problem. [Provide time for participants to complete the problem, and if time allows, share their strategies used with the room.] Lesson 20 / Example 2

Mathematical Area Problems
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Mathematical Area Problems Examine the change in dimension and area of squares as the lengths and widths are increased by the given amount. In this lesson, we use an area model to make sense of a process that many students struggle to understand with regard to algebraic reasoning. The main objective of the lesson is to generalize a formula for the area of rectangles that result from adding to the length and width. [Use the slide to introduce the problem, but then allow participants time to complete the opening exercise in their packets.] Following the Example, ask participants to complete Example 3 in their packets. It is important to realize that negative areas do not make mathematical sense, so it is important that this be used as a foundational strategy for making sense of algebraic properties, however students must be scaffolded up to the use of strictly algebraic strategies. Lesson 21 / Example 1

Area Problems with Circular Regions
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Area Problems with Circular Regions The diameters of four half circles are sides of a square with side lengths of 7 𝑐𝑚. Find the exact area of the shaded region. The problems in Lesson 22 are a step up from those experienced in Module 3. After completing several scaffolding examples for finding regions in the lesson, students are asked to complete this problem. Lesson 22 / Problem Set #4

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Surface Area Determine the surface area of the right prism. Use composition and/or decomposition strategies to find the surface area of the figure shown on the screen. [Allow participants time to complete this exercise. There are several strategies that will work. Share strategies] Lesson 23 / Exercise 4

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Surface Area A right rectangular pyramid has a square base with side lengths of 10 inches. The surface area of the pyramid is 260 𝑖 𝑛 2 . Find the height of the four lateral triangular faces. Students also use the relationship of the area of a figure to its dimensions to determine missing dimensions. This was also done in Module 3, however only with the right rectangular prism. Again, allow time for participants to complete this problem, then share strategies aloud.] Lesson 24 / Example 3

Topic E: Problems Involving Volume
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Topic E Opener Topic E: Problems Involving Volume 7.G.B.6 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. The problems included in Topic E require students to exercise the additive property of volume for determining the volume of composite figures, determining the volume of an irregular figure via indirect measurement, as well as solving problems involving rates of flowing liquids.

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Volume of Right Prisms Two containers are shaped like right triangular prisms, each with the same height. The base area of the larger container is 200% more than the base area of the smaller container. How many times must the smaller container be filled with water and poured into the larger container in order to fill the larger container? Students take a slightly different approach to generalizing the formula V=bh in this module, based on a comparison of the area of a prism’s base to the area of a related base where the figure has been decomposed and rearranged into a familiar rectangle or triangle. It is seen that the total volume of a composite solid is equal to the volume of the decomposed pieces, all of which have the same height. Therefore the total volume of the prism is equal to the area of the composed base times the height of the prism. [Give participants time to complete the problem then share strategies with their table and the room.] Lesson 25 / Example 3

Volume of Composite 3-Dimensional Objects
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Volume of Composite 3-Dimensional Objects Find the volume of the right prism shown whose base is the region between two right triangles. Use two different strategies. Objects are more complex in lesson 26 including composite figures that include holes in the shapes of familiar figures. This was see to some degree in Module 3, and is continued and extended in Module 6. Lesson 26 / Example 2

Real-World Volume Problems
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: Participant Work Packet X Real-World Volume Problems Jim wants to know how much his family spends on water for showers. Water costs $1.50 per 1000 gallons. His family averages 4 showers per day (total). The average length of a shower is 10 minutes. He places a bucket in his shower and turns on the water. After one-minute, the bucket has 2.5 gallons of water in it. About how much money does his family spend on water for showers in a month (30 days)? In this lesson, students bring all learned knowledge about volumes and rates of flow to solve problems in real world and mathematical contexts. They use formula V=rt to solve problems. [Provide time for participants to complete the problem, then share out solutions and strategies] Lesson 27 / Exit Ticket

End-of-Module Assessment
Grade 7 – Module 6 Module Focus Session May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: End-of-Module Assessment X End-of-Module Assessment Assessment Progression Toward Mastery Rubric Exemplary Student Responses

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Module 6 in Review Take a few moments to discuss, with those around you, the major highlights of the Module. Likes What was and what is Concerns Major Takeaways

May 2014 Network Teams Institute TIME ALLOTTED FOR THIS SLIDE: ELAPSED TIME minutes MATERIALS NEEDED: X Key Points Student transitioning to more complex levels of geometry Using algebra to solve unknown angle problems Explore the structure of triangles Determine the conditions that determine unique triangles Identify identical triangles and use their relationship to solve problems Describe the slices of three-dimensional figures Apply strategies for area, surface area, and volume in solving real world and mathematical problems You (participants) have identified most, if not all, of the key points listed on this slide. Students are transitioning to more advanced levels of geometry skills where diagrams are much more complex, several geometric relationships are present within nearly any diagram, and problems are taking on sequences of steps (more than previous) where justification is necessary to follow the sequence.

Grade 7 – Module 6 Module Focus Session

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Grade 7 – Module 6 Module Focus Session

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