Geometry Chapter Opener for Chapter 1: Basics of Geometry and
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Geometry Chapter Opener for Chapter 1: Basics of Geometry and
Geometry Chapter Opener for Chapter 1: Basics of Geometry and Lesson 1.1 – Day 1: Points, Lines, and Planes Opener Objective: To review skills necessary for the upcoming chapter. Vocabulary Review: base, height Pacing: 20 minutes TEKS Standards 6.2.B 6.3.D 6.8.D INTRODUCTION (5 minutes) Other Resources Scaffolding in the Classroom Graphic Organizers: Concept Circle: A Concept Circle can be used to organize information about a concept. Students write the concept above the circle. Then students write associated information in the sectors of the circle. Associated information can include (an explanation of the) Concept, Apply, Solve, Check, Example, and Justify. Concept Circles can have any number of sectors. Students can place their concept circles on note cards to use as a quick study reference. Explain how the idea is useful for the opener. • • • • Student Journal Skills Review Handbook Dynamic Classroom Lesson Tutorials Other Resources PART 1 (5 minutes) Finding Absolute Value Review the example with the students. Check for understanding. Have students work through the exercises. • • • • Student Journal Skills Review Handbook Dynamic Classroom Lesson Tutorials Other Resources PART 2 (5 minutes) Finding the Area of a Triangle Review the example with the students. Check for understanding. Have students work through the exercises. MATHEMATICAL THINKING (5 minutes) • • • • Student Journal Skills Review Handbook Dynamic Classroom Lesson Tutorials Other Resources • Dynamic Classroom Mathematical Thinking Review the example with the students. Check for understanding. Have students work through the Monitoring Progress questions. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0100_opener_0101_lessonDay1 1 of 3 Big Ideas Math Texas Geometry Lesson Plan Essential Question: How can you use dynamic geometry software to visualize geometric concepts? Lesson Objective(s): Students will name points, lines, and planes. Students will name segments and rays. Students will sketch intersections of lines and planes. Students will solve real-life problems involving lines and planes. Previous Learning: Students will be familiar with the basic geometric terms of point, line, segment, ray, and plane. They should also be familiar with how each is represented, meaning the notation used to identify these geometric objects. New Vocabulary: undefined terms, point, line, plane, collinear points, coplanar points, defined terms, line segment, segment, endpoints, ray, opposite rays, intersection Materials for Teacher: coffee stirrers, straws, spaghetti, wooden dowels, chopsticks, file folders, cardboard dividers from a box, posterboard Materials for Students: dynamic geometry software Pacing: Day 1 – 25 minutes TEKS Standards Mathematical Process Focus G.2.A G.2.B G.1.D Other Resources • Dynamic Classroom • Start Thinking, Warm Up, and Cumulative Review Warm Up • Homework Check • Answer Presentation Tool • Laurie’s Notes INTRODUCTION (5 minutes) Warm Up Have students answer Warm Up questions. Review the answers as a class. Motivate from Laurie’s Notes Write two lists of words on the board – those beginning with “geo-” and those ending with “-metry.” For example, geothermal, geopolitics, geophysical, geology, geoid, geometry; and asymmetry, symmetry, trigonometry, optometry, densitometry, geometry. Ask students to discuss the two lists with partners, specifically deciding what the prefix “geo-“ means and the suffix “-metry” means. The prefix “geo-“ is derived from the Greek word geo, which means earth. The suffix “metry” means the process or science of measuring. It is derived from the Greek word metria, which means to measure. EXPLORATION 1 (5 minutes) Other Resources Using Dynamic Geometry Software • Dynamic Classroom • Student Journal • Laurie’s Notes In this exploration, students draw points, lines, line segments, and rays with dynamic geometry software. • Have students work in pairs to complete the exploration. • Students should be familiar with the terms point, segment, ray, and line. The goal is for students to familiarize themselves with the software. • Ask the students if these geometric objects could be drawn without the coordinate grid. • Discuss the difference between synthetic and coordinate geometry. Geometry will be explored in both environments this year. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0100_opener_0101_lessonDay1 2 of 3 Big Ideas Math Texas Geometry Lesson Plan EXPLORATION 2 (5 minutes) Other Resources Intersections of Lines and Planes • Dynamic Classroom • Student Journal • Laurie’s Notes In this exploration, students sketch lines and planes, describing how they intersect. • Have students work in pairs to complete parts (a) – (c). • The goal of this exploration is to get students talking and observing the geometry around them. • In addition to the physical components of the classroom, you should have available items that can serve as models of lines and planes. For lines, you could use coffee stirrers, straws, spaghetti, wooden dowels, or chopsticks. For planes, you could use file folders, cardboard dividers from a box, or poster board. • Ask for volunteers in each group to model the various possibilities. Other Resources EXPLORATION 3 (5 minutes) • Dynamic Classroom • Student Journal • Laurie’s Notes Exploring Dynamic Geometry Software In this exploration, students explore dynamic geometry software. • If time permits, allow students to use the software to explore terms with which they might not be familiar. • Have students work in pairs to complete the exploration. Student Focus on Mathematical Process Standard G.1.D Discuss the Mathematical Process statement with students. Other Resources CHECK FOR UNDERSTANDING (5 minutes) • Dynamic Classroom • Student Journal • Laurie’s Notes Closure (as time permits) Communicate Your Answer • Give students time to answer Question 4. • You could extend this question to include the objects and physical components of the classroom. Homework Assignment • Abstract Reasoning Exercise (Opener) and 3 – 6 Suggestions for Leveling • Basic: Abstract Reasoning Exercise (Opener) and 3, 5 • Advanced: Abstract Reasoning Exercise (Opener) and 4, 6 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0100_opener_0101_lessonDay1 3 of 3 Other Resources • • • • • • • Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Dynamic Assessment & Progress Monitoring Tool Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.1 – Day 2: Points, Lines, and Planes Essential Question: How can you use dynamic geometry software to visualize geometric concepts? Lesson Objective(s): Students will name points, lines, and planes. Students will name segments and rays. Students will sketch intersections of lines and planes. Students will solve real-life problems involving lines and planes. Previous Learning: Students will be familiar with the basic geometric terms of point, line, segment, ray, and plane. They should also be familiar with how each is represented, meaning the notation used to identify these geometric objects. New Vocabulary: undefined terms, point, line, plane, collinear points, coplanar points, defined terms, line segment, segment, endpoints, ray, opposite rays, intersection Materials for Teacher: file folders, wooden dowels, Internet resources Materials for Students: dynamic geometry software, colored pencils Pacing: Day 2 – 45 minutes Mathematical Process Focus G.2.A G.2.B G.1.A, G.1.C Other Resources INTRODUCTION (5 minutes) Warm Up Have students answer Start Thinking questions. Review the answers as a class. Review previously assigned homework, if necessary. Copyright © Big Ideas Learning, LLC All rights reserved. TEKS Standards TXGeom_lessonplans_0101_lessonDay2 1 of 4 Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 1 (8 minutes) Other Resources Core Concept – Undefined Terms: Point, Line, and Plane In this Core Concept, students learn the undefined terms in geometry, along with their notation. Introduce the undefined terms – point, line, and plane. Ask students how many lines can be drawn through one point and how many can be drawn through two points. Ask students how many planes can be drawn through one point, how many can be drawn through two points, and how many can be drawn through three points not on the same line. Use these questions to help students visualize geometric relationships and to understand necessary conditions to establish a line and a plane. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Naming Points, Lines, and Planes In this example, students name points, lines, and planes in a diagram. Have students Think-Pair-Share for parts (a) and (b). Have students work in pairs to answer Monitoring Progress Question 1. Review the answers together, with students presenting their work to the class. Focus on Mathematical Process Standard G.1.A Make a three-dimensional physical model for Example 1. Use a file folder for the plane and wooden dowel for line n. EXAMPLE 2 (7 minutes) Other Resources Core Concept – Defined Terms: Segment and Ray In this Core Concept, students learn some of the defined terms in geometry, along with their notation. Introduce the defined terms – segment and ray – and how each is represented. For the diagram shown, be sure students understand that ሬሬሬሬሬԦ ܤܣand ሬሬሬሬሬԦ ܣܤare not the same ray, but ሬሬሬሬሬԦ ܣܤand ശሬሬሬሬሬ ܤܣare the ሬሬሬሬሬԦ ശሬሬሬሬሬ same ray. However, it is not common to write ܣܤas ܤܣ. Use physical models in the discussion of the terms. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Naming Segments, Rays, and Opposite Rays In this example, students name segments, rays, and opposite rays in a diagram. Have students work with a partner for parts (a) and (b). Have students work independently to answer Monitoring Progress Questions 2 and 3. Then have neighbors check each other’s work. Review the answers together, with students presenting their work to the class. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0101_lessonDay2 2 of 4 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 3 (5 minutes) Other Resources Sketching Intersections of Lines and Planes In this example, students sketch intersections of lines and planes. Sketching intersections of lines and planes can be challenging for students. Share the explicit steps for sketching. Offer colored pencils for shading if available. Hand a file folder (model of a plane) and a wooden dowel (model of a line) to a student. Ask the student what the intersection possibilities are for a line and a plane. Students should notice that the non-rectangular parallelogram is used in sketching planes. It gives the perspective of continuing to cover the plane in a way that a rectangle does not. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Focus on Mathematical Process Standards G.1.A and G.1.C For each case, the student should use the tools to model the intersection. Modeling with physical objects is easier than sketching! EXAMPLE 4 (7 minutes) Other Resources Sketching Intersections of Planes In this example, students sketch planes that intersect in a line. Have students use Think-Pair-Share to work through this example. Have students work in pairs to answer Monitoring Progress Questions 5 – 7. Review the answers together, with students presenting their work to the class. EXAMPLE 5 (8 minutes) Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Other Resources Modeling with Mathematics In this example, students use the Four-Step Approach to Problem Solving to solve a real-life problem. Sulfur hexafluoride is a non-toxic, invisible gas that you can use to perform interesting chemistry demonstrations. Breathe it in and make your voice much deeper when you talk. Pour it into a container and float light objects on “nothing.” If time permits, search the Internet for “sulfur hexafluoride” and show a quick video demonstrating properties of sulfur hexafluoride. Display an image of the sulfur hexafluoride molecule and pose the problem. Have students work with a partner to Understand the Problem and Make a Plan. Ask the students why the plane must contain line r and a point not on line r. Have the students work independently to finish the example. Review the answer. Have students work in pairs to answer Monitoring Progress Questions 8 – 10. Review the answers together, with students presenting their work to the class. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0101_lessonDay2 3 of 4 Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Big Ideas Math Texas Geometry Lesson Plan ASSESS (5 minutes) Other Resources Closure (as time permits) Sketch and label the cube shown on page 7 in the Teaching Edition. Ask students to identify segments, intersecting lines, intersecting planes, a line not on a plane, and so on. Homework Assignment 1, 2, 10 – 46 even, 50 – 54 even, 55, 65 – 72 Other Resources Suggestions for Leveling Basic: 1, 2, 3 – 19 odd, 25 – 43 odd, 50, 55, 65 – 72 Advanced: 1, 2, 12 – 26 even, 34 – 54 even, 55 – 72 Copyright © Big Ideas Learning, LLC All rights reserved. Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool TXGeom_lessonplans_0101_lessonDay2 4 of 4 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.2 – Day 1: Measuring and Constructing Segments Essential Question: How can you measure and construct a line segment? Lesson Objective(s): Students will use the Ruler Postulate. Students will use the Segment Addition Postulate. Students will use the Distance Formula Students will copy segments and compare segments for congruence. Previous Learning: Students have prior knowledge of congruent segments from middle school. New Vocabulary: postulate, axiom, coordinate, distance, between, construction, congruent segments Materials for Teacher: standard paper clip, unsharpened pencil, marker, 1-liter bottle Materials for Students: rulers, standard paper clips, straightedges, 3-inch by 5-inch index cards Pacing: Day 1 – 45 minutes TEKS Standards Mathematical Process Focus G.2.B G.5.B G.5.C G.1.B, G.1.D, G.1.G INTRODUCTION (10 minutes) Other Resources • Dynamic Classroom • Start Thinking, Warm Up, and Cumulative Review Warm Up • Homework Check • Answer Presentation Tool • Laurie’s Notes Warm Up Have students answer Start Thinking questions. Review the answers as a class. Review previously assigned homework, if necessary. Motivate from Laurie’s Notes Display four items at the front of the room, such as a standard paper clip, an unsharpened pencil, a marker, and an empty 1-liter bottle. Divide the class into four groups. Each group is assigned one of the objects and asked to estimate the length of one classroom wall using their nonstandard unit. Gather estimates and record. Do a quick vote to decide on the best estimate. Ask the students if it is possible to measure the wall using any of the four objects as the unit of measure. It would take quite some time to perform the actual estimates, so do this in advance using the efficient method of measuring in inches and doing a conversion. Share results. Discuss The goal of the exploration is to help students recognize that we have standard and metric units of measures that many are familiar with; however, any length could be established as the base of a measurement system. The challenge, of course, is that if you use a nonstandard unit of measure – say a paper clip – not everyone would be familiar with it. In the world of commerce, it is necessary to communicate in units that are understood by all. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0102_lessonDay1 1 of 3 Big Ideas Math Texas Geometry Lesson Plan EXPLORATION 1 (12 minutes) Other Resources Measuring Line Segments Using Nonstandard Units In this exploration, students use a paper clip to measure line segments. • Have students work in pairs to complete parts (a) – (d) • Students should have sharp pencils when measuring a segment with paper clips. • Most students will have a measure of 1.25” for 1 paper clip. This conversion factor, like any conversion factor, can be written two ways: 1.25 inches = 1 paper clip or 1 inch = 0.8 paper clip. • Ask students to show a Thumbs Up to determine whether they understand the problem posed in part (d). It may not be obvious to students that they should use the straightedge to start with a segment they know looks longer than 6 inches. • Dynamic Classroom • Student Journal • Laurie’s Notes Student Focus on Mathematical Process Standard G.1.B Discuss the Mathematical Process statement with students. EXPLORATION 2 (10 minutes) Other Resources Measuring Line Segments Using Nonstandard Units In this exploration, students use a paper clip to measure line segments. • Have students work in pairs to complete parts (a) – (d). • If you do not have 3-inch by 5-inch index cards available, you could use a different sized index card or cut scrap paper to a 3-inch by 5-inch size. • After the students find the length of the diagonal, ask students what fraction of an inch is 0.83 inch. • In part (d), students will recognize that while the Pythagorean Theorem works for any nonstandard unit of measure, it is difficult to be accurate when measuring with paper clips. • Dynamic Classroom • Student Journal • Laurie’s Notes Focus on Mathematical Process Standard G.1.G The exact length of the diagonal is √34 inches, which is approximately 5.83 inches. EXPLORATION 3 (8 minutes) Other Resources • Dynamic Classroom • Student Journal • Laurie’s Notes Measuring Heights Using Nonstandard Units In this exploration, students find their heights using nonstandard units. • Have students work with a partner to complete this exploration. Focus on Mathematical Process Standard G.1.D Students will likely convert their known height in inches to a new unit called 1 diag. Ask volunteers to explain how they found their height in diags. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0102_lessonDay1 2 of 3 Big Ideas Math Texas Geometry Lesson Plan ASSESS (5 minutes) Other Resources Closure (as time permits) I Used to Think… But Now I Know: Take time for students to reflect on their current understanding of units of measure. Homework Assignment • Communicate Your Answer Other Resources • • • • • • Suggestions for Leveling • Basic: Communicate Your Answer • Advanced: Communicate Your Answer Copyright © Big Ideas Learning, LLC All rights reserved. • Mini-Assessment • Extra Practice (A and B) • Dynamic Assessment & Progress Monitoring Tool • Answer Presentation Tool TXGeom_lessonplans_0102_lessonDay1 3 of 3 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.2 – Day 2: Measuring and Constructing Segments Essential Question: How can you measure and construct a line segment? Lesson Objective(s): Students will use the Ruler Postulate. Students will use the Segment Addition Postulate. Students will use the Distance Formula Students will copy segments and compare segments for congruence. Previous Learning: Students have prior knowledge of congruent segments from middle school. New Vocabulary: postulate, axiom, coordinate, distance, between construction, congruent segments Materials for Teacher: dynamic geometry software Materials for Students: graph paper, compasses, straightedges Pacing: Day 2 – 45 minutes TEKS Standards Mathematical Process Focus G.2.B G.5.B G.5.C G.1.C INTRODUCTION (5 minutes) Other Resources Warm Up Have students answer Warm Up questions. Review the answers as a class. Review previously assigned homework, if necessary. EXAMPLE 1 (5 minutes) Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes Other Resources Postulate 1.1 – Ruler Postulate Discuss the postulate with the students. Ask students what the verb postulate means and what the noun postulate means. Discuss the difference between AB and തതതത ܤܣ. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Using the Ruler Postulate In this example, students find the length of a segment. Work through the example with the students. തതതത . This would be less likely if the coordinates were negative Students may know by inspection the length of ܧܤ rational numbers. Have students work in pairs to answer Monitoring Progress Questions 1 – 4. Review the answers together, with students presenting their work to the class. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0102_lessonDay2 1 of 3 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 2 (8 minutes) Other Resources Postulate 1.2 – Segment Addition Postulate Discuss the postulate with the students. Copy the postulate on the board. Draw a sketch to show what it would mean if point B is not between point A and point C. See Laurie’s Notes, 1.2 Lesson. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Using the Segment Addition Postulate In this example, students find the lengths of segments using the Segment Addition Postulate. Have students use Think-Pair-Share to answer parts (a) and (b). If students struggle with part (b), guide them to set up the equation where they will see that all they have to do is subtract. Have students work in pairs to answer Monitoring Progress Questions 5 – 7. Review the answers together, with students presenting their work to the class. Focus on Mathematical Process Standard G.1.C Demonstrate examples, such as Example 2, using dynamic geometry software. EXAMPLE 3 (8 minutes) Other Resources Core Concept – The Distance Formula In this Core Concept, students learn how to find the distance between two points in a coordinate plane. Using the Distance Formula Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool In this example, students solve a real-life problem using the Distance Formula. Complete the example with the students. Students should see the connection between the Pythagorean Theorem and finding the length of a segment. Have students Turn and Talk. Have partner A explain the connection between the Pythagorean Theorem and the Distance Formula. Choose a partner B to explain the connection to the class. Have students work in pairs to answer Monitoring Progress Question 8. Review the answer together, with students presenting their work to the class. Focus on Mathematical Process Standard G.1.C Choosing to locate your apartment at the origin is a convenient way to draw a model for Example 3. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0102_lessonDay2 2 of 3 Big Ideas Math Texas Geometry Lesson Plan CONSTRUCTION (5 minutes) Other Resources Dynamic Classroom Laurie’s Notes Copying a Segment In this construction, students use a compass and straightedge to copy a segment. Work through the construction with the students. Have students draw their own segments and construct copies of the segments. EXAMPLE 4 (7 minutes) Other Resources Core Concept – Congruent Segments In this Core Concept, students learn the definition of and notation for congruent segments. Copy and discuss the Core Concept. Students should be familiar with congruent segments (and angles) from middle school. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Comparing Segments for Congruence In this example, students compare segments in a coordinate plane to see if they are congruent. Pose the example and ask students how they will decide whether the segments are congruent. Have students work independently to solve and then check with neighbors. Have students work independently to answer Monitoring Progress Question 9. Then have neighbors check each other’s work. Review the answer together, with students presenting their work to the class. ASSESS (7 minutes) Other Resources Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Closure (as time permits) Writing Prompt: A postulate is . . . Writing Prompt: An example of a postulate is . . . Homework Assignment 1, 8 – 36 even, 40, 42, 45 – 52 Other Resources Suggestions for Leveling Basic: 1, 2, 3 – 33 odd, 40, 42, 45 – 52 Advanced: 1, 2, 10 – 42 even, 43 – 52 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0102_lessonDay2 3 of 3 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.3 – Day 1: Using Midpoint Formulas Essential Question: How can you find the midpoint of a line segment on a number line or in a coordinate plane? Lesson Objective(s): Students will find segment lengths using midpoints and segment bisectors. Students will partition a segment on a number line. Students will find the midpoint of a segment in a coordinate plane. Previous Learning: Students are familiar with the Pythagorean Theorem, which is related to the Distance Formula. Students also know how to solve equations with variables on both sides. New Vocabulary: midpoint, segment bisector Materials for Teacher: none Materials for Students: centimeter graph paper Pacing: Day 1 – 45 minutes TEKS Standards Mathematical Process Focus G.2.A G.2.B G.1.F INTRODUCTION (10 minutes) Other Resources Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes Warm Up Have students answer Start Thinking questions. Review the answers as a class. Review previously assigned homework, if necessary. Motivate from Laurie’s Notes Ask students what common distances they know. Discuss what it means to find a distance: for instance, not all of the examples provided in Laurie’s Notes, 1.3 Exploration are segments. Refer to a distance in the classroom, such as the distance between yourself and a particular student and ask where the point that is halfway between the two of you is. Then ask what the distance is from you to the point halfway between the two of you if you are 16 feet 4 inches from the student. Hold up a piece of paper with a line segment drawn. Have students Turn and Talk to describe how they could find the middle of this line segment. Students may describe a paper folding method or using a ruler to measure the segment. Ask if a similar method could be used if you wanted to divide the segment into three equal parts, four equal parts, and five equal parts. Tell students that in this lesson they will partition a segment on a number line. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0103_lessonDay1 1 of 3 Big Ideas Math Texas Geometry Lesson Plan EXPLORATION 1 (10 minutes) Other Resources Dynamic Classroom Student Journal Laurie’s Notes Finding the Midpoint of a Line Segment In this exploration, students find the midpoint of a line segment on a number line. Have students work in pairs to complete parts (a) – (e). As students begin this exploration, listen to be sure that they understand the word bisect. A formal definition is not necessary, simply the concept of what it means to bisect something. Students can find the midpoint by inspection. Remind them that they are looking for a rule that will work for any coordinates: both positive, both negative, rational, and so on. Ask for a volunteer to share his or her rule. Student Focus on Mathematical Process Standard G.1.F Discuss the Mathematical Process statement with students. EXPLORATION 2 (8 minutes) Other Resources Finding the Midpoint of a Line Segment In this exploration, students find the midpoint of a line segment on graph paper. Have students work in pairs to complete parts (a) – (e). The goal of this exploration is for students to discover how to determine the midpoint of a segment when the coordinates of the endpoints are known. Solicit answers in part (b) as to how students bisected the segment. Some students may fold their paper. Others may use a slope approach and say, “Starting at point B go right 4, up 3, and repeat again to get to point A.” Listen for valid approaches. Have students Turn and Talk. Ask partner A to explain how the coordinates of point M relate to the coordinates of points A and B, and then ask a partner B to state the relationship to the class. CHECK FOR UNDERSTANDING (7 minutes) Other Resources Communicate Your Answer Have the students Turn and Talk about Question 3. Ask partner B to explain how to find the midpoint and length of a line segment in the coordinate plane. Then ask a partner A to summarize for the class. Give students time to answer Questions 4 and 5. Review the answers together. Copyright © Big Ideas Learning, LLC All rights reserved. Dynamic Classroom Student Journal Laurie’s Notes TXGeom_lessonplans_0103_lessonDay1 2 of 3 Dynamic Classroom Student Journal Laurie’s Notes Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 1 (5 minutes) Other Resources Core Concept – Midpoints and Segment Bisectors In this Core Concept, students learn the definitions of midpoint and segment bisector. Write the Core Concept. Ask students how midpoints and segment bisectors are similar and how they are different. Ask students if a segment can have more than one midpoint and if a segment can have more than one bisector. Then ask if a line can have more than one midpoint. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Finding Segment Lengths In this example, students find the length of a segment using the Segment Addition Postulate and definition of midpoint. Have students use Think-Pair-Share to work through this example. Have students work in pairs to answer Monitoring Progress Questions 1 and 2. Review the answers together, with students presenting their work to the class. Other Resources ASSESS (5 minutes) Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Closure (as time permits) Writing Prompt: A midpoint is… Writing Prompt: A segment bisector is… Homework Assignment 3–6 Other Resources Suggestions for Leveling Basic: 3, 5 Advanced: 4, 6 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0103_lessonDay1 3 of 3 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.3 – Day 2: Using Midpoint Formulas Essential Question: How can you find the midpoint of a line segment on a number line or in a coordinate plane? Lesson Objective(s): Students will find segment lengths using midpoints and segment bisectors. Students will partition a segment on a number line. Students will find the midpoint of a segment in a coordinate plane. Previous Learning: Students are familiar with the Pythagorean Theorem, which is related to the Distance Formula. Students also know how to solve equations with variables on both sides. New Vocabulary: midpoint, segment bisector Materials for Teacher: none Materials for Students: graph paper Pacing: Day 2 – 45 minutes TEKS Standards Mathematical Process Focus G.2.A G.2.B G.1.C, G.1.D, Other Resources INTRODUCTION (5 minutes) Warm Up Have students answer Warm Up questions. Review the answers as a class. Review previously assigned homework, if necessary. EXAMPLE 2 (5 minutes) Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes Other Resources Using Algebra with Segment Lengths In this example, students find the length of a segment using the definition of midpoint. Pose the problem and have students work with their partners to solve. Solving an equation with variables on both sides should be a secure skill for students. Have students work independently to answer Monitoring Progress Questions 3 and 4. Review the answers together, with students presenting their work to the class. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool CONSTRUCTION (5 minutes) Other Resources Bisecting a Segment Dynamic Classroom Laurie’s Notes In this construction, students use paper folding to find the midpoint of a segment. Ask students to discuss with a partner why this technique works. Focus on Mathematical Process Standards G.1.C and G.1.D If students have paper-folded to find the midpoint in the explorations, change the context. You want to find the midpoint of something that cannot be folded, like the diagonal of a picture frame. You have a straightedge (not a ruler) and paper. Explain how to find the midpoint. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0103_lessonDay2 1 of 3 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 3 (7 minutes) Other Resources Core Concept – Partitioning a Segment on a Number Line In this Core Concept, students learn how to partition a segment on a number line into a given ratio. Ask students how they could divide a segment into three equal parts if they know the endpoints of the segment on a number line. Write the Core Concept, noting the order in which the ratio is stated, namely b : a. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Partitioning a Segment In this example, students find the coordinate of the point that partitions a given segment on a number line into a given ratio. The values used in this example help students make sense of how to calculate the coordinate of the partitioning point. Students may think of the calculation as a weighting process, meaning 2 : 1 means the partitioning point is closer to the right endpoint so that coordinate is weighted more. Other Resources EXAMPLE 4 (8 minutes) Finding the Midpoint In this example, students find the coordinate of the midpoint of a given segment on a number line. Pose the problem and have students work independently to solve. Have students work in pairs to answer Monitoring Progress Questions 5 – 7. Review the answers together, with students presenting their work to the class Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0103_lessonDay2 2 of 3 Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Big Ideas Math Texas Geometry Lesson Plan Other Resources EXAMPLE 5 (10 minutes) Core Concept – Midpoint of a Segment in a Coordinate Plane In this Core Concept, students learn how to find the midpoint of a segment in a coordinate plane. Write the Core Concept. Note the use of the word average in describing the coordinates of the midpoint. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Using the Midpoint In this example, students use the Midpoint Formula to find the coordinates of the midpoints of line segments in a coordinate plane. Have students work through part (a) with a partner. Students may take the absolute value of coordinates that are negative. Finding distance involves absolute value, but determining the midpoint does not. Complete part (b) together. Ask students if they can find a shortcut for finding the missing coordinates of an endpoint rather than writing out the algebraic steps. One way to do this is to double the coordinates of the midpoint, and subtract the coordinates of the given endpoint. Have students work in pairs to answer Monitoring Progress Questions 8 – 11. Review the answers together, with students presenting their work to the class. If students are secure with the Midpoint Formula, you may want to omit Example 5 and check for understanding with Monitoring Progress Questions 8 and 10, or 9 and 11. ASSESS (5 minutes) Other Resources Closure (as time permits) Exit Ticket: Given A(–3, 5) and B(4, –1), find the coordinates of the midpoint of തതതത ܤܣand the length of segment AB. Homework Assignment 1, 2, 8 – 12 even, 16 – 34 even, 35, 40, 44 – 51 and Mathematical Thinking questions (page 27) Suggestions for Leveling Basic: 1, 2, 7 – 33 odd, 40, 44 – 51 and Mathematical Thinking questions (page 27) Advanced: 1, 2, 8 – 12 even, 16 – 32 even, 33, 38 – 42, 44 – 51 and Mathematical Thinking questions (page 27) Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0103_lessonDay2 3 of 3 Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Other Resources Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Dynamic Classroom Big Ideas Math Texas Geometry Lesson Plan Geometry Sections 1.1 – 1.3 What Did You Learn? and Quiz and Lesson 1.4 – Day 1: Perimeter and Area in the Coordinate Plane What Did You Learn? and Quiz Objective: To review and administer the Quiz. Vocabulary: undefined terms, point, line, plane, collinear points, coplanar points, defined terms, line segment, segment, endpoints, ray, opposite rays, intersection, postulate, axiom, coordinate, distance, between, construction, congruent segments, midpoint, segment bisector Pacing: 25 minutes Other Resources INTRODUCTION (5 minutes) • Skills Review Handbook • Dynamic Classroom • Lesson Tutorials What Did You Learn? Ask students to review any Core Vocabulary or Core Concepts they do not remember. ASSESS (20 minutes) Other Resources Quiz Administer the quiz from the textbook or Assessment Book Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0100_midwdyl_quiz_0104_lessonDay1 1 of 3 • • • • Assessment Book Skills Review Handbook Dynamic Classroom Lesson Tutorials Big Ideas Math Texas Geometry Lesson Plan Essential Question: How can you find the perimeter and area of a polygon in a coordinate plane? Lesson Objective(s): Students will classify polygons. Students will find perimeters and areas of polygons in the coordinate plane. Previous Learning: Students have a prior understanding of perimeter and area of polygons, as well as the names of the polygons in the lesson. They will also use the Distance Formula from earlier in this chapter. Previous Vocabulary: polygon, side, vertex, n-gon, convex, concave Materials for Teacher: Popsicle Sticks Materials for Students: centimeter graph paper, straightedges Pacing: Day 1 – 20 minutes TEKS Standards Mathematical Process Focus G.11.B G.1.D, G.1.F, G.1.G Other Resources INTRODUCTION (5 minutes) • Dynamic Classroom • Start Thinking, Warm Up, and Cumulative Review Warm Up • Homework Check • Answer Presentation Tool • Laurie’s Notes Warm Up Have students answer Start Thinking questions. Review the answers as a class. Review previously assigned homework, if necessary. Motivate from Laurie’s Notes Draw a parallelogram and divide it into a rectangle and two triangles, as shown in Laurie’s Notes, 1.4 Exploration. Ask students to show how they could divide a hexagon and octagon into familiar polygons. EXPLORATION 1 (5 minutes) Other Resources Finding the Perimeter and Area of a Quadrilateral • Dynamic Classroom • Student Journal • Laurie’s Notes In this exploration, students find the perimeter and area of a quadrilateral in a coordinate plane. • Have students work in pairs to complete parts (a) – (d). • This exploration is an application of the Distance Formula, so students should need little guidance in finding the perimeter. • Ask students how they can tell whether adjacent sides are perpendicular. • Ask students if they can tell a quadrilateral has four right angles if they find one right angle (a pair of adjacent sides that are perpendicular). • It is quite possible that instead of subtracting coordinates to find the vertical and horizontal distances, students will count with their fingers on the diagram. • Use Popsicle Sticks to select a student to share his/her justification for part (d). Focus on Mathematical Process Standards G.1.D, G.1.F, and G.1.G Students should construct a viable argument using definitions to determine whether the quadrilateral is a square. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0100_midwdyl_quiz_0104_lessonDay1 2 of 3 Big Ideas Math Texas Geometry Lesson Plan EXPLORATION 2 (5 minutes) Other Resources Finding the Area of a Polygon • Dynamic Classroom • Student Journal • Laurie’s Notes In this exploration, students find the areas of triangles and a square. • Have students work in pairs to complete parts (a) – (c). • If students have worked with geoboards or dot paper, deconstructing the quadrilateral into four triangles and a square will seem familiar. • Students should recognize the 3-4-5 right triangles in this diagram. Students may also have the spatial skills to see that two of the triangles could be transformed to form a 3-by-4 rectangle. • Ask the students if the sum they found for quadrilateral ABCD is the same as their answer in the previous exploration. Student Focus on Mathematical Process Standard G.1.F Discuss the Mathematical Process statement with students. CHECK FOR UNDERSTANDING (5 minutes) Closure (as time permits) Other Resources • Dynamic Classroom • Student Journal • Laurie’s Notes Communicate Your Answer • Give students time to answer Question 3. • To find the perimeter of a polygon in a coordinate plane, you need to find the length of each side. To find the area of a polygon, it is necessary to deconstruct the polygon into polygons of which you know how to find the area. • Ask students if they can think of a polygon that they would not be able to deconstruct into simpler polygons. Homework Assignment • Communicate Your Answer Question 4 Other Resources Suggestions for Leveling • Basic: Communicate Your Answer Question 4 • Advanced: Communicate Your Answer Question 4 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0100_midwdyl_quiz_0104_lessonDay1 3 of 3 • • • • • • Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.4 – Day 2: Perimeter and Area in the Coordinate Plane Essential Question: How can you find the perimeter and area of a polygon in a coordinate plane? Lesson Objective(s): Students will classify polygons. Students will find perimeters and areas of polygons in the coordinate plane. Previous Learning: Students have a prior understanding of perimeter and area of polygons, as well as the names of the polygons in the lesson. They will also use the Distance Formula from earlier in this chapter. Previous Vocabulary: polygon, side, vertex, n-gon, convex, concave Materials for Teacher: Popsicle Sticks Materials for Students: graph paper, whiteboards Pacing: Day 2 – 45 minutes INTRODUCTION (5 minutes) Mathematical Process Focus G.11.B G.1.B, G.1.G Other Resources Warm Up Have students answer Warm Up questions. Review the answers as a class. Review previously assigned homework, if necessary. Copyright © Big Ideas Learning, LLC All rights reserved. TEKS Standards TXGeom_lessonplans_0104_lessonDay2 1 of 4 Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 1 (7 minutes) Other Resources Core Concept – Polygons In this Core Concept, students review vocabulary about polygons. Write the Core Concept. Students should be familiar with the terminology. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Classifying Polygons In this example, students classify polygons by the number of sides and then tell whether the polygons are convex or concave. The (mostly Greek) numerical prefixes for polygons are likely familiar to students. They may have heard the 7-sided polygon called a septagon, from the Latin prefix “septua-.” “Hepta-“ is the Greek prefix for seven. The suffix “-gon” is from the Greek meaning angles. Interestingly, trilateral and quadrilateral are Latin. Note concave polygons are defined as not being convex. This may be students’ first experience with a definition that, instead of defining what it is, defines what it is not. Students should be familiar with concave lenses and spoons. Have students use Think-Pair-Share to complete parts (a) and (b). To quickly check for understanding, ask students to sketch a convex hexagon, concave octagon, and concave triangle (not possible). Use Popsicle Sticks to select students to share their work at the board. Have students work in pairs to answer Monitoring Progress Questions 1 and 2. Review the answers together, with students presenting their work to the class. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0104_lessonDay2 2 of 4 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 2 (5 minutes) Other Resources Discuss Discuss the Perimeter and Area box with the students. Have the students copy the formulas for perimeter and area into their notes. Ask students if they can sketch a rectangle with an area of 8 inches. The purpose of this question is for students to recognize the units. It is impossible because the units are not squared. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Finding Perimeter in the Coordinate Plane In this example, students find the perimeter of a triangle in a coordinate plane. This example is similar to Explorations 1 and 2, except that a calculated distance is irrational. Have students work in pairs to complete the example. Have students work in pairs to answer Monitoring Progress Questions 3 – 6. Review the answers together, with students presenting their work to the class. You could also assign odds to some students and evens to the rest of the students. Have students present their work on whiteboards. Focus on Mathematical Process Standard G.1.G Discuss the difference between exact measures (√61) and approximate measures (7.81). EXAMPLE 3 (10 minutes) Other Resources Finding Area in the Coordinate Plane In this example, students find the area of a triangle in a coordinate plane. Have students work with a partner. Have students Turn and Talk to discuss what information they will need to find the area of the triangle. You may also hear other students discuss how the triangle could be rearranged to form a rectangle. One possibility is shown in Laurie’s Notes, 1.4 Lesson. Left alone, students may follow different routes. Have students work in pairs to answer Monitoring Progress Questions 7 – 10. Review the answers together, with students presenting their work to the class. You could also assign odds to some students and evens to the rest of the students. Have students present their work on whiteboards. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Focus on Mathematical Process Standard G.1.B Allow time for different solution methods to be shown. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0104_lessonDay2 3 of 4 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 4 (8 minutes) Other Resources Modeling with Mathematics In this example, students use the Four-Step Approach to Problem Solving to solve a real-life problem. Students should have little difficulty finding the area of the rectangle in this example. Differentiate instruction for this example by having different polygons of which students find the area. Parallelograms, hexagons, and octagons should provide an appropriate challenge for students. Have the students use Think-Pair-Share to Understand the Problem and Make a Plan. Then have students work independently to Solve the Problem and Look Back. Have students use Think-Pair-Share to answer Monitoring Progress Question 11. Review the answer together, with students presenting their work to the class. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Other Resources ASSESS (10 minutes) Closure (as time permits) I Used to Think . . . But Now I Know: Summarize the skills and strategies you are confident in as a result of today’s lesson. Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Homework Assignment 1, 2 – 28 even, 31, 32, 36, 38 – 44 Other Resources Suggestions for Leveling Basic: 1, 2, 3 – 25 odd, 31, 32, 36, 38 – 44 Advanced: 1, 2 – 32 even, 33, 34, 36 – 44 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0104_lessonDay2 4 of 4 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.5 – Day 1: Measuring and Constructing Angles Essential Question: How can you measure and classify an angle? Lesson Objective(s): Students will name angles. Students will measure and classify angles. Students will identify congruent angles. Students will use the Angle Addition Postulate to find angle measures. Students will bisect angles. Previous Learning: Measuring and classifying angles should be familiar to students from previous grades. They found the sum of interior angles and the sum of exterior angles of a triangle in grade 8. New Vocabulary: angle, vertex, sides of an angle, interior of an angle, exterior of an angle, measure of an angle, acute angle, right angle, obtuse angle, straight angle, congruent angles, angle bisector Previous Vocabulary: protractor, degrees Materials for Teacher: protractor, Popsicle Sticks Materials for Students: rulers, protractors, scissors, heavyweight paper Pacing: Day 1 – 45 minutes TEKS Standards Mathematical Process Focus G.5.B G.5.C G.1.C, G.1.G INTRODUCTION (5 minutes) Other Resources Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes Warm Up Have students answer Start Thinking questions. Review the answers as a class. Review previously assigned homework, if necessary. Motivate from Laurie’s Notes Draw and label ሬሬሬሬሬԦ ܻܺ and ሬሬሬሬሬԦ ܻܼ with an angle of about 80° on the board. Label ∠XYZ. Ask the students what the measure is of ∠XYZ. Ask them how they know for sure. Then extend one ray of the angle, and ask what the measure of the angle is now. Students should recognize that the measure of the angle has not changed. Model how to use a protractor and measure the angle. Discuss The common error or confusion that students have in using a protractor is reading the wrong scale. A relatively simple strategy is to think first about whether the angle is acute or obtuse. Certainly looking at which scale reads 0° when one ray of the angle is aligned with the edge of the protractor is another strategy. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0105_lessonDay1 1 of 4 Big Ideas Math Texas Geometry Lesson Plan EXPLORATION 1 (5 minutes) Other Resources Dynamic Classroom Student Journal Laurie’s Notes Measuring and Classifying Angles In this exploration, students measure and classify angles. Have students work in pairs to complete parts (a) – (h). This exploration should be a review and should not take long. Solicit answers for each of the angles listed using Popsicle Sticks. Focus on Mathematical Process Standards G.1.C and G.1.G These problems serve as a quick review of reading a protractor to find the measure of an angle. You will also hear students use the vocabulary associated with angles. EXPLORATION 2 (10 minutes) Other Resources Drawing a Regular Polygon In this exploration, students draw a regular hexagon and find the sum of the angle measures. Have students work in pairs to complete parts (a) – (d). You should have scissors and heavyweight paper available for students to use. If students are accurate with their tracing, they should end up with a regular hexagon. Students found the sum of the interior angles of a triangle in grade 8. They also found the exterior angles sum as well. The Big Idea in this exploration is that when a polygon is decomposed into smaller polygons, the interior angle sums still add up to the expected sum. For instance, when the hexagon is decomposed into two trapezoids, four of the angles are still 120°. Two of the 120° angles have each been bisected into two 60° angles. Note that 4(120°) + 4(60°) = 720°. When the regular hexagon is decomposed into six equilateral triangles, you cannot find the sum of the interior angles of the hexagon by multiplying 6(180°). There are six 60° angles at the center of the hexagon that must be subtracted from the sum because they are not part of the interior angles of the hexagon. Dynamic Classroom Student Journal Laurie’s Notes Student Focus on Mathematical Process Standard G.1.G Discuss the Mathematical Process statement with students. CHECK FOR UNDERSTANDING (5 minutes) Other Resources Communicate Your Answer Give students time to answer Question 3. Ask for volunteers to share their answer. Students should be comfortable using the protractor to measure angles and then classify them. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0105_lessonDay1 2 of 4 Dynamic Classroom Student Journal Laurie’s Notes Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 1 (5 minutes) Other Resources Naming Angles In this example, students name angles in a figure. Have students Turn and Talk. Sketch a figure similar to Example 1. Have partner A share what he/she knows about naming angles. Use Popsicle Sticks and ask a partner B to share with the class how to name angles. Discuss the common technique of using numbers in the interior of an angle. When a diagram includes several rays, numbers are a convenient way to refer to various angles. Have students work in pairs to answer Monitoring Progress Questions 1 – 3. Review the answers together, with students presenting their work to the class. EXAMPLE 2 (5 minutes) Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Other Resources Postulate 1.3 – Protractor Postulate Discuss the postulate with the students. Write the postulate and connect it to the work students did in the explorations. The lengths of the rays forming the angle have no bearing on the measure of the angle. Refer to the Motivate in Laurie’s Notes, Overview of Section 1.5. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Core Concept – Types of Angles In this Core Concept, students learn the names of the different types of angles. Explain to students that angles are classified according to their measure Have students draw and label an example of each type of angle in their notebooks. Measuring and Classifying Angles In this example, students measure and classify angles. If students completed the explorations, you may want to omit Example 2. You can go straight to the Monitoring Progress Questions 4 – 6. Have students work in pairs for this example. Have students work independently to answer Monitoring Progress Questions 4 – 6. Then have neighbors check each other’s work. Review the answers together, with students presenting their work to the class. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0105_lessonDay1 3 of 4 Big Ideas Math Texas Geometry Lesson Plan ASSESS (10 minutes) Other Resources Closure (as time permits) Writing: Describe the Protractor Postulate in your own words. Homework Assignment 4 – 14 even Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Other Resources Suggestions for Leveling Basic: 3 – 13 odd Advanced: 6, 8, 12, 14 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0105_lessonDay1 4 of 4 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.5 – Day 2: Measuring and Constructing Angles Essential Question: How can you measure and classify an angle? Lesson Objective(s): Students will name angles. Students will measure and classify angles. Students will identify congruent angles. Students will use the Angle Addition Postulate to find angle measures. Students will bisect angles. Previous Learning: Measuring and classifying angles should be familiar to students from previous grades. They found the sum of interior angles and the sum of exterior angles of a triangle in grade 8. New Vocabulary: angle, vertex, sides of an angle, interior of an angle, exterior of an angle, measure of an angle, acute angle, right angle, obtuse angle, straight angle, congruent angles, angle bisector Previous Vocabulary: protractor, degrees Materials for Teacher: none Materials for Students: compasses, straightedges, whiteboards Pacing: Day 2 – 45 minutes TEKS Standards Mathematical Process Focus G.5.B G.5.C G.1.D, G.1.G INTRODUCTION (5 minutes) Other Resources Warm Up Have students answer Warm Up questions. Review the answers as a class. Review previously assigned homework, if necessary. Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes CONSTRUCTION (10 minutes) Other Resources Copying an Angle Dynamic Classroom Laurie’s Notes In this construction, students use a compass and straightedge to copy an angle. Work through the steps in copying an angle. The construction can be proven valid by the Side-Side-Side Congruence Theorem. You may wish to refer back to this construction in Section 5.5. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0105_lessonDay2 1 of 3 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 3 (5 minutes) Other Resources Identifying Congruent Angles In this example, students identify congruent angles. Discuss how to represent the fact that two angles have the same measure and that two angles are congruent. Have students work independently on the example. Have students work in pairs to answer Monitoring Progress Question 7. Review the answer together, with students presenting their work to the class. EXAMPLE 4 (5 minutes) Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Other Resources Postulate 1.4 – Angle Addition Postulate Discuss the postulate with the students. The Angle Addition Postulate is similar to the Segment Addition Postulate. Ask a student to summarize what the postulate means. Listen for something like, “When a ray is drawn in the interior of an angle and has an endpoint at the vertex, the measures of the two smaller angles add to the measure of the larger angle.” Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Finding Angle Measures In this example, students use the Angle Addition Postulate. Have partners solve this example on whiteboards and quickly check for understanding. Have students work in pairs to answer Monitoring Progress Questions 8 and 9. Review the answers together, with students presenting their work to the class. You could also have some students solve Question 8 and other students Question 9. Then ask two volunteers to share their work at the front of the class. Ask the students what the relationship is between the two angles in Question 8 and what the relationship is between the two angles in Question 9 to review supplementary and complementary angles. CONSTRUCTION (10 minutes) Other Resources Bisecting an Angle Dynamic Classroom Laurie’s Notes In this construction, students use a compass and straightedge to bisect an angle. There are several figures that can bisect a segment. Rays bisect angles. Work through the steps in bisecting an angle. The construction can be proven valid by the Side-Side-Side Congruence Theorem in Section 5.5 and corresponding parts of congruent triangles in Section 5.7. You could also have students paper-fold the angle bisector of an angle. Focus on Mathematical Process Standards G.1.D and G.1.G Ask the students if the following statement is always, sometimes, or never true: “When you bisect an obtuse angle, the result will be two obtuse angles.” Have students justify their reasoning and critique the reasoning of others. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0105_lessonDay2 2 of 3 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 5 (5 minutes) Other Resources Using a Bisector to Find Angle Measures In this example, students use the definition of an angle bisector to find the measure of an angle. Have students use Think-Pair-Share to complete the example. Have students use Think-Pair-Share to answer Monitoring Progress Question 10. Review the answer together, with students presenting their work to the class. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Other Resources ASSESS (5 minutes) Closure (as time permits) Exit Ticket: Draw the diagram on page 42 in the Teaching Edition. Given that m∠ABC = 143°, find m∠ABD and m∠DBC. Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Other Resources Homework Assignment 1, 2, 16 – 44 even, 49, 52, 54, 58 – 65 Suggestions for Leveling Basic: 1, 2, 15 – 27 odd, 43, 52, 54, 58 – 65 Advanced: 1, 2, 16, 28 – 32 even, 40 – 44 even, 47 – 49, 52, 54, 58 – 65 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0105_lessonDay2 3 of 3 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.6 – Day 1: Describing Pairs of Angles Essential Question: How can you describe angle pair relationships and use these descriptions to find angle measures? Lesson Objective(s): Students will identify complementary and supplementary angles. Students will identify linear pairs and vertical angles. Previous Learning: Students may have a previous understanding of angle pair relationships. New Vocabulary: complementary angles, supplementary angles, adjacent angles, linear pair, vertical angles Previous Vocabulary: vertex, sides of an angle, interior of an angle, opposite rays Materials for Teacher: aerial view of the runways at an airport Materials for Students: protractors Pacing: Day 1 – 45 minutes TEKS Standards Mathematical Process Focus G.6.A G.1.D, G.1.F, G.1.G INTRODUCTION (5 minutes) Other Resources Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes Warm Up Have students answer Start Thinking questions. Review the answers as a class. Review previously assigned homework, if necessary. Motivate from Laurie’s Notes Show an aerial view of the runways at an airport. Number the angles formed by different runways to facilitate students being able to reference them more easily. Ask students to make a list of pairs of angles and then state the relationships between each pair of angles. Do not specify the measures of the angles. You should have a sense of what relationships students recall and are familiar with. EXPLORATION 1 (10 minutes) Other Resources Finding Angle Measures In this exploration, students find the values of various angle measures. Have students work in pairs to complete parts (a) and (b). The measure of one interior angle of a regular pentagon is given. That is all that will be needed in order for students to determine the remaining angles. The important part of this exploration is hearing students give their explanations. Often students will use eyesight to state the angle measure, versus being able to state the underlying relationship. Dynamic Classroom Student Journal Laurie’s Notes Focus on Mathematical Process Standards G.1.D, G.1.F, and G.1.G A sample explanation might be, “Angle x is supplementary to the 108° angle, so it measures 72°. Angle z would be 72° for the same reason. Angle y is supplementary to angles x and z, so it is 108°. It is also a vertical angle. Angle w is supplementary to another interior angle of the regular pentagon, so it is 72°. Finally, the angles of a triangle sum to 180°, so angle v = 180° – 2(72°) = 36°.” Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0106_lessonDay1 1 of 3 Big Ideas Math Texas Geometry Lesson Plan EXPLORATION 2 (10 minutes) Other Resources Finding Angle Measures In this exploration, students find the values of various angle measures. Have students work in pairs to complete parts (a) and (b). Students come in knowing that a square has four right angles. In this exploration, what you are listening for is how students describe the angle pair relationships. Do they have the vocabulary to describe an angle being bisected? Two angles being congruent? Two angles being vertical or supplementary? CHECK FOR UNDERSTANDING (5 minutes) Other Resources Communicate Your Answer Dynamic Classroom Student Journal Laurie’s Notes Give students time to answer Questions 3 and 4. Ask for volunteers to share their answers. Listen for student understanding of common angle pair relationships such as complementary, supplementary, and vertical. Dynamic Classroom Student Journal Laurie’s Notes Student Focus on Mathematical Process Standard G.1.G Discuss the Mathematical Process statement with students. EXAMPLE 1 (5 minutes) Other Resources Core Concept – Complementary and Supplementary Angles In this Core Concept, students learn the definitions of complementary, supplementary, and adjacent angles. Write the Core Concept and draw sketches to support the definitions. Note that complementary and supplementary angles do not need to be adjacent. Their relationship is a matter of measurement, not position. Ask students for the complement and supplement of a 72° angle and a 142° angle. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Focus on Mathematical Process Standard G.1.G While students may be familiar with the concepts, they need to be aware of the definitions. For instance, adjacent angles share a common vertex and common side, but have no interior points in common, meaning the angles cannot overlap. Identifying Pairs of Angles In this example, students identify the three different types of pairs of angles. Have students use Think-Pair-Share to complete this example. Students should have a good understanding of these terms from the explorations. Note the Common Error in the sidebar column. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0106_lessonDay1 2 of 3 Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 2 (5 minutes) Other Resources Finding Angle Measures In this example, students use the definitions of complementary and supplementary angles to find the measures of angles. Have students work with a partner to complete the example. This example could be done by asking students to sketch or use a protractor to draw each of the angle pairs. Have students work independently to answer Monitoring Progress Questions 1 – 4. Then have neighbors check each other’s work. Review the answers together, with students presenting their work to the class. ASSESS (5 minutes) Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Other Resources Closure (as time permits) Think Aloud: Can two angles form a pair of both complementary and supplementary angles? Complementary and adjacent? Adjacent and supplementary? Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Homework Assignment 3 – 10 Other Resources Suggestions for Leveling Basic: 3 – 9 odd Advanced: 4 – 10 even Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0106_lessonDay1 3 of 3 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Lesson 1.6 – Day 2: Describing Pairs of Angles Essential Question: How can you describe angle pair relationships and use these descriptions to find angle measures? Lesson Objective(s): Students will identify complementary and supplementary angles. Students will identify linear pairs and vertical angles. Previous Learning: Students may have a previous understanding of angle pair relationships. New Vocabulary: complementary angles, supplementary angles, adjacent angles, linear pair, vertical angles Previous Vocabulary: vertex, sides of an angle, interior of an angle, opposite rays Materials for Teacher: Internet resources or math history books, Concept Cards Materials for Students: whiteboards Pacing: Day 2 – 45 minutes TEKS Standards Mathematical Process Focus G.6.A G.1.D, G.1.F, G.1.G INTRODUCTION (5 minutes) Other Resources Warm Up Have students answer Warm Up questions. Review the answers as a class. Review previously assigned homework, if necessary. EXAMPLE 3 (5 minutes) Other Resources Real-Life Application In this example, students apply pairs of angles to a real-life situation. A ball-return net may not be familiar to all students. Ask for examples of similar structures: ramps, mousetraps, windshield wipers, diagonal paths off a sidewalk, and so on. On whiteboards, have students work with a partner on this example. Have students work in pairs to answer Monitoring Progress Question 5. Review the answers together, with students presenting their work to the class. Copyright © Big Ideas Learning, LLC All rights reserved. Dynamic Classroom Start Thinking, Warm Up, and Cumulative Review Warm Up Homework Check Answer Presentation Tool Laurie’s Notes TXGeom_lessonplans_0106_lessonDay2 1 of 3 Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Big Ideas Math Texas Geometry Lesson Plan EXAMPLE 4 (10 minutes) Other Resources Core Concept – Linear Pairs and Vertical Angles In this Core Concept, students learn the definitions of linear pairs and vertical angles. Write the Core Concept. Students should be familiar with vertical angles, but have likely not heard the phrase linear pairs. Ask students what the relationship is between linear pairs and supplementary angles. A common question from students will be why vertical angles are called vertical instead of opposite. Also confusing to students is the use of vertical to describe the orientation of being upright, and yet the position of vertical angles can be horizontal! Searching online or in math history books, there are different explanations for the origin of the term vertical angles. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Identifying Angle Pairs In this example, students find all of the linear pairs and vertical angles in a diagram. Have students work independently on the example. Then find a partner to compare answers. Review the answers together. EXAMPLE 5 (15 minutes) Other Resources Finding Angle Measures in a Linear Pair In this example, students find angle measures from a description of a linear pair of angles. Students should have previous knowledge of writing expressions from a description from Algebra 1. Have students work with a partner to write the equation, and then solve independently. Compare answers with your partner, and share with the class. The Monitoring Progress Questions integrate skills from algebra. Students are expected to translate words into symbols. Have students work in pairs to answer Monitoring Progress Questions 6 – 8. Review the answers together, with students presenting their work to the class. Dynamic Classroom Laurie’s Notes Extra Example Lesson Tutorials Answer Presentation Tool Concept Summary – Interpreting a Diagram In this Concept Summary, students summarize things they can and cannot conclude from a diagram. Draw the diagram in the Concept Summary. Have students use Think-Pair-Share to make a list of those things that they can conclude from the diagram and those which they cannot conclude on whiteboards. Focus on Mathematical Process Standards G.1.D, G.1.F, and G.1.G Share as a whole class and have students give explanations for what statements were written for each list. Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0106_lessonDay2 2 of 3 Big Ideas Math Texas Geometry Lesson Plan Other Resources ASSESS (10 minutes) Concept Card Mapping: See the description of this activity in Laurie’s Notes, Overview of Section 1.6. Mini-Assessment Extra Practice (A and B) Dynamic Assessment & Progress Monitoring Tool Answer Presentation Tool Homework Assignment 1, 2, 12, 20 – 42 even, 46, 48, 49, 52 – 59 Other Resources Closure (as time permits) Suggestions for Leveling Basic: 1, 2, 11 – 27 odd, 46, 48, 52 – 59 Advanced: 1, 2, 14, 21, 22 – 46 even, 48 – 59 Copyright © Big Ideas Learning, LLC All rights reserved. TXGeom_lessonplans_0106_lessonDay2 3 of 3 Puzzle Time Student Journal Lesson Tutorials Skills Review Handbook Enrichment and Extension Differentiating the Lesson Big Ideas Math Texas Geometry Lesson Plan Geometry Sections 1.4 – 1.6 What Did You Learn? and Chapter Review Review Objective: To review for the chapter assessment. Vocabulary: angle, vertex, sides of an angle, interior of an angle, exterior of an angle, measure of an angle, acute angle, right angle, obtuse angle, straight angle, congruent angles, angle bisector, complementary angles, supplementary angles, adjacent angles, linear pair, vertical angles Pacing: 45 minutes INTRODUCTION (5 minutes) Other Resources What Did You Learn? Ask students to review any Core Vocabulary or Core Concepts they do not remember. • Skills Review Handbook • Dynamic Classroom • Lesson Tutorials MATHEMATICAL THINKING (5 minutes) Other Resources Mathematical Thinking • Dynamic Classroom • Lesson Tutorials Complete the Mathematical Thinking questions independently. Ask for volunteers to share answers. CHAPTER REVIEW (30 minutes) Other Resources Chapter Review Have students complete the review independently, with a partner, or in groups. ASSESS (5 minutes) Other Resources Homework • Chapter Test from the textbook or Assessment Book Copyright © Big Ideas Learning, LLC All rights reserved. • Resources by Chapter • Dynamic Classroom • Lesson Tutorials TXGeom_lessonplans_0100_endwdyl_review 1 of 1 • • • • Assessment Book Skills Review Handbook Dynamic Classroom Lesson Tutorials Big Ideas Math Texas Geometry Lesson Plan