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I. Triangle Inequality II. Grade Level: Geometry 8 by Kim Dinwiddie III. Length of Lesson: 2‐3 days IV. Overview V. Context of the Lesson VI. Connections to State and National Standards Common Core State Standards: In this inquiry lesson, students will investigate the question, What is the relationship between the longest side and the sum of the remaining sides of a triangle? Students will use materials to create models of triangles and non‐triangles, and they will measure the sides and angles of a scalene triangle. They will investigate the relationship between the longest side of a triangle and the sum of its other two sides, as well as the relationship between the largest angle and the longest side of a triangle. This unit should be taught during a unit on triangles. Before this lesson, students should already have explored how to classify triangles by angles and/or sides into acute, obtuse, right, scalene, equilateral, and isosceles triangles. This lesson serves as a good transition into congruence theorems. During this guided inquiry lesson, some students will also investigate the relationship between an angle of a triangle and its opposite side: this is a great segue into a lesson (that should immediately follow this one) on ordering the triangle angles based on the lengths of the sides and vice versa. Before the lesson, give a pre‐assessment that addresses students’ measuring skills with a ruler and a protractor and their understanding of the relationship between sides of a triangle. Use the pre‐ assessment results to group students of similar levels of skill and understanding together. This lesson uses a variety of adaptations for diverse learners (differentiation strategies): kinesthetic learners can use materials to create triangles (or non‐triangles); visual learners can present data in a visual way; and auditory learners can listen to class and group discussion. National Mathematics Standards for Grades 6‐8 Geometry: In grades 6‐8 all students should • understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects, • create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship. • use coordinate geometry to represent and examine the properties of geometric shapes. • draw geometric objects with specified properties, such as side lengths or angle measures. • 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. Virginia Standards of Learning (SOLs) for Mathematics: Triangle Inequality 1 • • SOL 8.6 The student will a) verify by measuring and describe the relationships among vertical angles, adjacent angles, supplementary angles, and complementary angles; and b) measure angles of less than 360° SOL G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will c) determine whether a triangle exists; and d) determine the range in which the length of the third side must lie. VII. Unit Goals and Lesson Objectives a. Know (facts) o Angle types (acute, obtuse, right) o Degree o Triangle parts b. Understand (big idea) Patterns help us predict. c. Do (skills) VIII. o o o o o o Use the triangle inequality to solve problems involving triangles. Measure triangle side lengths with rulers. Measure triangle angles with protractors. Given the lengths of three sides, determine whether a triangle is formed. Given the lengths of two sides of a triangle, determine the range of the third side’s length. Use mathematical terminology/vocabulary for angles: acute, obtuse, right. Pre‐assessment of students’ prior knowledge and/or skills Before this lesson, students should be comfortable with and skilled at using a ruler and a protractor to measure lengths and angles. Use the pre‐assessment (see Resources) to assess this skill in order to see which students may need some extra guidance during the lesson. The pre‐assessment will also give you an idea of how well students already understand the relationship between the longest side and the sum of the other two sides of a triangle. Given three side lengths, students will answer whether a triangle can be constructed: while they may be able to answer this correctly, they will likely have difficulty justifying their answers, which indicates real comprehension. Give the pre‐assessment the day before the inquiry lesson, then use the results to group students of similar skill and understanding levels together, in order to encourage collaboration, enable more advanced students to further challenge themselves and each other, and give you the opportunity to give more focused guidance to those students who are struggling. IX. Materials Pre‐assessment: • Pre‐assessment paper/pencil worksheet • Pencils Triangle Inequality 2 Guided Inquiry Lesson: • Supplies to construct triangles: uncooked spaghetti noodles; straws; etc (Place 15 or so of these in plastic baggies to cut down on material‐gathering time; keep some in reserve in case students need more than is in the baggie.) • Scissors • Rulers • Protractors • Pencils • Paper • Formative Assessment Data Notes Taking Chart (for you) • Day 1 Homework handout • Day 2 Exit Card Post‐assessment: • Post‐assessment paper/pencil worksheet • Pencils X. Level of Inquiry: Guided This is a guided inquiry lesson. You will present the question, but students will work together in groups to plan and carry out their own investigations, collect and analyze data, and present data to the class. You should act as a facilitator rather than an instructor and ensure that your students are actively engaged. XI. Teaching Strategies Pre‐Assessment Give this the day before beginning the lesson in order to form groups of similar skill and understanding levels. Day 1 Divide students into the groups you have determined based on pre‐assessment results. Pass out two sets of three sticks to each group, one set that will form a triangle and one that won’t. After giving them time to manipulate the sticks, explain that not all sets of three sticks (line segments) can form a triangle. WHY? Tell them that this is what they will be exploring in class. Write the investigation question on the board: What is the relationship between the longest side and the sum of the remaining sides of a triangle? Give students 10‐15 minutes to discuss hypotheses and procedures for finding the answer. As they discuss, circulate through the room, jotting down notes and offering help and guiding questions as needed. In their procedures, students may include only sides of an already‐complete triangle. If this happens, have a brief class discussion of the experimental benefits of using lengths that do not create triangles as well. Remind students that they will present their methods, results, and conclusions to the class tomorrow. Do not tell them how to present their data: leaving this up to them will likely result in a range of data recording and presenting methods that should yield a rich class discussion of data collection and presentation methods. Have students work in their groups for the remainder of class to conduct their experiments. Monitor their progress by observing their experiments and jotting down notes, using the Formative Assessment Data Note Taking Chart (see Resources). If they need help, do not tell them what to do or what they Triangle Inequality 3 need to know, but ask probing questions to guide them through. Make sure that they are using more than one trial (more than one triangle) to answer the question. Some of the more advanced groups may finish early; ask these groups to use a procedure similar to the one they have been using to investigate the relationship between the length of a side and the size of the angle opposite it. Pass out the homework assignment for Day 1 (see Resources). Day 2 Have each group present their findings to the class. Ideally, they should all have reached the conclusion that regardless of type of triangle, the sum of the two smaller sides must be greater than the longest side of the triangle. Take notes on the presentations as part of your formative assessment. Discuss as a class the different procedures groups followed. Some groups may have measured different types of triangle than others did; discuss this and how it relates to the investigation question. Discuss as a class the findings of the groups that investigated the relationship between angle and opposite side. If students who did not investigate this question challenge the conclusion (that the largest angle is opposite the longest side and the smallest opposite the shortest side), invite everyone to measure their own triangles with rulers and protractors to discover this relationship for themselves. At the end of the day, give students the Day 2 Exit Card (see Resources). Post‐assessment Give the post‐assessment a day after the lesson. XII. Assessment Plan XIII. Resources As stated in section VII, the goals for this guided inquiry lesson include learning about the relationship between the longest side of a triangle and the other two sides, as well as the relationship between size of angle and length of opposite side. Summative assessments include the pre‐ and post‐assessment. These test student understanding of these relationships in various ways, including justifying their answers, which will give you a good sense of how successfully the lesson reached the learning objectives. Formative assessments include your informal notes on the note‐taking chart; your notes on the student presentations; the homework assignment from Day 1, and the Exit Card from Day 2. These will all give you a sense not only of how well the students learned the relationships that were the learning objectives, but also of how well they take to and like inquiry‐based learning. Books: Llewellyn, D. (2007). Inquire Within: Implementing Inquiry‐Based Science Standards in Grades 3‐8, 2nd Edition. Corwin Press. Websites: Johnson, H. L. (n.d.). Illuminations: Inequalities in Triangles. Retrieved August 20, 2014 from http://illuminations.nctm.org/Lesson.aspx?id=2339 Triangle Inequality 4 Pre‐assessment 1. Measure the length of the line segment to the nearest centimeter. 2. Measure the angle to the nearest degree. For questions 3 – 5, can you construct a triangle with the given lengths? Why or why not? 3. 3cm, 8 cm, 14 cm 4. 12 cm, 12 cm, 12 cm 5. 8 cm, 10 cm, 14 cm Triangle Inequality 5 Formative Assessment Data Notes Chart Student Name What’s the relationship? Organize Data Largest Side Opposite Largest Angle Notes Put check mark if observed ~ include some details if necessary Triangle Inequality 6 Homework, Day 1: In the box below, write 3 inequalities that are always true for a triangle with side lengths a, b, and c. ________ + ________ > _________ ________ + ________ > _________ ________ + ________ > __________ Triangle Inequality 7 Exit Card, Day 2 1. 2. 3. 4. 5. 6. Can you construct a triangle with side lengths 3 cm, 8 cm, and 14 cm? Why or why not? Can you construct a triangle with side lengths 12 cm, 12 cm, and 12 cm? Why or why not? Can you construct a triangle with side lengths 8 cm, 10 cm, and 14 cm? Why or why not? Two sides of a triangle are 6 cm and 10 cm. Determine a range of possible measures for the third side. Two sides of an isosceles triangle measure 3 cm and 7 cm. Which of the following could be the measure of the third side? a. 9 b. 7 c. 3 In triangle ABC, the measure of angle A is 30 degrees and angle B is 50 degrees. Which is the longest side of the triangle? a. Triangle Inequality AB b. AC c. BC 8 Post‐assessment 1) Hannah is putting a border around her triangular garden. Two sides of the garden have lengths of 12 feet and 17 feet. What is the range in which the third side of the garden must fall? 2) In ΔABC , with BC > AC , which of the following statements must be true? A. m ∠C is greater than m ∠B B. m ∠B is greater than m ∠A C. m ∠ A is greater than m ∠B D. m ∠C is greater than m ∠A 3) In ΔDEF, side DE = 8 inches, FE = 6 inches, and FD = 10 inches. Which lists the angles in order from smallest to largest? A. ∠E, ∠F, ∠D B. ∠F, ∠D, ∠E C. ∠D, ∠E, ∠F D. ∠D, ∠F, ∠E 4) John wants to make a triangular garden. Which of the following are possible dimensions? A. 4 ft by 5 ft by 10 ft B. 6 ft by 8 ft by 10 ft C. 8 ft by 12 ft by 20 ft D. 6 ft by 6 ft by 12 ft 5) Two sides of a triangle measure 14 inches and 8 inches. Which cannot be the length of the r emaining side? A. 21 in. B. 6 in. C. 14 in. D. 8 in. Triangle Inequality 9 6) Jennifer made these measurements on ΔABC . BC must be — A. between 12 and 22 inches B. between 10 and 12 inches X C. greater than 22 inches D. less than 10 inches 50 °° B C 60 10 12 A 7) Josh is planning a trip and the path of his top three destinations forms triangle ABC. Order the angles from smallest to largest. 8) Kari needs to create a triangular structure for her science project. Her teacher has allowed her to pick three pieces of wood from a large pile to use for her project. Which three pieces of wood could create a triangle? A. 2.7 in, 4.3 in, 6.8 in B. 6.7 in, 4.2 in, 2.4 in C. 1.3 in, 8.1 in, 6.4 in D. 2.3 in, 5.1 in, 7.4 in Triangle Inequality 10