8.2.7Ma


 * Grade: 8 Unit: 2 Week: 7 Dates: 11/12-11/16 **
 * Content: Using the Pythagorean Theorem and Proving the Pythagorean Theorem **


 * Theme Essential Question: **
 * How does mathematical reasoning apply to the attributes of two- and three- dimensional figures as seen in everyday life? **


 * Essential Questions: **
 * ** How can you use the Pythagorean Theorem to solve problems? **
 * ** How can you prove the Pythagorean Theorem and its converse? **


 * Standards **
 * ** 8.G.6 ** Explain a proof of the Pythagorean Theorem and its converse.
 * ** 8.G.7 ** Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
 * ** 8.G.8 ** Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.


 * Objectives **
 * The student will apply the Pythagorean Theorem to solve real-world two dimensional problems.
 * The student will become familiar with the common Pythagorean triplets.
 * The student will apply the Pythagorean Theorem to solve real-world three dimensional problems.
 * The student will use the Pythagorean Theorem to calculate distance between two points in the Coordinate Plane.
 * The student will generate a geometric argument for the proof of the Pythagorean Theorem.
 * The student will predict if a triangle is a right triangle by using the converse of the Pythagorean Theorem.


 * Reflections and/or Comments from your PCSSD 8th Grade Curriculum Team **

As you read and review the Background Information below, you will realize the concept that the sum of two side measures is greater than the third side measure might not be familiar to our students. This concept will be important as they apply the converse of the Pythagorean Theorem. It would be advisable to provide the students with this learning experience. (Glencoe Geometry The Triangle Inequality Theorem 5-4 (p. 261-262))

@http://www.azed.gov/wp-content/uploads/PDF/MathGr8.pdf
 * Background Information **
 * Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials. **
 * (Taken from Ohio Dept of Education Mathematics Model Curriculum 6-28-2011 ) **

Previous understanding of triangles, such as the sum of two side measures is greater than the third side measure, angles sum, and area of squares, is furthered by the introduction of unique qualities of right triangles. Students should be given the opportunity to explore right triangles to determine the relationships between the measures of the legs and the measure of the hypotenuse. Experiences should involve using grid paper to draw right triangles from given measures and representing and computing the areas of the squares on each side. Data should be recorded in a chart such as the one below, allowing for students to conjecture about the relationship among the areas within each triangle.

Students should then test out their conjectures, then explain and discuss their findings. Finally, the Pythagorean Theorem should be introduced and explained as the pattern they have explored.

Time should be spent analyzing several proofs of the Pythagorean Theorem to develop a beginning sense of the process of deductive reasoning, the significance of a theorem, and the purpose of a proof. Students should be able to justify a simple proof of the Pythagorean Theorem or its converse.

Previously, students have discovered that not every combination of side lengths will create a triangle. Now they need situations that explore using the Pythagorean Theorem to test whether or not side lengths represent right triangles. (Recording could include Side length //a//, Side length //b//, Sum of //a2// + //b2//, //c2//, //a2// + //b2// = c//2//, Right triangle?

Through these opportunities, students should realize that there are Pythagorean (triangular) triples such as (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41) that always create right triangles, and that their multiples also form right triangles. Students should see how similar triangles can be used to find additional triples. Students should be able to explain why a triangle is or is not a right triangle using the Pythagorean Theorem.

The Pythagorean Theorem should be applied to finding the lengths of segments on a coordinate grid, especially those segments that do not follow the vertical or horizontal lines, as a means of discussing the determination of distances between points.

Contextual situations, created by both the students and the teacher, that apply the Pythagorean Theorem and its converse should be provided. Challenge students to identify additional ways that the Pythagorean Theorem is or can be used in real world situations or mathematical problems, such as finding the height of something that is difficult to physically measure, or the diagonal of a prism.


 * Assessment **
 * Product **
 * Students should continue reviewing the problem solving connections on page 137-140 in the Houghton Mifflin OnCore Mathematics Middle School Grade 8 book. The final presentation of the “Where in the park is Xander?” can be presented as a PowerPoint, poster, etc. This will conclude the unit.


 * Key Questions **
 * How is the Pythagorean Theorem, a2 + b2 = c2, associated with the sides of the right triangle?
 * What are the two type of calculation associated with the Pythagorean Theorem?
 * How is a2 + b2 = c2 extended in order to be used in a three dimensional application?
 * How is the Pythagorean Theorem applied to find the length of a segment plotted on the coordinate plane?
 * What is a viable argument to justify the Pythagorean Theorem?
 * How is a triangle tested using the converse of the Pythagorean Theorem to show it is a right triangle?


 * Observable Student Behaviors **
 * The student can appropriately use the Pythagorean Theorem to solve real-world two and three dimensional problems.
 * The student can find the length of a line segment that has been plotted on the coordinate plane.
 * The student can provide a viable argument to prove the Pythagorean Theorem.

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||
 * ** Mathematical Practices **

Leg and hypotenuse of a right triangle, theorem, converse
 * Vocabulary **

Gizmo Lessons Explore the properties of complementary, supplementary, vertical, and adjacent angles using a dynamic figure. Derive the sum of the angles of a polygon by dividing the polygon into triangles and summing their angles. Vary the number of sides and determine how the sum of the angles changes. Dilate the polygon to see that the sum is unchanged. (Student Exploration Sheet & Teacher Guide Available) Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity. Measure the angles of a triangle and find the sum. Then reshape and resize the triangle and confirm that the sum of angle measures is the same for all triangles. Highly Recommended: The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
 * Suggested Activities ** [see Legend to highlight MCO and HYS]
 * Houghton Mifflin OnCore Mathematics Middle School Grade 8 Unit 5-5 and 5-6, p. 127-132
 * ABC Mastering the Common Core in Mathematics Chapter 11-3 through 11-7, p. 169-179
 * ** Investigating Angle Theorems-Activity A **
 * ** Polygon Angle Sum-Activity B **
 * ** Similar Polygons **
 * ** Triangle Angle Sum-Activity A **
 * __ Teaching the Common Core Math Standards with Hands-On Activities __ by Judith Muschla
 * 8.G.6- Activity 1 p. 227, Activity 2 p. 228
 * 8.G.7- p. 230
 * 8.G.8- p. 232
 * JBHM-8th GP3 (p. 105-152)
 * Glencoe Pre-Algebra The Pythagorean Theorem 9-5 (p. 460-464)
 * Glencoe Algebra 1 The Pythagorean Theorem 11-4 (p. 605-610)
 * Glencoe Geometry Activity (p. 349)
 * Glencoe Geometry The Pythagorean Theorem and Its Converse 7-2 (p. 350-356)
 * @http://illustrativemathematics.org/standards/k8 (Overall Objective)
 * @http://illustrativemathematics.org/illustrations/112 (G.7)


 * Diverse Learners **
 * Odyssey (teacher discretion)
 * Skills Tutor (teacher discretion)
 * Math’scool: Unit B Module 8

See appropriate Glencoe, Houghton Mifflin OnCore Mathematics Middle School Grade 8, JBHM, and ABC Mastering the Common Core in Mathematics materials
 * Homework **


 * Terminology for Teachers **

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 * Resources **


 * Professional Texts **


 * Literary Texts **

@http://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htm
 * Informational Texts **
 * See New York Common Core Aligned Task (other resources)


 * Art, Music, and Media **


 * Manipulatives **
 * http://nlvm.usu.edu/ National Library of Virtual Manipulatives


 * Games **
 * www.sumdog.com


 * Videos **
 * Discovery Learning @http://www.discoveryeducation.com/

SMART Board Resource WebsiteSMART Board lesson search engine
 * SMART Board Lessons, Promethean Lessons **


 * Websites **
 * @http://illustrativemathematics.org/standards/k8 PARCC Online (released items)

In this student interactive, from Illuminations, students watch a dynamic, geometric "proof without words" of the Pythagorean Theorem. By clicking on a button, students can see the theorem in action; they are then challenged to explain the proof. @http://www.nctm.org/standards/content.aspx?id=26776 The Pythagorean relationship, a2 + b2 = c2 (where a and b are the lengths of the legs of a right triangle and c is the hypotenuse), can be demonstrated in many ways, including with visual 'proofs' that require little or no symbolism or explanation. The activity in this example from Illuminations presents one dynamic version of a demonstration of this relationship. e-Math Investigations are selected e-examples from the electronic version of the Principles and Standards of School Mathematics (PSSM). The e-examples are part of the electronic version of the PSSM document. Given their interactive nature and focused discussion tied to the PSSM document, the e-examples are natural companions to the i-Math investigations. @http://illuminations.nctm.org/LessonDetail.aspx?ID=L451 The Pythagorean relationship, //a//2 + //b//2 = //c//2 (where //a// and //b// are the lengths of the legs of a right triangle and //c// is the hypotenuse), can be demonstrated in many ways, including with visual "proofs" that require little or no symbolism or explanation. The activity in this example presents one dynamic version of a demonstration of this relationship. Visual and dynamic demonstrations can help students analyze and explain mathematical relationships, as described in the Geometry Standard. The interactive figure in this activity can help students understand the Pythagorean relationship and gives them experience with transformations that preserve area but not shape. @http://illuminations.nctm.org/ActivityDetail.aspx?ID=30
 * Other Activities, etc. **
 * ** Understanding the Pythagorean Relationship Using Interactive Figures **
 * @http://www.nctm.org/standards/content.aspx?id=26776 **
 * **Understanding the Pythagorean Relationship Using Interactive Figures**
 * ** Understanding the Pythagorean Relationship Using Interactive Figures **
 * ** Proof Without Words: Pythagorean Theorem **
 * Why is //c//2 = //a//2 + //b//2? ** Watch a dynamic, geometric "proof without words" of the Pythagorean Theorem. Can you explain the proof?

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