Unit 6: Pythagorean Theorem


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1 Approximate Time Frame: 2 weeks Connections to Previous Learning: In Unit 1, students learned to evaluate expressions and equations with exponents and solved equations of the form worked with triangles and classifying triangles.. In previous grades, students have Focus of this Unit: Students will apply their prior knowledge of triangles to the specific qualities of right triangles and find the missing side lengths of right triangles in various realworld 2D and 3D situations. They will also apply the concepts of squares and square roots. Connections to Subsequent Learning: In Unit 7, students will apply the concept of the Pythagorean Theorem to find the height or radius of a cone given the slant height and either the height or radius during the study of volume. In addition, they will extend these concepts to Trigonometry. Desired Outcomes Standard(s): Understand and apply the Pythagorean Theorem 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 realworld 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. Supporting Standards: 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., ). For example, by truncating the decimal expansion of, show that is between and, then between and, and explain how to continue on to get better approximations. Adapted from UbD framework Page 1
2 B Transfer: Students will apply concepts and procedures regarding the Pythagorean Theorem to represent, interpret, prove, and solve realworld and mathematical problems in both two and three dimensions and in the coordinate system. Ex: A rectangular prism has a height of 12 cm, length of 6 cm, and a width of 4 cm. What is the length of the interior diagonal? Ex: You live at point A and your friend lives at point B. What is the shortest distance between your house and your friend s house? A Understandings: Students will understand that Right triangles have a special relationship among the side lengths which can be represented by a model and a formula. The Pythagorean Theorem can be used to find the missing side lengths in a coordinate plane and realworld situations. The Pythagorean Theorem and its converse can be proven. Essential Questions: Why does the Pythagorean Theorem apply only to right triangles? How does the knowledge of how to use right triangles and the Pythagorean Theorem enable the design and construction of such structures as a properly pitched roof, handicap ramps to meet code, structurally stable bridges, and roads? How can the Pythagorean Theorem be used for indirect measurement? How do indirect measurement strategies allow for the measurement of items in the real world such as playground structures, flagpoles, and buildings? Adapted from UbD framework Page 2
3 Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) * 1. Make sense of problems and persevere in solving them. Students have the opportunity to make sense of more complicated, multistep Pythagorean problems. For example, consider a rectangular prism where the length, width, and height are known. Students can determine the interior diagonal length. To do so, they must first determine the diagonal of the base and then use that value to find the length of the interior diagonal. 2. Reason abstractly and quantitatively. * 3. Construct viable arguments and critique the reasoning of others. Students will use the Pythagorean Theorem to prove whether a triangle is a right triangle or not. This should be done algebraically as well as using measurement. For example, students may be given three lengths of popsicle sticks, be asked to create a triangle with them, and then determine whether or not it is a right triangle. * 4. Model with mathematics. Students should verify, using a model, that the sum of the squares of the legs is equal to the square of the hypotenuse in a right triangle. For example, students may take a paper model of,, and and cut apart the and pieces to make it fit within the piece. 5. Use appropriate tools strategically. Students use various forms of technology appropriately when solving problems. For example, they identify situations when it is appropriate to use a calculator and other situations that can be solved using mental math. * 6. Attend to precision. Students will need to attend to precision when using irrational numbers. If students round and truncate decimals at a low level of precision and continue to use these values as part of subsequent steps, final values could be compromised. The degree of precision will vary depending on the circumstances. For example, in baseball if the 2 nd baseman throws a ball to home plate, the specific distance of the throw, while irrational, is typically reported as a rational approximation. 7. Look for and make use of structure. Students can use their knowledge of the structure of equations to solve problems with unknown side lengths or unknown hypotenuse lengths. * 8. Look for and express regularity in repeated reasoning. One example of this is the Pythagorean Theorem applied to the distance formula. Adapted from UbD framework Page 3
4 Prerequisite Skills/Concepts: Grade 8: Students should already be able to Use the properties of similarity, congruence, and right triangles. Calculate square roots and squares. Represent numbers in radical form (irrational) and to approximate these numbers as rational. Evaluate linear equations in one variable with one solution using the real number system. Use the properties of exponents and real numbers (commutative, associative, distributive, inverse, and identity). Solve equations of the form using the square root as the inverse operations of squaring. Advanced Skills/Concepts: Some students may be ready to Derive (and use) the distance formula from the Pythagorean Theorem using the hypotenuse of a triangle. Explore trigonometric ratios. Knowledge: Students will know The Pythagorean Theorem. When to apply the Pythagorean Theorem. Skills: Students will be able to Explain a proof of the Pythagorean Theorem and its converse. (8.G.6) Use the Pythagorean Theorem to solve for a missing side of a right triangle given the other 2 sides in both 2D and 3D problems. (8.G.7) Apply the Pythagorean Theorem to solve problems in realworld contexts. (8.G.7) Apply the Pythagorean Theorem to find the distance between two points in the coordinate system. (8.G.8) WIDA Standard: (English Language Learners) English language learners communicate information, ideas, and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: Attention given to visual representations of all concepts and vocabulary whenever possible. Vocabulary will be taught explicitly using tactile and virtual tools (e.g. software tools). Real world examples to reinforce vocabulary. For example, use the book What s your Angle, Pythagoras? Adapted from UbD framework Page 4
5 Critical Terms: Legs of a triangle Hypotenuse Right triangle Pythagorean theorem Pythagorean triple Converse of Pythagorean theorem Square root Academic Vocabulary: Supplemental Terms: Distance formula Irrational Perfect squares Radical Assessment PreAssessments Formative Assessments Summative Assessments SelfAssessments Is it Right? 8.G.6 PostTest PT in 2D 8.G.78 PostTest PT in 3D Part I PT in 3D Part II Distance Between Two Points 8.G.6 PreTest 8.G.78 PreTest Sample Lesson Sequence 1. 8.G.6 Pythagorean Theorem Proof and Converse (model lesson) 2. 8.G.7 Applying the Pythagorean Theorem in 2D and 3D Situations 3. 8.G.8 Applying the Pythagorean Theorem on Coordinate Plane Adapted from UbD framework Page 5
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