Lines, Triangles and Pythagorean Theorem
Grade 8 Math
Unit Length and Description:
In Unit 3, students use exploration and deductive reasoning to determine relationships that exist between angle sums and exterior angle sums of triangles, as well as angles created when parallel lines are cut by a transversal. Students construct various triangles and find the measures of the interior and exterior angles. Students make conjectures about the relationship between the measure of an exterior angle and the other two angles of a triangle. Students construct parallel lines and a transversal to examine the relationships between the created angles. Students recognize vertical angles, adjacent angles and supplementary angles from 7th grade, and build on these relationships to identify other pairs of congruent angles. Using these relationships, students use deductive reasoning to find the measure of missing angles.
In this unit, the students have their first exposure to the Pythagorean Theorem as well as its converse. Using models, students explain the Pythagorean Theorem, understanding that the sum of the squares of the legs is equal to the square of the hypotenuse in a right triangle. Students also understand that given three side lengths with this relationship forms a right triangle. Allowing the students to discover for themselves the Pythagorean Theorem by exploring the proof numerically through perfect squares, geometrically through area, and then with right triangles will promote a deeper conceptual understanding and a greater transfer of knowledge to future courses and applications. Students will use the Pythagorean Theorem to find unknown side lengths in right triangles in two and three dimensions. This will reinforce and allow the students an opportunity to apply their understanding of rational and irrational numbers in a new context. When applying the Pythagorean Theorem to solve real-world and mathematical problems, the students will use rational approximations to create a more meaningful solution within the context of the problem. In addition, students will apply the Pythagorean Theorem to find the distance between two points on the coordinate plane. The expectation is that students will be able to explain a proof of the Pythagorean Theorem and be able to apply it to solve geometric and real-world problems.
8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form 𝑥2 = 𝑝 and 𝑥3 = 𝑝, where 𝑝 is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.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.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Focus Standards of Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:
· I can recall the Pythagorean Theorem and its converse and relate it to any two distinct points on a graph.
· I can determine how to create a right triangle from two points on a coordinate graph.
can use the Pythagorean Theorem to solve for the distance between the two
· When parallel lines are cut by a transversal, corresponding angles, alternate interior angles, alternate exterior angles, and vertical angles are congruent.
· A relationship exists between angle sums and exterior angle sums of triangles, as well as angles created when parallel lines are cut by a transversal.
· The measure of the interior angles of any triangle is 180°.
· The measure of a right triangle is always 90°.
· A relationship exists between the measure of an exterior angle and the other two angles of a triangle.
· The number system consists of numbers that are rational and irrational.
· Irrational numbers can be represented on a real number line.
· Every number has a decimal expansion.
· How can algebraic concepts be applied to geometry?
· How is deductive reasoning used in algebra and geometry?
· Why does the Pythagorean Theorem apply only to right triangles?
· How do I know when an estimate, approximation, or exact answer is the desired solution?
· 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?