Unit 3

Lines, Triangles and Pythagorean Theorem

 

Grade 8 Math 

Unit Length and Description:

 

20 days

 

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.

 

Standards:

 

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.

 

  • Standards Clarification: Students will use square roots to solve for unknown measures in right triangles. It is not the intent of the standards to teach students how to simplify a radical through factoring the radicand. The emphasis throughout the standards beginning in grade 8 and continuing through high school is to be able create a rational approximation for irrational numbers.  Also note that students will work with radicals throughout grade 8 when writing and solving equations for an unknown measurement of a three-dimensional figure and again when working with the Pythagorean Theorem.

 

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.

 

  • Standards Clarification: The angle-angle criterion for similarity of triangles will be visited in Unit 4.

 

8.G.B.6  Explain a proof of the Pythagorean Theorem and its converse.

 

  • Standards Clarification: This standard does not require the students to prove for themselves the Pythagorean Theorem or its converse rather explain a proof of it. To ensure mastery of this standard, multiple proofs of the Pythagorean Theorem and its converse should be used.

 

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.

 

  • Standards Clarification: This unit begins the learning of the Pythagorean Theorem.  Students begin the work of finding the length of a leg or hypotenuse of a right triangle using . 

 

8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

 

  • Standards Clarification: This unit begins the connection of distance to the Pythagorean Theorem. 

 

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.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

8.EE.A.2

  • I can use square roots to represent solutions to equations of the form x2 = p, where p is a positive rational number.

 

8.G.A.5

  • I can define and identify transversals.
  • I can identify angles created when a parallel line is cut by transversal (alternate interior, alternate exterior, corresponding, vertical, adjacent, etc.)
  • I can justify that the sum of the interior angles equals 180. (For example, arrange three copies of the same triangle so that the three angles appear to form a line.)
  • I can justify that the exterior angles of a triangle is equal to the sum of the two remote interior angles.

 

8.G.B.6

  • I can define key vocabulary: square root, Pythagorean Theorem, right triangle, legs a and b, hypotenuse, sides, right angle, converse, base, height, proof.
  • I can identify the legs and hypotenuse of a right triangle.
  • I can explain a proof of the Pythagorean Theorem.
  • I can explain a proof of the converse of the Pythagorean Theorem.

 

8.G.B.7

  • I can recall the Pythagorean Theorem and its converse in order to apply it to real-world and mathematical problems using 2 and 3 dimensional figures.
  • I can solve basic mathematical Pythagorean Theorem problems and its converse to find missing length of sides of triangles in two and three dimensions.
  • I can apply the Pythagorean Theorem in solving real-world problems dealing with two and three dimensional shapes.

 

8.G.B.8

·        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.

·        I can use the Pythagorean Theorem to solve for the distance between the two points.
 

Enduring Understandings:

 

·        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.

•   The Pythagorean Theorem can be used both algebraically and geometrically to solve problems involving right triangles

•   There is a relationship between the Pythagorean Theorem and the distance formula.

•   Both the Pythagorean Theorem and distance formula can be used to find missing side lengths in a coordinate plane and real-world situation.

•   Finding the square root of a number is the inverse operation of squaring that number.

•   Finding the cube root of a number is the inverse operation of cubing that number.

•   Right triangles have a special relationship among the side lengths which can be represented by a model and a formula.

•   Pythagorean Triples can be used to construct right triangles.

•   Attributes of geometric figures can be used to identify figures and find their measures.

 

Essential Questions:

 

·        How can algebraic concepts be applied to geometry?

·        How is deductive reasoning used in algebra and geometry?

·        What angle relationships are formed by a transversal?

·        What is the length of the side of a square of a certain area?

·        What is the relationship among the lengths of the sides of a right triangle?

·        Why does the Pythagorean Theorem apply only to right triangles?

·        How can the Pythagorean Theorem be used to solve problems?

·        What is the correlation between the Pythagorean Theorem and the distance formula?

·        How can I use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle?

·        How do I use the Pythagorean Theorem to find the length of the legs of a right triangle?

·        How do I know that I have a convincing argument to informally prove the Pythagorean Theorem?

·        What is the Pythagorean Theorem and when does it apply?

·        How can I determine the length of a diagonal?

·        How can I find the altitude of an equilateral triangle?

·        How could I find the shortest distance from one point to another if there was an obstacle in the way?

·        Where can I find examples of two and three-dimensional objects in the real-world?

·        How do I simplify and evaluate algebraic equations involving integer exponents, square roots, and cube roots?

·        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?