12-Math 7a-CC3 Ch9

This chapter builds several core geometry concepts related to parallel lines and triangles. In Section 9.1, students will begin by learning about three angle relationships for parallel lines intersected by a transversal – corresponding, alternate interior and same side interior angles. Then students look at triangles, learning about the Triangle Angle Sum Theorem and that any given exterior angle is equal to the sum of the two remote interior angles (essentially a corollary of the Triangle Angle Sum Theorem). The development of these concepts is done informally using tracing paper to compare angles and then creating a conjecture rather than through formal proof. Lastly, students revisit similarity, which was studied in Chapter 6, learning that the angles in similar shapes are always equal. They then use the idea of the Triangle Angle Sum Theorem to determine that only two pairs of corresponding angles need to be equal to decide that two triangles are similar.

Section 9.2 then turns the attention to the relationships between the side lengths of individual triangles. Students will explore when three lengths form a triangle and when they do not. When three lengths form triangles, students will determine whether those triangles are right, obtuse, or acute triangles. Then, using the Pythagorean Theorem and the square root operation, students will solve for the length of the unknown third side of a right triangle. Students will also be introduced to irrational numbers and will contrast them with some of the rational numbers that they have worked with in this course.

Section 9.1 is an opportunity for students to attend to precision as they learn new angle pair relationship vocabulary. When investigating these angle pair relationships, students will be looking for and expressing regularity in repeated reasoning. They will move from reasoning quantitatively as the measure angles, to reasoning abstractly as the generalize angle pair relationships.

In Section 9.2, students look for and make use of the structure of square root and the Pythagorean Theorem. In Lessons 9.2.5 and 9.2.6, students make sense of Pythagorean Theorem application problems by drawing models, and persevere in solving the problems.

8.G.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.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.

8.EE.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.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., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.