Teaching Geometry in Grade 8 and High School According to ...

Teaching Geometry in Grade 8 and High School According to the Common Core Standards

H. Wu

c Hung-Hsi Wu 2013

October 16, 2013

Contents

Grade 8

6

1. Basic rigid motions and congruence (page 8)

2. Dilation and similarity (page 42)

3. The angle-angle criterion (AA) for similarity (page 57)

4. The Pythagorean Theorem (page 61)

5. The angle sum of a triangle (page 66)

6. Volume formulas (page 68)

High School

73

1. Basic assumptions and definitions (page 80)

2. Definitions of basic rigid motions and assumptions (page 95)

3. Congruence criteria for triangles (page 110)

1

4. Some typical theorems (page 126) 5. Constructions with ruler and compass (page 144) 6. Definitions of dilations and similarity (page 149) 7. Some theorems on circles (page 175)

2

Preface

This is the companion article to Teaching Geometry According to the Common Core Standards.

The Common Core State Standards for Mathematics (CCSSM) have reorganized the geometry curriculum in grade 8 and high school. Because there are at present very few (if any) ready references for such a reorganization, this document is being offered as a stopgap measure.

In terms of the topics covered, there is hardly any difference between what is called for by the CCSSM and by the other curricula. The change occurs mainly in the internal (mathematical) reorganization and the change of (mathematical) focus. For example, transformations are usually taught as rote skills in middle school with no mathematical applications or relevance, and the concepts of congruence and similarity are talked about but never defined except in the case of polygons. By contrast, the CCSSM develop all these topics on the foundation of transformations, thereby giving them coherence and purposefulness. The "coherence" of the CCSSM has been much bandied about in recent discussions, but it is time to realize that the coherence of the CCSSM is not an educational slogan but a mathematical fact, and one of its manifestations is the coherence of the geometry curriculum embedded in the CCSSM. For the benefit of students' learning, this change is a welcome development. However, it is unfortunately the case that while these basic topics are routinely discussed in the mathematics literature, not much of this information can be found in the education literature except perhaps H. Wu, Pre-Algebra. The intentions of the CCSSM have thus become hidden for the time being. If the detailed account given in this document is at all successful, it will furnish a bridge across this mathematical chasm for the time being.

My specific targets are middle and high school mathematics teachers as well as the publishers of textbooks. I hope that teachers will find this account helpful in their preparations for the implementation of CCSSM by year 2014. If, in addition, their school district can offer professional development, then maybe they can make use of this document to articulate the kind of professional development they want. We are entering an era when teachers must take an active role in their own professional life. The CCSSM are charting a new course, and district administrators and professional developers have to work together with teachers to find their new bearings in the

3

transitional period. As for the publishers, my contact with them in the past fifteen years has made me

aware that their claim of not having the needed resources to improve their books is indeed entirely legitimate. Our educational system has been broken for a long time and we have to find ways to forge a new beginning. At a time when the CCSSM are initiating a significant change in the teaching of geometry, it would be unconscionable--as in the days of the New Math--to once again ask for change without providing the necessary support for this change. It is hoped that this document will provide some temporary relief in the present absence of this support.

This document is essentially a compendium of selected topics from the lecture notes for the annual summer professional development institutes (MPDI) and upper division courses (Math 151?153) at Berkeley that I have given since 2006. I have been advocating this transformations-based approach to the teaching of middle school and high school geometry because, in terms of student learning, it is a more reasonable alternative to the existing ones (see the discussions on page 79 ff. and page 125 ff. for part of the reason). By a happy coincidence, the CCSSM agreed with this judgment. (The reference, Wu, H., Lecture Notes for the 2009 Pre-Algebra Institute, September 15, 2009. on page 92 of the CCSSM is the same as H. Wu, Pre-Algebra.) In any case, the detailed development of this approach to middle school and high school geometry, together with exercises, will be found in the following textbooks by the author: From Pre-Algebra to Algebra (for middle school teachers, to appear in late 2014), and Mathematics of the Secondary School Curriculum (a two volume set for high school teachers, to appear probably in late 2015).

It may also be mentioned that I expect to post detailed student lessons for grade 8 according to the CCSSM by the fall of 2014.

Acknowledgements. I wish to thank Wolfgang Buettner for his interesting contributions, Mark Saul for his willingness to read through the grade 8 portion and make suggestions, and Larry Francis for his usual excellent editorial assistance. To Angelo Segalla and Clinton Rempel, I owe an immense debt for numerous corrections.

4

Conventions

A turquoise box around a phrase or a sentence (such as H. Wu, Pre-Algebra) indicates an active link to an article online.

The standards on geometry are listed at the beginning of each grade in sans serif fonts.

5

GRADE 8

Geometry 8.G

Understand congruence and similarity using physical models, transparencies, or geometry software.

1. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines.

2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

3. Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.

4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

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.

Understand and apply the Pythagorean Theorem.

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

6

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. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

u

Goals of eighth grade geometry

1. An intuitive introduction of the concept of congruence using rotations, translations, and reflections, and their compositions (page 8)

2. An intuitive introduction of the concepts of dilation and similarity (page 42) 3. An informal argument for the angle-angle criterion (AA) of similar triangles

(page 57) 4. Use of AA for similarity to prove the Pythagorean Theorem (page 61) 5. An informal argument that the angle sum of a triangle is 180 degrees (page

66) 6. Introduction of some basic volume formulas (page 68)

These six goals are intended to be achieved with an emphasis on the intuitive geometric content through the ample use of hands-on activities. They will prepare eighth graders to learn about the geometry of linear equations in beginning algebra. They are also needed to furnish eighth graders with a firm foundation for the more formal development of high school geometry.

7

1. Basic rigid motions and congruence

Overview (page 8) Preliminary definitions of basic rigid motions (page 10) Motions of entire geometric figures (page 18) Assumptions on basic rigid motions (page 23) Compositions of basic rigid motions (page 25) The concept of congruence (page 38)

Overview

The main new ideas are the concepts of translations, reflections, rotations, and dilations in the plane. The first three--translations, reflections, rotations--are collectively referred to as the basic rigid motions, and they will be the subject of inquiry in this section. Dilation will be explained in the next.

Before proceeding further, we note that the basic rigid motions are quite subtle concepts whose precise definitions require a bit of preparation about more advanced topics such as transformations of the plane, the concept of transformations that are one-to-one and "onto", separation properties of lines in the plane, distance in the plane, and other concepts that are necessary for a more formal development. Such precision is neither necessary nor desirable in an introductory treatment in eighth grade. Rather,

it is the intuitive geometric content of the basic rigid motions that needs to be emphasized.

In the high school course on geometry, more of this precision will be supplied in order to carry out the detailed mathematical reasoning for the proofs of theorems. For eighth grade, however, we should minimize the formalism and emphasize the geometric intuition instead. Fortunately, the availability of abundant teaching tools makes it easy to convey this intuitive content. In this document, we will rely exclusively on the use of transparencies as an aid to the explanation of basic rigid motions. This expository decision should be complemented by two remarks, however.

8

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download