Hung-Hsi Wu 2013 January 1, 2012. Third revision: October 10, 2013.

Contents Preface

2

Grade 4

7

Grade 5

19

Grade 6

31

Grade 7

47

Grade 8

62

High School Geometry

110

1

Preface Le juge: Accus´e, vous tˆacherez d’ˆetre bref. L’accus´e: Je tˆacherai d’ˆetre clair. —G. Courteline1 This document is a collection of grade-by-grade mathematical commentaries on the teaching of the geometry standards in the CCSSM (Common Core State Standards for Mathematics) from grade 4 to high school. The emphasis is on the progression of the mathematical ideas through the grades. It complements the usual writings and discussions on the CCSSM which emphasize the latter’s Practice Standards. It is hoped that this document will promote a better understanding of the Practice Standards by giving them mathematical substance rather than adding to the verbal descriptions of what mathematics is about. Seeing (correct) mathematics in action is a far better way of coming to grips with these Practice Standards but, unfortunately, in an era of Textbook School Mathematics (TSM),2 one does not get to see mathematics in action too often. Mathematicians should have done much more to reveal the true nature of mathematics, but they didn’t, and school mathematics education is the worse for it. Let us hope that, with the advent of the CCSSM, more of such efforts will be forthcoming. The geometry standards in the CCSSM deviate from the usual geometry standards in at least two respects, one big and one small. The small one is that, for the first time, special attention is paid to the need of a proof for the area formula for rectangles when the side lengths are fractions. This is standard NF 4b in grade 5: Find the area of a rectangle with fractional side lengths by tiling it with [rectangles] of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 1

Quoted in the classic, Commutative Algebra, of Zariski-Samuel. Literal translation: The judge: “The defendant will try to be brief.” The defendant replies, “I will try to be clear.” 2 A turquoise box around a phrase or a sentence (such as Textbook School Mathematics ) indicates an active link to an article online.

2

The lack of explanation for the rectangle area formula when the side lengths are fractions is symptomatic of what has gone wrong in school mathematics education, or more precisely, in TSM. Often basic facts are not clearly explained, or if explained, it is done incorrectly. Because the explanation in this case requires a full understanding of fraction multiplication and the basic ingredients of the concept of area (see (a)–(d) on page 21 ), and because the reasoning is far from routine and yet very accessible to fifth graders, this explanation is potentially a high point in students’ encounter with geometric measurements (length, area, volume) in K–12. Let us make sure that it is so this time around. The major deviation of the CCSSM from the usual geometry standards occurs in grade 8 and high school. There is at present an almost total disconnect in TSM between the geometry of middle school and that of high school. Congruence and similarity are (vaguely!) defined in middle school as “same-size-and-same-shape” and “same-shape-but-not-necessarily-same-size”, respectively, while middle-school geometric transformations (rotations, reflections, and translations) are taught seemingly only for the purpose of art appreciation, such as appreciating the internal symmetries of Escher’s famous designs and medieval Islamic art without any reference to their mathematical relevance. In the high school geometry of TSM