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Deepening Mathematics Instruction for Secondary Teachers Deepening Mathematics Instruction for Secondary Teachers Lance Burger Fresno State Preliminary Edition Contents Preface ix 1 Continued Fractions 1 1.1 Two Definitions of the Rational Numbers . 1 1.2 UnitFractions........................... 2 1.3 Continued Fractions . 6 1.3.1 At the Primary level - The Proportion Area Model . 7 1.3.2 Reversibility and Teaching Mathematics . 10 1.4 Chapter 1 Exercises . 12 v Preface This text is for students who want to become secondary mathematics teach- ers, or for teachers already in the field seeking professional development. One aim of the book is to show the structural connections between arith- metic and algebra. With this in mind, students should understand that although they might only teach middle school after their university degrees and credentials, a focus of this approach is to teach the deep connections and importance of early arithmetic concepts learned in the primary grades, as they form the building blocks for the algebraic concepts needed as far as the college level, preparing their students one day for Calculus and higher mathematics. Another important role for this workbook, is to introduce the prospective teacher to approaches to lesson design which emphasize the cre- ation of rich problems encompassing many standards, as opposed to a linear build-up from basic standards to more complex problems. In the author’sex- perience, it is better to have students practice solving of a lot of similar hard problems, with repetition, rather than build up to a more limited amount of hard problems, not conducive to the seeing of patterns and generalized knowledge for problem solving strategies. Lance Burger January, 2014 ix Chapter 1 Continued Fractions Introduction to Rich-Problem Topics It is the main theme of this book to teach prospective mathematics teach- ers how to design lessons less vertically, and more horizontally. What is meant by this is that most mathematics instruction begins with the basics of a topic and then builds on the basics so that they can be applied in what are typically known as word problems. These types of problems most often occur at the end of a Chapter or section of a text. One problem with this approach is that when building blocks are introduced without much context or connection with each other, students often can not see their relevance or use, and thus quite logically ’tune out’on the subject. The central theme of this book then, is that it is better to practice doing ’many’complex prob- lems which require a lot of sub-procedures, concepts and problem solving strategies, than it is to learn and drill the basics first, with less treatment on problem solving later. In light of recent ’Common Core’trends in mathe- matics which focus on problem solving, this more complex horizontal/holistic repetition approach hopefully can help students as they will hopefully need less memorization of procedures which have been learned by working with richer conceptual structures. 1.1 Two Definitions of the Rational Numbers The origins of continued fractions are diffi cult to pinpoint exactly, but histor- ical records date their use to at least 2000 years ago. In simplified language, 1 2 CHAPTER 1 CONTINUED FRACTIONS continued fractions historically represent numerical quantities by having only a 1 in the numerator of a fraction. Just some basics, fractions are elements of the Rational numbers, denoted by the script symbol Q, we have two im- portant definitions: p 1. Q = : p and q are integers, and q = 0 . f q 6 g 2. Rational numbers are numbers expressible as repeating decimals. What are the integers? Do you know the symbol for them? • Why must the definition for the rational numbers have b = 0? Could • you explain why if a student asked? 6 What are the names for the different parts of the fraction, p and q in • the first definition above? How does the number 5.2 fit in with the second definition for rational • numbers? Is 2.4 in the form of a rational number, by the definitions above? • 5 Problem 1 Is 0.999 . < 1? Problem 2 What fraction is 1.24682468 ...? Problem 3 What fraction is 23.11333333 ...? 1.2 Unit Fractions In the distant past, it did not make much sense to people to divide a smaller number by a larger one - which is one reason continued fractions were ap- pealing. Also, peoples such as the Egyptian scribes, preferred to represent fractions as sums of ’unit fractions,’which were written as the sum of pro- gressively DECREASING fractions . One reason for this, which is also an excellent pedagogy tool for primary students, is that different fractions can be compared, as in the following example. 1.2 UNIT FRACTIONS 3 Example 1 Which is larger, 4/5 or 7/10? One way to compare the fractions is to cross multiply, but this can be a diffi cult process to understand. 4 7 4 7 ? 4 10 > 5 7 40 > 35 ? > 5 10 ! · · ! ! 5 10 Another way to compare fractions is to convert them to the same denom- inator. In this case, 10 would suffi ce as a suitable denominator> 4 7 2 4 8 7 > because = > 5 10 2 · 5 10 10 But, instead of using the lcm(5, 10) = 10, one could use the common mul- tiple, 5 10 = 50, which is not necessarily the smartest common denominator to use,· but it explains the process: 10 4 7 5 40 35 40 35 ? > 50 > 50 40 > 35 10 · 5 10 · 5 ! 50 50 ! · 50 · 50 ! Remark 1 Understanding WHY processes and formulas in mathematics work is an important component to encouraging a culture of instruction that teaches students to take time to think, reflect and solve problems; since, if we want people to create NEW things in mathematics and science, they need to learn how to analyze and understand current knowledge, be it abstract or concrete. 4 As Egyptian fractions, we see much more concretely that 5 is larger than 7 1 10 by exactly 10 : 4 1 1 1 = + + 5 2 5 10 7 1 1 = + 10 2 5 Example 2 Egyptian fraction decompositions are NOT unique, for, as we will see next using the Greedy algorithm: 4 1 8 5 3 = = 5 2 10 10 10 4 CHAPTER 1 CONTINUED FRACTIONS 3 1 1 = + 10 2 5 There are several methods for writing a fraction as the sum of unit frac- tions. We will focus on two of these methods here: 1. Sylvestor’sMethod (Also known as the ’Greedy Algorithm’) 2. Divisor Decomposition Method Sylvester’sMethod Originally developed by Fibonacci (1175-1250), Sylvester rediscovered it in 1880. Strategy: Subtract from the given non-unit fraction the largest unit fraction pos- • sible. If the result is not a unit fraction, repeat the procedure as many times • as necessary to obtain all unit fractions. 1.2 UNIT FRACTIONS 5 Example 3 3/4 : What is the largest unit fraction less than 3/4? 1/2 is less, so subtract: 3 1 3 2 1 = = 4 2 4 4 4 3 1 1 = + 4 2 4 Example 4 1 : What is the largest unit fraction less than 1? 1/2 is less, so subtract: 1 1 1 = 2 2 1 1 3 2 1 = = 2 3 6 6 6 1 1 1 1 = + + 2 3 6 Divisor Decomposition Method Strategy: Make the denominator of the original fraction large enough so that the • numerator can be decomposed in such a way as to produce cancella- tions. 6 CHAPTER 1 CONTINUED FRACTIONS Example 5 11/15 : the key to this method is to multiply top and bottom by a number such that the new numerator can be decomposed into numbers which divide the new denominator. 2 11 22 = 2 · 15 30 30 has divisors 1, 2, 3, 5, 6, 10, 15 22 = 15 + 6 + 1 11 22 15 + 6 + 1 15 6 1 1 1 1 = = = + + = + + 15 30 30 30 30 30 2 5 30 Example 6 13/24 : 3 13 39 = 3 · 24 72 39 = 36 + 3 13 39 36 + 3 36 3 1 1 = = = + = + 24 72 72 72 72 2 24 Problem 4 Use the Divisor Decomposition Method to write a three-term 7 unit fraction decomposition of 8 . 1.3 Continued Fractions 1 Definition 1 An expression of the form a0 + 1 is a simple con- a1+ 1 a2+ a + 3 ··· tinued fraction where the ai can be either real or complex numbers, however for this text, they will be taken as positive integers. As a short-hand, the continued fraction defined can be written as: [a0, a1, a2, a3, ...]. If the fraction does not contain a ’whole number’part, then a0 = 0. 1.3 CONTINUED FRACTIONS 7 What is another name and the symbol for the positive integers? • If the number 0 is added to the set of positive integers, what is the • name and symbol for this new set? 1 2 What if a student asked, why does 3 = 3 ? Could you provide a math- • 2 ematical justification? 4 4 1 1 Example 7 The fraction 5 = [0, 1, 4]... since: 5 = 0 + 5 = 0 + 1 . 4 1+ 4 7 Example 8 The fraction 5 = [1, 2, 2]... since upon using the division algo- 7 2 1 1 rithm two times: 5 = 1 + 5 = 1 + 5 = 1 + 1 . 2 2+ 2 1.3.1 At the Primary level - The Proportion Area Model The topic of continued fractions is a rich topic spanning from the primary grades all the way to the highest levels of mathematical thought. From the common core standards for 3rd grade: (4) Students describe, analyze, and compare properties of two dimensional shapes.
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