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 teachers, or for teachers already in the ﬁeld seeking professional development. One aim of the book is to show the structural connections between arithmetic 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

Chapter 1

Continued Fractions

Introduction to Rich-Problem Topics It is the main theme of this book to teach prospective mathematics teachers 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 problems 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 mathematics 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 important deﬁnitions:

p 1. Q = : p and q are integers, and q = 0 . { q 6 } 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 denominator. 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 fractions. 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. They compare and classify shapes by their sides and angles, and connect these with deﬁnitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.

Example 9 Using the proportion area model, ﬁnd the continued fraction of 45 16 .

45 16 is ﬁrst represented as a rectangle having he following proportions: 8 CHAPTER 1 CONTINUED FRACTIONS

Figure 1

Next, by simply making the largest possible square within the rectangle, Figure 2 is obtained:

Figure 2 The division algorithm using squares.

This process of making squares in the rectangle, if you think about it, is the same as seeing how many times 16 divides into 45, which is twice with remainder 13. Notice that the two 16 16 squares are each in a proportion × 13 of 1 and the remaining rectangle is in a proportion of 16 , which corresponds 45 13 to: 16 = 2 + 16 . To ’continue’to use the division algorithm until the last 13 1 fraction is a unit fraction, it is necessary to change 16 into 16 , which is kind 13 of like now turning one’shead sideways and doing the previous ’largest square’process on the orange 13 16 rectangle. × 1.3 CONTINUED FRACTIONS 9

Figure 3

45 1 3 The representation in Figure 3 corresponds to: 16 = 2 + 3 , so since 13 1+ 13 is not a unit fraction, the process must continue on the red 3 13 rectangle, resulting in Figure 4: ×

Figure 4 The ﬁnal diagram for the continued fraction of 45/16.

1 Since the last yellow rectangle is 1 3 the process MUST stop, since 3 × 1 1 would keep resulting in the same answer of 3 , and we see the correspondence of counting the largest to the smallest squares to the representation:

45 1 = 2 + 1 = [2, 1, 4, 3]. 16 1+ 1 4+ 3

Problem 5 Use the Proportion Area Model to make a continued fraction 49 rectangle for 16 . 10 CHAPTER 1 CONTINUED FRACTIONS

1.3.2 Reversibility and Teaching Mathematics In Piaget’stheory of Cognitive Development, the Concrete Operational Stage refers to the ability reason logically on abstract objects such as numbers, fractions and variables. A process that aides a learner to develop this capability is known as ’Reversibility,’ which is the ability to recognize that numbers, objects, or expressions can be changed and then returned to their original conditions. As an example for how to model instruction using reversibility, after making the continued fraction diagrams in the previous section, students can be given a ﬁnal diagram, such as in Figure 4, and work backwards to ﬁnd the original fraction.

Example 10 Find the original fraction which resulted in the continued fraction diagram in Figure 5 below:

Figure 5 Working backwards.

1 This diagram reads as: [2, 5, 2] = 2 + 1 . Working from the bottom, 5+ 2 1 2 5 1 10 1 11 1 2 11 2 2 5 + 2 = 2 1 + 2 = 2 + 2 = 2 ; thus, [2, 5, 2] = 2 + 11 = 2 + 11 = 11 1 + 11 = · 2 · 22 2 24 11 + 11 = 11 . As seen in the previous example, many aspects of fraction arithmetic were needed in order to solve the problem. This idea is a common theme of this book, which is to teach rich problems which involve many standards and procedures, as opposed to working too much on the basics in isolation.

Problem 6 What fraction is [1, 2, 3, 4, 5, 6]? 1.3 CONTINUED FRACTIONS 11

Algebraic Connections-Finding Square Roots Example 11 Prove √2 = [1, 2, 2, 2, 2, ...]. 1 Proof. Let x = 1 + 1 then the key is to look at the continued fraction 2+ 2+ and decompose a number··· so that the entire fraction is again seen, as in the following: 1 1 1 x = 1 + 1 = 1 + 1 = 1 + 2 + 2+ 1 + 1 + 2+ 1 + x ··· ··· 1 x = 1 + x(1 + x) = (1 + x) 1 + 1 1 + x → · x2 + x = x + 1 + 1 = x + 2 x2 + x = x + 2 x2 = 2 x = √2 → → ± But, since the continued fraction is obviously positive, then it has been shown that x = √2

Problem 7 Prove that √40 = [6, 3, 12]. 12 CHAPTER 1 CONTINUED FRACTIONS 1.4 Chapter 1 Exercises

Exercise 1 Use the Proportion Area Model and graph paper to make a 29 colored continued fraction rectangle for 17 .

Exercise 2 Use the Proportion Area Model and graph paper to make a col- 26 ored continued fraction rectangle for 19 .

3 Exercise 3 Write 7 as a a sum of unit fractions using Sylvester’s method.

Exercise 4 What fraction is [3, 4, 1, 1, 2, 5]?

11 Exercise 5 Write 18 as a sum of unit fractions using the Divisor Decompo- sition Method.

Exercise 6 Without graph paper, use the division algorithm to sketch a Pro- 1885 portion Area Model for 102 .

Exercise 7 Recall that the Fibonacci sequence is generated by letting F1 = 1, F2 = 1 and Fn = Fn 1 + Fn 2. − − 1. Generate the ﬁrst 20 terms of the Fibonacci sequence. 2. Convert successive ratios of Finonacci numbers into continued fractions. In other words, given 1/1, 2/1, 3/2, 5/3, 8/5,....ﬁnd the continued fractions for the ratios listed. Describe the pattern.

Exercise 8 Prove √5 = [2, 4, 4, 4...].

Exercise 9 Prove √7 = [2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4...].

Exercise 10 Write a 1-2 page reﬂection on the ’Polished Stones’ video. What aspects of mathematics instruction in Taiwan and Japan do you like or dislike? In what ways can instruction in the US be informed by what you saw? In what ways could instruction in Taiwan and Japan beneﬁt by methods used in the US?