Monday, December 14, 2015

Math What?

Nature of Math

The saga continues! Is math a science? Is math discovered or invented? I thought I had a solid answer on the former question, but Alas! Mathematics Strikes Back!

The debate on these two questions were brought front and center in my History of Math class. The class was divided on this issue with a large portion of the class believing math was a science. A small portion believed math is completely separate from science. A small few were still agnostic on the issue. The class was less divided over whether math is discovered vs invented. So, I'll start there.

Math Discovered or Invented?
Discovered. Definitely. As our class has discussed and from what Pickover has discussed in The Math Book (blog post), there are math milestones that were worked out by different people simultaneously, but independently from each other (1). Credit for Calculus has been given to both Isaac Newton and Gottfried Wilhelm Leibniz (1). Another example was when hyperbolic geometry was being worked out by three different mathematicians at the same time: Janos Bolyai, Nikolai Lobachevskii, and Carl Friedrich Gauss (4). If math was invented then it would just be a random collection of concepts that were pulled out of thin air by the imaginations of great mathematicians, and it would be hard to make the case that these simultaneous creations by different mathematicians occurred by coincidence. How could they just happen to have the (almost) exact same creation at the same time? That would be like Leonardo DaVinci and Michelangelo painting the exact same Mona Lisa at the same time. Don't Even.  

Furthermore, mathematics is far too ordered, structured, and interconnected to be randomly invented. In BBC Horizon's documentary, The Last Theorem, a connection was established between three seemingly unrelated math topics: elliptic curves, modular functions, and Fermat's Last Theorem (5). To prove Fermat's Last Theorem, Andrew Wiles had to prove that modular forms and elliptic curves are the same, because when someone believed that Fermat's Last Theorem was false, those equations looked like an elliptic curve that wasn't modular (5). In the end, Andrew Wiles proved Fermat's Last Theorem was true by showing elliptic curves and modular forms were the same (5).

The reason behind the order, structure and interconnectedness in mathematics is logic. Jordan Ellenberg in his book, How Not to Be Wrong: The Power of Mathematical, describes math as using common sense, and applying rational thinking or logic to problems (the problems don't have to be mathematical) (4). Eugenia Cheng, in a presentation at Grand Valley State University, explained that "mathematics is the logical study of logical things" (6). Logic is the underlying pinnings that unites all of mathematics together.

A part of mathematics logical structure are axioms. Ellenberg describes axioms as self-evident rules that form a system of mathematics (4). They are used to manipulate elements in the system to prove logical statements about the system, and these logical statements are based on truth tables, the language of logic. When Janos Balyai decided to remove the Parallel Postulate from Euclid's Axioms, he worked out a new geometry - hyperbolic geometry (4). He found that the parallel postulate was neither contradictory, which would make Euclid's geometry false, nor mandatory, which would make it the only possible geometry (4). Hyperbolic geometry is just another geometric world in addition to Euclid's geometry or, more broadly speaking, mathematical space in mathematics (3). When another world was discovered with a different set of axioms, the nature of axioms changed (3). Axioms are more like hypotheses that lay the framework for the world one is in (3).  But even though there is an important difference between hyperbolic and Euclidean geometry, there is overlap because there is overlap of axioms between the two worlds and the logic that is used to prove statements in all of mathematics is the same. Another example is with integers and rings. Integers are a specific type of ring, and depending on which axioms are used, proven statements (theorems) in the integers can be used or slightly modified to prove similar results in rings.

Furthermore, when I think of something that is invented, I think of art and games. When children come up with a new game, they create their own rules. Most of the time, these rules aren't logical. Take Monopoly for example. Why do players get free money whenever they pass Go? That doesn't make sense or even apply to the real world. It's a whimsical rule, and mathematics are not governed by whimsical rules.

Mathematics are governed by logical rules, specifically truth tables and axioms. Axioms, which are the foundation of proof and mathematical theorems, can be compared to rules of a game. With games, rules can be changed and the game can still be played. To continue with the Monopoly example, if the rule of getting money when passing Go disappears, all the other elements of the game remain. But in math, the rules, or axioms can't just change or disappear and leave the theorem or concept fundamentally different. If they do, all the mathematical concepts that relied on those axioms and theorems crumbles (4 and 5). This is the key difference between games (rules without logic) and math (rules with logic). Axioms and truth tables, the logical rules, makes math a discovered endeavor, and rules without logic are of human invention.

Lastly, mathematics is not a solely human invention. If math is truly a human invention, then only humans could do it, understand it, and come up with it. But primates and other animals can be trained to count, while a type of Saharan Ant knows the number of steps and the lengths of their stride to find their way back to their nest (1). But as someone in the class mentioned, if math is discovered, then it has to be found in nature, which would make it a science. And that brings us to the next question.

Is Math a Science?
I was a firm believer that math is not a science. Now, there is room for doubt. I have written a detailed blog post on this topic, so I will only give a brief description of what I already discussed. The pursuit of science is to discover how natural processes work, while the pursuit of math is discovering relationships with numbers. Math is found and can be applied to nature. In fact discoveries in math started with the ancient Greeks describing our 3-D world with geometry (3). But even though math can be found or be used to describe nature, there are abstract ideas that transcends the physical world. The example I gave before was a one-dimensional line expanding to infinity. Furthermore, math's capability to describe our natural world with accuracy does not make it the reason why it works (2). Just because math is accurate in modeling natural phenomenon, it does not mean it's the explanation behind the process (2). There could be a completely different reason that makes the process act that way (2). Accuracy does not necessarily make something true (2).

But then I came across the Mathematical Universe Hypothesis. Max Tegmark is the man who came up with the hypothesis (1). According to Clifford Pickover's explanation of the hypothesis, "...our physical reality is a mathematical structure and that our universe is not just described by mathematics - it is mathematics" (1). If this hypothesis is proven to be true, then mathematics is not just something that can be applied and describe our natural world with accuracy - it is the foundation, the reason, and the process behind how everything in our natural world works. And that would make math a science....and it would make science provable with mathematical proof. But at this point, the Mathematical Universe Hypothesis is just a hypothesis, and the divide between science and math, nature and number, clear. And I can continue believing, a little shakily, that math is not a science.

(1) Pickover, Clifford A. The Math Book. Sterling Publishing Co., Inc. New York, NY. 2009.

(2) Wigner, Eugene. "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." College Course 9. 

(3) Space (mathematics). Wikipedia. 

(4) Ellenberg, Jordan. How Not to Be Wrong: The Power of Mathematical Thinking. The Penguin Press. New York, NY. 2014. 

(5) Simon Singh and John Lynch. Fermat's Last Theorem Documentary. BBC Horizon. 1996. 

(6) Eugenia Cheng's Presentation at Grand Valley State University. Quotation from MathHombre's Tumblr page:

Monday, December 7, 2015

The Spider and the Fly: An Exploration in 3-D Geometry and Graphing

Doing Math

The NRICH asks an interesting math question at this Link. Let's say you're a spider, and you see a fly across a room. What would be the shortest, and therefore fastest, route you would have to crawl to catch the fly?

The spider is in a room, so it's 3-D. The usual 2-D graph of x and y, rise over run, doesn't setup the problem accurately. There is width, height, and length: an x, y, and z coordinates respectively. But even though the spider and the fly lives in a 3-D world, we can break down the problem into a 2-D one, since the spider can only crawl on the wall in a 2-D way - side-to-side (x), up-and-down (y).

We are given that the room is 4 meters wide, 2.5 meters high, and 5 meters long. The spider is in the middle of the smallest wall on one side of the room. The fly is on the opposite, smallest wall and it's position is 1.5 meters high and 0.5 meters away from the adjacent wall next to the window. Below is how I visualized the room, and how I labeled the coordinates. The room is not drawn to scale.

Then, I thought how to move the spider from wall to wall to get to the fly in the shortest distance. If the spider goes towards the back wall, then the spider would have to cross the window to get to the fly, which is 3.5 m away from the fly (4-0.5 = 3.5). So, the better way for the spider to travel is to move towards the front wall. Next, I imagined the fly on the spider's wall in it's (0.5, 1.5) position. Eventually, the spider has to get to the height of 1.5 meters, so I figured it should do so while traveling to the edge of the front wall. That way, the spider will only need to crawl in a straight line to get to the fly. Below is the route indicated by the arrows.
Since the spider is crawling a straight line on the front wall and the opposite wall, then the spider will be crawling a distance of 0.5 + 5 = 5.5 meters for those two walls. I needed to use the distance formula to calculate the diagonal distance the spider crawls to get to the edge of the front wall at the height of 1.5 meters.
Distance Formula

The coordinate of the spot I want the spider to crawl to is (0, 1.5, 5), since x=0 is the edge of the front wall, 1.5 is the height of the fly, and 5 is the z coordinate for end wall. I don't need to use the z-coordinate, 5, because the spider is moving in 2-D, from side-to-side and up-and-down. My calculation using the distance formula is below.
This distance is approximately 2.02 meters. Therefore, the total distance traveled for the spider to reach the fly is 0.5+5+2.02= 7.52 meters. Then the problem asks another question: If the fly starts moving towards the floor (I'm assuming the fly  moving straight down), then at what point should the spider change it's route to catch the fly?

Not knowing when the fly starts moving downwards, and where the spider is on it's route, I decided to redo the problem with the fly already at the bottom of the wall. Then I thought I would do the problem in the same way as above. But then I had an Eureka! moment. I was already breaking down spider's movements or the walls of the room into 2-D. The two end walls had (x, y) coordinates and the front and back walls had (z, y) coordinates. So, I decided to remove the back wall, and flatten the 3 remaining walls into one large wall. Then, the spider just needs to travel diagonally to get to the fly, instead of changing direction to go straight once the spider reaches an edge. Above is how I visualized the new problem (the fly is moving straight downward).

Now, we have just have (x,y) coordinates to deal with. The fly's new coordinate is (9.5, 0) and the spider's new coordinate is (2, 1.25). I used the distance formula again to calculate the distance the spider will travel on the diagonal line. Below is the calculation.

The answer is approximately 10.68 meters. Since, I thought of a new way to look at the problem I decided to re-calculate the distance of the first problem using this method. So, the spider will, again, move diagonally only, and not change direction to go straight. Below is the new visual for the first problem.
So, the spider has the coordinates of (2, 1.25) and the fly has the coordinates of (9.5, 1.5). The calculation for distance using the distance formula is below.
The answer is approximately 10.61 meters. So, it would seem that the best route was my first one. Going diagonally all the way across adds distance instead of shortening it. So, going diagonally over a shorter distance and then going in a straight line to the final destination point is the best route. But the fastest route, instead of going around the room, is to go through the room. So, what if, instead of a spider that has to crawl over a 2-D space (side-to-side/up-and-down), there is a dragonfly that can fly through a 3-D one (side-to-side/up-and-down/forward-and-backward)?

3-D Extension
I was very interested in exploring 3-D geometry, and the first thing I wanted to learn was how to obtain a line in 3-D space. So, I watched David Butler's "Example: Finding the equation of the line in 3D through two points" YouTube Video from Maths Learning Centre UofA channel, and applied it to this problem. My work is below.

Then, I wanted to find the distance the dragonfly would have to travel to reach the fly. My professor explained I could use the distance formula with the z-coordinates added. My work is below:

The square root is about 5.226. So, the dragonfly would have to fly about 5.226 meters to get to the fly.

A great problem, like this one, inspires the imagination. The original problem, even though the setup is in 3-D, asks a 2-D question. The concept that needs to be understood is that a spider can only move in two ways on the wall or plane. The problem can be thought of and solved in different ways, as shown above. So, the original problem has lots to explore, but it also offers more. As soon as I read the problem, my mind went straight to "But what if the spider could fly?", turning the 2-D question into a 3-D one, and then I sought out ways to answer my question. Curiosity is an indication of interest and readiness to learn something new. Offering opportunities for students to imagine "what if" is the best way to engage and encourage their learning. It will motivate them to learn on their own, fostering a sense of responsibility for their education. And as a teacher, that is the most important skill I can encourage in my students.

Figures created using GeoGebra and Microsoft Word. I do not have any affiliation with NRICH, GeoGebra, or David Butler, and none have endorsed my work. I am a college student. 

Sunday, November 1, 2015

Arctan and Euler

 Communicating Math

Leonard Euler has made many contributions to math, and one of them is a formula for arctan. To see and investigate Euler's formula, I used GeoGebra's An Equation of Euler by timteachesmath (Link). There are 5 sets of examples of the formula in each of the 3 options in the GeoGebra exploration. Below is my explanation of the equation in relation to the GeoGebra activity.

Option 1

First, at T=0, there is a square with a diagonal line (Figure 1). As I move T along the slider, the green square becomes longer to create a rectangle with the original square. I also get a blue square and blue rectangle that has been rotated. The reason the blue square/rectangle gets rotated is so the two corners of the blue rectangle remain touching the corners of the green rectangle.

Figure 1: Option 1 - T=0

When T=1, the formula appears (Figure 2). Essentially, it's saying the angles b and c will add up to equal angle d. Since tangent of an angle = opposite/adjacent, the first arctan corresponds to angle b (green line to the red diagonal) in the green rectangle, since tan (b) = 1/3. The second arctan corresponds to angle c (from blue line to red line) in the blue rectangle, since tan (c) = 1/2. The final arctan corresponds to the angle d (from green line to blue line) in the green square, since tan (d) =1/1.
Figure 2: Option 1 - T=1

Option 2

Again, Option 2 starts with the same square as Option 1 when T=0. When T=1 though, the first square gets divided into fourths, there are 3 large green squares and 1 large rotated blue square. Again, the two corners of the blue square remain on the two corners of the green rectangle. Now, the equation asserts that angle b of the green rectangle and angle c of the blue square add up to angle d of the green rectangle comprised of the 2 smaller squares (Figure 3). This is because tan(b)=1/3 , tan (c ) = 1/1, and tan (d) = 2/1.
Figure 3: Option 2 - T=1

Option 3

Here, Option 3 starts with 2 green squares stacked vertically at T=0 (Figure 4).
Figure 4: Option 3 - T=0

 At T=1, the 2 green squares expand to 6 (3 columns and 2 rows). There are 5 blue rotated squares, and again the 2 corners of the bigger blue rectangle remain on the two corners of the bigger green rectangle (Figure 5). Because we have two rows of squares the numerator of the first arctan is 2. In other words, tan (b) = 2/3 (the big green rectangle), tan (c ) = 1/5 (the blue rectangle), and tan (d) = 1/1 (the green square).
Figure 5: Option 3 - T=1

The key for this equation is keeping the two opposite corners of the blue square/rectangle inline with the expanding green square/rectangles. This makes the blue squares rotate and expand with the addition of the green squares. This creates a blue diagonal that goes through the green squares and rectangles. Thus, the red and blue lines divide the green rectangle in different ways to make the equation possible.

There were two other interesting things I noticed. The first arctan term was the answer for the next formula in Option 1, and all the numerators and denominators are numbers in the Fibonacci Sequence in all the options. Also, the Fibonacci Sequence was in reverse order in Option 1. As the blue rectangle rotates and expands with the green rectangle, a relationship between the green squares and rectangles and the blue squares and rectangles is maintained. So, the equation only works when the original square (T=0) can divide the green rectangle equally and the blue rectangle can be divided equally by a different size square.

Update (11/9/2015): Eureka!  The rotated blue rectangle creates a parallelogram with the green rectangle. The red diagonal splits the parallelogram in half, creating two congruent triangles. We can also see the blue lines that create the sides of the parallelogram as two congruent transversals that cut the green rectangle. So, the properties of parallel lines and parallelograms is why Euler's arctan formula works.

Screen shots are of timteachesmath's An Equation of Euler on GeoGebra. I added the angles b, c, and d. I do not have any affiliation with GeoGebra or timteachesmath, and neither has endorsed my work. I am a college student. 

Sunday, October 18, 2015

From Primitive to Modern: The Evolution of Mathematics

Book Review: Clifford A. Pickover's The Math Book

From the very beginning in the Introduction, Pickover makes it clear that this book is not the complete, definitive list or the final word on what the milestones of math should be. As he mentions, these are his selections that cover a wide range of mathematical topics, from games to important theorems, and are written in such a way that an average person could understand, in a general sense, what the topic is about. And to his credit, Pickover does just that, and does it well.

The list is in chronological order and the topics have one-page summaries with an accompanying picture that can help the reader visualize the math concept or milestone featured. The breadth of time Pickover covers is from BC to the early 2000s. The list starts, surprisingly, not with a human mathematical milestone, but with an animal one. In fact, the first two milestones describe how animals such as ants and monkeys have mathematical capabilities. And the next milestone describes a mathematical pattern in Cicada (insect) behavior. The first human mathematical milestone, which is the fourth milestone in the book, is the presence of knots. Then the milestones become more historical with the mathematical advancements from ancient civilizations, like the ancient greeks. Famous games, equations, numbers, and texts are featured in this book, as well as technological advancements, like computers that can help prove conjectures.

The reason Pickover showcased the milestones chronologically, as he states in his introduction, is to have the reader see the evolution of mathematical thought, and how one milestone is connected and/or spurred another milestone. Pickover delivers on this goal, since I saw how mathematics is ultimately a compounding network of many topics, each connected - influencing, advancing and creating others. He not only points out these connections, he even lists the related topics at the bottom of the page. Furthermore, Pickover also points out how a milestone is important or useful outside the field of mathematics like in economics or the sciences. The best example of this are knots. Pickover mentions knots as the first primitive human milestone, and the topic continues to progress in complexity with advances in knot theory like Jones Polynomial, which has applications regarding the DNA molecule and proteins (pg. 24, 478, 490).

But why start the book, which is about mathematical milestones, with a mathematical capability of an animal, instead of a human? Why would it even be considered a milestone? Pickover discusses how mathematics is not solely human, and it implies how human mathematical capability has evolved from our animal ancestors.

All in all, this book is a great read for those who want a condensed history of mathematical milestones. Since this is my first major exposure to the history of mathematics, it's hard for me to compare this list to other milestones and say "Why isn't such-and-such in this book?!?" However, 3 out of my 5 milestones that I could barely come up with in this blogpost were featured in Pickover's book (Zero, Calculus, and Alan Turing). The first one that didn't show up was Albert Einstein's E=mc2, even though Pickover mentions Einstein in the book, and specifically that equation in the Introduction. The second was a milestone that happened after the publication of the book, which was the first woman to win the Fields Medal (1).

None of the people or organizations below have endorsed my work, nor am I affiliated with any organization or any person. I am a college student.

Pickover, Clifford A. The Math Book. Sterling Publishing Co., Inc. New York, NY. 2009.

(1) Ball, Phillip. "Maryam Mirzakhani Becomes First Woman To Win Prestigious Fields Medal." The Huffington Post.

Sunday, October 11, 2015

Math: Making Science Possible

Nature of Mathematics

Math has always been taught as a separate subject from science throughout my education. Furthermore, math was taught first. Addition and subtraction were 1st through 3rd grade concepts. Multiplication and division were in 4th and 5th grade. And that's when science was introduced as a subject - not until the 4th and 5th grade. Based on the structure of my education alone, it seems that the concepts of math (numbers, operations etc.) had to be taught first before the concepts of science could be learned, since science uses a lot of mathematics. But could math be considered a science field because of this connection? 

In my class the other day, we conducted an experiment where we dropped objects from a bridge that were attached to a certain number of rubber bands. We were supposed to determine the right number of bands that would make our object come closest to the ground without touching it. For me, I felt like I was doing both science AND math while doing this experiment. The scientific component: forming a hypothesis, factoring gravity, weight, and elasticity to make that hypothesis, and testing that hypothesis with an experiment. The mathematical component: the acceleration of gravity is a mathematical constant, the mass of the object, and the force of the weight, which is described in a mathematical equation. Even though they overlap, I still see them as two separate fields.

To help illustrate this point further, I've made a Venn Diagram showing how I see math and science as two separate fields that do overlap. 

Math/Science Venn Diagram

So why did I organize the Venn Diagram the way I did? I won't be able to explain my decision-making process for each subtopic (my blogposts are long enough), but I will try with a few examples. I put geometry squarely in Math, and Astronomy in Science, even though I can think of ways they overlap. For example, orbits around the sun in astronomy can be described as ellipses and math was applied to the data to discover this. But shapes, like ellipses, exist and have mathematical properties in geometry independent of astronomy. Furthermore, there are concepts in astronomy that are independent of math, like which planets have or don't have atmospheres and how that impacts that planet's environment. So, that's why geometry and astronomy, and other subtopics, are not in the overlap part of the diagram.  

So, what connects the two? What is in the overlap? Data most importantly. Science collects data during experiments. Math and Statistics - a field I consider separate from Math, but is included as a subfield of math for simplicity - are used to make sense of the data. Data and math were used to discover that planets orbit the sun in an ellipses shape. Physics, even though it's usually referred to as a science, is so dependent on math, I put it in the overlap. I can't think of physics (gravity, force, acceleration etc) without thinking of math, like trigonometry and quadratics, which are all in the overlap.

There are two concepts in Math and Science that are similar but have different processes and functions that help separate Math and Science from each other, and they are Proofs and Experiments, respectively (look at the bottom of the circles). Scientists conduct experiments to test hypotheses, which are predictions based on observations of the natural world. Mathematicians, when they observe a pattern with numbers or shapes, make a conjecture and then they try to prove it. When they do, the conjecture becomes a theorem and it is considered fact in the mathematical community. This is where the line between math and science is most visible for me. In science, hypotheses are never considered fact. When the data is inline with the hypothesis, scientist say the hypothesis is supported, never proven. It takes multiple experiments done repeatedly before a hypothesis or hypotheses are considered to have enough supporting evidence to be a theory, which is still not considered a fact.

And there are other fundamental differences that separate math and science. Science is about understanding how things in the physical world/universe work. As my class discussed, math is about quantifying the world/universe and understanding numerical patterns, which are often found in nature, like the Fibonacci Spiral in pine cones (1). But finding patterns in the natural world that can be described using numbers doesn't make mathematics a science. Math exists without science, as illustrated with subtopics like geometry. In geometry, there are abstract concepts that don't exist in nature, like a one-dimensional line that expands to infinity. Also, discoveries or advancements in math are concerned with patterns in numbers, not physical processes in the world/universe, like in science. Furthermore, it seems that math makes science possible since math is used to interpret scientific data. And even though, math describes gravity with quadratics, science would not perish if the connection between science and math was less strong. For example, what if gravity was random, and couldn't be represented by a quadratic? (Let's not think about the fate of humanity and all life for the moment). Scientists would still be trying to find out how things work, but it would have less of a mathematical connection.

None of the people or organizations below have endorsed my work, nor am I affiliated with any organization or any person. I am a college student.


(1) Vi Hart. "Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]." YouTube Video. Date of Access: October 12, 2015.

Figure created using GeoGebra.

Monday, September 28, 2015

The Value of Nothing: How Did We Get From One to Zero?

History of Mathematics

At the very beginning of my K-12, and maybe even my preschool education, zero has always been a number. The concept of zero - as a number that stands for something that isn't there - is as acceptable, and natural to me as the numbers that represent things that are there (1 thing, 2 things etc.). Mathematics wouldn't make sense without it. For counting, I could either start at nothing, 0, or 1. For arithmetic, 2-2=0. Two things exist. Take those two things away and those two things aren't there anymore. Nothing is left.

But the concept of zero as a number isn't natural at all, especially in Western Civilization (Europe) (0). The concept of zero started in the East, most notably in India, and it took until the Renaissance for Western Civilization to accept a representative number for nothing (0 + 1). So how did Western Civilization finally place a value for nothing? How did we get from one to zero? The time period and from what civilization zero was born depends on how one defines a number.  

According to the MacTutor History of Mathematics Archive, the expression of zero throughout history can be separated into two categories (1). The zeroeth category (see what I did there) is zero as an indicator for empty space in a number system or as a positional notation in context with other numbers (1). The first category is zero as an abstract, independent number with value and properties, which is the way we think of zero today (1+ 2).

We will first discuss the first category: zero as an abstract number. Brahmagupta, an Indian mathematician living from 598 to 665 AD, is considered to be the first to think of zero this way (0 + 3). In his work titled Brahmasphutasiddhanta, he developed properties for negative and positive numbers, and for the number zero (1 + 4). The idea of zero as an actual number spread to China and the Middle East (1). Fibonacci in the 1200s tried to get Europe to replace Roman numerals with the 0-9 number system that we use today, but met opposition (0 + 1 + 7). In Christianity,  God was associated with infinity and the idea of nothingness, the opposite of infinity, was therefore associated with the devil (0). Furthermore, Fibonacci was campaigning for the new number system during the Crusades, and the new number system was associated with Islam, since they had already taken the same number system from the Hindus (6 + 7). Finally, in the 15th century , Europe converted to the 0-9 number system (1).

But before zero was seen as an abstract number, the concept of zero started as a positional value, which is the zeroeth category (1). In an interview on BBC's In Our Time Radio program, Robert Kaplan, Ian Stewart, and Lisa Jardine describe the importance of the number zero in history, and the significance of it's use in positional notation. The concept of zero started as a symbol to signify no-number about 5,000 years ago in the Sumerian number system, which used positional notation (0). Eventually, zero as a positional value became common in bookkeeping in different civilizations over time because of trade, except in Western Civilization (0). The ancient Greek mathematicians considered basic arithmetic that was used in trade not intellectually stimulating enough to think about (0). Furthermore, the ancient Greeks decided that the concept of nothing couldn't exist in the real world (5). So, the zero that meant no-number in trade didn't cross over to become an abstract mathematical concept for the ancient Greeks, nor the Romans, who based their number system off of the ancient Greeks (0).

So when in mathematical history should the advent of the number zero be marked? Was Sumerian Civilization the first when they used zero has a positional value? Or was it Indian Civilization the first when they treated zero as an abstract, independent number?

As I discussed in a previous post, I believe counting to be a mathematical concept, and therefore consider assigning numbers and values to the physical, tangible world as one of the first mathematical activities humans did. Historically speaking, humans did not start counting at zero, as indicated by the number systems of many ancient civilizations, since nothing isn't tangible and therefore has no value (1).

Yet, the concept of zero was used as a positional value in number systems, and arithmetic was used in trade, even though people of these civilizations didn't treat zero as a number that had value, and therefore didn't compute it in their calculations like we do today (0). They probably skipped right over the symbol, considering that the symbol for zero in position notation was often represented as an empty space (1). But they were inadvertently placing a mathematical property on that empty space when they did arithmetic, a property that Brahmagupta would attribute to the number zero: any number plus or minus zero is the same number (4). With that said, this action of skipping over the symbols for zero as a positional value is equivalent to us using the number 0 in 101+190 = 291.

Therefore, I believe the number zero was born when it was just a positional value at the time of Sumerian Civilization. Very humble beginnings indeed, especially when we look ahead in time and see all that was achieved during the Renaissance when zero was finally accepted as a number in Western Civilization, like Newton discovering calculus (0 + 2). Who knew nothing could amount to so much?

Sources: *No person or organization has endorsed my work, nor am I affiliated with anyone or organization. I am a college student.*

0. Podcast. Interviewees: Kaplan, Robert. Stewart, Ian. Jardine, Lisa. Interviewer: Bragg, Melvyn. "In Our Time: Zero." BBC Radio 4. Date of Access: September 28, 2015.

1. O'Connor, John J. and Robertson, Edmund F. "A History of Zero." MacTutor History of Mathematics Archive. Date of Access: September 28, 2015. 

2. "Who Invented the Zero?" Ask History. Date of Access: September 28, 2015.

3. Hayashi, Takao. "Brahmagupta: Indian Astronomer." Encyclopedia Britannica. Date of Access: September 28, 2015.

4. Mastin, Luke. "Indian Mathematics - Brahmagupta." The Story of Mathematics. Date of Access: September 28, 2015.

5. "A Brief and Early History of Zero (ca. 2nd C BC - Onward)." The Ancient Standard. Date of Access: September 28, 2015.

6. Mastin, Luke. "Islamic Mathematics - Al-Khwarizmi."  The Story of Mathematics. Date of Access: September 28, 2015.

7. Mastin, Luke. "Medieval Mathematics - Fibonacci."  The Story of Mathematics. Date of Access: September 28, 2015.

Sunday, September 13, 2015

Riddle Me This, Diophantus

Blog post for Doing Math

Diophantus was a great, ancient greek mathematician, who played a very important role in the subject of algebra ( To honor this man, my MTH 495 professor gave us a riddle - a riddle of algebra - to solve that would give Diophantus' age when he died. Here it is, as given on the class handout, below:

" 'Here lies Diophantus,' the wonder behold. Through art algebraic the stone tells how old: 'God gave him his boyhood one-sixth of his life, one-twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; in five years, there came a bouncing new son. Alas, the dear child of master and sage after attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' "

The language seems clear but riddles are tricks of the mind. What seems straightforward, isn't, and the challenge is finding where the riddle deliberately leads the mind astray, so it would make an erroneous assumption. In other words, how has the riddle tricked us into thinking one interpretation, when it really meant something else?

To demonstrate how challenging the riddle was for me, I went through 7 attempts or 7 different interpretations of the riddle before I got the correct answer. For the first attempt, I tried solving the riddle without using algebra, and that meant picking an age for Diophantus, and doing the series of computations laid out in the riddle to see if it was correct. Basically, guess-n-check. I picked 50 because people rarely lived for very long back then.

On this first attempt, I thought the challenge, or the trick in the riddle, was in the line "And then one-seventh ere marriage begun." The word ere means before, according to ( So, I had to subtract 1/7 from his boyhood and youth before continuing on with addition. So, the sequence of events laid out in the riddle is not in chronological order. Everything else seemed straightforward. I interpreted "God gave him one-sixth of his life," as 1/6 of his age. "One-twelfth more as youth," I interpreted as his youth was 1/12 more than his boyhood, or 1/6 of his age. "One-seventh ere marriage begun," I took as 1/7 of youth subtracted from his boyhood and youth. My interpretation for the rest: then 5 years after his marriage, a son was born, who grew to half the age that Diophantus eventually reached, 4 years after his son's death. As mentioned before, I picked 50 for Diophantus' age. So, I did the following computations:

Since the ages don't match, my guess was wrong. But the closeness between the two numbers gives a false impression that 50 was close. However, the fractions would divide any number in such a way that would make the two numbers seem close. At this point, I decided that I could keep on blindly guessing or try to solve the riddle in the most efficient way possible, with algebra.

When I continued with my other attempts, I realized the challenge was no longer with the chronological wording of the riddle. The trick lay with the wording of the fractions. In my middle school teaching course, the class went over fractions, and whenever someone said "Take 1/2," our professor would always say "1/2 of what?" The fraction needs to be applied to another number, but sometimes the riddle doesn't tell explicitly what the fraction is being applied to. For example, nowhere in the line "One-twelfth more as youth" does it say what 1/12 is being taken from. Is it 1/12 more of his entire life that is being added to his boyhood, or is it  1/12 of his boyhood, or is it just simply 1/12 of his life was his youth, and not in addition to his boyhood?   

I interpreted the riddle differently, with different algebraic equations, 6 more times. On the last attempt, I got it right. I will show the different combinations or ways I interpreted the riddle with the variations for each section of Diophantus' lifespan below. The straightforward interpretations are on top and the multiple interpretations are on the bottom:

Finally on the last, overall 7th attempt, I got the answer right, at least according to Wikipedia ( The interpretation necessary was youth equaling his boyhood plus 1/12 of his entire life, and his marriage was 1/7 of his boyhood before the end of his youth. My solution is below:

I enjoyed this riddle very much. It reminded me how the process of mathematics should be like. Doing math is about looking at problems or situations with different interpretations to see how it plays out and what's discovered in the process. After all, learning requires making mistakes. Figuring out what went wrong is necessary to find out what will make it right.

Figures were created using Microsoft Word and Geogebra.

Wednesday, September 2, 2015

Mathematics: It Started With Numbers

I’ve made a harrowing discovery. There is a gaping hole in my understanding and knowledge of mathematics. For most, my statement doesn’t seem concerning. Mathematics isn’t a popular subject. But for a future math teacher, like myself, it’s a confession of a big, terrible shortcoming.

My revelation came when my class was asked a very simple question, “What is mathematics?,” and I couldn’t think of a satisfactory answer. The confidence I had in the very subject I was preparing to teach was gone. How can someone teach a subject, when they can’t even define it in words?

Before that question, I believed I knew what mathematics was. I’ve been doing it for years - adding, subtracting, multiplication, matrices, algebra, geometry, and proofs! But those words are either what people do in mathematics or a specific area within mathematics. Not one of them encapsulates what mathematics is, and listing everything in mathematics does not describe what it is either. So, what do all those words have in common to help define mathematics in a clear, precise way?

At first, I thought numbers. Recalling my elementary years, learning math was learning adding, subtracting, and other operations. But before I could do or be taught math, I needed to know numbers. To learn numbers, I counted physical objects before and during kindergarten.

Others in the class shared my view, stating that math is about describing the physical world with numerical values. Some expanded on this concept and said putting numerical values in an order is math. Others mentioned measurement, explaining how trade spurred the use of mathematics, while others mentioned astronomy and how mathematics was used to find the length of a year and day. Finding patterns was also mentioned, and a particular person said, “mathematics is the science of patterns.”

“But is that enough?,” our professor asked and continued to ask with each new claim of what math is. Is having a pattern enough for something to be labeled math? How about counting? Are people doing math when they count? How about labeling a quantity with a specific value, like 4 or 9? Does knowing you have 9 cows make it math? Or could any conceptual form of measurement make it mathematical? A student used an example of a stick as an example of the first rudimentary means of measurement - walk 4 stick lengths and then turn left. How about time? Does knowing the position of the sun in the sky or time of the year make it mathematical? Does assigning that position with a numerical value like 1:00 pm make it mathematical? Just because it has a number or value or quantity, does it make it math?

Since the class period, I’ve been thinking a lot about these questions the professor posed to the class, and I’ve come to the tentative conclusion that yes, just because something has a number or could be described or substituted with a number, it is math. To do the simplest math, like adding, numbers needed to exist or at the very least some form of measurement had to be in place to describe and order our world. To figure out time, the concept of a start and end needed to be established, which could then be described more specifically with numbers. Then operations could have been conceptualized from there. Imagine two people looking at a sundial. One says to the other, “Meet me in two sun-lengths time from now.” The other person sees the shadow on the sundial, and then imagines the position of the shadow in two sun-lengths time. That person has just done addition. Mathematics started with numbers.

Lastly, I have another confession to make. My knowledge of the history of mathematics is very and utterly limited to pop culture. Another embarrassing fact about the math teacher to-be. When asked to think of five top moments or milestones in mathematics, I could barely think of five at all. The first one I thought of was Albert Einstein’s E=mc2. My knowledge of that equation is very lacking. Do not ask me to explain it or tell you what it is for. My embarrassment is bad enough.

The second milestone was the discovery of the number Zero, since it took civilizations a long time to create a number to describe nothing. A student in my group had also mentioned this when we were discussing the “What is mathematics?” question, and I remembered hearing that fact in high school when I learned about how certain civilizations influenced others when they traded or were conquered in war. Again, don’t ask me which civilization came up with the concept of Zero first.

The third milestone was Sir Isaac Newton discovering or creating Calculus. The fourth was Alan Turing’s use of mathematical theory (I’m assuming from the movie) to create a computer during World War II - thank you very much, The Imitation Game! The fifth…didn’t the first woman receive the Nobel Prize in mathematics for something last year? I remember reading an article about it. The math portion was completely over my head.  

Luckily, someone acknowledging they have a problem is the first step towards a solution, and my remedy to fix my lack of knowledge is to read Clifford Pickover’s The Math Book. According to, the book gives brief explanations to many historical moments in mathematics. Just the book I need.

Sunday, April 5, 2015

Repeating Decimals Repeated

In the previous post, I discussed the connection decimals have with fractions, and specifically went over when fractions have repeating decimals. I explained how we can tell when a fraction is a repeating decimal, and that concept is so important and fundamental to mathematics, I decided to bring up repeating decimals again to talk about it more deeply. And that concept is finding patterns. We know when we have a repeating decimal, and what those repeating decimals are, by finding repetition in the remainder during long division.

One of the best fractions to illustrate this point of finding and using patterns in math are the ones with 7 as the denominator, x/7. For a homework assignment, I explored repeating decimals and fractions. So I started with fractions with 3 in the denominator, then 6, then 7, and then 11. The fractions with 3 or 6 in the denominator had a simple pattern with one or two repeating decimals. With 7 as the denominator, the pattern was more complex with 6 repeating decimals. But an even more interesting pattern came to light. Usually when the numerator changes, so do the numbers in the fraction's decimal form. That wasn't the case when the numerator changed with 7 as the denominator. Instead, the numbers were the same, they were just in a different order that could be determined by the sequence or pattern of the remainders, as shown in the figure below.

To show how the fraction x/7 is a special case to students, I would start with 1 in the numerator, or x=1. In the previous post about repeating decimals, I showed the long division process using both decimals and whole numbers. But in the this post, I only used whole number because I wanted to keep the number of figures down. But during a class, I would do both. Now, the number 7 can't go into the number 1. So we add a zero, shown in red, to get 10. 7 goes into 10 once, with a remainder of 3. 7 can't go into 3, so I drop down a zero, shown in purple, to get 30. 7 goes into 30 four times, with a remainder of 2. I need to drop down a zero again, shown in turquoise, to get 20. Then 7 goes into 20 twice, with a remainder of 6.
At this point, a student might say that this fraction doesn't have a repeating decimal. This is another good reason to explore this example with students. It shows that a repeating decimal can have any number of repeating digits in it's answer, not just 1 or 2. So, I show the students we need to continue by dropping down another 0, shown in green, to get 60. 7 goes into 60 eight times, with a remainder of 4. Drop down another zero, shown in orange, to get 40. 7 goes into 40 five times, with a remainder of 5. Drop down another 0, shown in pink, to get 50. 7 goes into 50 seven times, with a remainder of 1, which is the number we started out with in the numerator. And thus, we finally have the repeating reminder that indicates which numbers are repeating in the decimal form of this fraction. This should help illustrate that fractions will either have a finite number of digits or an infinite number of repeating digits, whether it's 1 or 12, in their decimal form.

Next, I would ask the students to change the numerator and divide by 7 to see if they notice any patterns when they do so. As shown in the examples below, I have compared 1/7 to the other fractions that have 7 as the denominator. I've color coordinated the drop-down-zeros to show how the order of the remainders, and consequently the order of the decimal numbers, changes when the numerator changes. I have also labeled the remainder and decimal order based on 1/7 to help illustrate this pattern even more. 
In 1/7, there are 7 numbers, 0-1-4-2-8-5-7, before the remainder 1 repeats itself again in the long division process. This indicates that there are 6 repeating digits in the decimal form: 1-4-2-8-5-7, as indicated by the long bar over these 6 numbers.  As we can see, whenever the numerator changes, that numerator corresponds to one of the remainders when we divided 1 by 7. For example, the numerator 2 corresponds with remainder #3 from 1/7, and the numerator 3 corresponds with remainder #2. Consequently, this shifts the order of the remainders, which shifts the order of the digits in the decimal. So, for 2/7, the remainder and decimal order is 3, 4, 5, 6, 1, 2. For 3/7, the remainder and decimal order is 2, 3, 4, 5, 6, 1 as shown in the figure below. 

 For 4/7, the remainder and decimal order is 5, 6, 1, 2, 3, 4. And for 5/7, the remainder and the decimal order is 6, 1, 2, 3, 4, 5, as shown in the figure below.

Lastly, we have 6/7. And the remainder and decimal order for this fraction is 4, 5, 6, 1, 2, 3, as shown in the figure below.

Hopefully, the students will notice a pattern at the very least, and the class can share their reasons for why the pattern happens the way it does. The figures above should help them along in their reasoning and understanding. By the end of this example, I hope they achieve a better understanding of repetition patterns with remainders to determine a repeating decimal. Also, I hope they gain a better ability to notice patterns in math in general, since patterns greatly help in understanding math concepts and solving problems. 

Figures were created using GeoGebra.

Sunday, March 22, 2015

Repeating Decimals

There is a direct relationship between fractions and decimals. All fractions can be rewritten as decimals, and those decimals (but not all) can be rewritten as fractions. However, the relationship between fractions and decimals may not be clear to students. The first exposure students have with using decimal notation is with money, since American money is written like this: $1.43. With the dollar sign written in front of it, people know 1.43 is representing money and people say that value as 1 dollar and 43 cents. However, money doesn't give students the sense that decimals can represent fractions. But without the dollar sign, the money context is removed, and $1.43 becomes 1.43, which is communicated as one and forty-three hundredths. In this context, the relationship between decimals and fractions becomes clearer, since 1 43/100 is communicated in a similar way: one and forty-three one-hundredths.

To illustrate this point further for my students, I would show them how fractions are hidden division problems. For example, 3/4 can be said as three-fourths or 3 divided by 4. The answer to 3 divided by 4, which is 0.75, is pretty straightforward and the answer doesn't have a remainder. The main focus of this blog post is introducing students to the concept of the repeating decimal, which is when a fraction in decimal form doesn't end neatly, but instead the answer has repeating digits that goes on indefinitely.

The first example I would use to illustrate repeating decimals to my students is 5\6, as shown in Figure 1. Now, I would tell my students that dividing by a number that is bigger than the number being divided, has a similar process as having a smaller number dividing a bigger number. So, I would use the same language that the students are familiar with when doing long division. So, 6 goes into 5 zero times. So, I would put a 0 as the first digit. Since 5 is in the ones place, then I  must use the tenth place (I've designed this lesson assuming I've already discussed place values with decimals with my students). So, I need to place a decimal point at the time when I use the tenth place to show where the whole digits stop and the decimal digits begin. And that spot is right after the 0, as shown in Figure 1. Now, when I was being taught decimals using long division, the teacher had used whole numbers when subtracting. But my professor, John Golden, showed my class a way using decimals. So I will illustrate both ways in my blog post and to my students. That way, students who prefer one method over the other can choose for themselves. In this blog post, the left side of the figures will use whole numbers and the right side will use decimals.
Figure 1: Dividing 5/6

To continue, we need to add a zero, as shown in Figure 2. This concept might be easier to grasp when using decimals, since 5 and 5.0 has the same value, while 5 and 50 does not. But either way, the value for tenth place in our answer will be the same. 6 goes into 50, 8 times, since 6 x 8 = 48. Then I subtract and get 2. Now, I will use decimals. 6 goes into 5.0, 0.8 times, since 6 x 0.8 = 4.8. Then I subtract and get 0.2. This also serves as a good example to show students the trick of shifting the decimal point over one place when multiplying with tenths.
Figure 2: Dividing 5/6
Again, I need to add another 0, so 50 becomes 500, and 5.0 becomes 5.00. Then, I can drop down the 0 to get 20 or 0.20, as shown in Figure 3. Either way the number in the hundredth place of the answer will be the same. 6 goes into 20, 3 times, since 6 x 3 =18. Then I subtract to get to 2. For the decimal side, 6 goes into 0.20, 0.03 times, since 6 x 0.03 = 0.018. Then I subtract to get 0.02. As we can see, especially with the whole number example, we got the number 2 again (the remainder) after we subtracted the two previous numbers. So, I would ask the students, "do you think we will get the same number for the thousandths place in the answer?" To check, I would go through the steps of long division one last time.
Figure 3: Dividing 5/6

To begin one more round of long division, I need to add another 0 again. So, 500 become 5,000 and 5.00 becomes 5.000. I drop the zero down, to get 20 and 0.020. Again, 6 goes into 20, 3 times, since 6 x 3 = 18. And 6 goes into 0.020, 0.003 times, since 6 x 0.003 = 0.018. So, 3 goes into the thousandth place, and when I subtract, we will get the number 2 as the remainder again, as shown in Figure 4. This will show the students that the answer will have a repeating decimal that will continue indefinitely. It will also show the students how to recognize a repeating decimal when it happens, and that's by seeing remainders repeating during long division. I would then show them the proper notation for a repeating decimal, and that is placing a bar over the number or numbers that are repeating, as shown in Figure 4.
Figure 4: Dividing 5/6

In my next example, I will show the students how a series if numbers can repeat. So, I will use the fraction 1/11 to show this. I'm dividing 1 by 11, so 11 goes into 1, zero times. So, 0 goes in the ones place. Again, since I need to use the tenth place to continue, I need to place the decimal point next to the 0 to indicate where the decimal values start, as shown in Figure 5. Again, I will be using whole numbers for long division on the left side of the figure and decimals on the right.
Figure 5: Dividing 1/11
Now, I need to add a 0. So, 1 becomes 10 on the left, and 1 becomes 1.0 on the right, as shown in Figure 6. 11 goes into 10 and 1.0, zero times again. So, 0 goes in the tenth place. Then I subtract, and get 10 on the left side, and 1.0 on the right side, as shown in Figure 6.
Figure 6: Dividing 1/11
Again, I need to add another 0, so 10 becomes 100 and 1.0 becomes 1.00. I drop the 0 down, and get 100 and 1.00 at the bottom. Now, 11 goes into 100, 9 times, since 11 x 9 = 99. I subtract the previous two numbers and get 1 as the remainder. For the right side, 11 goes into 1.00, 0.09 times, since 11 x 0.09 = 0.99 times. I subtract the previous numbers, and I get 0.01 as the remainder. Since I have the number 1 in the remainder again, I know I will have to repeat 0 and 9 again indefinitely. So, 09 is my repeating decimal and I show that by placing the bar above those two numbers in the answer, as shown in Figure 7.

Figure 7: Dividing 1/11

Hopefully by the end of both of these examples my students will have a better understanding of the relationship between fractions and decimals. Furthermore, my students should be able to recognize when they will have a repeating decimal by noticing when they have a repeating remainder. 

The images were created using GeoGebra.

Sunday, March 8, 2015

Instrumental and Relational Understanding with Fractions

For my teaching class, I read an article "Relational Understanding and Instrumental Understanding," by Richard R. Skemp from Mathematics Teaching in Middle School, September 2006, Vol. 12 No. 2, pages 88 - 95. It defines instrumental understanding as the ability to follow rules and know when to use them, while relational understanding is defined as the ability to know and explain the concept. When I learned multiplying and dividing fractions, it was just a bunch of rules. For multiplication, just multiply the numerator and denominator across to get the answer, as shown in Figure 1. For division, the rule is flip the numerator with the denominator of the second fraction, and then multiply across, as shown in Figure 1. But learning math shouldn't be memorizing a bunch of rules without knowing how they work. It should be about true understanding; knowing the how behind the rule. So, I've designed a lesson showing how the rules work with visual explanations, which will incorporate student instrumental understanding with relational understanding.
Figure 1: Multiplication and Division of Fractions
 First, I would start with dividing 1 by 2 using one circle as the unit. The unit defines what the whole is. In Figure 2, we have 1 circle divided by 2. Students usually know what a half of something is, since division is associated with splitting an amount into groups. So, instinctively they should say that the answer of 1 divided by 2 is 1/2 because the circle is being split by a green line into two groups of 1/2, as shown in Figure 2. Then I would ask the students: does the division rule of fractions fit in with this example and how? Some might see the connection to the flip rule. But if not, then I would show them how dividing by 2 is the same as multiplying by 1/2, which gives us the division flip rule.
Figure 2: Dividing 1 by 2
 Next, I would take the answer from the last problem and have 1/2 divided by 2, which is 1/4. According to Figure 2 above, the circle is already split into halves. Then, I would split the two halves of the circle by 2 as demonstrated with the red line in Figure 3. This will give us 1/4. To show that the division flip rule still works, I will flip the 2 to make it 1/2 and then multiply 1/2 with 1/2 to get 1/4.
Figure 3: Dividing 1/2 by 2
In the past 2 problems, I've only been dividing one circle repeatedly. Now, I'm going to show how the division flip rule still works when dealing with numbers greater than 1. In Figure 4, there are 11 circles, which are being divided by 2. So, the circles will be split into two equal groups, where 5 circles and a half of one circle go into each group which is 5 1/2, or 11/2. Again, dividing 11 by 2 is the same as 11 being multiplied by the fraction, 1/2.
Figure 4: Dividing 11 by 2
Then, I would take the answer from the previous problem, and have 5 1/2 divided by 2 to get 2 3/4, or 11/4. 4 circles get divided equally, so both groups get 2 circles. 1 circle gets split into two halves, so each half goes into a group. And the one-half gets split into two one-fourths, so each fourth goes into a group. Add one group up (2+1/2+1/4), and we get 2 3/4 circles, as shown in Figure 5. And the division flip rule still applies, since 11 x 1 = 11, and 2 x 2 = 4, which gives us 11/4, or 2 3/4.
Figure 5: Dividing 5 1/2 by 2
I have shown how the division flip rule works for whole numbers. Now, I'm going to show that the division flip rule is the same when dividing by fractions. So, again we will take the answer from the previous problems and have 2 3/4 divided by 1/2, which will get 11/2 or 5 1/2 as shown in Figure 6. This is the same as multiplying 2 3/4 by 2, which is the opposite of multiplying by 1/2. So, instead of the answer getting smaller, like in the previous problems, the answer will get bigger. Specifically, it will be the number that got multiplied by 1/2 in the previous problem, which is 5 1/2.
Figure 6: Dividing 2 3/4 by 1/2
So far, I've been multiplying and dividing by unit-fractions. Next, I will show the students how the rules for multiplying and dividing fractions hold for any fraction. So, I will give them the problem 2/3 x 3/4 = 1/2. So, we take the circle and divide it into thirds, and then divide each third into fourths, which gives us a circle split into twelfths, as shown in the Figure 7. In the numerator, 2 x 3 = 6, so we count 6 pieces, which will give us 6/12 or 1/2, which is the purple line in Figure 7. The rule for multiplication holds, since 2 x 3 = 6 and 3 x 4 = 12, giving us 6/12 or 1/2. The red lines indicate the thirds, the blue line indicate the fourths, and the purple line indicate the half of the second circle in Figure 7.
Figure 7: Multiplying 2/3 by 3/4
Since division is the opposite of multiplication, I would again show how dividing the previous answer will get the number that was divided before. So, the problem is 6/12 divided by 3/4 = 2/3. So, we have a circle that is split into twelfths. Then, we split each twelfth into thirds, and then we use the division flip rule. This makes the problem 6/12 x 3/4. And since 6 x 4 = 24, we count 24 pieces in the second circle which gives us 2/3, as shown in Figure 8. The red lines in the second circle indicate thirds. Again, the division flip rule still holds, since 6 x 4 = 24 and 12 x 3 = 36, which gets 24/36 or 2/3.
Figure 8: Dividing 6/12 by 3/4
Hopefully, by the end of this lesson, my students will have the visual explanation that will help them understand how these fraction rules work. That way, the students will have the relational understanding to go with the instrumental understanding.

Figures were created using GeoGebra

Sunday, February 22, 2015

Distribution Dilemma

Students at a middle school I was observing were simplifying equations, and I noticed a consistent issue among many of them; they did not know how to distribute properly. Many would distribute to the first number inside the parenthesis, but not the second. Others would distribute the outside number to both sets of parenthesis if there were more than one.  So, if I were a teacher and saw this issue with my students, how would I fix it? What is it about the concept the students aren't getting?

The underlying issue might be that the students don't understand the function of the parenthesis and how the number outside each parenthesis got there. Since distribution involves factors and multiplies of numbers, teaching that connection could help the students understand how distribution works, and understand the purpose of the parenthesis. As a way of teaching this concept, I will use a great problem solving method I learned from my professor, John Golden, called the work-backwards method. For example, lets take the problem 8x = 48. Even though 8 is being multiplied by x to get 48, 48 needs to be divided by 8 to get the solution for x, which is 6. In other words, we start with the answer and work backwards. And that's how I approached designing this lesson. I'm going to start with an expression in distributed form and then take out the common factor to show how the parenthesis groups together a set of numbers that has a common factor. This will show how distribution is factoring in reverse.

So lets start with a simple expression as shown in Figure 1. One common factor between 40 is 8. The number 4 is also correct, as well as 2. So, 8 is taken or factored out, as indicated by the arrows, and after dividing 40 and 72 by 8, 5 and 9 are left. The next step is crucial, which is explaining the parenthesis. Since I took out the 8, I need a way to show that the 5+9 was originally 40+72. So, I place the parenthesis around 5+9, right after the 8, which will indicate that the 8 gets distributed and gets multiplied by 5 and 9 to get 40+72. This setup tells you that the number outside the parenthesis gets distributed to the group of numbers inside that parenthesis only. In other words, the parenthesis groups different sets of numbers together respective to their distributor.
Figure 1: Factoring out an 8
The next expression will have 3 terms instead of 2. That way the students will see that more than 2 numbers can be within a parenthesis. As shown in Figure 2, we start with 20+45+55. To help the students visually with the parenthesis, I put them in at the first step. Then the 5 gets factored out, and 4+9+11 is left in the parenthesis.
Figure 2: Factoring out a 5
The next expression will show that not all numbers in the expression can be factored. As shown in Figure 3, a 3 gets factored out of 27 + - 60 - 9, but not 11. So the parenthesis group the left over 9 -20 -3 right after the factored out 3.

Figure 3: Factoring out a 3, but not 11

The next expression has 4 terms, which are factored out by 2, which is placed outside the parenthesis, and the left over 4+6+25+5 is inside the parenthesis.
Figure 4: Factoring out a 2
As discussed before some of these can be factored out in multiple ways. So, I repeated the expression in Figure 4, to show how the same expression can be re-written by re-grouping different sets of numbers by their different common factors. 4 can be factored out of 8 and 12, resulting in a 2+3 in one grouping (as indicated by the parenthesis), while 10 can be factored out of 50 and 10, which results in another grouping of 5 +1 (as indicated by the parenthesis).

Figure 5: Factoring out a 4 and a 10

As an evaluation after this mini-lesson on distribution and factors, I would give the students an expression with the variable x, like the one in Figure 6, for them to distribute and simplify. Hopefully, as Figure 6 indicates, they would distribute correctly. The simplified answer to Figure 6 is 1 - 13x. A good indication of how well this lesson worked at increasing student understanding would be seeing less errors with distribution.

Figure 6: Distribute 5 and 2

The figures in this blog post were created using GeoGebra.