VISUALIZING FRACTIONS FOR ADULTSAn Approach Using Colored Pieces

by

F. Richard Singer III and Jennifer McCleary & Angela Lovettmailto:richardsinger3@sbcglobal.net:

Copyright ©2002 By Domestic Planner 2012 edition

Most people learn how to calculate with fractions primarily through drill and practice. Because of this, most adults do not have the kind of understanding of fraction concepts that allows them to help children obtain a good basis for understanding fractions. This book provides materials and ideas that relate fraction concepts to ordinary skills in visualization. It is intended primarily for adults who want to help younger children obtain a foundation for the work they will latter do in calculating with fraction. This book ignores drill, except to say drill is easy to find abounding elsewhere and we do not recommend it. Instead we believe better learning occurs when students invent their own activities. For example looking back at whole numbers we know that 57 = 12. It should be easy for anyone to make a situation to which this equation applies. Altho this may be easy, making up such a situation reinforces the use of arithmetic as well as numerical ideas.

This book is a free sample advertising our main text Understanding Fractions. It is almost identical to the first five lessons from the more extensive book. The appendix of both books contains a detailed account of the use of these materials with a ten year old student. The mentor was her older sister, a dance major with no previous experience in teaching mathematical concepts. The appendix also contains commentary on the use of these materials.

At the end of our book there is a template for making the colored pieces to be used.We recommend saving it as a separate file for making these pieces.

Also it can be re-paged to save the cost of laminating.

We welcome all comments on this book.We also ask you to share this book with as many friends as you can imagine.

ACKNOWLEDGEMENTSRichard is the grandfather of Jennifer and Angela. Richard is a Mathematics Professor Emeritus at Webster University. Jennifer was twenty and Angela was ten when we were first writing this book in 2002. Jennifer used the materials in teaching her sister. Her account of there is a major component of this book. We would like to add additional account of adults using material with children, and we will add any that you or your friends send to newer additions of the book. Incidentally Jennifer has not studied math beyond the college freshman level and has no earlier experience in teaching mathematics. In fact working with Angela is her only experience in teaching math. Her college work centered on Dance. Angela majored in Psychology and almost has a masters degree in counseling. She has now added her perspective as an adult descriptive psychologist.

While many people could be acknowledged for contributions to our efforts, we wold especially like to acknowledge Brian Koenig for his efforts to promulgate our books. We also want to acknowledge Octavia Randolph and Jim Pelech. Octavia used Understanding Fractions as an adult to obtain and master fraction concepts, providing an extra incentive for us to attempt finding a broader audience for our ideas. Jim used our book with considerable success in his course on teaching methods

Octavia website is http://www.octavia.net/: It is devoted primarily to Anglo-Saxon England. This site contains a wealth of information about the people and customs of that time. It also contains an excellent novel. There is a trilogy set Anglo-Saxon England, the first Book being entitled The Circle of Ceridwen. There is also an epic about Lady Godiva. Turning to a later historical setting there is a novel about John Ruskin.

Jim (JPelech@ben.edu) is a tenured professor of mathematics education at Benedictine University. He has experience in teaching mathematics at the secondary level and has used this experience in teaching teachers. He has published books and articles on constructivist learning, namely The Comprehensive Handbook of Constructivist Teaching: From Theory to Practice which is published by Information Age Publishing Charlotte NC. Jim is also president elect of the Association for Constructivist Teaching, the international organization for advancing this type of teaching and learning.

Comments by Octavia Randolph

Understanding Fractions and its smaller versions Visualizing Fractions by F. Richard Singer III taught me that it is never too late to overcome a math deficiency. Years of assuming that I could never truly "get" fractions evaporated in one sitting with this book. I took it with me on a coast to coast flight and was immediately intrigued with the simplicity and clarity of the text and visually pleasing graphic demonstrations. I had one "Eureka" moment after another as I grasped concepts that had always eluded me with other teaching techniques. I actually felt I was learning an entirely different way to think about fractions - a way that made sense to me. On top of all of this, book written in a way that makes it completely accessible and fun, even for those of us with serious math deficiencies. I am looking forward to his Singer’s book on Algebra so I can tackle that next!

Comments by Jim Pelech

I have used Understanding Fractions and its smaller versions Visualizing Fractions in a number of colleges classes at Benedictine University. Richard’s ideas on fractions and how to delivery instruction on fractions certainly aligns with Constructivist theory; combining hands-on and visual activities with a dialogue-type platform, this approach will be beneficial to all teachers and students. As we all know, every student learns differently, and comes into our classrooms with different experiences and different prior knowledge. These ideas by Richard provide the teacher with the flexibility for meeting different student needs and understanding.

Preface

To understand involves more than just gathering information. It also involves having a network of related concepts. Unlike information, which a person can receive directly from another, a person must acquire conceptual relationships by augmenting and transforming ideas they already understand. To expand understanding, we need experiences that challenge us to construct new ways of thinking. As a preliminary illustration of this, consider the question below and some answers that might be given. In a traditional classroom setting this question might be given to see if the students could multiply ½ times ½ to obtain the answer. While people learn to do this, it often does not fit with their concept of multiplication, and the result does not fit with their idea that a product should not be smaller than the numbers being multiplied. Instead of posing this question after they have studied fraction multiplication, this question can be posed early in their experience with fractions. The more a person can learn to think about fraction concepts before being asked to calculate, the easier it will be to understand and use fraction concepts.

Question:: You have been driving all day and have completed half of a trip you are taking. You want to take it easy tomorrow, so you plan to only go half as far as you went today. What part of this trip are you expecting to complete tomorrow?

Roy: ½ times ½ is ¼.

Kay: If you went ½ of the trip today, then the other ½ of the trip is still left. Since ½ of ½ is ¼, they are expecting to complete ¼ of the trip tomorrow.

Jan: I imagined the whole trip as 100 miles. Half of this is 50 miles. Half of this is 25 miles. This is one fourth of 100. It also works for trips of other sizes.

Bob: I used some white blocks to picture the trip. I replaced half of them by red blocks for half the trip. I then used half as many blues to picture the next day’s trip. This is one fourth the trip.

Jan: I love literature, with a special interest in historical novels. My favorite author is Octavia Randolph. Solutions I and my friends just gave are likely to occur because we think in terms that we find familiar. Each of us understands the concepts of a half and a fourth. I am comfortable with whole number concepts and can imagine a number of other questions that I could think about in this manner.

Roy: I love biology and I am especially interested in DNA. I am also a sports fan. I somehow know that I can use multiplication for this problem, but it is not clear to me how I know this. In fact I would not even think about knowing how I know if I had not encountered Descriptive Psychology. We all have different characteristics. However Declaration of Independence says we are all created equal.

Kay: I love math and logic. I am very comfortable with concepts and how they are related. I can easily relate fraction concepts to the concept of multiplication.

Bob: I love art and art history. I find visualization extremely helpful. The constructivist approach we are taking towards learning a conceptual net for fractions is builds on my ability to visualize. I concur with Roy, and would we all remain equal. Differences make for community, not for superiority.

Comment: These learners are somewhat atypical. They are idealized not in the sense that there are no such learners, but in the sense that our ideal is to have many more like them. We believe

that this ideal can be realized.

THIS PAGE IS IN CASE YOU WANT INSERT ANYTHING

CHAPTER 1 BASIC CONCEPTS

LESSON 1 INTRODUCTION

Three children remark on what they see in this picture.

Jan: There are three glasses with juice in them.

Roy: There are two and a half glasses of juice.

Kay: There are two full glasses of juice and one which appears about half full.

Whole numbers (like 1,2,3 etc) enable us to COUNT objects. Hence they are also called, ‘counting numbers’. When we count, we consider whether each object is there in its entirety.

Jan has used counting to focus on how many glasses are needed for this amount of juice, and for this purpose her observation is sufficient. On the other hand, Roy and Kay are focusing on the amount of juice. The glass appears half full to Roy. Kay realizes that perhaps it is not exactly half full, but for many purposes the difference between Roy’s and Kay’s observation would not matter.

For measuring quantities of something that can vary smoothly from one value to another, we need number concepts that include but go beyond the concept of counting.

The numbers between 0 and 1, such as ½, are called fractional numbers.

Numbers larger than 1, having both a whole and fractional part, are called mixed numbers.

(such as 2½ used by Roy and Kay)

Fractional numbers, whole numbers, and mixed numbers are extremely useful for measuring.

In order to focus on how these numbers are used we call them measuring numbers.

It is these measuring numbers that are the subject of this book.

Side Remark: Mathematicians refer to measuring numbers as ‘positive rational numbers’.

The simplest measuring numbers are the fractional numbers. These numbers include those that are larger than 0 and smaller than 1.

There are many ways of naming measuring numbers. Decimals provide one useful way of naming them. For example, ‘.5’ is the decimal name for the number we also name as one half.

The names most often used in this book are called fractions.

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Activity: The fractional number most of us first learned is ½. There are uses of ½ other than for half a glass of juice, such as the examples below. Think of some that you use.

There is only ½ of this pie left.

There are 2 pints in a quart, so a pint is half a quart. 1 pint = ½ quart

Exactly ½ of the tiles on this wall are red.

You can probably tell by just looking.

How could you tell by counting?

There are 12 inches in a foot, so half a foot is 6 inches.

6 inches = ½ foot

There are 60 minutes in an hour,

So half an hour is 30 minutes.

30 minutes = ½ hour

One useful way to picture fractions is in relation to pieces of a pie. However rectangular shapes marked by small cells make for easier comparisons. We will use such shapes in this book. To help with rapid recognition of fractions, we

associate exactly one color with each of the fractions we most often use.

2

A Fraction Is A Type Of Name

People have used many ways of naming numbers.

For example a long time ago people named numbers using marks likeI, II, III, IIII, IIIII, etc.

Of course this is tedious for large numbers. The Romans invented a system of names that we now call

Roman Numerals. They used special symbols, such as V for IIIII.

While we still use Roman Numerals for some purposes, the names we normally use for whole numbers are the modern numerals 1, 2, 3, etc.

We have many ways for naming measuring numbers. Decimals provide one useful way of naming these numbers.

For example, ‘.5’ is the decimal name for one half.

The names in this book are called fractions. As you already know, ‘½’ is a fraction name for one half.

We want to stress that the word fraction refers toa type of name rather than to a type of number,

and an understanding of fractions dependson understanding how measuring numbers are named.

Thought Question: Why would it matter if someone did not understand that a fraction was a name of a number rather than the number itself?

Because they are easy to picture, we have chosen colored pieces to help us imagine how the fraction names we use for numbers can be related to their numerical sizes.

A 4 inch by 6 inch white rectangle is used to imagine the numeral 1. For short we refer to a white piece as a white.

This rectangle is built from 24 small cells, primarily, because this allows for a variety of common fractional parts. The reason for this is explained more fully in the commentary.

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A Red Is Used To Imagine ½

A white takes up as much space as 2 reds.

So a red is half of a white.

We use a red to imagine the fraction ½.

Another way to picture this is by pulling 2 reds out of a white.

Another way to recognize that a red is used to imagine ½ is to focus on the 12 small cells inside the red and to think of the white as a 4 by 6 rectangle with 24 small cells.

In this lesson and the next we use dotted lines to help you focus on the number of cells in the pieces we are using. However this is only a temporary device. What is fundamental, is how many pieces of each color it takes to make a white. Focus on this and keep color pieces handy. There is a file named “Fraction Pieces” on our website that has a template for making them.

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Remember that a 4 by 6 white pieceis used to imagine the numeral 1.

It takes 4 blues to make a white.

What fraction do we use a blue to imagine?

Focus on the small cells provides one way to recognize the fraction that a blue is used to imagine.A blue contains 6 small cells. A white is a 4 by 6 rectangle with 24 small cells.

A blue can be any shape, as long as it contains 6 small cells.

This is the blue we use most of the time.

We will sometimes find this shape useful for a blue.

As you have probably already observed we use a blue to imagine the fraction ‘¼’.

The fraction ‘¼’ is called one fourth. We also call it a quarter.

Question: Why do you think a 25 cent coin is called a quarter?

Question: It takes 2 quarts of milk to make a half gallon. Why do you think we use the word quart for half of a half gallon?

5

A Fraction Is A Name Using Two Numerals Separated By A Line

The top numeral in a fraction is called its numerator. The numerator tells us how many parts to imagine.

The bottom numeral in a fraction is called its denominator. The denominator tells us how many of these part fit into a white.

To visualize many of our fractions we need pieces having more than one part.

Recall that a 4 by 6 white is use to imagine 1

This piece is called a 3blue (read as three blue).

It is made by dividing a white into 4 equal sized parts,and taking 3 of these 4 parts.

What fraction do you think we use this to imagine?

You can also think of a white this way.

This gives a convenient way to show a 3blue.

So this is another way to picture ¾.

Question: Recall which fraction was called a fourth or a quarter. What fraction do you think is called three fourths or three quarters? How do three quarters relate to a dollar? What decimal name do we use for three quarters?

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Different Fractions Can Name The Same Fractional Number

This piece has a heavy line to separate it

into 2 parts. This piece is called a 2blue.

Since a blue represents 1/4,

A 2blue represents 2/4.

A 2blue is the same size as a red.

Since a 2blue represents 2/4,

And a red represents 1/2,

2/4 and 1/2 name the same fractional number.

Since fractions are names, the fractions ‘1/2’ and ‘2/4’ are two different fractions.

These two different fractions name the same fractional number.

Fractions that name the same number are called equivalent fractions

When fractions are pictured by the same size pieces these fractions name the same number.

Important Activity: In the “Fraction Pieces” file you will find patterns for the colored pieces we use. Some of these pieces contain dotted lines, but most do not. The dotted lines show how a piece is composed of 1 inch squares, making it easy to see what fraction is being imagined. However they focus on counting squares, and once the shapes and colors become familiar this can be distracting. Make some of the pieces without the dotted lines before starting the next lesson. See if you can tell what fractions they represent. The choice of 1 inch squares is mathematically irrelevant. This size was chosen because this made the pieces both easy to produce and easy to manipulate. This book make extensive use of pictures of these pieces, which for typographical reasons are smaller than the template pieces. These pictures may be sufficient for understanding all the concepts we present. However we recommend using the template pieces, at least occasionally, as you read this book. A combination of seeing and handling should provide a lasting intuitive utilitarian understanding for fractions.

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LESSON 2 THE COLOR PIECES MODEL FOR FRACTIONSWe have used a red to represent the unit fraction 1/2. We have used a blue to represent the unitfraction 1/4. We also use 5 other colors for unit fractions.

We call this the color pieces model for fractions

Recall that a fraction is a name for a fractional number, and that this name uses two numerals separated by a line. The top numeral in a fraction name is called its numerator. The bottom numeral is called its denominator. A fraction whose numerator is the numeral 1 is called a unit fraction.

We can use unit fractions to measure any part that fits into a whole an exact number of times.

It takes exactly 3 feet to make a yard. 1 ft. = 1/3 yd.

It would take 4 quarts of this lemonade to fill this gallon jug.

So 1 quart is 1/4 of a gallon.

It takes exactly 7 daysto make a week.

So 1 day is 1/7 of a week.

One dime equals 10 pennies,

so a penny is 1/10 of a dime.

A day has 24 hours, so an hour is 1/24 of a day.

There are 26 letters in the alphabet, so one letter is 1/26 of all the letters. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Activity: Think of more examples which involve unit fractions.

8

Unit Fractions With Larger Denominators Name Smaller Numbers.

Activity: Determine which colors pieces represent which unit fractions.

Bob: We already know that 4 by 6 white piece is used to imagine 1.

Jan: It takes 3 yellows to make a white. So we all knowwhat fraction we imagine with this yellow?

Roy: We already used a blue for 1/4 , I think we should call this color aqua.

Bob: This is another way we can make an aqua.

Kay: It is half a yellow.

Roy: Either way, an aqua is used to picture 1/6.

Bob: We also have a pink, a green, a violet.

Can you tell what fractions they represent? Explain your answers.

Each of the pieces that we have given a color can be used to divide a white into an exact number of equally sized parts, and each is used to imagine a fraction. These pieces will represent the unit fractions we most often use in this book.

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This glass is one third full of grape juice.

Here is another glass of grape juice. It has twice as much juice in it.

We say it is two thirds full.

We use a yellow to imagine 1/3.

1/3 is a unit fraction

We picture 2/3 with a 2yellow.

We separate 2 yellows by a solid line.

This indicates that it is a single piece,

used to picture one fraction name.

Recall that a fraction is a name for a fractional number that uses two numerals separated by a line.

The ‘2’ in the fraction ‘2/3’ is the numerator. It tells us how many parts to imagine.

The ‘3’ in ‘2/3’ is the denominator. It tells how many of these parts fit into a white.

On the previous page we saw that ittakes 8 pinks to make a white.

Can you tell what fractionwe imagine with a 3pink?

Why is a 5green used to picture 5/12?

10

We Can Use An Inequality Symbol To Compare Two Numbers.

The symbol ‘<’ means ‘is smaller than’. We call ‘<’ an inequality symbol.

A comparison written using ‘<’ is called an inequality.

A 2green is smaller than a 3green.

So 2/12 is smaller than 3/12

2/12 < 3/12Roy: It is easy to compare fractions with the same denominator, just compare numerators.

3/8 < 5/8

5/24 < 7/24

1/4 < 3/4

3/6 < 5/6

Bob: It is also easy to compare unit fractions.

A blue is smaller than a red.

So 1/4 is smaller than 1/2.

1/4 < 1/2

Activity: An aqua is 1/6. A yellow is 1/3. Write the inequality that compares 1/6 and 1/3. Compare some other unit fractions. Can you tell how to compare unit fractions without looking at pictures?

Jan: 1/6 < 1/31/8 < 1/6

1/12 < 1/31/6 < 1/4

Roy: For unit fractions the one with the larger denominator is the smaller.

Kay: This is because the larger denominator tell us that it takes more of them to make a white.

11

Jan: There is an unlimited number of other ways to divide a rectangle into equally sized parts, but not if we only use squares in a 4 by 6 grid.

Bob: Below I imagine 1/5 using a rectangle separated into 5 parts. We can use gray for fractions that do not fit the color pieces model.

Gray Model Picture For 1/5

Kay: We can use this gray model to observe that 1/5 = 3/15. Just draw lines across to make 15 parts.

Gray Model Picture For 3/15

Question: Frank cleaned 2/5 of the tiles on this wall. Bill cleaned another 1/3 of them. Ronda did the rest. What fraction of the wall did she do? Who did the most? Who did the least?

Jan: Frank cleaned 6 tiles, and Bill cleaned 5 tiles. That left 4 for Ronda. So she cleaned 4/15 of them.

In the color pieces model for fractions, a 4 by 6 white is always used to imagine the number 1. In the gray model, the size of the piece used to imagine the number 1 will vary,

depending on our purpose.

Gray Model Picture For 3/7

What fraction of these tiles are clean?

What fraction of these tiles are a little dirty?

What fraction of these tiles are very dirty?

12

LESSON 3 VISUALIZING EQUIVALENT FRACTIONS

In lesson 1 we saw that a 2blue was the same size as a red.

Changing the fraction ‘2/4’ to the fraction‘1/2’ is called reducing the fraction. Changing the fraction ‘1/2’ to the fraction ‘2/4’ is called raising the fraction.

Some books use different names for these processes.

The focus in this lesson is on using pictures to imagine reducing and raising fractions. Fractions can also be raised or reduced using an arithmetic rule. Hopefully such pictures will make this rule seem reasonable. Since you may want to see if you can discover and explain the rule for yourself, we leave our discussion of it until the end of the next lesson. If you prefer seeing the rule before the pictures used to imagine it, you can look ahead.

From now on we omit the lines that show cells within a piece. We represent reducing 2/4 to 1/2 by a picture which trades a 2blue for a red.

2/4 1/2

We represent raising 1/2 to 2/4 by a picture which trades a red for a 2blue.

1/2 2/4

Raising a fraction is the opposite of reducing it.

Thought Question: Reducing a fraction gives a simpler name for a number. Since raising a fraction does not simplify the name of a number, you may wonder why we ever want to raise a fraction. Think about this as you continue this lesson.

Why would anyone want to raise a fraction?

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Fractions That Name The Same Number Are Said To Be Equivalent.

We have just seen that the fractions ‘1/2’ and ‘2/4’ name the same number.

So we say that the fractions ‘1/2’ and ‘2/4’ are equivalent.

We express this by either of these equations: 2/4 = 1/2 or 1/2 = 2/4

Reducing ‘2/4’ to ‘1/2’ gives an equivalent fraction with a smaller numerator and denominator.

Raising ‘1/2’ to ‘2/4’ gives an equivalent fraction with a larger numerator and denominator.

We usually write ‘2/4 = 1/2’ if we want to focus attention on reducing. We usually write ‘1/2 = 2/4’ if we want to focus attention on raising.

Question: Four months is what fraction of a year?

Roy: One month is 1/12 of a year so four months is 1/3 of a year.

Kay: This would be true if each month had 30 days and a year had 360 days.

Jan: Well lets agree that what Roy said is close enough for many purposes. I would rather say that four months is a third of a year than try to be exact.

Bob: I can picture this with color pieces. Since a month is 1/12 of a year, 4 months is 4/12 of a year. This reduces to 1/3.

We use a 4green to imagine 4/12.

We can trade a 4green for a yellow.

A yellow represents 1/3.

This shows that 4/12 reduces to 1/3.

4/12 = 1/3

We can also trade a yellow for a 4green.

This shows that 1/3 can be raised to 4/12.

1/3 = 4/12

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Question: How many greens make a white? What fraction do we imagine with a 9green? How many pinks make a white? What fraction do we imagine with a 6pink? What equations for reducing fractions is imagined in the pictures below?

Roy: The first is a picture for 9/12 = 3/4, the second for 6/8 = 3/4.

Bob: I agree it takes 12 greens to make a white so a 9green represents 9/12, and it takes 8 pinks to make a white so a 6pink represents 6/8.

Kay: Since both 6/8 and 9/12 reduce to 3/4 these three factions are equivalent.

Jan: This means that these fractions all name the same number: 3/4 = 6/8 = 9/12

Kay: Many fractions equivalent to 3/4.

Bob: I can only find one more for which we have a color piece. We can use the gray model to see that 3/4 is equivalent to 15/20.

Kay: One reason to raise fractions is just for the fun of finding lots of names for the same number.

Roy: That is not my idea of fun.

15

Fraction Without Equivalent Simpler Names Are Called Reduced Fractions

Thought Question: All unit fractions are reduced. The fraction ‘3/4’ is reduced. So is 5/6 and 7/8. They are not unit fractions. What do they have in common?

Can you find any other general criteria for reduced fractions?For instance can you tell which fractions with ‘2’ as numerators are reduced?

Kay: The decimal name .75 seems simpler than the fraction name 3/4, and they are equivalent.

Jan: You are just acting clever. You know that by a simpler name we mean another fraction name with a smaller denominator.

Roy: 3/4 and 5/6 and 7/8 all have denominators that are one more than their numerator.

Bob: 2/3 does not reduce, but 2/4 reduces to 1/2, 2/6 to 1/3,

2/8 to 1/4, 2/12, to 1/6,

2/24 to 1/12. All the fractions with numerator 2 for which we have color pieces reduce.

Roy: That is because their denominators are all even. 2/5 will not reduce and 2/7 will not reduce. A fraction with numerator 2 is reduced if its denominator is odd.

Jan: I know some more fractions that will not reduce. Those with 4 as their numerator and odd denominators.

Kay: I know a rule for telling whether or not a fraction is reduced. It works for all fractions.

Question: In a 2 mile race, Barbara ran 3/4 of the first mile and Martin ran 5/8 of the first mile. Who was ahead at this point? The race ended in a tie. How much of a lead did the runnerwho was behind overcome?

Roy: 3/4 of a mile is more that 5/8 of a mile so Barbara is ahead. Martin overcame a lead of 1/8 of a mile.

Bob: I can picture this. Just raise 3/4 to 6/8 to see that Barbara was ahead. Martin needs to over come a lead of 1/8 of a mile.

Barbara Martin

Barbara’s is ahead

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Raising Can Help Us Compare Fractions With Different Denominators

Roy could tell that 3/4 is larger than 5/8 without bothering to raise 3/4 to 6/8. It may not always be that easy.

We use a 2yellow to imagine 2/3. We use a 5pink to imagine 5/8. Can you tell from the picturewhich is larger 2/3 or 5/8?

One way to see

that 5/8 is smaller than 2/3

is to arrange

the 5pink in a different way.

5/8 < 2/3We could also trade color pieces to see which is larger.

Each pink part is the size of 5 violets, so a 5pink trades for a 15violet. Each yellow part is the size of 8 violets so a 2yellow trades for a 16violet.

A 15 violet is smaller than a 16 violet.

15/24 < 16/24 so 5/8 < 2/3

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LESSON 4 USING EQUIVALENT FRACTIONS

This lesson contains the main visualizations which can be used to picture fraction addition, subtraction, multiplication, division. These are given as a preview of the arithmetic of these operations, which are developed in later chapters.

Question: If you mix a half gallon of grape juice with a quart of apple juice, how much juice will you have?

Jan: A half gallon is 2 quarts, so you will have 3 quarts.

Kay: We could also express this using fractions. A quart is 1/4 of a gallon and a half gallon is1/2 of a gallon. Raising 1/2 to 2/4, we get the mixture as 3/4 of a gallon.

Bob: I can picture this with fraction pieces. This gives a red&blue piece. I don’t think we have a fraction name for mixed color pieces, but a red&blue has the same size as a 3blue.

combine

For many other sums involving ordinary situations, the results are not as easy to picture directly,

but we can picture them with color pieces.

Roy: I can see that mixing 1/4 of a gallon with 1/2 a gallon makes 3/4 of a gallon without a picture and without raising 1/2 to 2/4.

Kay: We all can but there are situations in which raising would really help?

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Question: Helen is running a mile along a country road from her house to Lin’s house. She starts at her house. It is half a mile to a bridge over the creek. It is a third of a mile from the bridge to Lin’s driveway where the road turns. How far is it from Helen’s house to Lin’s driveway?

Jan: There are 5280 feet in a mile. A half mile is 2640 feet and a third of a mile is 1760 feet. So it is 3400 feet from Helen’s house to Lin’s driveway.

Bob: We can do this using fractions. I picture combining 1/2 and 1/3 by combining a red and a yellow. First I trade them for a 3aqua and a 2aqua. I get a 5aqua for 5/6.

Kay: We raise 1/2 to 3/6 and 1/3 to 2/6 so we can see how to combine a half mile and a third of a mile I would use a less detailed picture.

trade andcombine

Roy: My older brother showed me how to do this byusing common denominators for adding fractions.

1/2 + 1/3 = 3/6 + 2/6 = 5/6

Bob: We haven’t studied that yet. For now I will stick with color pieces.

19

Jan: Using my way we can see that Lin’s driveway is 880 feet long. Just subtract.

5280 3400 = 880Bob: We could also use fractions. From Helen’s house to Lin’s driveway is 5/6 of a mile. We need to add 1/6 of a mile to make a mile.

Jan: And 880/5280 is 1/6. I never expected to reduce a fraction with such a large denominator. There are lots of equivalent fractions involving measurements. I have already been using fractions to think about pints and quarts and gallons. From running track, I know that 880 yards is a half mile. We could get this taking 1/3 of 5280 to see that there are 1760 yards in a mile. Reducing 880/1760 to 1/2 shows that 880 yards is 1/2 a mile. Likewise reducing 440/1760 shows a 440 is a quarter mile, and reducing 220/1760 shows that this is 1/8 of a mile.

Kay: We could get this by just taking halves. A mile is 1760 yards, a 1/2 miles is 880 yards,1/2 of 1/2 is 1/4 so 1/4 a mile is 440,

1/2 of 1/4 is 1/8 so 1/8 of a mile is 220 yards.

Bob: I can see how to picture 1/2 of 1/2 and 1/2 of 1/4. Taking 1/2 of something is just dividing by 2. So take a red for a 2blue and trade a blue for a 2pink.

take one part

take one part

Jan: We could also use 1 / 2 of 1 / 4 to compare a pint to a gallon. A quart is 1 / 4 gallon and a pint is 1 / 2 a quart. Bob’s idea would work for taking half of any fraction. His way of trading raises the fraction to one that we can picture as having 2 equal parts.

1/2 of 1/3 is 1/2 of 2/6.

so 1/2 of 1/3 is 1/6

1/2 of 5/12 is 1/2 of 10/24.

so 1/2 of 5/12 is 5/24

Roy: I can take 1/2 of a fraction without a picture. But Jan and Bob probably would not see this without first raising the fraction.

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Question: A foot is 12 inches so 4 inches is 4/12 of a foot, which reduces to 1/3 of a foot. Can you find what fraction 4 inches is of a yard?

Jan: A yard is 36 inches so 4 inches is 4/36 of a yard, which reduces to 1/9 of a yard.

Kay: We get the same answer by taking 1/3 of 1/3.

Bob: I can see that 1/3 of 1/3 is 1/9, but what does this have to do with the question?

Kay: A foot is 1/3 of a yard, and 4 inches is 1/3 of a foot.

1 yd

1 ft 1 ft 1 ft

4 in 4 in 4 in 4 in 4 in 4 in 4 in 4 in 4 in

Roy: I know 1/3 of 1/3 is 1/9 by multiplying, but I bet that Bob used a picture like this one. If we use his picture we would take 1/3 of 1/3 by taking 1/3 of 3/9. My way is easier.

Bob: To take 1/3 of 1/3 we separate 1/3 into 3 equal parts and take 1 of them.

Kay: Taking 1/3 of 3/4 gives 1/4. We can easily imagine this using color pieces. I think Roy would multiply and get 3/12. He would then reduce this to 1/4. It think its fun to have more than one way to think about something. It can also be useful in finding shortcuts.

Roy: I can speak for myself, and this is not the way I would do it. My brother showed me how to multiply fractions by canceling.

Bob: You always want to use arithmetic and calculate, but you never explain why what you do works. I need to understand why something works.

Jan: I have several half-pint cartons for ice cream. A quart is 1/4 of a gallon and a half-pint is 1/4 a quart. So a half-pint is 1/16 of a pint. We could use a picture to show that 1/4 of 1/4 is 1/16.

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Question: Sue has a half gallon of chocolate milk. She drinks a pint of it for lunch. How much does she have left?

Jan: That is easy. A half gallon is 4 pints. So she has 3 pints left. I can also describe this using fractions, and I can use color pieces. A pint is 1/8 of a gallon. So we start with 1/2 gallon and remove 1/8 gallon leaving 3/8 gallons. Of course this is the same as 3 pints.

start

drink remainder

Bob: I can see this better if I trade a red for a 4pink. I guess that I am raising 1/2 to 4/8 so it is easy to see how to take away 1/8.

start

Question: In order to get an advance on her royalties, Mary agreed to complete her book in six months. She wasted one month, leaving 5 months to complete her book. How could you think about this using fractions?

Jan: She had 1/2 a year and wasted 1/12 of a year. Raising 1/2 to 6/12 leaves her with 5/12 of a year. Here is the expanded picture using color pieces.

start

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Question: James has 1/2 pound of dog food. He shares this among his dogs, giving each of them 1/8 of a pound. How many dogs does he have?

Jan: He has 8 ounces and gives each dog 2 ounces. He must have 4 dogs.

Roy: Just use fractions we can separate 1/2 into 4/8 just like we did before.

Bob: Yes we can raise 1/2 to 4/8 and divide this into 4 parts

Jan: I can make up a similar question. Suppose his brother has larger dogs that eat twice as much. He shared 1 1/2

pounds of dog food among his dogs, giving each of them 1/4 of a pound. How many dogs does he have?

Kay: There are lots of case where we might want to raise a fraction in order to separate it into parts of the same size. For example suppose we want to separate 3/10 into parts of size 1/20. Raise 3/10 to 6/20, so we get 6 parts.

Jan: Thinking about dollars this is the same as trading 3 dimes to get 6 nickels.

Bob: We can picture raising 3/10 to 6/20. It is easy to see that this makes 6 parts of size 1/20.

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LESSON 5 USING ARITHMETIC TO CHANGE FRACTION NAMES

Major Thought Question How can we use arithmetic to raise or lower fractions?

Why does this process do the job?

The focus of this section is on the questions above. At the end of the section we provide our answers. As a prelude we suggest a number of activities that we hope will help you answer the questions before reading our answers. We begin with activities that involve raising fractions, since we feel that it might be easier to discover the arithmetic for this.

Activity: The first picture below can be used to imagine 2/3 = 4/6. Tell what equations the other pictures would be used to imagine. Observe anything these equations have in common. Relate this to the thought question above.

Bob: These pictures show that 2/3, 4/6,

8/12, and 16/24 all name the same fractional number.

Kay: In raising a fraction we are only changing the name of a number. Of course this does not change the number.

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One Way To Raise A Fraction Is To Double Both The Numerator And The Denominator.

Recall reducing 6/8 to 3/4. Reversing the picture gives a way to imagine raising 3/4 to 6/8.

Just separate each blue part into 2pink parts.

Observe that in raising 3/4 to 6/8 the numerator doubles and the denominator doubles.

WHY IS THIS?

Jan: Drawing a line changes each blue part into 2pink parts, giving twice as many pink parts as there were blue ones. So the numerator becomes twice as large. Each blue is twice as large as a pink. So it takes twice as many pinks as blues to make a white. This makes the denominator also become twice as large.

Kay: Recall the pictures used to imagine raising 2/3 to 4/6, 4/6 to 8/12, and 8/12 to 16/24. What Jan said also applies to these.

In each case we can double the numerator and denominator.

Bob: We can also use the gray model. Below I imagine raising

3/5 to 6/10.

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Question: We have just seen that one way to raise a fraction name to another fraction name for the same number is to double both the numerator and the denominator. In the first picture below we use lines to separate each blue part into 3 green parts. This raises 3/4 to 9/12. The next picture is for raising 5/8 to what fraction?

Roy: This pictures raising 5/8 to 15/24.

Bob: Earlier we raised 2/3 to 4/6 and 4/6 to 8/12. Thus we could raise 2/3 to 8/12. Just separate each yellow into 4 parts.

Question: What do these suggest as an arithmetic way to raise fraction names? Give some examples.

Roy: Multiply both its numerator and its denominator by the same number.

Kay: Here are some for which we do not have color pieces.

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We can use the gray model to picture raising 2/3 to 10/15

Observe that in raising 2/3 to 10/15 the numerator is 5 times as large and the denominator is 5 times as large.

Likewise 3/4 can be raised to 27/36.

3/4 = (93)/(94) = 27/36

General Rule For Raising A FractionMultiply Both Its Numerator And Its Denominator By The Same Number.

Kay: Remember when we talked about reasons for raising fractions? I can think of another reason to raise fractions. Consider a fraction like this one:

Jan: That is weird. Are you sure you can have a fraction like that?

Kay: Sure! I know a math major in college and you should see the fractions shewrites. If we raised this by doubling both the numerator and the denominator, the raised fraction would be 5/10.

Roy: Hey! That reduces to 1/2. That makes it much easier.

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Recall raising 2/3 to 4/6. We can reverse this to reduce 4/6 to 2/3.

A 4aqua trades for a 2yellow so the reduced name of 4/6 is 2/3.

We can also use arithmetic to reduce fractions. Just reverse the process used to raise fractions.

Observe that in reducing 4/6 to 2/3 we can think of 4 as 22 and 6 as 23. Now remove the factor 2 from both the numerator and denominator.

WHY IS THIS?Removing a factor of 2 from the numerator makes it half as large so we have half as many pieces. Removing a factor of 2 from the denominator also makes it half as large. With a smaller denominator we have larger pieces. So we have half as many pieces which are twice as large. This keeps the total size the same.

If the numerator and denominator of a fraction are both even numbers we can reduce the fraction by removing a factor of 2 from both the numerator and the denominator. You may want to think of this as dividing both the numerator and the denominator by 2.

General Rule for Reducing a Fraction:To reduce a fraction, do the reverse of what we do to raise a fraction.

Find a number that is a factor of both its numerator and its denominator. Then unmultiply, that is remove this factor.

You may want to think of this as dividing the numerator and denominator by the same number.

For example, to reduce 6/9 to 2/3, divide the numerator and the denominator by 3.

We do not have a color piece for fractions whose denominator is 9.

If you want a picture for reducing 6/9 to 2/3 use the gray model.

Activity: Use both pictures and arithmetic to reduce the following fractions: 10/12, 6/8,

4/10, 3/15

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APPENDIX 1 GENERAL COMMENTARY

This commentary serves two main purposes.

It presents additional perspectives on fractions in order to help adults broaden their own understanding. This is presented from an adult viewpoint and does not directly relate to how children may think. However with a better understanding of fractions, adults will be able to come up with ideas for helping children understand fraction concepts.

Some of the commentary also contains illustrations of fraction applications taken from the experience of adults working with children. While these are intended to give ideas for ways to apply fraction concepts with a child, applications for a particular child depend on a multitude of that child’s characteristics. If you work with children, you will have to tailor what you do to fit their individual characteristics and needs.

Understanding: This book focuses a utilitarian way of understanding fractions. i.e. understanding how operations on measuring numbers must be as conceptualized if they are to be applicable to a variety of situations. However we have chosen one specific application as our central model. This model is described in the first lesson. It uses a rectangle to picture the number one and portions of this rectangle to picture various fractions. The rectangle most frequently used has 24 cells, each of which is a one-inch square. This size is mathematically irrelevant. It was chosen to make the pieces fairly easy to handle. Even larger pieces might be convenient for young children.

A more abstract way to understand fractions is in terms of basic algebraic laws such as the associative law, the commutative law, the distributive law, etc. For example the rule for adding fractions with like denominators can be seen as a specific case of the distributive law. While such an algebraic understanding of fractions provides further perspective, a utilitarian understanding based on a model is sufficient for many purposes. In fact, until the emergence of contemporary mathematics, it was the only significant type of understanding available. Thus we have not included any materials in this book that are designed to introduce an algebraic understanding of fractions. For anyone interested in an algebraic understanding of fractions we are working on an ordinary algebra unit entitled “Basic Algebra for Rational Numbers”. It will soon be available on our website.

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Cast of Characters:

Our characters are imaginary and are not intended as typical. One reason we did not choose to use typical persons is because they are hard to characterize. Each person is different; what is typical for one may not by typical for another. However our characters at least represent some of the major attitudes or styles that influence the learning of mathematics. Another reason these characters are not typical is that we are using them to communicate ideas to adults. In this book, we talk to adults through them. Our examples are often better related to adults than to children. When constructing examples for a child, use things that this child can relate to. For example, most children would relate better to a situation dealing with candy than one dealing with acres of land.

Jan: good with whole numbers, likes counting, not sure about fractions Bob: needs things manifest, likes hands on and pictures, is unsure of abstractionsRoy: fair at concepts, likes and is very good with algorithmsKay: good at concepts, makes detailed conceptual distinctions

Like Jan, many people who are comfortable with whole numbers are not comfortable with fractions. However Jan knows how to convert questions about fractions to a question about whole numbers. She knows that a pound is 16 ounces, a foot is 12 inches, a gallon is 8 pints, a yard is 36 inches. Given problems involving these units of measure she can use whole numbers instead of fractions.

In the book we talk as if Bob thinks in pictures. Actually he manipulates color pieces, and we abbreviate what he has done by a picture. To get a better feeling for the activity we recommend using these pieces. This will provide another alternative for understanding fractions that is more sensory than the numerical perspective Jan uses.

Roy may seem better at arithmetic than Jan or Bob, but he has a limitation that they do not have. They have ways of thinking that allow them to answer questions that involve fractional concepts prior to leaning the arithmetic. Roy relies on arithmetic rules with little concern on understanding why they work. Not knowing a rule he tends to feel lost. However when he can turn his attention away from rules and rely on his own ability to think, he is just as capable as Jan or Bob.

While Kay grasp ideas easily and often goes beyond the thinking of most adults. She will probably develop an appreciation and understanding of mathematical ideas regardless of the way she is taught. However with the help of an adult who understands concepts Kay will develop an even deeper appreciation. Some with Kay’s potential are taught a way that stifles their imagination. An adult who does not have the understanding needed to help Kay should at least stay out of her way.

In general, an adult who wants to help any child learn mathematics needs to be able to think about mathematics in various ways. The more understand about how fraction problems can be converted to whole number problems, the more I could help Jan. I also need to be able to formulate questions that can lead her from thinking in terms of whole numbers to thinking in terms of fractions. Similar remarks apply to helping Bob. Helping Roy is more difficult. While my versatile understanding of why the algorithms are useful, a large part of Roy’s difficulty is based on his attitude that learning mathematics means learning algorithms. I would have the same problem with Jan and Bob, if they were more like students who thought the way they did outside of mathematics class but carried an attitude like Roy’s into learning situations.

You will learn more about our cast of characters as you read this book. In the process you will find that as the characters learn more about fractions some of their characteristics may change. We often discuss new ideas through their perspectives.

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COMMENTARY ON EACH OF THE LESSONS

Overview( by Richard) The commentary is divided into separate sections for each lesson in the text. Each section begins with general remarks. These are sometimes followed by some additional remarks, which can be considered as optional. For a perspective on the use of these materials and concepts with children I have included illustrations of adults working with children. For Chapter 1 we focus on one child Angela who is being taught by her sister Jennifer. These are written by Jennifer in first person.

Jennifer and Angela are my granddaughters. To help in developing this book, Angela agreed to participate as a subject for learning fraction concepts. Jennifer worked with Angela on the ideas in chapter 1 for two hours a week for about 12 weeks. Angela is ten years old and prior to this experience had at best a minimal understanding of fraction concepts. Mathematical thinking has never been an easy matter for her. Jennifer is a twenty year old senior at Webster University majoring in business with a minor in dance. Her only teaching experience has been in the area of dance and her only college math course an independent study with me on developing materials for teaching fraction concepts. Other than this she has had no special preparation for teaching mathematical ideas to children.

Lesson 1 (by Richard)

The focus of this first lesson is on presenting an overview of the most basic concepts for thinking about fractions. At this point there is no need to compute with fractions. Instead the main purpose is to imagine situations and describe it in terms of fractions. This lesson also introduces the idea of using color pieces to picture fraction concepts. Some of the main ideas are sketched below.

Whole numbers can be used for counting but we need additional numbers for measuring. A fraction is a type of name, a type that is used to name measuring numbers. Different fractions may name the same measuring numbers.

I decided to use the term ‘measuring numbers’ rather than the standard mathematical term ‘positive rational numbers’ primarily because the terminology suggests their relationship to a larger mathematical structure which also contains negative numbers. I wanted terminology that did not suggest this relationship. However calling them measuring numbers also suggests the kinds of situations from which such numerical concepts have been abstracted.

Pie pieces are often used to represent fractions. They easily show fractions such as 1/2, 1/3, 2/3. However it is hard to distinguish others such as 3/7 and 5/8. It becomes even harder to compare fractions as the denominators become larger. We chose to use rectangular pieces in this book because the rectangular shape has many possibilities. The rectangular shape allows for easy comparison. When we break the rectangle into parts, they are symmetrical. We use a 4 by 6 rectangle. Having 24 parts allows for the creation of a variety of commonly used fractions. We also color-code these pieces, which makes for even easier comparison. There are some limitations. The 4 by 6 rectangle does not allow for certain fractions such as 1/5 and 1/7. For this purpose we continue to use a rectangle with a heavily lined border for 1. We shade some of the parts gray to represent the fraction.

For example, to illustrate 3/5 we separate a rectangleinto 5 parts and shade 3 of them gray.

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Additional Remarks (by Richard)

This remark refers to the thought question about names. Since this is a thought question I leave it to you to find as many answers to it as you can. I will merely make some initial thoughts of my own.

Ordinary language distinguishes between numbers and their names by using the word ‘numeral’ to refer to the name of a number. At one time many elementary math books stressed this distinction but this never seemed very important for the purposes that most teachers had when teaching arithmetic. I suspect that the distinction between a name and what is being named is just too ordinary to matter to most of us most of the time. As long as we implicitly understand a distinction we need not make it explicit to avoid confusion. Consider the following example.

Example: The name of my son’s dog is Bear. I hear this name when my son tells his dog to sit. I see this name when it is written. Which of these sentences below refers to the dog and which one refers to the dog’s name? How can you tell?

Bear begins with the letter B. Bear weighs about 100 pounds.

This name does not look or sound like a dog. Would anyone confuse a dog with the dog’s name?

However confusion of a number and its name is fairly common. Perhaps it is because numbers are imagined objects and are not the kind of objects that can be seen. You can see five apples or five pennies. But how can you see five? You can see the word ‘five’ or the numeral ‘5,’ but numerals and words are names. They are not numbers. Numbers are imagined objects. But is it important to realize this? Does the confusion of a number with one of its names matter? The simple answer to this is that any type of confusion may cause subtle problems that limit our ability to effectively use our concepts. However the thought question asks for more pointed answers. I give an example so elementary that you may think no confusion could occur.

Which is larger? 4 or 5

While this example may seem artificial it is not totally different from the example of a child who says that 3/7 is bigger than 1/2 because 3 is bigger than 1 and 7 is bigger than 2.

We want to make it clear that the word fraction refers to a type of name, and not to a type of number. One way to stress that we are referring to a name rather than using it to talk about a number is to use quotes. The sentence on the left is about two fractional numbers. The one on the right is about particular names of these numbers.

1/4 is smaller than 1/2. ‘1/2’ and ‘1/4’ have the same numerator.

Confusion might result if someone said 2/3 and 4/6 are different because their numerators and their denominators are different. Do they really mean that ‘2/3’ and ‘4/6’ are different?

Since the use of quotes is tedious we usually rely on context to tell whether we are talking about a name or a number.

32

Jennifer working with Angela (by Jennifer)

When working with Angela on fractions, I tried to incorporate fractions into everyday life. To her fractions were an abstract math concept used only during math class in some textbook. I wanted her to see that fractions are often used and can be helpful. Here are some of the activities we did:

We walked around the house finding fractions in everyday items. I would ask things such as “What fraction of people in that picture are smiling?” or “What fraction of the towels in the bathroom are pink?” or “What fraction of the flowers in this pot are blooming?”

Angela plays piano, so she is somewhat familiar with musical notes. However, she had no concept of how fractions relate to playing music. We worked with these concepts and created measures of music. We figured out how to write a measure given a certain timing, such as 4/4 time. She then had to figure out what notes would fit into the measure given whole notes, half notes, quarter notes, and eighth notes. Instead of just having her count out the notes to make sure they were the right value, we made it more interesting and sang the counts.

Both Angela and I dance. Since dancing is interesting to her, I knew that this was a topic she would relate well to. Dance combinations are often made up of counts of eight. I used this concept to work on fractions. I taught her a simple combination and then I asked her to do 1/4 of the combination or to do the last 1/8 of the combination. I would ask what fraction of the combination a certain step was. I also had her make up a combination and ask me about it.

I know that these activities have helped Angela think of fractions in a new way. The other day, three of us were eating French fries and she told us “Two thirds of us have finished our fries” She said it just as a passing comment, like it was a completely normal thing to say.

Angela and I have also been working on some questions that involve fraction concepts.

Question:: There are twelve cars. A third are red. How many cars are red?

Angela writes: “1/3, 1/3, etc.” (she writes “1/3” twelve times) and says, “The answer is 12/3.”

Looking at this she says, “No it is wrong, but I don’t know how to do it.” Then after thinking some time she writes: “3+3+3+3 = 12” and says that the answer is 4.

Jennifer: “How did you get that answer?”

Angela: After thinking some time, “No it’s not the right answer. The answer is 2.”

Jennifer: “How did you get 2?” I’m not sure why she so quickly disregarded her first answer as completely wrong. I think she is just afraid of being wrong and is very concerned with getting the right answer. When I ask a question like “How did you get that answer?” she immediately assumes that she is wrong. She is not used to having to explain her thinking process of how she gets to an answer. In the past the only task she has concerned herself with is getting to the right answer and then moving on to the next problem.

Angela erases all her work and starts over. She writes 3+3+3+3 = 12. She decides that the answer really is 4.

Jennifer: “Show me how you figured out that the correct answer is 4.”

Angela makes twelve dots, four groups, with each group containing 3 dots

= 12

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Question: Samantha baked 3 dozen sugar cookies. She burnt 1/4 of them. How many cookies were not burnt?

Angela first added 12+12+12 to get 36 cookies total.

Then she drew 36 marks in groups of four, finding that it made nine groups.

I’m not sure why she chose to make groups of four. I think maybe because the problem contained “1/4”, and her initial reaction to this number was to create groups containing four items. This is a typical reaction from her. She sees a number and decides to use it, without really thinking about why she should use that number.

After looking at the picture Angela said: “My grouping won’t work.”

She reasoned that it would work if the question used the fraction 1/3 instead of 1/4. Then she could have three equal groups, each containing twelve marks. I was impressed with this observation because she usually does not make connections like this. She tends to follow a straight path of just finding the correct answer. She doesn’t like to talk about things that interfere with this or take her on another path.

She erased all her marks and started over. She figured that each group had to have more than eight marks, but less than twelve. She guessed nine and it worked.

Angela: “1/4 of the cookies were burnt, so that was nine burnt cookies.”

I reminded her that the question asked how many were not burnt and she went on to count all the “non-burnt” cookie marks. I stopped her and asked if there was an easier way, a faster way than counting all those marks.

Angela: “369 = 27. So the answer is 27.”

Jennifer: “Is there another way to figure that out?”

With some prompting she concluded that she could also do 9+9+9 which was the same as 93.

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Lesson 2 (by Richard)

The focus of this lesson is to present all of the colored pieces as a way of picturing fractions. The numeral 1 is pictured by a 4 by 6 white rectangle with 24 cells. Each of the unit fractions 1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 1/24 is pictured by a different colored piece. Each piece contains the number of cells that is the appropriate fraction of this white piece. A fraction that is not a unit fraction is pictured by a piece consisting of a number of parts separated by heavy lines. The number of parts indicates the numerator of the fraction and their color indicates its denominator. We also introduce the gray model for fractions. This allows us to represent both fractions for which we do and do not have color pieces.

Success in using color pieces to understand fraction concepts depends on the ability to automatically associate each fraction with its colored piece. One type of activity that might be helpful is to use a variety of strangely shaped pieces, and see how they can be measured using the fraction pieces. All these shapes should use the same one-inch squares as were used for the fractions pieces. I made some of them as multiple violets to make them consistent with the fraction pieces we had made. Others had no color are markings. If possible this activity should be done before the concept or arithmetic of equivalent fractions.

Jennifer only used these two shapes with Angela. I recommend using many more and returning to this type of activity until the fraction pieces are extremely familiar.

While the writing style in this book is for adults, the ideas are suitable for most children by the time they are twelve. The non-arithmetic ideas of Chapter 1 can be mastered by even younger children. Most of them could also learn to do some of the arithmetic given applications to which they can relate and which only use fractions with small denominators. There is nothing magic about the size and colors we chose to use for our fractions. You can make up your own fractions and colors as long as they stay consistent. We used a 4 by 6 rectangle, but other rectangles would work well too. A 6 by10 rectangle would be good if you want to work with fractions such as 1/5, and 1/10. A 15 by 16 rectangle would be more elaborate and supply many possibilities.

In working with children the use of color pieces should be supplemented with other activities. Below is an example of a type of word puzzle that some children may find interesting. You can alter these examples for different aged children. Younger children would have difficulties with this, but you could choose topics other than names of states such as names of their friends, farm animals, colors, fruits, etc.

I am thinking of the name of a state. Half of the letters in this name are vowels. Half of the vowels are i’s. Half of the consonants are s’s.

I am thinking of the name of a state. Half of the letters in this name are vowels. 3/4 of the vowels are i’s. Half of the consonants are l’s.

I am thinking of the name of a state. Half of the letters in this name are vowels. 2/5 of the vowels are i’s and 2/5 of them are a’s.

I am thinking of the name of a state. Half of the letters is this name are vowels. 2/3 of the vowels are a’s.

35

Jennifer working with Angela (by Jennifer)

The first activity Angela and I worked on was to make her own color pieces by using 4 by 6 index cards and crayons. The experience of making her own color pieces was important. If I had simply handed her color pieces that were already made, she would have missed the learning experience that came from making them. As she colored and cut the fraction pieces, she was learning what fraction each represented. I frequently asked her questions about the fraction pieces while we were making them. I asked her to make comparisons between different fractions. For example I asked her what other piece was the same size as a 2blue. She was informally using the concept of equivalent fractions without needing the arithmetic to do the problem. I used this time as a visual prelude to the idea of equivalent fractions. Here is the first shape I gave her.

Some of the fractions Angela came up with were:

1/4 2/8 6/24 3/12.

To find these answers, she laid color pieces atop this piece.

She also came up with combinations: 1/6 and 1/12 1/8 and 1/12 and 1/24 1/8 and 3/24.

She is adding fractions but does not realize it. After she had explored this shape for a while, I asked her about her last answer.

Jennifer: “Tell me about what 1/8 and 3/24 are.

Angela: “This funky looking piece.”

Jennifer: “Can you think of a number for that piece?’

Angela: “Can I use the fraction pieces?”

Jennifer: “Yes, please do.”

Angela: “It equals 1/4. That’s the most reduced answer, but they all are the same.”

It was clear to me that Angela had seen the connection between equivalent fractions. She had made the connection that while all these numbers were the same, 1/4 was the most reduced answer. She could have answered my question by using any of the fractions she had named.

Another shape I gave her was made from 8 one-inch squares. This shape did not have any of the inside lines drawn because I did not want it associated with any colors.

I am also using this activity as a prelude for working on equivalent fractions. However even though Angela had done some work on reducing fractions in school I think this is the first time she understood the concepts involved.

She put 4 green pieces on top of this shape anddetermined that this shape was 4/12 of a whole.Then she came up with other equivalent fractionswithout putting them on top of this shape:

8/24 1/3 2/6.

I was impressed that she had confidence and didn’t feel she had to place each equivalent fraction piece on top of this

one.

36

Lesson 3 (by Richard)

The main purpose of this lesson is to obtain a good visual picture for the concept of equivalent fractions. Raising and reducing a fraction is pictured by trading a piece for another piece of the same size but of a different color. The use of the gray model for this purpose gives another alternative.

A common stumbling block in the study of equivalent fractions is that children often see no purpose in raising fractions. In traditional classrooms, students are told that they will use this concept later in their school career. This response is taken by conscientious students as something that they have to do. So they learn it, but many of them just end up forgetting it later. Less conscientious students take it as something that they don’t really have to do and they don’t really care about it. When introducing a new concept, students need to be involved in the learning process. When introducing new concepts, we need to find situations that students can relate to, giving real-life examples. However at the beginning of this lesson I merely pose a thought question, giving no immediate indication of any answer. Instead the lesson proceeds with these concepts but with the primary emphasis on reducing. Only at the end do I give an example of why one might want to raise a fraction. The entire next lesson is devoted to situations in which raising a fraction is important. This strategy of briefly comparing raising to reducing, but to only focus on reducing, is intended as a way to introduce an important concept in a setting in which its acquisition does not matter.

Additional Remarks

Both reducing and raising fractions are often called renaming. We will use the more specific terms of raising and reducing in order to stress which we are doing. Simplifying is used in some books instead of reducing, but it is also used for a variety of other processes. In order to be more descriptive, we will not use the term simplify when we want to reduce.

Many books rejected the term reducing because they were concerned that students might think that reducing made the number smaller. To avoid this we stress that the fraction name is what is getting smaller when reducing. The actual number does not become smaller. This is an important idea to understand when reducing or raising fractions. As an ordinary analogy consider a boy named Johnny. We could reduce his name to John. His name became smaller, but the boy did not get smaller.

We will omit the dotted lines within the pieces from now on. It is important to focus on the actual pieces instead of counting the individual cells within each piece. It is also important to become familiar with the colors and the fraction each one represents.

Kay’s focus on mathematical exactness needs to be balanced with the idea that we often use exact mathematics for situations that are not that exact. To use mathematics to model a situation we only need an approximate match that is good enough for the purposes at hand. For the others this will never be a concern. They do not feel a need for an exact match.

The most powerful motivation for studying mathematics is a curiosity about mathematical concepts and their relationships. Most people can acquire some degree of pure mathematical curiosity if they experience success in doing mathematical thinking that they find challenging.This is one of the main reasons for using applications that children already have some means to solve, but which they can solve more efficiently with extended mathematical understanding. For a child like Kay there is no need to use an application as a prelude to conceptual considerations, and such children should often be challenged by purely mathematical questions. However they can and should also be motivated by applications.

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Jennifer working with Angela (by Jennifer)

In order to reinforce the habit of thinking about fractions in ordinary situations and also work on the idea of equivalent fractions I did some more observation activities. I also had her ask me questions about what we were observing. Sometimes I would purposely give the wrong answer and she would have to guide me to the correct fraction. I would give her the most reduced fraction and she would have to figure out what fraction that came from.

One question came from a picture that had 2 girls and 2 boys. She asked me “What fraction of the people in this picture are girls?” I responded with “1/2 of the people in this picture are girls.” She told me I was wrong and insisted that the correct answer was 2/4. She wouldn’t believe me until she figured out for herself that this could be true by raising 1/2 to 2/4. She then realized that I had given the reduced fraction as my answer. At this point I wanted her to visualize the relationship between 1/2 and 2/4, rather than to do arithmetic whose rationale she did not understand. In spite of having earlier seen that a 2blue was the same size as a red, she did not visualize this until we looked at it again. Later Angela confronted a similar situation. Angela was riding in the car with her uncle, granddaddy, her grandmother, and her 2 older brothers. Granddaddy asked her what fraction of the people in the car are female. She easily responded 2/6. He then asked her to give a reduced answer. When she had difficulty he asked her to think about color pieces. It took a while, but she came up with the correct answer. Granddaddy said that he thought most of her problem was focus.

When working with my sister on equivalent fractions, I found that she had trouble thinking abstractly about them. She could find equivalent fractions by matching the color pieces, but when it came time to talk about the fraction names or do it with only imagining the pieces, she would become confused. She had trouble visualizing concepts in her head. She had to see it or touch it, in order for her to understand the concept. For example, I asked Angela what was equivalent to 4/4. She knew that any fraction that had the same numerator and same denominator equaled one, but when I asked her if any of the other fraction pieces were also equivalent, she had to take each fraction set and place them atop the whole piece to see if they matched. I assumed that after she did this a few times, she would tell me that all of them (2/2, 6/6, 8/8, etc.) would be equivalent to one whole. I found that especially in this instance, using the fraction pieces was a great tool for her. Following up later, I went back to these concepts to see what she remembered and found that she had no trouble with concepts after she was able to visually understand them.

Some additional comment on working with Angela (by Richard)

Jennifer and I talk regularly about her work with Angela. However I have left a large part of the responsibility to Jennifer. I wanted to see how well these ideas would work when an adult with no special preparation for teaching fractions used them with a child. One thing I observed about Jennifer’s work with Angela came as no surprise. What Angela needs to do is to relax and use her own understanding rather that think about what is expected of her. This is the case not only in learning about equivalent fractions, but in all of her learning about mathematics. For Angela, the habits she has already acquired in school are her greatest barriers to learning. She often takes more time to learn than some of her teachers think is appropriate. She wants to be right and she worries about being scolded for not being able to do something. As a result she focuses on doing what is expected rather than on thinking. This fits with characteristics she exhibits in other situations. I have taught her a multitude of games. She seldom learns strategy in the way I would expect. So I must take care in suggesting strategy ideas too early. Often she seems to suddenly go from almost no strategy to a high level in a single jump that surprises me. With a child like Angela, provide her enough relevant experience and no external pressure and her own expectations will produce quality results.

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Lesson 4 (by Richard)

The primary purpose of this lesson is to provide a number of situations for which raising fractions might be useful. The questions about these situations can all be answered by manipulating color pieces. This is an important prelude providing a visual basis for the arithmetic of raising and reducing fractions, which is the topic of Lesson 5.

A closely related secondary purpose is to stimulate thinking about how to apply fraction concepts to a variety of situations and to answer questions about these situations without using arithmetic. This should provide a basis for understanding all the other arithmetic calculations that will be the topics of future the larger book. The more experience that a person has in using fraction concepts prior to calculating with fractions the more likely it will be that they will obtain a good understanding of why fraction arithmetic works.

As you will see Angela still needs considerably more non-arithmetic problem solving. At times she is able to handle questions that she could solve using fraction pieces. Often her success depends primarily on focus. She needs to focus on imagining the situation as if it is really happening. It is also important in teaching her to expect her to think about the situation rather than about arithmetic operations. Jennifer tends to think as if she were teaching primarily as a prelude to understanding the arithmetic. While this is of some importance, what Angela primarily needs is experience in problem solving and in visualizing equivalent fractions.

Sometimes children learn things from their parents or siblings without understanding the reasoning behind it. Bob is uncomfortable if he does not understand the reasoning so he pictures it with the pieces. On the other hand, Roy likes using algorithms and may not always understand what he is doing. We need different strategies for dealing with different children. For children like Bob, encourage this thinking and when they understand the pictures then encourage them to use algorithms, but do not push them to do this. For children like Roy put them in situations where they feel a need to explain what they are doing. Putting Roy in a group with Bob will require him to explain his thinking. If you are working with a child like Kay, you do not have to worry about this. Just let her explore anything she wants to. Encourage her unusual thinking. For children like Jan let them think in whole numbers. Find as many fractional situations as possible that they can use whole numbers to solve. Slowly, get them to think in fractional terms as they become comfortable with these fractional situations.

Jan has made an observation that was not asked for in one of the questions. This is to be encouraged. For all children, encourage them to go beyond what a problem asks for and explore many possibilities.

Jennifer working with Angela (by Jennifer)

Angela has not been too eager to work with the fraction pieces. I wondered if it was because she had never worked with manipulatives like these before and was not confident using them. After talking with her about this, I found that this was only a small part of why she did not want to work with them. She told me something which I found interesting. She said that she did not like using the fraction pieces because she did not like setting them up. She said it takes too long to separate the fraction pieces. (We have been storing them in one big envelope). It only takes about thirty seconds to set them up, but if this was discouraging her from using them, I thought that we should find a better way to store them. So we separated the different fraction pieces into different envelopes, thus making for easier set-up. She seemed much happier with this system and has never complained about using the fraction pieces since. Sometimes when working with children we think that there is a large, complicated problem, but it turns out to only be a minor inconvenience. Make sure you talk with the child to see what is really bothering them before you try to solve a big problem that may not even exist.

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Question: Sam’s mom baked a cake. Sam ate 1/6 of the cake. He fed his dog 1/12 of the cake. Sam’s dad ate 1/2 of the cake. How much cake is left for Sam’s mom to eat?

I did not draw a cake for her, hoping she could focus with only her imagination. Perhaps a picture of a cake cut into 12 pieces might have helped. I could then have had her mark how much each one had.

Sam dad dad mom

Sam dad dad mom

dog dad dad mom

Angela: Angela pulls out the pieces for 1/6, 1/12, and 1/2. “The answer is 1/12.”

Jennifer: “Where did you get 1/12?” I have no idea how this occurs to her.

Angela: “Oh, I’m silly! Did she bake one whole cake?”

Jennifer: “Yes.”

Angela lays out a white. Over that she puts the “eaten part” 1/6, 1/12, and 1/2. Then she lays 3 greens over the “uneaten part” She says: “3/12 is what is left.”

Jennifer: “That answer uses 3 fraction pieces. Is there a way so that it has less than 3 pieces?”

Angela pulls out 2 pink pieces, 2/8.

Jennifer: “Where did you get 2/8? What made you decide to do that?”

Angela: “1,2,3,4,5,6” She counted out the units in the three green pieces. “There are also 6 units in two pink pieces.”

Jennifer: “Now you have two pieces instead of three. Is there a way you can make it into one?”

Angela: “I need 6 units.” She looks around at the fraction pieces and chooses 1/4. “1/4! I did it!”

Jennifer: “So what does that number represent?”

Angela: “The mom ate 1/4 of the cake.”

Jennifer: “Who ate the most cake?”

Angela lays out the pieces 1/2, 1/4, 1/6, and 1/12 and tells me the order from greatest to least is: Dad ate 1/2, Mom ate 1/4, Sam ate 1/6, and the dog ate 1/12.

At another session I asked Angela to come up with an addition or subtraction problem for me to solve using the fraction pieces. This is what she came up with:

Fred had some candy. He had ½ of a piece of candy. He split the candy with 4 of his friends and it came up even. How much did each friend get?

Jennifer: To clarify the question I ask, “Did Fred eat any of the ½ piece?”

Angela: “No.”

Jennifer: “So what happened to the other ½ piece of candy.”

Angela: “That’s what Fred ate. He was giving the other half to his friends because he wasn’t hungry for it anymore.”

Jennifer: This seemed like a division problem to me. However I feel it is best to focus on the question rather than ask her why she thinks of this as a subtraction problem. This is the solution gave. I pulled out a 4 pink to use for 4/8. “Each friend gets 1/8 of a piece of candy.”

Angela: “No. Yes.” She seems unsure.

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Jennifer: “So am I done? Is that an okay answer?”

Angela: “No. Yeah, you are done.” She doesn’t look convinced that this is the answer she wants.

Jennifer: “Should I reduce?”

Angela: “Yes, reduce it.”

Jennifer: Hoping she’ll see a connection, I reduce 4/8 back to 1/2.

Angela: “Yes. Now raise it.”

Jennifer: “Ok, I’ll raise it back to 4/8. Is that okay with you?”

Angela: “Yes, now raise it more.”

Jennifer: “I could raise it to 12/24. Then each friend would get 3/24. (I do this with the fraction pieces) But why would I want to do that?”

Angela: “Because I want to see if you can do it.” She likes being able to make me give answers.

Jennifer: “So what would the best answer be for the question you asked? And tell me why.”

Angela: “1/8 is the easiest. It is only one fraction piece. If you chose 3/24 that would be three fraction pieces for each friend. They are both the same, but 1/8 is easier because it’s more reduced than 3/24.”

Question:: Kim is taking a walk around town. She walks to the bakery which is 1/6 of a mile. Then she walks to the library which is 1/12 of a mile. Finally she walks to the pet store which is 2/24 of a mile. How far did Kim walk in total? To help focus her attention, this question is presented with a simple picture.

Angela goes straight to the fraction pieces and easily pulls out an aqua, a green and a 2violet. Reverting to arithmetic, she looks lost and says: “I don’t know how to add 1/6, 1/12, and 2/24.” She has not learned the process to add fractions with unlike denominators, and she still feels she must use arithmetic rather than rely on her own thinking.

Jennifer: “That’s why we are using the fraction pieces. I bet you could use those pieces to help you figure out how to do the problem.”

She lays the green and the 2violet atop the aqua and discovers that putting the green with the 2violet is the same size as the aqua. She trades for another aqua and says her answer is 2/6.

Jennifer: “Can you do something with 2/6? Do you want this as your final answer?” I hope that she will be able to explain her thinking to me.

She shows me again how the pieces for 1/12 and 2/24 fit on top of the piece for 1/6. She says that it is best to trade for fewer pieces. She looks unsure so I ask her if she could do anymore trading to make it an easier fraction.

She quickly trades the 2aqua for a yellow. We talked about why 1/3 is an easier answer, but that 2/6 was not wrong. She seemed more comfortable with trading after doing this problem.

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Question:: Peg painted 1/2 of her house. Lin painted 1/3 of Peg’s house. How much of the house was painted?

Angela uses the fraction pieces, pulling out a red piece for 1/2, and a yellow piece for 1/3. She lays the red piece down and then places the yellow piece on top of the red piece. Then she picks up 2 green pieces and places them next to the yellow piece, on top of the red, covering the red piece with one yellow and two greens.

Angela: “2/12 is the answer.”

It seems to me that she has subtracted to get 2/12. I’m not sure what made her subtract. After talking with her I don’t think she even knew she was subtracting. You can tell from our following conversation that Angela is very unsure of her answers and what the problem is actually asking.

Jennifer: “Explain to me what the problem is asking for. What could you do to solve it?”

Angela: Not answering my question she says, “Oh, I see it. The answer is not 2/12”

“Good” I think. She has now realized that she shouldn’t be subtracting.

Jennifer: “Why did you lay the yellow piece on top of the red piece? What does that mean? Could you lay a red on top of a yellow?”

Angela: “You can also lay a red on top of a yellow.” She lays a yellow down, then a red on top of it, then another yellow on top of that. “The answer is 1/3.”

Jennifer: “How did you get 1/3?”

She reads through the problem, writing it out and drawing pictures. “Peg paints 1/2.” She draws a square and colors in 1/2 of it. “Lin paints 1/3.” She draws another square and colors in 1/3 of it.

Next, she goes back to the fraction pieces and begins piling pieces like crazy. She lays out 4 reds, then on top of that she puts 6 aquas, then 2 blues, and finally 2 yellows. To me this looks like a big messy pile of fraction pieces. To my surprise she looks at her pile and tells me that the answer is 5/6.

Now I’m really confused. She came up with the right answer, but I have no idea how, or what process she used to get to that answer. Hoping that she will shed some light on my questions, I ask her to give another explanation.

Jennifer: “How did you get 5/6?”

Angela: “You add 1/3 and 1/2.”

Jennifer: “You add them?!” I am excited that she finally sees this connection.

Angela: “Well, I’m not sure if you add them or not.”

Jennifer: “Well, do you? Look at the problem.”

Angela: “No, you don’t add them.”

Jennifer: “Why not?” I’m not as excited anymore, just more confused.

Angela: She tries to add them by writing the equation out 1/2 + 1/3. “It doesn’t work.” I see what is confusing her now. She can’t add them with different denominators; that’s why she thinks it’s not addition. Now that I have a better grasp on her thinking I ask her to try it with the fraction pieces.

She places the 1/2 piece next to the 1/3 piece. Then she picks 5 aquas and makes a trade, coming up with the answer 5/6. It works!

Jennifer: “So you do add?”

Angela: “Yes.”

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I think that after completing this problem, Angela has a better grasp on how to add fractions and is more aware of adding fractions with unlike denominators. Now she can do it with fraction pieces. What still confuses me is the fact that by looking at the word problem she didn’t know that it was an addition problem. Even after she gave an answer, she still wasn’t convinced that it was a problem dealing with addition. I also don’t know how she came up with the answer 5/6. Did she guess? Or in her mind did she add and just not understand how she did it?

To help Angela visualize fractions in another way, we worked with measuring cups and water. The measuring cups we used were: 1/2 cup, 1/4 cup, 1/3 cup, and 1 cup. We used regular measuring cups and also liquid measuring pitchers. These held up to 4 cups, and had the measurement marks all the way from 1/4 cup to 4 cups.

Question: You are baking a recipe which calls only for water. The ingredients are 1/2 cup water and 1/4 cup water. How much water is used in this recipe?

Angela fills up the 1/2 cup and the 1/4 cup with water. After looking at the cups for a while, she begins to write. Before I can see what she has written, she erases her work and tells me 3/4 cup.

Jennifer: “How did you get 3/4?”

Angela: “I figured it out in my head. I pictured the fraction pieces. I pictured one red and one blue, which is 1/2 and 1/4.”

Jennifer: “So 3/4 is the correct answer?”

Angela: Once again she doubts her answer. “No.”

Jennifer: “Why not?”

Angela: Thinking she asks herself “How many units are in a red? Twelve. How many are in a blue? Six. So that’s a total of 18 units. A whole has 24 units, so it’s less than one whole, but more than one half. I think it is 3/4.”

Jennifer: “How can you check your answer?”

Angela: She pours water into the 1/4 and 1/2 cups. Then she pours both into the 1 cup, telling me that that’s how you check the answer.

Jennifer: “That proves that it’s bigger that 1/2 and less than 1 cup, but how do you know that it is 3/4? It could be anything between 1/2 and 1. I suggest she try using the pitchers to measure it.

She pours 1/2 and 1/4 into the pitcher and reads the measurement as 3/4. It works!

Question: This recipe calls for 1/2 cup cold water, 2/3 cup cool water, 1/2 cup warm water, and 1/3 cup hot water.

Angela pours all the “ingredients” into the large pitcher and reads the measurement as 11/2 cups. (She had done it correctly, however she wasn’t very accurate in her measurement of the water, so I asked her to try it again, just be more careful.)

This time it worked, giving us 2 cups as the answer. We talked about why it should be 2 cups and not 11/2 cups. We grouped all the thirds together and all the halves together. Angela determined that there were three thirds, which was equivalent to 1. There were two halves which also equaled 1. When you put these together, you came up with 2 cups.

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Lesson 5 (by Richard)

This is the only lesson in Chapter 1 which focuses on the use of arithmetic. In particular it focuses on the arithmetic way to raise and reduce fractions. For children who can easily multiply and divide whole numbers, the calculations involved are fairly simple. However a conceptual basis for these calculations is far from apparent to most children. Thus it is not unusual for a child to seem to have mastered the arithmetic of equivalent fractions and yet not be able to apply it at appropriate times.

Kay’s thinking about a fraction whose numerator is a mixed number may seem strange, but it is not unusual for gifted children when they are exposed to such ideas. If you are working with gifted children, pay attention to what Kay has to say. To keep such children interested, encourage them to expand their numerical imaginations. They can often be more easily motivated this way than through the use of ordinary situational problems.

Jennifer working with Angela (by Jennifer)

Question:: Who Am I? I am equivalent to 2/3. My denominator is 6.

Angela wrote 63 = 22 = 4. She said she was done and the answer was 4.

Jennifer: “I am a fraction.”

Angela erases all her work and starts over. She writes:

Jennifer: “Were you wrong when you wrote 63 = 22 = 4? Angela: “No, I was mostly right. I could have just put the 6 for the denominator.”

What concerns me is that when she saw that this problem dealt with equivalent fractions, she just plugged numbers into a formula she knew. Angela has already been exposed to the arithmetic of raising fractions. She simply followed a routine process of finding an answer without understanding why she should do it this way. I don’t even think she realized that the process (63 = 22 = 4) was a method to figure out the numerator. To her she was just figuring out some number.

My Comment on the use of Mathematical Language (by Richard)

While I do not totally agree with the saying “sloppy language is a sign of sloppy thinking”, there are times when this is the case. As a case in point here is one of the things Angela wrote and Jennifer excepted. This is certainly sloppy language. In this case I doubt that it involves sloppy thinking. What ‘63 = 22 = 4’ actually says is that 6 divided by 3 gives the same result as 2 multiplied by 2 and that this result is 4. What is intended is that 6 divided by 3 is 2 and that 2 multiplied by 2 is 4. The grammatically correct way to say this is more cumbersome than what was actually written:

63 = 2 and 22 = 4

How serious is this poor use of mathematical language? On scratch paper or in the process of solving a problem it seems harmless enough, so I would let this go at such times. However it is a sign suggesting future activities on the use of mathematical language. A student who does not know the correct grammar of mathematical language will at some point misread this language. In particular they are likely to think of an expression like 1+23 naming 9 rather than 7.

Many students never learn to make a totally automatic use of the order of operation conventions that are so necessary for understanding equations.

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Back to Jennifer working with Angela (by Jennifer)

I asked Angela to pick a fraction and then come up with as many equivalent fractions as she could. One fraction she chose was 1/3. She came up with many equivalent fractions, such as 2/6, 3/9, 4/12 etc. I wanted to see what reducing/raising concepts Angela really understood. I wasn’t sure if she realized that all these numbers she had come up with were really just raised versions of 1/3. So I put this to the test by asking her to reduce 4/12.

At first she seemed really confused as if I had asked her to do an entirely different problem. It was obvious to me that she did not see the immediate connection between this question and what she had just been doing with the fraction pieces.

After some thinking she said that the answer was 1/9.

Jennifer: “How did you get that answer?”

Angela: “I subtracted 3 from the top and bottom of 4/12.”

I asked her to show me this with the fraction pieces, but she could not do this. So I asked her to write it out for me. She begins to write it, but then realizes that she had the wrong sign and wrong number.

Angela: “I shouldn’t use a subtraction sign. It is supposed to be division. And it should be divided by 4, not 3”

She performed the traditional algorithm to reduce 4/12 to 1/3 by dividing both the numerator and denominator by 4.

I asked her to show me this with the fraction pieces, since this was the connection I wanted her to make. She picked up several of the pieces and gave me a very confusing explanation, one which I couldn’t make any sense of. She knew that it was off the wall too because she was even confusing herself. I finally had to show her how the 4/12 pieces traded for a 1/3 piece. I was really surprised that this concept confused her so much especially since she was the one who had chosen 4/12 as an equivalent fraction to 1/3. She has raised and reduced fractions before and seemed to understand it. However I think in the past she has relied on algorithms and just followed a process without understanding how it worked.

Richard’s comment on helping Angela understand why the arithmetic works

Angela has been exposed in school to the arithmetic of reducing fractions without understanding why it works. This was typical of Angela’s work with arithmetic. She tried to learn what she was supposed to do, and she had never developed confidence in her own ability to just think about a question and use whatever she understood to find an answer. Most of the summer’s work has been devoted to the understanding of concepts. The main barrier Jennifer had to overcome in helping Angela to learn was to help her rely on understanding. Without this understanding she may learn to do certain calculations with fractions, but she will quickly forget them. More important, only an understanding of concepts will give her the functional mastery that will allow her to apply fractions in appropriate situations. At the end of the summer we finally reached a point where she had a basis for understanding the concepts that relate to the arithmetic of equivalent fractions. However this understanding was not deep enough to give her confidence in her own competence. We only had one session left to help her begin to see that the arithmetic of equivalent fractions was not just some arbitrary process, but something that she could figure out on the basis of her own understanding. The final session which Jennifer next describes is a first step in this direction.

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Back to Jennifer working with Angela (by Jennifer)

I wanted to work some more with Angela on reducing by using arithmetic. We didn’t use the color pieces in this exercise, instead we used the gray model. I gave the example below to help her understand how we were approaching these problems:

2/3

Raise by 2To do this, draw a line to split gray piece into 2 pieces.

You will get the raised fraction 4/6.

4/6

Jennifer: Angela will you raise 3/4 by 2. First she draws a 3/4.

She draws a line only partly across two columns of the picture.

I’m not sure why she did this. I don’t think she understands why we draw the line across the picture. She is just following a procedure that she saw me do. I don’t know how to help her understand. I ask her to check her picture by using arithmetic.

She checks by multiplying both the numerator and denominator by 2.

Angela realizes her mistake. She extends the line across the entire picture. We talk about why this works and she seems to have a better understanding.

I try another picture. I ask her to raise 7/8. She looks at me confused and asks, “What do I raise it by?” I tell her by two. It encouraged me that she picked up on this. She was aware of the fact that you could raise the fraction by other numbers this way. We also worked on raising fractions by numbers other than two. We reversed roles and she made up fractions that I could raise by different numbers. After she got past the first problem and figured out what she did wrong, she was able to do all the other questions correctly and seemed to have a better understanding of why it worked.

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APPENDIX CONSTRUCTIVIST IDEAS AND EXAMPLESConstructivist Learning: In many traditional classrooms, children learn by absorbing and accumulating information, with active thinking and reasoning seldom emphasized. Constructivist learningm goes beyond this type of learning, realizing that children must actively think and reason in order for learning to be rich. Learners must be involved in the learning process. With guidance, they can construct concepts that both fit with and go beyond the concepts they already understand. Learning this way they are able to interpret and apply these concepts. For persons to learn on their fullest, they must be encouraged to explore many possibilities of solving problems. One limitation in much of the teaching is that it often focuses on getting the “right” answer. It does not sufficiently concern itself with the process of getting to the solution. As constructivists we encourage students to explore and understand various ways of solving problems, but not to worry about being right and wrong in their initial attempts. We want them to think of mistakes as a natural part of learning, something to learn from rather than be concerned about. We want learners to feel free to say what they are thinking and then build discussion on their thoughts. They then will be able to work their way through similar problems and be able to understand and explain how that problem works.

Persons in a constructivist learning environment often acquire new concepts by exploring questions that they can understand but have not been taught how to answer. This allows them to pull ideas from their own understanding of the situation and think for themselves. Word problems can be viewed as a prelude to skills. In this book, we will be introducing new concepts with word problems before the arithmetic needed to do the problem is taught. You do not have to know how to do something well before you do something real. Consider a team of young children just learning to play basketball to illustrate this point. Most would probably be more enthusiastic about playing a basketball game rather than running basketball drills over and over again. Some people would argue that drill should come first, at least until certain skills are acquired. This can be useful, but if they hate this routine, little skill is achieved thru the drills. Most children would probably be much better off by simply playing the game badly for a while. This can provide the kind of experience that will help them realize what they are good at and what they need to worked on. When this is discovered, they will realize that if they work on the particular aspect that their coach says needs improvement then they will be a better team. This discovery will motivate them to practice particular skills, allowing for even greater learning to take place.

Deliberate Bridging: Bridging involves connecting concepts that a person finds remote to anything that seem manifest to that person. People repeatedly bridge such gaps in the course of ordinary living. With experience, concepts that once seemed remote become manifest. As a toddler Joe finds the concept ‘mother’ manifest, but it applies only to one person. The broader mother concept is too remote for Joe to grasp. Yet in a few years and with no deliberate effort the broader concept emerges as manifest. How does this happen? Joe learns that his mother is also his sister’s mother. He learns that his friends have mothers. He learns that his mother has a mother. Gradually his mother concept is not restricted to those he knows. The gap between a highly manifest concept and one that once was too remote to grasp has been bridged. This bridge may be built from a variety of smaller bridges, each of which has a small span between what is highly manifest to something slightly more remote. Of course how many small bridges are needed will vary from person to person. For many concepts that a person is less likes use in the ordinary course of living, deliberately designed bridging activities can be used to help make these concepts a part of a person’s world. Deliberate bridging activities can also be used to speed up the acquisition of concepts that might have eventually been obtained thru ordinary experience.

Note: Descriptive Psychology is a network of theory neutral conceptual tools. One of these tools is called parametric analysis. A parameter merely enables a person to focus on some aspect that seems useful for some purpose.

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Concept Parameters: Concept Parameters discusses the role that a concept may have in a person’s world, with special emphasis on ways to think about a person’s mastery of a concept. The role a concept has in a person’s world is conceptualized as complex personal state of affairs, using five parameters for thinking about its various features. Bridging relates most closely to what I call the proximity parameter. This parameter indicates and describes how close a concept is to a person’s common experiences. Proximity can vary from highly manifest to extremely remote. A concept is manifest to a person P to the extent that it is close to experiences that P finds it easily accessible and easy to identify and understand. It is remote to P to the extent that P finds it removed from such experiences. The chair a person sits at for dinner will be a highly manifest concept. A person’s concept of a chair may be somewhat less manifest, but since the function of a chair is easy to understand in terms of ordinary experience, it will also be highly manifest to people who use chairs. Altho initially not as manifest as the concept of a chair, the concept of furniture also easily becomes highly manifest to most people. In fact, all concepts in a person’s network of concepts for ordinary matters will seem manifest that person. The concept of a galaxy is likely to be at least somewhat remote to many people, altho some people find it less remote than others do. Bridging also involves what I call the integration parameter, since a person’s mastery of a concept normally depended on how it is integrated with that person’s other concept. A concept is adequately integrated if it is appropriately connected to other concepts and if these connections are those that the person would most commonly need in order to understand and communicate with others. The utility parameter indicates the uses that a person my have for a concept. Bridging is normally more effective when it enable a person to recognize the utility that the concepts will have for that person. Activities like 2b and 2c are intended to illustrate the utility that fraction concepts might have for some persons. However, what utility a person’s world, so bridging activities that allow some people to recognize utility may not serve that purpose for others.

Let’s use the topic of division and remainders to illustrate the differences between traditional and constructivist learning. In most traditional classrooms, the topic of division would be presented first, then the concept would be drilled through the use of worksheets. Finally, word problems might be given. The constructivist method approaches the introduction of this concept differently. We illustrate this method with a scenario of young children. These children are younger than the age group this book focuses on.

The teacher breaks a classroom of children into several groups of 4. Each group is given a bag containing 15 cookies. The teacher asks each group how they would distribute the cookies among themselves. Students have done a number of division problems, but they have never encountered remainders.

One group counts the cookies this way “One for me, one for you, one for you, and one for you.” They continue this process until one of the children yells out, “Hey, there’s not enough. It doesn’t come out even! It won’t be fair!” The teacher encourages the children to discuss this problem among themselves. She asks how they might go about solving the problem. One child declares, “We could cut up the 3 extra cookies, but how would that work?” Another student says, “We could just leave them. They are extras.”

Another group quickly sets up the problem ‘15 divided by 4’. However they are puzzled when the problem does not come out even. They too comment on there being extra cookies. One child says, “ Maybe our teacher gave us the wrong amount of cookies. It should have been 16 cookies. 16 divided by 4 equals 4. That would be right.”

A third group sees the problem a little differently. “Just because we’ve never seen a problem with things left over, that doesn’t mean that it’s wrong. I think it’s okay to have things left over. It’s just something different.”

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This type of learning stimulates learners to put their minds at work and figure things out for themselves. Sharing cookies is something they can grasp much easier than theories and laws written in textbooks.

Our book focuses primarily on a wide variety of questions about situations involving fraction concepts. These questions are explored primarily thru conceptual reasoning in relation to visual models, with no focus on the mastery of algorithms for calculating with fraction. This does not mean that arithmetic calculation is considered unimportant. It merely means that such algorithms are usually better mastered when they follow, rather than precede, conceptual understanding and the ability to relate concepts to meaningful questions about various situations. In this book the only important algorithm is that of raising and reducing fractions. However the goal here is primarily to introduce rather than master that algorithm. Mastery for some students will automatically come later as they use the process in adding and subtracting fractions. For others it might be advisable to do more work on this process before going on to addition of fractions. Routine drill can be used. However more situations including a wide variety of questions about situations to which the student can easily relate will probably do more to enhance mastery. For visually oriented students I also recommend supplementing the arithmetic with manipulating color pieces.

With students for whom fraction concepts come easily, only occasional use of the color model for fractions is advised. Rely instead primarily on the gray model. Also frequently use other visual representations initiated by the student. The color model is good for introducing concepts, but limits the fractions that can be used. The gray model overcomes this limitation. However manipulating color pieces, or even drawing the gray model, in order to answer questions can be tedious. Thus it is wise to encourage students to use any appropriate visualization. Furthermore when a student initiates a way of looking at the question, the quality of the student's involvement will often be of a higher level than when the model is given by someone else.

Algorithms: One critic of constructivist teaching claims that it ignores the hard work involved in becoming good at the algorithms used for arithmetic calculations. He compares the acquisition of arithmetic skill to the acquisition of athletic skill, remarking about the hours of tedious drill and practice needed to master such skills. There are a number of comments to make in response, first on the general idea of functional mastery and then on the role of drill and practice.

The main constructivist principle is that unlike information, which a person can receive in a fairly direct manner from another, a person must acquire a conceptual understanding by augmenting and transforming ideas they already understand. In this sense concepts are constructed from within the conceptual net of the person learning the concept. This does not mean that practicing the concept once it is acquired is unimportant. Constructivist teaching should not focus only on initial acquisition of a concept, and thus ignore functional mastery. Functional mastery of many mathematical concepts involves both an understanding of related algorithms and skill in doing them. The difference between a constructivist approach and a non-constructivist one is in how mastery of algorithms is obtained.

Again consider the analogy of drill and practice in the acquisition of an athletic skill such as basketball, distinguishing between the concept of drill and the concept of practice. While there is no sharp line between these concepts, drill is routine but practice is not. Drill is useful and can be tedious, but the real hard work comes in practice and the highest form of practice involves playing the game. Furthermore the utility of tedious drill depends highly on attitude and would be of little use to the athlete who did not want to play the game. The analogy to drill and practice in mathematics should be obvious. In fact when it comes to developing useful conceptual skill the role of attitude and real use may be even more crucial than it is for physical skills. Those most competent in conceptual skill are those that use them because they enjoy using them.

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UNIT: UNDERSTANDING DIVISION

Developed by the Association for Conceptual Studies

Presented by: Richard Singer

website: www.fractions-plus.com email: richard-acs@worldnet.att.net

Edition date 3/2003

Overview: Section 1 focuses on the concept of division. Section 2 focuses on algorithms for doing what has been called long division. Section 2 can be read without reading Section 1, as long as concept of division has been understood in relation to subtraction.

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The concept of division: We now focus on the types of situations to which we apply the concept of division. There are three basic types of these, which we refer to these as dividing between, dividing for size, dividing to compare.

Dividing for size: to divide some quantity to obtain parts of a specified size.

Dividing between: to divide some quantity equally between a specified number of parts.

Dividing to compare: to divide quantities of the same type in order to obtain a ratio.

To divide 8 padlocks for sizes of 4, i.e. so each person gets 4 padlocks; we can share them among 2 person. 8 4 = 2. With 8 padlocks to divide between 2 people, gives each person 4 padlocks. for which we use 8 2 = 4. The same picture can be used to show dividing 8 padlocks between 2 people and dividing 8 padlocks for sizes of 4.

While the same picture shows both dividing between and dividing for size, in first learning to divide, these picture would be obtained in different ways. To physically divide 8 padlocks between 2 people we start with 8 padlocks or tokens for them.

We then partition them. The most primitive method of dividing between is move one at a time to each pile.

Send one to each person.

Send one to each person.

Send one to each person.

Send one to each person.

To obtain parts with size 4 keep is a process of taking away 4 padlocks until none are left.

Send 4 into a pile which gives 1 pile and some padlocks left.

Send 4 more into a pile giving 2 piles and none left.

Dividing 8 padlocks between 4 people has a different picture than dividing 8 padlocks between2 people. It has the same final picture as dividing 8 padlocks for sizes of 2.

Even before people had names for numbers, they could divide 150 sheep equally between 3 shepherds. Merely separate them by giving one to each shepherd, another to each shepherd, etc. However to divide 150 sheep to see how many groups of size 50 can be obtained, they would need some way to represent 50, perhaps the ability to count to fifty either symbolically or with objects such as stones. First remove 50 sheep; then 50 more; etc. This is still the way children first solve problems of dividing between and dividing for size.

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Quantity Types: When dividing with whole numbers, we can imagine many situations that involve dividing between and many situations that involve dividing for size. However, they are not always stated this way. For example, if we can seal 500 envelopes in 5 minutes then how many envelopes can we seal in a minute? This is of dividing between because we are sharing 500 envelopes between 5-minute intervals. In dividing between we think of dividing envelopes by 5 to obtain envelopes, but our divisor is minutes rather than envelops. In general when we divide between, the dividends and the quotient are quantities of the same type but the divisor is a quantity of some other type.

Between: 500 envelopes sealed in 5 minutes, do 100 envelopes in each minute: 5005 = 100.

We can also divide envelopes for sizes. Below we have specified the size job for each minute and are looking for how many jobs of this size. Here we divide envelopes by envelopes and the result is how many minutes. In general when we divide for size the divisor and dividend are of a similar type but the quotient is not.

For size: 500 envelopes sealed at 100 envelopes each minute, taking 5 minutes: 500100 = 5.Using ordinary language, we might describe dividing 500 objects for size 100 as dividing 500 objects into groups of 100. We then talk about the arithmetic of how many 100s go into 500. Using the phrase ‘dividing into’ in these situations can cause confusion for some children. This is why we recommend initial use of the phrase, ‘dividing for size’ instead of ‘dividing into’. Once concepts are well mastered the duality in the use of ‘into’ should be clear from context.

Dividing To Compare: Dividing to compare is pictured differently than dividing for size or dividing between. We picture both the divisor and dividend, then look. The quotient is a pure number rather than a quantity of any type.

To compare 8 padlocks to 4 padlocks picture both but separate the larger into groups of the smaller and count these groups. Comparing 8 to 4 and comparing 8 to 2 have different pictures.

Comparing 8 to 4, we see that 8 padlocksis twice as many as 4 padlocks.

8 4 = 2

Comparing 8 to 2, we have 8 padlocks is four times as many as 2 padlocks.

8 2 = 4

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Long Division: The primitive way of dividing for size relates to subtraction and understanding the algorithm for long division depends on this relationship. The algorithm to the right below shows the usual way of writing a solution to the problem of dividing 8475 by 23. Altho this algorithm is concise, many of the underlying ideas are left implicit. Later we show alternatives that make ideas more explicit. If one goal is to master an algorithm for dividing with multi-digit numbers these alternatives can be used instead of the standard algorithm or as a prelude to understanding it. This is discussed later.

The process in the first step is often described as follows:

Take 23 into 84, 3 times 23 is 69, subtracting 69 from 84 gives 15, now bring down the 7.

The novice may not be aware of the fact that the 3 is really 300 and that the 69 is really 6900.

3 36 36823 8475 23 8475 23 8475

69 69 69 157 157 157

138 138 195 195

184 11

368To make the underlying idea more explicit we can include zeros in the standard algorithm. However, the fact that the 3 in the quotient is really 300 may not be apparent until the final step. The modified algorithm on the far right makes this explicit. It also stresses the fact that what we are doing is seeing how many 23’s we can subtract from 8475.

23 8475 23 8475 300 6900 6900 1575 1575 60 1380 1380 195 195 8 184 184 11 11 368

The intermediate steps for the modified algorithm can be indicated as follows:

23 8475 300 23 8475 300 6900 6900 1575 1575 60

1380 195

When using the standard algorithm, it is important to estimate the partial quotients accurately. This is not the case when using the modified one. How much to subtract each time is a matter of choice. Some choices involve more steps than others do. However, for anyone who is not comfortable with all the multiplication fact, taking more steps might have an advantage.

We do not need to start with a multiple of 100. For example, we could start by taking away 303. This would leave 1506. Of course, there may be no practical advantage in such a choice.

23 8475 100 2300 6175 100 2300 3875 100 2300 1575 40 920 655 20 460 195 5 115 80 3 69 11 368

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306 306Some students have trouble when bringing down a single number give something smaller than the divisor.

23 7038 23 7038 23 7038 300 69 6900 6900 13 138 138 6 0 138 138 138 306 138

Why Learn About Long Division: Given the availability of calculators, the actual need to master the skill of long division may not seem very useful. This raises the question of whether or not it is useful to master an algorithm for long division. My answer is that this depends on the reason a person P might have for mastering it.

Suppose P’s only reason for mastering algorithms is to be able to find numerical answers. Most persons are unlikely to encounter situations in which they have no calculator and such answers to long division problems are needed. In the cases that they do, then merely understanding how division relates to subtraction will allow them to get an answer. The work involved is likely to be less than the work involved in learning and remembering the standard algorithm. The modified algorithm will do just about as well and it is easier to master. There is a standard algorithm for extracting square roots, but very few people master it, and those who do almost never use it. From the perspective of finding answers, I can see no more reason to master the standard algorithm for long division than to master this algorithm for extracting square roots.

Utility is relative to purpose, and the preceding comments apply only to a very limited purpose. P’s purpose might include being able to apply mathematics to various situations. It might be also include a desire to understand mathematical ideas primarily because they seem interesting. With either or both of these purposes, merely being able to use a calculator is far from sufficient. To do either, it is necessary to understand the concepts involved well enough to be able to chose which concepts are relevant. Altho algorithms can be learned without understanding concepts, they cannot be appreciated without such an understanding. Thus examining or creating algorithms is one way to gain a greater understanding of concepts. Since one important type mathematical activity involves designing algorithms, mastery of some algorithms is a crucial factor in understanding mathematics. This does not imply that either the standard or the modified algorithm for division is necessarily important. However it an algorithm that relates division to subtraction, and this is certainly an important conceptual relationship.

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Bridging to Venn Diagrams: This is one of several examples for educators illustrating the use of bridging. Some bridging strategies center on a common principle that has some highly manifest instances and then bridges to a type of instance that is more remote. However a bridging strategy may use any means to relate something that might remote to something manifest. To bridge to the concept of a Venn diagram we start with a basic attribute game and focus on how it can be used with elementary school children. Similar attribute games can also be used at a higher level to bridge to additional concepts. Below are pictures of the tokens that are used for this game. Items vary by the attributes types {size, color, shape}. Sizes are large and small. Colors are blue and red. Shapes are circle and diamond. A larger set of attribute items can also be used, but even college students can benefit by starting with the basic game.

Note: There is a template for making attribute tokens at the end of this appendix.

This game uses 6 labels for sizes and colors and shapes. It is played on a board with 2 intersecting circles. These circles partition the attribute items into 4 regions. Each will have 2 elements, depending on the values chosen for the labels. As a prelude to the game, place the labels as in the sample diagram to the left. Ask how the tokens for these items should be placed. Expect some doubts, but you are likely to find some who will place in the regions indicated below. Make sure that everyone understands why each item can only be placed in this manner.

You might choose two other labels (of different attribute types) and again ask how the tokens should be placed.

You might place a token and ask what the label possibilities are.

SAMPLE DIAGRAM

Challenge Question: If lbc (large blue circle) goes in the left region of the left circle, which other item can you locate no matter what the labels are? Later we give an analysis.

Attribute Game Rules: The game involves 2 teams, with 2 or 4 people on each. To start, Team 1 selects labels of different attribute types and places them face down beside the circles. Team 2 chooses any token. Team 1 must put this token in the correct place. Team 2 must place the

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remaining tokens in the 4 regions. On each trial, Team 1 either verifies the choice or removes the token if it is in the wrong region and gives it back to Team 2. Team 1 collects one point each time they return a token, however if they return a token incorrectly or allow it to be placed incorrectly, they forfeit the game. Team 2 may try a rejected token elsewhere or try a different token. Once Team 2 has placed all the tokens they must either correctly identify the labels or forfeit the game. The goal of the game for Team 2 is to identify the labels giving Team 1 as few points as possible in the process. If they correctly identify the label then Team 1 and Team 2 reverse roles and play is repeated. If there is no forfeit then the team with the most points is the winner. The same rules apply to games using 3 circles. The game can also be played with more attribute items.

Suggested Use: The purpose of playing and discussing the game is to make the use of the regions extremely manifest. This is also the purpose for either playing or discussing the 3-circle version. For elementary children such activities should probably be spread over more than one session. When using the game in a class, divide the class into groups of six to the extent possible, as teams of size 3 seem to work best. However opponent teams need not be the same size. Have each group play the game and record any observations. Then bring the groups together for a discussion. Then have them play the game once more. Discuss strategies used game playing. For instance, when a token placement is rejected some people try it some other place but others try a different token. Ask questions (such as the sample question) to help reinforce the way items are placed.

If there is an interest then play a more challenging game, using 3 circles. Otherwise at least have the class chose labels and place pieces. If you only use attributes of different types then each region will have exactly one token. However you can use a larger set of attribute items and more labels. You can also allow using labels of the same type. Introduce the term ‘Venn Diagram’ and discuss the concepts of union and intersection. Illustrate the use of Venn Diagrams with the other situations given later.

Vegetable Situation: A survey was conducted about the role vegetables in family diets. The most frequently used cooked green vegetables were {Peas, Spinach, Broccoli}. One question asked which of these vegetable the family used on the average at least once a week. The following information

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was gathered: Broccoli 34, Spinach 43, Peas 56. Of those, 7 used both Broccoli and Spinach, 11 used both Broccoli and Peas, 19 used both Spinach and Peas. Moreover 4 of these used all three vegetables. In addition, 8 who never used any of these vegetable at least once a week. How many said only Broccoli? How many only Spinach? How many only Peas? How many families were surveyed? How many families used exactly two of theses vegetables at least once a week?

To construct a Venn diagram for the number of families for this information, begin by placing the 4 in the middle region. Since there are 7 in BS place a 3 in the other part of BS. The 7 and 15 can be placed in a similar fashion. Since there are 34 in B, there are 20 in broccoli only. There are 21 in spinach only, 34 in peas only. Adding the numbers in all regions gives 112 families in the survey. Taking 3715 gives 25 families who use exactly two of theses vegetables at least once a week.

Notation: The notation BS is short for BS, i.e. for the intersection of the sets B and S.

Reference: For more examples of this type, along with explanations use any of the following links.

http://www.chaselink.com/tune/:

http://regentsprep.org/Regents/math/venn/PracVenn.htm:

http://www.purplemath.com/modules/venndiag4.htm:

http://www.beva.org/maen503/week2/venn_diagram_examples.htm:

http://www.math.tamu.edu/~kahlig/venn/venn.html:

Other Bridges: The attribute game can be a starting point to bridge to some topics, including some that can be used in college level courses in Boolean Algebra and Informal Reasoning. Below are some ideas that can be used to develop bridging strategies. See Our paper Constructivist Learning.

Set Concepts: The set of items in the outside part of the sample diagram can be named as LB, as it is the intersection of set of large items with the set of blue items. It can also be named in terms of the labels S and R. The set of items that are not small is called the complement of the set of small items. Using the symbol ‘/’ for complements, we can use /S/R to name this set. It can also be named in terms of the complement of the union of S with R, i.e. as /(SR). This gives an example of DeMorgan’s Law that says that /(SR) = /S/R. As another example the subsets of R are SR and /SR, i.e. we have R = SR/SR. This can be deduced using the distributive law along with the complement law and the multiplicative identity law.

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Strategy Design: Coupled with the well-known game Twenty Questions, this game can also be used as a bridge to one type of strategy design concepts. In Twenty Questions (with no reason to use less questions), an ideal question would eliminate half of the possibilities, regardless of whether the answer was yes or no. With even a million possibilities, using such ideal questions would yield the answer. Recall that in an attribute game only an incorrect placement gives the opposing team a point. So if the placement is incorrect, we want it to eliminate as many alternatives as possible. For the basic game, there is a strategy that will result in never giving the other team more than 3 points, no matter how much luck goes against your team.

To illustrate this strategy, denote the values of the circles as X and Y.

Suppose srd was initially place in XY. This gives the 6 indicated possibilities for the labels, where SR means X = S and Y = R, etc. Chose a piece that differs in one way from srd, say lrd. Use it until it is placed. It must go in XY or /XY or X/Y. If lrdXY then the zeros indicate that 2 possibilities have been eliminated. In the worst case, you may get 2 rejections before placing it X/Y. Now either sbdXY or lrcXY, so at most there will be 1 more rejection.

srdXY SR SD RS RD DS DR

lrdXY 1 1 1 0 1 0

lrd/XY 0 0 1 1 1 1

lrdX/Y 0 0 1 0 1 0

sbdXY 1 0 1 1 0 1

lrcXY 1 0 1 0 0 0

This can also provide a simple bridge to the concept of expected cost. Using this strategy the probability of it giving up 3 points is 1/6, the probability of it giving up 2 points is 1/3, the probability of it giving up 1 point is 1/6, the probability of it giving up 0 points is 1/6. Thus the expected cost can be calculated as 31/6+21/3+11/3+01/6 = 1½.

Analysis of the Challenge Question: If lbc goes in the left region of the left circle, which other item can you locate no matter what the labels are?

Jan: The labels might be Large and Red. This would put both lrc and lrd in the intersection. However they might be Blue and Small, putting sbc and sbd in the intersection. So different labels that put lbc in the left part of the left circle will put those items I mentioned in different regions.

Roy: I also checked that it couldn’t be lbd or src. S the only one left is srd.

Bob: The 6 possibilities for the label pairs with lbc in the left region of the left circle are indicated in the table below. With each label pair I used a 0 to indicate that an item could not be in the intersection and a 1 to indicate it must be in the intersection. For everything but srd, there was both a way it would be and a way it would not be. However in each case srd was in the right region of the right circle.

Possible labels

Large Red 0 1 0 1 0 0 0 0

Large Diamond 0 0 1 1 0 0 0 0Blue Small 0 0 0 0 1 0 1 0

Blue Diamond 0 0 1 0 0 0 1 0Circle Small 0 0 0 0 1 1 0 0Circle Red 0 1 0 0 0 1 0 0

Kay: We do not need to consider possible label pair or any items except srd. We can deduce that srd belongs in the right part of the right circle as follows. Items inside of a circle must share an attribute. Since lbc is in the left circle,

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srd cannot be in the left circle. Items outside of a circle must share an attribute. Since lbc is outside of the right circle, srd must be inside of the right circle. Thus srd is in the right region of the right circle.

Roy: We can generalize. A pair of items that do not share any attribute must be in regions that are not adjacent.

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Bridging to Boolean Satisfaction: For a conditional of the form ‘If X then Y’, ‘X’ and ‘Y’ are called the antecedent and consequent respectively. In traditional two-valued logic, a conditional is considered false only when its antecedent is true and its consequent is false. In all other cases the conditional is considered true. Since this requirement for being true involves a perspective that many people find artificial or puzzling, it is important to note that in ordinary discourse we do not always use conditionals in this two-valued manner, nor do we use them as precisely. This may be one reason many people have trouble with the truth table for conditionals in a logic course. However there is another logic concept called boolean satisfaction that is more closely related to our ordinary concept of satisfaction, and there may be less of a problem in bridging from our ordinary uses of conditionals to the boolean satisfaction criteria below for conditional.

A conditional fails to be satisfied only when its antecedent is satisfied and its consequent is not satisfied. In all other cases the conditional is satisfied.

Note: Satisfaction is actually a more basic concept than logical truth, altho since this concept is usually first introduced in first order logic, this is not apparent in the way logic is usually taught.

Satisfaction for Ordinary Conditionals: We before turning to boolean satisfaction in logic we will examine how the concept of satisfaction relates to various ordinary uses of conditionals, including several types where it does not seem to apply. Comments about each type will be followed by a sample of what some highly articulate students might say. Most actual responses are likely to be simpler and less articulate.

Conditional Rules: Suppose a rule says that if you go to Europe then you must obtain a passport. The only way to violate this rule is to go to Europe without a passport. You cannot violate this rule if you do not go to Europe, regardless of whether or not you obtain a passport. You also will not violate this rule if you obtain a passport, regardless of whether or not you go to Europe. To satisfy a rule is merely to not violate it. It does not involve doing anything else. Using this concept of ‘satisfy’ you can satisfy the rule either by obtaining a passport or by not going to Europe. In general, for any such rule ‘satisfy’ is taken in a passive sense. You only violate a conditional rule when the antecedent applies to you and you violate the consequent. In any other case you have satisfied the rule.

Roy: A rule in many card games is that if you have cards in the suit led then you must play a card from that suit. The only way to violate the rule is by playing a card from another suit when you have a card from the suit led. You cannot violate the rule if do not have a card from the suit led.

Kay: A law says if you’re going to park at this parking meter you have to put money in. The only time you’ll get a ticket (i.e., violate the law) is when you park there and don’t put any money in the meter. You can satisfy this law by putting money in the meter or by parking elsewhere.

Bob: A rule is satisfied when it is obeyed. The only way to violate ‘If you are not wearing a tie then you cannot be admitted’ is to be not wearing a tie and still be admitted. Not admitting someone with a tie does not violate this rule. Whatever happens for a person with a tie cannot violate this rule.

Jan: Only someone who is under 65 can violate ‘If you take a double deduction for age on your tax return. you must be at least 65 years old’. It might be unusual for someone who is 65 not to take a double deduction, but this would not violate this rule.

Roy: Likewise nobody who can swim over 100 yards can break the rule ‘If you cannot swim l00 yards then you may not go past the rope’.

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Conditional Claims: Suppose your mother claims that if you eat too much Halloween candy you’ll get sick. This is a claim rather than a rule, but it works the same way. The only way to check her claim is to eat too much of the candy. If you do not get sick her claim was wrong. If you get sick her claim was correct. Not eating too much is no guarantee that you won’t get sick, so whether or not her claim is true, it will be satisfied if you do not eat candy. In a claim like this, one alternative is to refine the satisfaction concept into strong and weak satisfaction. From this perspective, if you do not eat candy there is no way to check the claim, so it neither fails nor is it strongly satisfied. It is only weakly satisfied. In logic, satisfaction means being weakly satisfied. So we do not mean that the claim is correct. We merely mean that nothing has been done to test it. Boolean satisfaction can also be applied to general conditional principles. A general principle differs from a rule in that it is a statement rather than a command. Consider the principle that all crows are black. To state this as a conditional we might say that if it is a crow then it is black.

Roy: A red cardinal or a black horse does not count as an exception, but a red crow would.

Jan: If you receive a poor education then you will not find a good job. A school dropout who was president of IBM might count as an exception showing that this is not a general principle. However the concept of a poor education is rather vague, as is the concept of a good job. Someone might claim that this person received a good education thru experience.

Kay: A general a principle is considered correct when it has no exceptions. When stated as a conditional, an exception occurs only when the antecedent is satisfied but the consequent fails. If you stretch a spring beyond its elastic limit then it will not return to its original shape. The only way to get an exception is stretch it too far and have it return to its original shape

Deliberately Vacuous Conditionals: A deliberately vacuous conditional is one in which an antecedent considered incorrect is followed by an extravagant consequent that everyone is expected to consider totally implausible. For instance, suppose Bill claims the he can make 50 straight free throws. Jill might say that if can do this then I can make a thousand straight free throws. Supposing Bill cannot, Jill’s conditional will be vacuously satisfied. Rather than denying his claim, this is her way of scoffing at it.

Roy: This is a fairly common type of put down of taking an extravagant claim as the antecedent of a conditional with an even more extravagant consequent. The user of such a putdown expects it to be satisfied because the antecedent is sure to fail.

Bob: For 3 years running Al has been eliminated in the first round of his college chess tournament. Undaunted, Al maintains that he will be the chess champion in his senior year. Bo responds if you win the college championship then I will win the world championship.

Kay: Common expressions such as ‘If X then I am a monkey’s uncle’ are made because X is considered incorrect, rather than because of any connection between X and being a monkey’s uncle.

Biconditionals: One reason that boolean satisfaction may not seem to apply to some conditional is that conditional language is often used with an implicit intention of stating a biconditional. That is, a person may say ‘If X then Y’ when they mean this as well as ‘If Y then X’. In most ordinary contexts we are able to implicitly recognize when conditional language is to be treated as if it was a biconditional. However this mixed use of conditional language is one reason that some people forget the way conditionals are used in logic. Consider whether (1) and (2) below as might implicitly be biconditionals?

(1) If a person has a billion dollars then that person is wealthy.

(2) If a person has a large amount of money then that person is wealthy.

Roy: Under normal circumstances (1) is not a biconditional. Having a billion dollars certainly make you wealthy, but

you would also consider a person wealthy with much less.

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Jan: I can easily imagine a context in which the intended interpretation of (2) was a biconditional. It might even be used as a definition of wealthy, with the understanding that anyone who satisfied the condition of being wealthy must have a large amount of money, and conversely anyone who satisfied the condition of having a large amount of money is considered wealthy.

Bob: I have a more personal example. My mother said told me that if I did my homework then I would be allowed to go to the movies. I did not do the homework, but I went to the movies. When she reprimanded me I told her I did not violate this conditional. It would have been violated only if I had done my homework and she had not allowed me to go. My explanation was not convincing. She said that I knew that she also meant that if I was going to be allowed to go to the movies then I must do my homework.

Roy: Suppose it was announced that if it rains Sunday then the picnic will be held inside. Someone who was not in town that Sunday, and who was informed that the picnic was held inside might conclude that it rained that Sunday. This would be a reasonable conclusion, since this conditional would probably be used in a context in which the picnic was not intended to be held inside unless there was rain, and thus it implicitly suggests a biconditional. Of course there could have been another reason to hold the picnic inside, so it is not entirely clear whether this conditional was intended as a biconditional.

Predictions: Conditional used as predictions may not on the surface seem to be best captured by boolean satisfaction, altho we can adopt a perspective which does use boolean satisfaction. The only time we count a conditional prediction as incorrect is when the antecedent is satisfied but the consequent fails. However we do not usually think of it as otherwise being correct. Even an unlikely prediction such as ‘If the prime rate falls by 2% next quarter then the Dow Jones average will drop 1900 points during that quarter’ cannot be counted as incorrect if the prime rate does not fall by 2%.

Roy: Perhaps not, but it is still a crazy prediction.

Bob: Using what we said before, it would only be weakly satisfied. Most people would want it to be strongly satisfied if it is to be considered as a correct prediction.

Kay: The use of boolean satisfaction for predictions rests on the desire to assign only 2 values to a prediction. As Bob just suggested, there is another option. We can classify a prediction in which the antecedent fails as indeterminate.

Jan: Consider a prediction like ‘If Jim marries Jane then they will be divorced within a year’. It would be satisfied if Jim marries Jane and they are divorced in a year. If Jim does not marry Jane we merely say prediction is indeterminate.

Kay: Advocates of boolean satisfaction can still make it applicable. Merely weaken the concept of satisfaction to also include indeterminacy. Of course this may decrease the utility of the satisfaction concept, since it ignores a conceptual distinction that could be useful for certain purposes.

Roy: I predicted that if the Suns do not have injuries to key players then they will win over 60 games. They had a key injury and still won over sixty games. Using this indeterminate concept, this prediction was indeterminate. It still seems correct to me.

Counterfactuals: Roy’s comment shows that the use of conditionals can be complicated. This suggests the most debatable use of boolean satisfaction, namely that of counterfactual conditionals. A counterfactual conditional is one whose antecedent can be considered as possible even though in fact it is false. For instance consider the claim that if Grant had been in command of the Union Army in 1861 then the Civil War would have lasted less than a month.

Jan: Most people would say that this conditional is incorrect; even though the antecedent was not satisfied. It seems implausible that Grant could have made that much difference.

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Bob: If we modify the consequent of the conditional to lasting less than 3 years, some people might argue that the conditional is correct and others that it was incorrect.

Kay: Neither side on such an issue is likely to settle the question by a direct application of boolean satisfaction. Something more about circumstances at the time would be needed. In fact boolean satisfaction seems totally irrelevant to what is at issue.

Roy: If Jill had not placed a sign over the door saying that this place is protected from dragons the house would have been invaded by dragons. The fact that she did place such a sign does not convince me that this statement is correct.

Note: Advocates of boolean satisfaction do have a strategy for using it with counterfactuals. They regard what appears to be a very specific conditional as merely a form of speech indicating a more general conditional, indicating a connection between kinds of events rather than specific events. A discussion of this strategy goes beyond the scope of this paper. Moreover it may not seem convincing to most people.

Bridging To Fraction Concepts: Of course this whole book is about this topic but we will now relate it directly to bridging without first thinking much about fractions.

Familiar Example: Jan gives Roy 2 quarters and 5 dimes for a dollar. This exchange involves equal monetary value. There are some reasons both parties might have for making this exchange. Perhaps Jan was glad to get rid of some of her excess change. Maybe she just she likes being helpful. It is easy to imagine many reasons that the change could have been useful to Roy. Perhaps he needed it to make a phone call (cell phones not yet having made more traditional pay phones obsolete) or use a vending machine.

Side Remark: Personalized equivalent value is determined by the parties involved, as in trading baseball cards or other instance of bartering or even trading money for goods when bargaining occurs. Standardized equivalent value is publicly recognized, as in the case of monetary exchange or using money to make purchases of priced items. Altho application of the trading principle to equivalent fractions is of the standardized type, considering example of the other type may also be useful, especially if the trading principle is also used as a bridge to understanding the evolution of various social practices for meeting human needs. See Activity 0c below.

Some possible responses to these some activities are given latter. Activities 0a-0c use concepts that will already seem highly manifest to anyone who understands the value of money. Activities 1a-1c use the same concepts for an imaginary money system. The other activities bridge to fraction concepts. Altho the activities are designed for mentors rather than for students, hopeful they will provide ideas for activities that could be used with a variety of different students.

Activity 0a: After is wrapping coins for a bank deposit, Bob has 3 quarters and 8 dimes and 11 nickels. He exchanges the 3 quarters and 7 of the dimes with some of his friends for 29 nickels. Consider reasons for the exchange. Imagine more examples of ordinary exchanges and reasons for them. Also consider and whether or not the equivalent value is standardized or personalized. If any of them seem interesting to you then you might want to share them with me or with others.

Activity 0b: The trading is sometimes used merely to make comparisons, altho such trading is usually merely conceptual. Kay and Bob are working with some younger children who have a limited mastery of numerical concepts. Mike has 7 dimes and Sue has 3 quarters. Both think that they have the most money. Bob explains that 7 dimes are worth 70¢ and that 3 quarters are worth 75¢. This is a conceptual change into pennies. Kay has them (temporarily) trade their coins for nickels. Discuss this situation, especially as it relates to bridging.

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Activity 0c: Value equivalence stays fixed with American Money. This is not the case with international currency exchange. Altho equivalent values are public, they fluctuate. According to www.x-rates.com:, on 1/9/08 you could have exchanged $91.33 and for 1000 Mexican Pesos. On 5/16/08, you could have exchanged 1000 for Mexican Pesos for $95.88. Consider various reasons for making these or similar exchanges.

An Imaginary Money System: In this system all, the money consists of colored rectangular pieces of different sizes and colors (identical to our fraction pieces but with a different use). Pieces are named by their colors. Value is determined by size. A white has the same size and value as two reds. It also has the same size and value as 3 yellows. It takes 4 blues to make a white. It takes 6 aquas to make a white. There is also a pink, a green, a violet. Altho this is an imaginary money system, it resembles a monetary system based on gold in one significant way. In both, value is based on size, spatial in the imaginary one and weight in the gold.

Reasons for Trading: Thinking of these pieces as money, any of the reasons for trading might apply. However in the Activities 1a-1d, you may merely focus on trades and ignore such reasons.

Activity 1a: We can trade 4 pinks for 3 aquas. Using pieces, we could fit them together to see that they are of equivalent value. We could also see that both could be traded either for a red or for 6 greens. Discuss other examples of equivalent value and indicate how this could be demonstrated.

Activity 1b: Kay has 2 yellows and Bob has 3 blues. Both can easily visualize that Bob has more money. Jan agrees but wants to demonstrate this by trading for greens. She says that there are cases where seeing who has the most by visualizing may be difficult, especially when more pieces are involved. She says that she cannot tell by visualizing that 7 pinks are worth more than 5 aquas. She claims trading will always work, because we could always trade for violets. Roy says that he could tell this without trading since 4 pinks is equivalent to 3 aquas and 3 pinks are more than 2 aquas. Describe this situation, including the trades Jan might make. Discuss other such comparisons.

Activity 1c: The comparison above involved comparing several pieces of one color with several pieces of another color. In order to compare 2 yellows and a red with 3 blues and 3 pinks, Bob trades the first for 28 violets the other for 27 violets. He says this is like trading for pennies. Jan trades 2 yellows and a red for a white and an aqua. She trades 3 blues and two pinks for a white and a pink. Check his trades and explain why this also settles the comparison. Kay continues from the trade Bob made to obtain 1 white and 4 violets in one case and a white and three violets in the other case. She says that this is like trading for dollars and pennies and writes her results as 1&4 and 1&3. Indicate how you could numerically convert her numbers to violets.

Activity 1d: Suppose we use all the pieces from the above activity. Then using Kay’s notation, their value is 2&7. Suppose we take one piece of each type. Using Kay’s notation, indicate their combined value. Describe a general way to determine the value of a combination of pieces.

Fraction Concepts: Pieces can be thought of as a fraction of a white. For instance, since 2 reds are equivalent to a white a red is ½ of a white. Comparing sizes is related to comparing fractions. For instance, saying that 7 pinks are worth more than 5 aquas can be bridged to saying 7/8 > 5/6. Trading provides one way to see this. Trading 7 pinks for violets gives 21 of them, and so 7/8 = 21/24. Trading 5 aquas gives 20 violets, and so 5/6 = 20/24. This gives an initial reason for raising fractions that does not involve the more remote concepts of adding or subtracting fractions. Moreover comparison can be bridged to subtraction by asking how the difference between 7 pinks and 5 aquas.

Activity 2a: Comparing 2 yellows and a red with 3 blues and 3 pinks can be bridged both to adding fractions and to changing improper fractions to mixed numerals. Explain how to think about this. Give additional examples.

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Activity 2b: Trading can be purely conceptual, i.e. it need not be implement with actual pieces. A person with a plot that is ¾ of an acre and a plot that is ½ an acre does not have to make an actual trade to know that he has 1¼ acres. Adding fractions, is a form of making a conceptual trade of ¾½ for 1¼. Discuss reasons for making this conceptual trade or for making it in the opposite direction.

Activity 2c: Sue and Joe each have 3 plots of land. Sue’s plots are {12/3, 1, 1/2} acres. Joe’s plots are {11/3, 5/6, 3/4} acres. Without using either fraction pieces or numerical calculations try to decide whether Sue has more than 3 acres of land. Do the same for Joe. Also see if you can tell who has the most land. Using faction pieces determine how much land each of them has. Use this activity as a bridge to adding fractions. Add any additional comments on this situation that you want to make.

Note: A multitude of such activities involving fractions and these colored fraction pieces is given in my book entitled Understanding Fractions. This book provides materials and ideas that relate fraction concepts to a simple understanding of counting and ordinary skills in visualization. This book is free and can be downloaded from my website. I wish that all adults who can imagine some need to deepen their understanding of fraction concepts would use it. However I realize that while a desire to learn is an essential part of a person’s motivation, an incentive beyond this is also useful. The incentive I propose is that a deep understanding of fraction concepts is essential for adults who want to help children in constructing these concepts. Because of this, the book includes the types of materials, activities, and questions that might be used as a resource for helping children acquire fraction concepts.

Responses to Some of the Activities: There are many appropriate responses that could be made to the activities. Below are some samples of them.

Response 0a: Since 40 nickels are needed for a roll, it seems likely that Bob wants to make such a roll and thus reduce the amount number of lose coins hat he has. His friends are probably just trying to be helpful, altho some of them may also prefer having fewer nickels. It is easy to imagine more examples of ordinary exchanges. For instance, Mac gives Jim three comic books for one that Jim gives him. Mac is interested it fills a gap in his collection. Jim just wants to increase the number of books he has to trade. The value equivalence involved is personalized rather than standardized.

Response 0b: In neither case is there a reason for an actual permanent trade. Trading is a way of bridging from the knowledge of how to trade specific coins to how value can be measured. Bob wants them to see how we normally measure small values in cents. He does it conceptually, since trading for that many pennies is tedious. Kay’s goal is less ambitious. She uses actual nickels, making the trades more manifest. Kay tells Bob that what she is doing is like using the smallest common denominator to compare fractions.

Response 0c: A possible response the dollar-peso activity could involve a tourist going into Mexico and wanting to use the local currency. It could also involve a currency trader making about a 5% profit altho such a trader would not bother with such a small amount of money.

Response 1c: Using Jan’s trade, since an aqua is larger than a pink, a white and an aqua is larger than a white and a pink. To convert to 1&4 and 1&3, use 244 and 243. In general x&y converts to x24y.

Response 1d: Using all the pieces from the above activity and using Kay’s notation their value is 2&7. If we take one piece of each type the value is 2&12. For a general way to determine the value of a combination of pieces with w whites, r reds, …, v violets, calculate 24w12r8y6b4a3p+2g1. The value is x&y where x is the quotient and y is the remainder when this number is divided by 24.

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Response 2a: 2 yellows and a red can be traded for 4 aquas and 3 aquas, which give 7 aquas. Bridging to adding fractions a follows 2/31/2 = 4/63/6 = 7/6. Trading 6 of these for a white gives a white and aqua, which bridges to changing 7/6 to 11/6. Likewise 3 blues can be traded for 6 pinks, and combining give 9 pinks, which can be traded for a white and a pink. Bridging to fractions give 3/43/8 = 6/83/8 = 9/8 = 11/8. A short cut can be had by trading 2 pinks for a blue and 4 blues for a white giving a white and a pink. This corresponds to 3/43/8 = 3/41/4 1/8 = 11/8.

Response 2b: The simplest reason for the conceptual trade is to see how much land he has. We might trade in the other direction if we knew how much land he had and the size of one of the plots.

Response 2c: Sue clearly has 2 acres plus 2/3 and 1/2, and since 2/3 is greater that 1/2, she has more than 3 acres. Joe only has one plot of more than 1 acre. The other are missing 1/6 and 1/4 and the extra 1/3 acre is not large enough compensate for both of these smaller plots. Since Sue has more than 3 acres, she has more land. Sue’s total land is can be represented as a white and 2 yellow plus a white plus a red. Trading the yellows and reds for aquas and then trading 6 aquas for a white give 3 whites and an aqua. This bridges to fractions as 12/311/2 = 14/613/6 = 27/6 = 31/6. Joe’s total land can be given as a white and a yellow plus 5 aquas plus 3 blues. Trading all but the white for greens gives 4109, which is 23 greens. These can be traded for a white and 11 greens. Bridging to fractions give 11/35/63/4 = 14/1210/129/12 = 123/12 = 211/12. We could also use color pieces to bridge to the equation 11/35/63/4 =12/65/63/4 = 21/63/4 = 22/129/11 = 211/12.

Comment: Calculating with fractions is clearly more efficient than manipulation color pieces, at least if a person knows how to calculate with fractions. However merely knowing how to calculate with fractions does not automatically mean fractions concepts have been adequately mastered. Nor does it mean that a person will know when to use factions or that fraction concepts are well integrated into a person’s world. Unless this happens the ability to calculate with fractions is likely to fade and the utility of fraction concepts will be limited. The advantage of using colored pieces is not that they merely allow a way to obtain answers but they do so in a way that forms a bridge to making fraction concepts seem manifest and thus become a part of a person’s world. Of course many people master fraction concepts without using this particular bridge. I once worked with a young woman who was beginning a business that involved interior decorating. She was able to bridge to fraction concepts by relating them to practical problems of estimating cost of various projects. Many persons with a good grasp of whole numbers can use this understanding to bridge to fraction concepts. The use of color pieces is especially useful for persons who like a visual model.

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FRACTION PIECES

Since the pieces are all fractional parts of a 4 by 6 rectangle, they could all be made 4 by 6 cards. You could use colored cards or white cards and magic markers. However for convenience this present file contains patterns for making fraction pieces using a color printer. We have found that laminating the pages and then cutting makes pieces that are very easy to use. We also use card stock instead of regular paper. Making the pieces and cutting them out is an excellent way to introduce them to young children.

The four pages after this one have patterns for making what we recommend as a set of fraction pieces for one person. This same set would also be adequate for by a mentor with one person or a small group. These pages contain enough pieces of each type to make a white.

The fifth page after this includes some sample odd shaped pieces. Pieces like this can be used to help reinforce the concept of equivalent fractions, as indicated in the Commentary for Chapter 1.

The last page of the section contains a sample of each piece representing a unit fraction with 1inch cells separated by dotted lines. Most of the time we recommend using pieces that do not show the small cells, so this page is probably only needed if you are working with younger children.

If you want to make additional pieces, without making a whole set you can use the following procedure:

(1) Use the file menu to make a page setup with margins set as small as possible. One way to do this is to set them to 0 and choose the fix option when they say you are out of bounds.

(2) Beginning on a new page, use the table menu to insert a table with 8 columns and 10 rows.

(3) Use table menu to set column width to 1” and row height to 72pt.

(4) Select the table and use format window to Box the table with a 6pt border.

(5) Select cells needed to make the pieces you want, marking their borders with the 6pt borders and shading the interior with the appropriate color.

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To Make these copy print, laminate, cut apart.

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