Fractions are Hard!

2.2 Pies per boy

We like to say that a fraction is the answer to a sharing problem, that is, a division problem.

 

Example: Suppose 6 pies are to be shared equally among 3 boys. This yields two pies per boy.

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We write:

\(\dfrac{6}{3}=2\).

We might also write  or  \(6 \div 3 = 2\).

 

We also have:

sharing 10  pies among 2 boys yields: \(\dfrac{10}{2}=5\)  pies (per boy),

sharing 8 pies among 2 boys yields:  \(\dfrac{8}{2}=4\) pies,

sharing 5 pies among 5 boys yields: \(\dfrac{5}{5}=1\) pie,

and

the answer to sharing 1 pie among 2 boys is \(\dfrac{1}{2}\), which we call “one half.”

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OUR FIRST FRACTION BELIEFS

 

Suppose we share 20 pies equally among 20 boys. How many pies per boy does that yield?

 

\(\dfrac{20}{20}=1\).

 

Suppose we share 503 pies equally among 503 boys. How many pies per boy does that yield?

 

\(\dfrac{503}{503}=1\).

 

And so on. It seems we have

 

BELIEF 1:           \(\dfrac{a}{a}=1\)

(at least for any positive counting number , but perhaps for other types of numbers too!)

 

 

Suppose we share 5 pies among one (lucky) boy. How many pies per boy is that?

 

\(\dfrac{5}{1}=5\).

 

Suppose we share 202 pies among one (lucky) boy. How many pies per boy is that?

 

 \(\dfrac{202}{1}=202\).

 

And so on. It seems we have

 

BELIEF 2:           \(\dfrac{a}{1}=a\)

(at least for any positive counting number , but perhaps for other types of numbers too.)

 

 

Thinking Question: “I have no pies to share among seven boys.”

Use this to make a mathematical statement about a class of sharing problems.

 

These first two beliefs about the mechanics of fractions feel so natural and right. It seems they should hold for all types of numbers. Let’s go with the idea that they do.

 

ASIDE ON JARGON: In a sharing problem expressed as \(\dfrac{a}{b}\) we call \(a\) the numerator of the expression and \(b\) the denominator of the expression.

 

PICTURING FRACTIONS

We call \(\dfrac{1}{n}\), the result of sharing \(1\) pie among \(n\)  boys,  “one \(n\)th.”   And we can draw the pictures associated with these quantities.

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How should we think about a more complex fraction such as \(\dfrac{3}{5}\)?

 

In our sharing model this is the answer to sharing three pies equally among five boys:

f

 

A Moment to Think: How might one physically accomplish the task of sharing three pies equally among five boys?

 

Answer: One could divide each of the three pies into five equal pieces (fifths) and give one piece from each pie to each boy. This approach gives each boy three \(\frac{1}{5}\)s.  For this reason, we call \(\frac{3}{5}\)  “three fifths.” (And this matches the early-grade approach of regarding  \(\frac{3}{5}\) as three copies of \(\frac{1}{5}\).)

 

A Moment to Think: GOING BACKWARDS

Suppose we were given the picture of the result of a sharing problem first. For example, suppose we are shown this picture only.

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How do we explain clearly that this is the result of sharing three pies equally among five boys?

 

Some questions to mull on.

 

Question 1:

a)       Here is the answer to a sharing problem:

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This represents the amount of pie an individual boy receives if some number of pies is shared among some number of boys.

 

How many pies?  _________

How many boys?  _________

(Are answers here unique?)

 

b)      Here is the answer to another sharing problem:

f

How many pies?  _________

How many boys?  _________

(Are answers here unique?)

 

Question 2:  Leigh says that “\(\dfrac{10}{13}\) is two times as big as \(\dfrac{5}{13}\).” She argues:

In one room, ten pies are shared among thirteen boys.

In another room, five pies are shared among thirteen boys.

Each boy in the first room receives twice as much pie as each boy in the second room.

Do you agree?

 

 

A COMMENT ON MIXED NUMBERS

Here’s a picture of two and a half pies, that is, two pies and another half a pie: \(2 + \dfrac{1}{2}\).

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Even though we say the word “and” out loud, we tend to omit the plus sign in our mathematical writing. Thus \(2 + \dfrac{1}{2}\) is usually written \(2\dfrac{1}{2}\). (And do you see that “\(2\dfrac{1}{2}\)” is the result of sharing 5 pies among 2 boys?)

 

In the same way,  \(3\dfrac{4}{27}\)really means \(3+\dfrac{4}{27}\), and  \(200\dfrac{1}{200}\)really means \(200+\dfrac{1}{200}\), and so on.

[We can even write  \(7\dfrac{18}{5}\) if we like. Why not?]

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