{"id":265,"date":"2017-10-28T22:26:20","date_gmt":"2017-10-28T22:26:20","guid":{"rendered":"http:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/chapter\/adding-and-subtracting-fractions\/"},"modified":"2018-05-25T01:33:22","modified_gmt":"2018-05-25T01:33:22","slug":"adding-and-subtracting-fractions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/chapter\/adding-and-subtracting-fractions\/","title":{"raw":"Adding and Subtracting Fractions","rendered":"Adding and Subtracting Fractions"},"content":{"raw":"

[latexpage]\n\n

Here are two very similar fractions: $\\frac 2 7$ and $\\frac 3 7$. What might it mean to add them? It might seem reasonable to say:\n\n\\[ \\frac 2 7 \\text{ represents 2 pies shared by 7 kids.} \\]\n\n\\[ \\frac 3 7 \\text{ represents 3 pies shared by 7 kids.} \\]\n\n

So maybe $\\frac 2 7 + \\frac 3 7 $ represents 5 pies among 14 kids, giving the answer $\\frac 5{14}$.  It is very tempting to say that \u201cadding fractions\u201d means \u201cadding pies and adding kids.\u201d\n\n

The trouble is that a fraction is not a pie, and a fraction is not a child. So adding pies and adding children is not actually adding fractions. A fraction is something different. It is related to pies and kids, but something more subtle. A fraction is an amount of pie per child<\/i>.\n\n

One cannot add pies, one cannot add children. One must add instead the amounts individual kids receive.\n\n

Example: 2\/7 + 3\/7<\/h3>

Let us take it slowly. Consider the fraction $\\frac 2 7$. Here is a picture of the amount an individual child receives when two pies are given to seven kids:\n\n

\"\"\n\n

Consider the fraction $\\frac 3 7$. Here is the picture of the amount an individual child receives when three pies are given to seven children:\n\n

\"\"\n\n

The sum $ \\frac 27 + \\frac 37 $ corresponds to the sum:\n\n

\"\"\n\n

The answer, from the picture, is $\\frac 5 7$.\n\n<\/div> \n\n

Think \/ Pair \/ Share<\/h3>

Remember that $\\frac 5 7$ means \u201cthe amount of pie that one child gets when five pies are shared by seven children.\u201d Carefully explain why<\/i> that is the same as the picture given by the sum above:\n\n

\"\"\n\n

Your explanation should use both words and pictures!\n\n<\/div>

Most people read this as \u201ctwo sevenths plus three sevenths gives five sevenths\u201d and think that the problem is just as easy as saying \u201ctwo apples plus three apples gives five apples.\u201d And, in the end, they are right!\n\n

\"\"\n\n

This is how the addition of fractions is first taught to students: Adding fractions with the same denominator seems just as easy as adding apples:\n\n

4 tenths + 3 tenths + 8 tenths =  15 tenths.\n\n\\[ \\frac 4{10} + \\frac 3{10} + \\frac 8{10} = \\frac{15}{10}. \\]\n\n

(And, if you like, $\\frac{15}{10} = \\frac{5\\cdot 3}{5 \\cdot 2} = \\frac 3 2$.)\n\n

82 sixty-fifths + 91 sixty-fifths  = 173 sixty-fifths:\n\n\\[ \\frac{82}{65} + \\frac{91}{65}\n\\quad =\\quad\n\\frac{173}{65}. \\]\n\n

We are really adding amounts per child<\/b> not amounts, but the answers match the same way.\n\n

We can use the \u201cPies Per Child Model\u201d to explain why<\/i> adding fractions with like denominators works in this way.\n\n

Example: 2\/7 + 3\/7<\/h3>

Think about the addition problem $\\frac 2 7 + \\frac 3 7$:\n\n\\begin{align*}\n\\phantom{+}\\text{ amount of pie each kid gets when 7 kids share 2 pies}\\\\\n\\underline{+ \\text{ amount of pie each kid gets when 7 kids share 3 pies}}\\\\\n\\phantom{+} \\text{?????} \\qquad\\qquad\\qquad\\qquad\\qquad\n\\end{align*}\n\n \n\nSince in both cases we have 7 kids sharing the pies, we can imagine that it is the same 7 kids in both cases.  First, they share 2 pies. Then they share 3 more pies. The total each child gets by the time all the pie-sharing is done is the same as if the 7 kids had just shared 5 pies to begin with. That is:\n\n\\begin{align*}\n\\phantom{+}\\text{ amount of pie each kid gets when 7 kids share 2 pies}\\\\\n\\underline{+ \\text{ amount of pie each kid gets when 7 kids share 3 pies}}\\\\\n\\phantom{+} \\text{amount of pie each kid gets when 7 kids share 5 pies.}\n\\end{align*}\n\n \n\n\\[ \\frac 2 7 + \\frac 3 7 = \\frac 5 7.\\]\n\n<\/div>

Now let us think about the general case. Our claim is that\n\n\\[ \\frac a d + \\frac b d = \\frac {a+b} d.\\]\n\n

Translating into our model, we have $d$ kids. First, they share $a$ pies between them, and $\\frac a d$ represents the amount each child gets. Then they share $b$ more pies, so the additional amount of pie each child gets is $\\frac b d$. The total each kid gets is $\\frac a d + \\frac b d$.\n\n

But it does not really matter that the kids first share $a$ pies and then share $b$ pies. The amount each child gets is the same as if they had started with all of the pies \u2014 all $a+b$ of them \u2014 and shared them equally. That amount of pie is represented by $\\frac{a+b} d$.\n\n

Think \/ Pair \/ Share<\/h3>