Example: Pies per child<\/h3>\r\n
Suppose 6 pies are to be shared equally among 3 children. This yields 2 pies per kid. We write \\[\u00a0\\frac{6}{3} = 2. \\]<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/6-pies-3-children.png\")
<\/p>\r\n
The fraction\u00a0$\\frac 6 3$\u00a0is equivalent to the division problem $6 \\div 3 = 2$. It represents the number of pies one whole child receives when three kids share six pies equally.<\/p>\r\n
In the same way ...<\/p>\r\n\r\n
- \r\n \t
- Sharing 10 pies among 2 kids yields\u00a0$\\frac{10}2 = 5$\u00a0pies per kid.<\/li>\r\n \t
- Sharing 8 pies among 2 children yields\u00a0$\\frac 8 2 = 4$\u00a0pies per child.<\/li>\r\n \t
- Sharing 5 pies among 5 kids yields\u00a0$\\frac 5 5 =1$\u00a0pie per kid.<\/li>\r\n \t
- Sharing 1 pie among 2 children yields\u00a0 $\\frac 1 2$, which we call \u201cone-half.\u201d<\/li>\r\n<\/ul>\r\n<\/div>\r\n
This final example is actually saying something! It also represents how fractions are usually taught to students:<\/p>\r\n
If one pie is shared equally<\/em> between two kids, then each child receives a portion of a pie which we choose to call \u201chalf.\u201d<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/1-pie-2-children.png\")
<\/p>\r\nThus students are taught to associate the number \u201c$\\frac 1 2$ \u201d to the picture
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onehalf.png\")
.<\/p>\r\nIn the same way, the picture
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onethird.png\")
is said to represent \u201cone-third,\u201d that is, $\\frac 1 3$. (And this is indeed the amount of pie an individual child would receive if one pie is shared among three.)<\/p>\r\nThe picture
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onefifth.png\")
is called \u201cone-fifth\u201d and is indeed $\\frac 1 5$, the amount of pie an individual receives if one pie is shared among five children.<\/p>\r\nAnd the picture
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/threefifths.png\")
is called \u201cthree-fifths\u201d to represent $\\frac 3 5$, the amount of pie an individual receives if three pies are shared among five children.<\/p>\r\n\r\n\r\nThink \/ Pair \/ Share<\/h3>\r\n
Carefully explain why this is true: If five kids share three pies equally, each child receives an amount that looks like this:
.<\/p>\r\nYour explanation will probably require both words and pictures.<\/p>\r\n\r\n<\/div>\r\n
On Your Own<\/h3>\r\n
Work on the following exercises on your own or with a partner.<\/p>\r\n
1. Draw a picture associated with the fraction $\\frac 1 6$.<\/p>\r\n
2. Draw a picture associated with the fraction $\\frac 3 7$. Is your picture really the amount of pie an individual would receive if three pies are shared among seven kids? Be very clear on this!<\/p>\r\n
3. Let\u2019s work backwards! Here's the answer to a division problem:<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/two-fifths.png\")
<\/p>\r\nThis represents the amount of pie an individual kid receives if some number of pies is shared among some number of children. How many pies? How many children? How can you justify your answers?<\/p>\r\n
4. Here's another answer to a division problem:<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/four-fifths.png\")
<\/p>\r\nHow many pies? How many children? How can you justify your answers?<\/p>\r\n
5. Here is another answer to a division problem:<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/four-sevenths.png\")
<\/p>\r\nHow many pies? How many children? How can you justify your answers?<\/p>\r\n
6. Leigh says that \u201c$\\frac 3 5$\u00a0\u00a0is three times as big as\u00a0 $\\frac 1 5$.\u201d Is this right? Explain your answer.<\/p>\r\n
7. Draw a picture for the answer to the division problem $\\frac 4 8$. Describe what you notice about the answer.<\/p>\r\n
8. Draw a picture for the answer to the division problem $\\frac 2{10}$. Describe what you notice about the answer.<\/p>\r\n
9. What does the division problem\u00a0$\\frac 1 1$\u00a0represent? How much pie does an individual child receive?<\/p>\r\n
10. What does the division problem\u00a0$\\frac 5 1$\u00a0represent? How much pie does an\u00a0 individual child receive?<\/p>\r\n
11. What does the division problem\u00a0$\\frac 5 5$\u00a0represent? How much pie does an individual child receive?<\/p>\r\n
12. Here is the answer to another division problem. This is the amount of pie an individual child receives:<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/one-and-one-half-300x155.png\")
<\/p>\r\nHow many pies were in the division problem? How many kids were in the division problem? Justify your answers.<\/p>\r\n
13. Here is the answer to another division problem. This is the amount of pie an individual child receives:<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/two-and-two-thirds-300x115.png\")
<\/p>\r\nHow many pies were in the division problem? How many kids were in the division problem? Justify your answers<\/p>\r\n
14. Many teachers have young students divide differently shaped pies into fractions. For example, a hexagonal pie is good for illustrating the fractions:<\/p>\r\n\\[ \\frac{1}{6}, \\frac{2}{6}, \\frac{3}{6}, \\frac{4}{6}, \\frac{5}{6}, \\text{ and } \\frac{6}{6}.\\]\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/sixths-hexagon.png\")
<\/p>\r\n\r\n- \r\n \t
- Why is this shape used? What does\u00a0$\\frac 1 6$\u00a0of a pie look like?<\/li>\r\n \t
- What does\u00a0$\\frac 6 6$\u00a0of a pie look like?<\/li>\r\n \t
- What shape pie would be good for illustrating the fractions\u00a0$\\frac 1 8$\u00a0up to $\\frac 8 8$?<\/li>\r\n<\/ul>\r\n \r\n\r\n
Problem 1<\/h3>\r\n
Some rectangular pies are distributed to some number of kids. This picture represents the amount of pie an individual child receives. The large rectangle represents one whole pie.<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/rectangular-pie.png\")
<\/p>\r\nHow many pies? How many kids? Carefully justify your answers!<\/p>\r\n\r\n<\/div>\r\n
Pies Per Child Model<\/h1>\r\n
In our model, a fraction\u00a0$\\frac a b$\u00a0represents the amount of pie an individual child receives when\u00a0$a$\u00a0pies are shared equally by\u00a0$b$\u00a0kids.<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/pies-per-child-model.png\")
<\/p>\r\n\r\n\r\nThink \/ Pair \/ Share<\/h3>\r\n
- \r\n \t
- What is $\\frac 2 2$? What is\u00a0$\\frac 7 7 $? What is$\\frac {100}{100}$? How can you use the \u201cPies Per Child Model\u201d to make sense of\u00a0$\\frac a a$\u00a0for any positive whole number $a$?<\/li>\r\n \t
- What is $\\frac 2 1$? What is $\\frac 7 1$? What is $\\frac {1876}1$? How can you use the \u201cPies Per Child Model\u201d to make sense of\u00a0$\\frac b 1$\u00a0for any positive whole number $b$?<\/li>\r\n \t
- Write the answer to this division problem: \u201cI have no pies to share among thirteen kids.\u201d How can you generalize this division problem to make a general statement about fractions?<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\n
Definition<\/h3>\r\n
For a fraction $\\frac a b$, the top number\u00a0$a$\u00a0(which, for us, is the number of pies) is called the numerator<\/strong> of the fraction, and the bottom number\u00a0$b$\u00a0(the number of kids), is called the denominator<\/strong> of the fraction.<\/p>\r\n\r\n<\/div>\r\n
Most people insist that the numerator and denominator each be whole numbers, but they do not have to be.<\/p>\r\n\r\n
\r\nThink \/ Pair \/ Share<\/h3>\r\n
To understand why the numerator and denominator need not be whole numbers, we must first be a little gruesome. Instead of dividing pies, let\u2019s divide kids! Here is one child:<\/p>\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hairpants.jpg\")
<\/p>\r\n\r\n
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/1-over-1_2.png\")
<\/p>\r\n![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onepie-150x150.png\")
\u00a0\u00a0![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/halfpie-1.png\")
<\/p>\r\n![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onekid.png\")
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onekid2.png\")
![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/halfkid.png\")
<\/p>\r\n\r\n<\/div>\r\n![\"\"](\"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onepie.png\")
\u00a0 \u00a0![\"\[ \frac{6}{3} = 2. \]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-88fd45c636697f90d54ed5d40d09c4cb_l3.png\" "\\"Rendered")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/6-pies-3-children.png\")
<\/p>\n![\"\frac 6 3\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-b36ea58fe5822f7338d5e8de05c29d6e_l3.png\" "\\"Rendered")
\u00a0is equivalent to the division problem ![\"6 \div 3 = 2\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-86165b890713be3418e2d7f76e03f723_l3.png\" "\\"Rendered")
. It represents the number of pies one whole child receives when three kids share six pies equally.<\/p>\n![\"\frac{10}2 = 5\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-2d466ed8207a8b3556c33d9066ab5c21_l3.png\" "\\"Rendered")
\u00a0pies per kid.<\/li>\n![\"\frac 8 2 = 4\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-b36f9a296c0eec551986938f63a798da_l3.png\" "\\"Rendered")
\u00a0pies per child.<\/li>\n![\"\frac 5 5 =1\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-f27669efb99b15db9ea712adb2046533_l3.png\" "\\"Rendered")
\u00a0pie per kid.<\/li>\n![\"\frac 1 2\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-63c46470be12c9c4c061babce2a728b4_l3.png\" "\\"Rendered")
, which we call \u201cone-half.\u201d<\/li>\n<\/ul>\n<\/div>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/1-pie-2-children.png\")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onehalf.png\")
.<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onethird.png\")
is said to represent \u201cone-third,\u201d that is, ![\"\frac 1 3\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-384b22d79bb44c2080d24b4589d6a780_l3.png\" "\\"Rendered")
. (And this is indeed the amount of pie an individual child would receive if one pie is shared among three.)<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onefifth.png\")
is called \u201cone-fifth\u201d and is indeed ![\"\frac 1 5\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-66534f6ba14c95ecf412c64507e699d4_l3.png\" "\\"Rendered")
, the amount of pie an individual receives if one pie is shared among five children.<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/threefifths.png\")
is called \u201cthree-fifths\u201d to represent ![\"\frac 3 5\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-c56f96390877df10fb76e7f65e3d603c_l3.png\" "\\"Rendered")
, the amount of pie an individual receives if three pies are shared among five children.<\/p>\n![\"\frac 1 6\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-eeebb3cae6f68bc778df86ec3fd679b5_l3.png\" "\\"Rendered")
.<\/p>\n![\"\frac 3 7\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-97fab1cc685948da3fb5946932e3b06d_l3.png\" "\\"Rendered")
. Is your picture really the amount of pie an individual would receive if three pies are shared among seven kids? Be very clear on this!<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/two-fifths.png\")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/four-fifths.png\")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/four-sevenths.png\")
<\/p>\n![\"\frac 4 8\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-3824892d1cdb79150b27e381a1f7ebb4_l3.png\" "\\"Rendered")
. Describe what you notice about the answer.<\/p>\n![\"\frac 2{10}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-3057c8247f6301efa20f06cbab6868a7_l3.png\" "\\"Rendered")
. Describe what you notice about the answer.<\/p>\n![\"\frac 1 1\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-e8dbddeb3d603a035afbaf82feeba7f9_l3.png\" "\\"Rendered")
\u00a0represent? How much pie does an individual child receive?<\/p>\n![\"\frac 5 1\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-68fb7f511f43585bf211213640f79cae_l3.png\" "\\"Rendered")
\u00a0represent? How much pie does an\u00a0 individual child receive?<\/p>\n![\"\frac 5 5\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-cc8037ee73902cc74d6cfa8a994425f1_l3.png\" "\\"Rendered")
\u00a0represent? How much pie does an individual child receive?<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/one-and-one-half-300x155.png\")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/two-and-two-thirds-300x115.png\")
<\/p>\n![\"\[ \frac{1}{6}, \frac{2}{6}, \frac{3}{6}, \frac{4}{6}, \frac{5}{6}, \text{ and } \frac{6}{6}.\]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-fd8b7de613326b03bec63b6f2aa5c903_l3.png\" "\\"Rendered")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/sixths-hexagon.png\")
<\/p>\n![\"\frac 6 6\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-08a105be0497907cf2ebb6491c7e39eb_l3.png\" "\\"Rendered")
\u00a0of a pie look like?<\/li>\n![\"\frac 1 8\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-8e603035a2f2d53e89cd2678a9b48fba_l3.png\" "\\"Rendered")
\u00a0up to ![\"\frac 8 8\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-3780832cf03f78560f2272e91a227ac3_l3.png\" "\\"Rendered")
?<\/li>\n<\/ul>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/rectangular-pie.png\")
<\/p>\n![\"\frac a b\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-28830ac8601c33c720c177cbcd21d5b9_l3.png\" "\\"Rendered")
\u00a0represents the amount of pie an individual child receives when\u00a0![\"a\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-3a23485890e06c0725fcc9e23d5eec94_l3.png\" "\\"Rendered")
\u00a0pies are shared equally by\u00a0![\"b\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-f7b67a53daa139387638458dabce5423_l3.png\" "\\"Rendered")
\u00a0kids.<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/pies-per-child-model.png\")
<\/p>\n![\"\frac 2 2\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-5ce9e8c05a9f6d057811788a26448e88_l3.png\" "\\"Rendered")
? What is\u00a0![\"\frac 7 7\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-c5115adf46591c4d7fa672835a1909e1_l3.png\" "\\"Rendered")
? What is![\"\frac {100}{100}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-abda853c0b3bc34096ce147d9807e008_l3.png\" "\\"Rendered")
? How can you use the \u201cPies Per Child Model\u201d to make sense of\u00a0![\"\frac a a\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-4b086207b38e603c44025b9efe5c6c3a_l3.png\" "\\"Rendered")
\u00a0for any positive whole number ![\"\frac 2 1\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-0e5a6c9d478ffa727d8325573df12a81_l3.png\" "\\"Rendered")
? What is ![\"\frac 7 1\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-cddea5d317e9f9cb285fd202a7698ac8_l3.png\" "\\"Rendered")
? What is ![\"\frac {1876}1\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-dfed474a049e87f6440cf4a0bc1339b7_l3.png\" "\\"Rendered")
? How can you use the \u201cPies Per Child Model\u201d to make sense of\u00a0![\"\frac b 1\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-b07cd3dfd23f89b5b47c4f1e954a164d_l3.png\" "\\"Rendered")
\u00a0for any positive whole number ![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/hairpants.jpg\")
<\/p>\n![\"\[\frac{1}{\left(\frac{1}{2}\right)}\]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-7abea1d6299cbcbb156d2017821b714d_l3.png\" "\\"Rendered")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/1-over-1_2.png\")
<\/p>\n![\"\[\frac{1}{\left(\frac{1}{2}\right)} = 2.\]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-4502eb0cc80eb705346cdcf4c19924d0_l3.png\" "\\"Rendered")
<\/p>\n![\"\[ \frac{1}{\left(\frac{1}{3}\right)} \]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-e42c28f6dfa1c0153c6dfab4df1ef8ac_l3.png\" "\\"Rendered")
<\/p>\n![\"\[ \frac{1}{\left(\frac{1}{6}\right)}?\]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-ffe61157d17e05f87195c690f663cb6d_l3.png\" "\\"Rendered")
<\/p>\n![\"\[ \frac{5}{\left(\frac{1}{2}\right)} \]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-b417ac703db4e5a078acf3d93e8419b8_l3.png\" "\\"Rendered")
<\/p>\n![\"\[ \frac{4}{\left(\frac{1}{3}\right)}? \]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-34bab7314af3933f8e489a9a84d766ab_l3.png\" "\\"Rendered")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onepie-150x150.png\")
\u00a0\u00a0![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/halfpie-1.png\")
<\/p>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onekid.png\")
![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onekid2.png\")
![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/halfkid.png\")
<\/p>\n<\/div>\n![\"\frac{45}{58}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-a7f251f67a82b4d5e83119b3cf196527_l3.png\" "\\"Rendered")
\u00a0is \u201cproper.\u201d<\/p>\n![\"\frac 7 3\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-f1000d6758971eae9e9ce436b0102532_l3.png\" "\\"Rendered")
\u00a0is \u201cimproper.\u201d (In the 1800’s, these fractions were called vulgar fractions.<\/i>)<\/p>\n![\"{\color{white}7/3}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-b27fb48234e3646fce0ab843f3a17ff0_l3.png\" "\\"Rendered")
<\/h3>\n![\"\"](\"\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/uploads\/sites\/17\/2017\/10\/onepie.png\")
\u00a0 \u00a0![\"\frac{7}{3}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-8b1e601a5f1f44530ab60122d5a3cf9b_l3.png\" "\\"Rendered")
\u00a0equals\u00a0![\"2\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-da7e4c36c9cf2ba30ec4638d230f1efb_l3.png\" "\\"Rendered")
\u00a0plus ![\"\[ \frac{7}{3}](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-d078ee2d6afe73106255a4ccd5c2364f_l3.png\" "\\"Rendered")
<\/p>\n![\"\[ 2 + \frac{1}{3}, \]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-14f4bf7eacc6c203366f94e75296e04c_l3.png\" "\\"Rendered")
<\/p>\n![\"2\frac 1 3\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-9a6133270f279a81f3ebad02b2a45088_l3.png\" "\\"Rendered")
\u00a0really means. But most people choose to omit the plus sign.<\/p>\n<\/div>\n![\"{\color{white}23/4}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-55dd9cf8ebadcb2ede2152f9c9b78e16_l3.png\" "\\"Rendered")
<\/h3>\n![\"\[ \frac{23}{4} \ = \ 5 \tfrac{3}{4}. \]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-ae57bc6c1ccd664cced412d7a1f6ccda_l3.png\" "\\"Rendered")
<\/p>\n<\/div>\n![\"{\color{white}2\frac{1}{5}}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-13abcc7736738ba3006df55ad66250b8_l3.png\" "\\"Rendered")
<\/h3>\n![\"\[ 2 = \frac{10}5.\]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-e9115dc87f39e594ed78f885537d4fc2_l3.png\" "\\"Rendered")
<\/p>\n![\"\[ 2 \tfrac{1}{5}](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-6af5bbcdd98032fbc743a1ba31c75077_l3.png\" "\\"Rendered")
<\/p>\n![\"2 \tfrac{1}{5}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-49f884ec8564b1cc331aec62103b43ab_l3.png\" "\\"Rendered")
\u00a0as the improper fraction ![\"\frac{11}{5}\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-b067221493f2e25545330499f7c80f44_l3.png\" "\\"Rendered")
.<\/p>\n<\/div>\n![\"\[\frac{17}{3}, \qquad\qquad \frac{8}{5}, \qquad\qquad \frac{100}{3}, \qquad\qquad \frac{200}{199}.\]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-e2cbac4be3e7f04331b75c91205ea99c_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n![\"\[ 3\frac{1}{4}, \qquad\quad 5\frac{1}{6}, \qquad\quad 1\frac{3}{11}, \qquad\quad 200\frac{1}{200}.\]\"](\"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-content\/ql-cache\/quicklatex.com-c10edcf3dd887c01cf80e6d2b8f583ea_l3.png\" "\\"Rendered")
<\/p>\n<\/li>\n<\/ul>\n<\/div>\n