Converting Between Mixed And Improper Fractions Accounting Essay

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Let's say you had a Super Bowl party and ordered pizzas for all your friends to eat while you watched the game. The next day there was one full Hawaiian pizza and half an anchovy pizza left over. You would say you had "one and a half" pizzas left over from the party. "One and a half" is the typical way of expressing verbally the amount of pizzas leftover. Written as a number you would write it as a mixed number "1 1/2".

While mixed numbers are the natural choice for verbal expression as well as the answer to a word problem, mixed numbers are not the easiest type of fraction to calculate with. When performing mathematical operations with fractions you will find it easier to deal with improper fractions rather than mixed numbers

Improper Fractions [glossary term; a fraction where the top number is bigger than the top number. http://www.bing.com/Dictionary/search?q=define+improper+fraction&FORM=DTPDIA] and Mixed Numbers [glossary term; a fraction number consisting of a proper fraction and a whole number together http://www.bing.com/Dictionary/search?q=define+mixed+number&FORM=DTPDIA]

You can use either an improper fraction or a mixed fraction to show the same amount.

[Pictures may need to be redone. Taken from http://www.mathsisfun.com/improper-fractions.html]

1 3/4 = 7/4

=

1 1/4 = 5/4

=

2 3/8 = 119/8 [Divide the two whole pizzas into 8ths]

=

What if I had multiple pieces of different pizzas left over from the party.

How could I tell someone how much I had left over?

Before we learn how to formally do operations with fractions, we need to be able to convert our mixed numbers to improper fractions and before we can verbally discuss our results we need to be able to convert our improper fractions into mixed numbers.

Converting from Mixed to Improper

To convert a mixed number to an improper fraction we need to remember what a fraction means.

The numerator in a fraction tells us how many pieces of a whole we have. The denominator tells us how many pieces the whole has been broken in to.

Given the mixed number 1 ½, we see the proper fraction of ½ tells us we have one of two pieces. The whole number 1 says we have two of two pieces or 2/2. If you were given 2 of two pieces and 1 more of 2 pieces, you would have 3 of two pieces or 3/2.

That one was easy. Let's look at the mixed number 3 2/5.

Using the reasoning from above, we see the proper fraction of 2/5 tells us we have two of five pieces. The whole number 3 says we have five of five pieces 3 times. Five of five pieces 3 times would suggest we have 15 pieces of five or 15/5. If you were given 15 of five pieces and 2 more of five pieces, you would have 17 of five pieces or 17/5.

The mixed number 3 2/5 is the same amount as 17/5

[chalk talk another example in words. Don't do the math steps yet. We want a true understanding of the concept before we introduce the algorithm]

[BEGIN YOUR TURN this would be a good applet to begin before we do the algorithm of how to convert. http://www.visualfractions.com/MixedFraction.html ]

Once you understand the idea of mixed numbers and improper fractions are two different ways of stating fractions more than 1, then the math related to the conversion is a piece of cake, or pizza.

Let's look at the mixed number 2 5/8.

To convert the mixed number to an improper fraction first we must multiply the denominator by the "whole" number to determine how many pieces are in our wholes.

2 5/828=16

There are sixteen pieces in our two whole pizzas.

[Use the picture above but divide the two whole pizzas into 8ths please.]

We now see that we have 16 pieces of eight from our wholes and we have 5 more pieces of eight from our fraction.

Add the numerator of the proper fraction 5/8 with our 16 pieces from our wholes. This makes 21 pieces of eight or 21/8

Mathematically it looks like this:

2 5/8  28 = 16+5 = 21  21/8

Notice how the denominator does not change. Only the number of pieces in the numerator change. A fraction tells us how many pieces a whole was broken into and how many pieces of a whole we have.

To convert a Mixed Number to an Improper fraction, follow these steps:

Multiply the whole by the denominator of the proper fraction.

Add this product to the numerator of the proper fraction.

The proper fraction now becomes an improper fraction.

The denominator stays the same.

Let's do another conversion from a mixed number, 5 2/3 to an improper fraction.

5 2/3  53 = 15+2 = 17  17/3

5 2/3 = 17/3

[YOUR TURN drag and drop or a, b, c kind of thing. answers are directly across from each other now.]

Match the mixed number to its improper fraction equivalent.

3 4/5 19/5

4 3/5 23/5

5 ¾ 23/4

7 ¼ 29/4

5 7/8 47/8

7 5/8 61/8

8 5/7 61/7

5 3/7 38/7

[END YOUR TURN]

Converting from Improper Fractions to Mixed Numbers

Looking at the left over portions of the pizza above we can tell how many slices we have left just by counting them. There are 17 slices of pizza left. Each pizza was cut into 8 equally sized slices. We could state our left over pizza as an improper fraction 17/8.

If we were going to talk about the amount of left over pizza we wouldn't say seventeen eighths we would want to use a mixed number.

How do we convert improper fractions to mixed numbers? It would make sense to do the opposite of converting mixed numbers to improper fractions.

To convert an improper fraction to a mixed fraction, follow these steps:

Divide the numerator by the denominator.

This will help us decide how many wholes we have.

Write down the whole number answer

Use the remainder from the division of the numerator by the denominator as the numerator of the proper fraction.

The denominator stays the same.

To follow the steps for the conversion of an improper fraction 17/8 to a mixed number we first need to divide 17 by 8.

8 divides into 17 twice. This tells us we have 2 wholes.

The remainder of 1 becomes the numerator of our proper fraction 1/8 and the denominator stays the same.

Put it all together and 17/8 turns into 2 1/8

[Can you create an animation of the pizza slices combining to become 2 1/8 please?]

So there you have it. To convert from an improper fraction to a mixed number you simply divide the numerator by the denominator to determine the number of wholes. Then use the remainder of that division to represent the proper fraction portion of the mixed number.

[Chalk talk the steps one more time.]

[YOUR TURN this is a great game for improper fractions and figuring out what they mean in terms of a whole scoop of ice cream. http://www.mrnussbaum.com/icecream/index.html

END YOUR TURN]

Now that you can convert mixed and improper fractions back and forth easily, let's see if you can tell their relative size in either form.

Position the following mixed and improper fractions on the number line below.

8 ½, 7/2, 15/2, 2 ½

[Make this one animated. Convert the improper fractions to Mixed numbers then show the mixed fractions flying onto the number line in its proper position.]

Just as it's easier for us to use mixed numbers when talking about everyday items, it's easier for us to relate the size of a fraction when it's in mixed number form.

[BEGIN YOUR TURN create a set of 3 drag and drops onto the number lines, please.]

5 3/8, 24/8, 6 5/8, 37/8

8 5/7, 50/7, 6 3/7, 63/7

3 1/5, 17/5, 15/5, 2 4/5

[END YOUR TURN]

[BEGIN REVIEW]

1. Write the improper fraction 22/5 as a mixed number.

*4 2/5

3 7/5

5 3/5

10 2/5

Feedback: Divide the numerator by the denominator to find the wholes. Use the remainder as the numerator of the proper fraction. The denominator does not change.

2. Write the improper fraction 25/5 as a mixed number.

*5

5 1/5

5 0/5

5 2/5

Feedback: Divide the numerator by the denominator to find the wholes. Use the remainder as the numerator of the proper fraction in this case there is NO remainder so there is NO fractional part of the mixed number.

3. Write the mixed number 3 2/3 as an improper fraction.

*11/3

8/3

10/3

12/3

Feedback: 3 2/3  33 = 9+2 = 11  11/3 Multiply the whole by the denominator of the proper fraction then add this product to the numerator of the proper fraction.

4. Write the mixed number 3 2/7 as an improper fraction.

*23/7

21/7

12/7

22/7

Feedback: 3 2/7  37 = 21+2 = 23  23/7 Multiply the whole by the denominator of the proper fraction then add this product to the numerator of the proper fraction.

5. Order the mixed and improper fractions from least to greatest.

*, , ,

, , ,

, , ,

, , ,

Feedback: Since they are equal, they are interchangeable, ,

5. Order the mixed and improper fractions from greatest to least.

*, , ,

, , ,

, , ,

, , ,

Feedback: , , ,

7. To convert a mixed number to an improper fraction, multiply the whole by the denominator of the proper fraction and add this product to the numerator of the proper fraction.

*True

False

8. Choose the statement below that is incorrect.

The denominator never changes when you convert mixed numbers to improper fractions.

When converting an improper fraction to a mixed fraction, use the remainder as the numerator of the proper fraction.

To convert the mixed number to an improper fraction first we must multiply the denominator by the "whole" number to determine how many pieces are in our wholes.

*The numerator in a fraction tells us how many pieces a whole is broken into. The denominator tells us how many pieces we have of the whole.

Feedback: The numerator in a fraction tells us how many pieces of a whole we have. The denominator tells us how many pieces the whole has been broken in to.

[END REVIEW]

[BEGIN QUIZ]

1. Write the improper fraction 37/6 as a mixed number.

*6 1/7

6 2/7

5 6/7

5 5/7

Feedback: Divide the numerator by the denominator to find the wholes. Use the remainder as the numerator of the proper fraction. The denominator does not change.

2. Write the improper fraction 49/7 as a mixed number.

*7

7 1/7

6 6/7

6 7/7

Feedback: Divide the numerator by the denominator to find the wholes. Use the remainder as the numerator of the proper fraction in this case there is NO remainder so there is NO fractional part of the mixed number.

3. Write the mixed number 10 2/5 as an improper fraction.

*52/5

17/5

22/5

57/5

Feedback: 10 2/5  105 = 50+2 = 52  52/5 Multiply the whole by the denominator of the proper fraction then add this product to the numerator of the proper fraction.

4. Write the mixed number 3 2/7 as an improper fraction.

*23/7

21/7

12/7

22/7

Feedback: 3 2/7  37 = 21+2 = 23  23/7 Multiply the whole by the denominator of the proper fraction then add this product to the numerator of the proper fraction.

5. Order the numbers from least to greatest.

*

Feedback: , , ,

5. Order the mixed and improper fractions from greatest to least.

*

Feedback:

7. To convert an improper fraction to a mixed number, multiply the whole by the denominator of the proper fraction and add this product to the numerator of the proper fraction.

True

*False

Feedback: To convert an improper fraction to a mixed fraction, follow these steps:

Divide the numerator by the denominator.

This will help us decide how many wholes we have.

Write down the whole number answer

Use the remainder from the division of the numerator by the denominator as the numerator of the proper fraction.

The denominator stays the same.

8. Choose the statement below that is incorrect.

The denominator never changes when you convert mixed numbers to improper fractions.

*When converting an improper fraction to a mixed fraction, use the remainder as the denominator of the proper fraction.

To convert the mixed number to an improper fraction first we must multiply the denominator by the "whole" number to determine how many pieces are in our wholes.

The numerator in a fraction tells us how many pieces of a whole we have. The denominator tells us how many pieces the whole has been broken in to.

Feedback: When converting an improper fraction to a mixed fraction, use the remainder as the numerator of the proper fraction.

[END QUIZ]

[BEGIN TEST]

1. Write the improper fraction 25/6 as a mixed number.

*4 1/6

4 2/6

3 6/6

3 5/6

Feedback: Divide the numerator by the denominator to find the wholes. Use the remainder as the numerator of the proper fraction. The denominator does not change.

2. Write the improper fraction 121/11 as a mixed number.

*11

10 10/11

11 1/11

10 9/11

Feedback: Divide the numerator by the denominator to find the wholes. Use the remainder as the numerator of the proper fraction in this case there is NO remainder so there is NO fractional part of the mixed number.

3. Write the mixed number 9 3/8 as an improper fraction.

*75/8

20/8

35/8

27/8

Feedback: 9 3/8  98 = 72+3 = 75  75/8 Multiply the whole by the denominator of the proper fraction then add this product to the numerator of the proper fraction.

4. Write the mixed number 8 1/6 as an improper fraction.

*49/6

9/6

8/6

48/6

Feedback: 8 1/6  68 = 48+1 = 49  49/6 Multiply the whole by the denominator of the proper fraction then add this product to the numerator of the proper fraction.

5. Order the numbers from least to greatest.

*

Feedback:

5. Order the mixed and improper fractions from greatest to least.

Feedback:

7. To convert an improper fraction to a mixed fraction, divide the numerator by the denominator, write down the whole number answer then use the remainder from the division of the numerator by the denominator as the numerator of the proper fraction.

*True

False

8. Choose the statement below that is incorrect.

*The denominator always changes when you convert mixed numbers to improper fractions.

When converting an improper fraction to a mixed fraction, use the remainder from the division of the numerator by the denominator as the numerator of the proper fraction.

To convert the mixed number to an improper fraction first we must multiply the denominator by the "whole" number to determine how many pieces are in our wholes.

The numerator in a fraction tells us how many pieces of a whole we have. The denominator tells us how many pieces the whole has been broken in to.

Feedback: The denominator always stays the same.

[END TEST]

CONCLUSION:

You can use either an improper fraction or a mixed fraction to show the same amount.

Mixed Numbers

While mixed numbers are the natural choice for verbal expression as well as the answer to a word problem, mixed numbers are not the easiest type of fraction to calculate with

To convert a Mixed Number to an Improper fraction, follow these steps:

Multiply the whole by the denominator of the proper fraction.

Add this product to the numerator of the proper fraction.

The proper fraction now becomes an improper fraction.

The denominator stays the same.

Improper Fractions

When performing mathematical operations with fractions you will find it easier to deal with improper fractions rather than mixed numbers

To convert an improper fraction to a mixed fraction, follow these steps:

Divide the numerator by the denominator.

This will help us decide how many wholes we have.

Write down the whole number answer

Use the remainder from the division of the numerator by the denominator as the numerator of the proper fraction.

The denominator stays the same.

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