Ch.1+Lectures


 * Chapter 1 Lectures**

__Part I: Adding and Subtracting Fractions with Like and Unlike Denominators__ Adding and subtracting fraction is a challenging task for many students. Remember that fractions come in the form a/b. A is refereed to as the numerator and b is called the denominator. It is also important to know that b cannot equal zero because it is not possible to divide by zero.

Guided Practice: In your word document, please describe the numerator and denominator of the following fractions:

1. 3/2 2. 5

Once you have written your answer, please check them here.

The important part of adding and subtracting fractions is that the denominators of all fractions being added or subtracted must be the same. This is often known as having a common denominator.In the first two examples the fractions will already have common denominators, in examples 3 and four the fractions will not have common denominators.

Example 1: Add fractions that already have a common denominator. Watch the following example:

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**Written Transcript:**"So we're asked to add three fifteenths and seven fifteenths and then simplify the answer. So just the process when you add fractions is...well first of all if they if they are not mixed numbers and neither of them are, and then if they have the same denominator. In this example the denominators are already the same. The denominators are fifteen. So if you add these two fractions, your sum is going to have the same denominator:fifteen, and your numerator is going to be the sum of the numerators: three plus seven or it is going to be equal to ten over fifteen. Now if we want to simplify this we would look for the greatest common factor in both the ten and the fifteen, and as far as I can tell five is the largest number that goes into both of them. So divide the ten by five and divide the fifteen by five and you get two over three, you get two thirds.

Now to understand why this works lets draw it out. Lets split something up into fifteen sections. So that is one section. Now let me copy and paste it. That is a second sections. Third sections. Fourth section. Fifth section. Now let me copy and paste this whole thing. So that is ten sections. The let me do it one more time. That is fifteen sections. You could imagine this whole thing as a candy bar or something and we have split it up into fifteen sections. What is three fifteenths. Well it is going to be three of the fifteen sections. One, two, three. There is three fifteenths. Now to that we are adding seven of the one fifteenth sections or seven of the sections. One, two three, four, five, six seven. So you see if you count the orange and the blue, you get one,two,three,four,five,six,seven,eight,nine,ten - ten of the sections or ten of the fifteen sections. And then to see why this is the same thing as two thirds, you can just split this candy bar into thirds - so each third would have five sections in it. Lets do that. One, two three, four, five - that is one third right there. That is another third right there. And notice when you do it like this, we have filled out exactly one, two of the thirds. So ten fifteenths is the same thing as two thirds"

The key points of adding fractions with like denominators are... - Add the numerators -Keep the denominator the same -After adding, simplify if you can

Example 2: Subtract fractions that already have common denominators. Watch the following example:

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The key points of subtracting fractions with like denominators are very similar to adding with 1 difference: - Subtract the numerators -Keep the denominator the same -Simplify if you can

Guided Practice: Add or subtract the following fractions:

3. 1/6 + 3/6 4. 7/8 - 1/8

Check your answers here.

Often time fractions will have unlike denominators and you will need to determine the least common denominator in order to add or subtract the fractions. The least common denominator is the smallest number that both of the denominators go into.

If you were asked to add 1/8 and 3/4, you would need to find the least common multiple of the denominators. Lets list the multiples of 8 and 4: 4 goes into 4,8,12,16..... 8 goes into 8,16,32,64... The smallest number that they have in common is 8. The common denominator would be 8.

Guided Practice: Find the least common denominator of: 5. 2/5 and 5/6. Check your answer here.

Finding the least common denominator will be very useful in adding and subtracting fractions with unlike denominators. Let's take a look at a few examples:

Example 3: Adding fractions with unlike denominators. Watch the following video.

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There are more steps when adding fractions with unlike denominators. You need to:
 * Find a common denominator (for 2/3 + 3/8 the common denominator would be 24)
 * Use the common denominator for both fractions (something/24 + something/24)
 * Determine the numerators.
 * If the denominator of the first fraction changed from 3 to 24 then it was multiplied by 8. This means that the numerator should also be multiplied by 8. 2(8) = 16. The first fraction should be 16/24.
 * If the denominator of the second fraction changed from 8 to 24, then it was multiplied by 3. This means the numerator of 3 should be multiplied by 3. This would mean the fraction is now 9/24
 * Add the two fractions (16/24+9/24 = 25/24).

A similar process is true for subtracting fractions with unlike denominators.

Example 4: Subtracting fractions with unlike denominators. Watch the video:

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Guided Practice: Add or subtract the following fractions: 6. 1/5 + 1/2 7.2/3-1/9 Check your answers here.

You have completed the lecture. Return to the Unit 2 page.

__Part II: Adding and Subtracting Mixed Numbers__

After the first lecture we should be aware of how to add fractions with like and unlike denominators. This works for all numbers in the form a/b, such as 3/4 and even improper fractions where the numerator is larger than the denominator, such as 4/3. However, there are other forms of numbers that we need to be able to add and subtract. Mixed numbers are numbers that feature a whole number and a fraction, such as one and a half (1 1/2). This lecture will focus on adding and subtracting mixed numbers. Adding and subtracting mixed numbers requires converting mixed numbers (1 1/2) to improper fractions (3/2).

Example 5: Convert Mixed Numbers to Improper Fractions. Watch the following video:

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1 3/4 = (1X4) + 3 = 7/4
 * The basic procedure is to multiply the whole number by the denominator and then add the numerator. This answer is the new numerator over the old denominator.

Example 6: Convert improper fractions to mixed numbers. Watch the following video.

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6/4. For this improper fraction, the denominator goes into the numerator once. Therefore the whole number is 1. The remainder is 2. The mixed number would be 1 2/4 or 1 1/2 if it is simplified.
 * The basic idea here is to see how many times the denominator can go into the numerator. That becomes your whole number. Whatever is left is the remainder and becomes the new numerator.

Guided Practice: Rewrite the following mixed numbers as improper fractions. 8. 5 3/4

Guided Practice: Rewrite the improper fraction as a mixed number. 9. 13/2

Check your answers here.

Now that we know how to convert between mixed numbers and improper fractions, we should be able to solve a problem which asks us to add to mixed numbers.

Example 7: Add two mixed numbers. Watch the following video.

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-Add the whole numbers -Add the fractions -If you can rewrite the fraction part of the answer as a mixed number do so and add it to the whole number part of the answer

Subtracting mixed numbers can be a bit more difficult. You need to follow specific steps as well. -Convert both mixed numbers to improper fractions -Subtract the fractions -If the fraction is improper, convert it back to a mixed number

Example 8: Subtracting two mixed numbers 4 1/5 - 2 3/5 Remember that 4 1/5 can be converted to (4X5)+1 or 21/4 Remember that 2 3/5 can be converted to (2X5)+3 or 13/4 We than have 21/4-13/4 which equals 8/4 This can be reduced to 2

Guided Practice: Add or subtract the fractions: 10. 3 3/4 + 5 3/4 11. 3 1/6 - 1 5/6 Check your answers here.

You have completed the lecture. Return to the Unit 2 page.