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Addition of Fractions with unlike Denominators

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Where is the most difficulty in adding fractions? Knowing how to find a common denominator and changing the fraction to an equivalent fraction.

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Common Mistake with Adding Fractions Many students simply add across.

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Why students make this mistake with adding fractions Students do not understand what a fraction is. They just see the numbers and add. They don't understand that the 3 is a third and the 5 is a fifth or they do not understand the concept of different units.

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Example: The 2 represents a half The 8 represents an eighth You can't add eighths and halves together so you must find a unit that they both have in common. 8 would be the unit they have in common which is called the common denominator. The problem would then look like this: But how do you find the common denominator?

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A common example you can show your students why you must use a common denominator to add Can you add oranges and apples together? Not unless you rename the oranges fruit and the apples fruit. Then you would have 3 fruit + 4 fruit = 7 fruit. When you use common denominators you are doing the same thing. You are renaming the units to a number that both denominators have in common.

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Fraction Strips This is a way to show students visually the equivalence of fractions. http://theapple.monster.com/training/articles/4324-4th-grade-adding-and-subtracting-fractions

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Some other ways for students to visualize fractions Graham crackers-they are sectioned off into pieces you can break apart. M&M's- you can use the different colors to the total number of m&m's to represent different fractions. Playing cards-you can use two different cards to represent a fraction. Pizza slices-slices to the whole. Money- dimes, nickels, quarters, to make a dollar. You can almost use anything to represent fractions.

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http://www.inspiration.com/kidspiration-math-examples

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Visuals are a good way to show younger students the concepts of fractions. It gives them something concrete to look at instead of just following another math rule. Once students do understand what a fraction is, they need to learn how to find the LCD other than drawing it or looking at manipulatives.

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Different ways to find the common denominator Example: Find a common denominator for: Simply multiply the denominators. 3 x 8 = 24 This method will not always give you the LCD(least common denominator). You will get the correct answer, but you may have more steps to reduce your final answer. Write down all the multiples of each denominator until you find a common multiple. 3: 1,2,3,4,5,6,9,12,15,18,21,24 8: 1,2,3,8,16,24 You can use prime factorization. 3: 1 * 3 8: 2 * 2 * 2 1 * 2 * 2 * 2 * 3 = 24

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When you have the common denominator what do you do then? Example: You found that 20 is the LCD(least common denominator). You must change each fraction to the equivalent fraction with the denominator 20: and 20/5 =4 so you multiply the numerator and denominator of 3/5 by 4 giving you 12/20. 20/4= 5 so you would multiply the numerator and denominator of 1/4 by 5 giving you 5/20. You just rewrite the fraction and add across and just rewrite your common denominator:

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Theorem Addition of Fractions with Unlike Denominators Let and be any fractions. Then =

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Example using Theorem

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Exercises Try these addition problems using the theorem.

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Exercises Answers

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Now try the same problems by finding the LCD.

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Addition using LCD

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Do you find using the theorem is easier than finding a common denominator? Which method do you prefer in solving different denominator addition problems? Do you think that the theorem should be taught to students that have been just introduced to fractions?

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Sources Spector, Lawrence. "The Math Page." Adding and Subracting fractions and mixed numbers. N.p., 2001-2010. Web. 10 Mar 2010.. http://www.math-drills.com/fractions.shtml Mathematics for Elementary Teachers - Class Textbook