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Chapter 6 Rational Number Operations and Properties






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Misconceptions about Adding Fractions. Explain what is wrong with the following reasoning for :means 1 part from a total of 2 partsmeans 1 part from a total of 4 partsSo when you add you get 2 parts from a total of 6 parts. Addressing the Misconceptions. There are two main misconcept
Chapter 6 Rational Number Operations and Properties

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1. Chapter 6 Rational Number Operations and Properties Section 6.2 Adding and Subtracting Fractions

2. Misconceptions about Adding Fractions Explain what is wrong with the following reasoning for ? + ? : means 1 part from a total of 2 parts means 1 part from a total of 4 parts So when you add you get 2 parts from a total of 6 parts

3. Addressing the Misconceptions There are two main misconceptions here. First, by combining the totals of 2 parts and 4 parts, you would be thinking of two wholes rather than the one whole. Secondly, it is not valid to combine parts that are different sizes! When adding (or subtracting) fractions, each fraction and the resulting fraction refer to the same whole. Also, we cannot combine parts unless the parts are of equal sizes. The key to adding and subtracting fractions in general is to think of the whole as divided into a common number of same size parts that can be grouped to represent each fraction. The area model representation is well suited for this. Also, the final answer must be expressed as a part-whole relationship where the whole is the original whole.

4. Area Models Area models are useful when dealing with fractions, especially in regards to comparing, adding and subtracting fractions. Area Model representations make use of vertical and horizontal line partitions of a whole to create equivalent fractions (and common denominators). NOTE: Area models are not the only means by which operations and comparisons with fractions can be demonstrated.

5. Area Model Example Use an area model to show 1/2 + 2/4 = 1. Begin with two equal squares for each fraction. Each square represents the same whole. The first square is divided (vertical partition) fairly into two shares with one vertically shaded share representing ?. The second square is divided fairly into 4 shares (using both vertical and horizontal partitions) with two horizontally shaded shards representing 2 of the fourths. How do these models verify the addition statement?

6. Modeling Addition and Subtraction of Fractions using Area Models Model each of the following using an area model. 1.) 2/5 + 1/5 2.) 1/3 ? 1/4 3.) 3/4 + 1/3 4.) 2/5 + 3/7 5.) 2/3 ? 1/2 6.) 1 1/3 ? 5/6

7. IMPORTANT! When using area models, make sure that students know that each fraction is represented on a different copy of the whole divided into a common number of equally-sized parts (which represents the a common denominator). Use the modeling to write a mathematical statement demonstrating your actions and the final result.

8. Mixed Numbers and Improper Fractions Mixed number: I a/b where I is an integer and a/b is a fraction with b > a. Improper fraction: c/d where c = d

9. Procedure for Adding and Subtracting Rational Numbers Represented by Fractions For rational numbers a/b and c/d, a/b + c/d = ad/bd + cb/bd = (ad + cb)/bd, and a/b ? c/d = ad/bd ? cb/bd = (ad - cb)/bd. NOTE: This algorithm is based on the operations involved in area modeling!

10. Properties of Addition of Rational Numbers Closure Property For rational numbers a/b and c/d, a/b + c/d is a unique rational number. Identity Property A unique rational number, 0, exists such that 0 + a/b = a/b + 0 = a/b for every rational number a/b ; 0 is the additive identity number.

11. Properties of Addition of Rational Numbers Commutative Property For rational numbers a/b and c/d, a/b + c/d = c/d + a/b. Associative Property For rational numbers a/b, c/d, and e/f, (a/b + c/d) + e/f = a/b + (c/d + e/f). Additive Inverse Property For every rational number a/b, a unique rational number ?a/b exists such that a/b + (-a/b) = -a/b + a/b = 0.

12. Word Problems for Adding and Subtracting Fractions Recall that when adding and subtracting fractions, each fraction involved as well as the answer (result of the operation) refer to the same whole. Wording should always make very clear what the whole is!

13. Write a word problem for 1/2 + 1/3 Correct: John ate 1/2 of a Hershey's candy bar. Mary ate 1/3 of the same kind of Hershey's candy bar. How much of a Hershey's candy bar did John and Mary eat altogether? Note here that the "whole" for the 1/2 and for the 1/3 is the same whole of one Hershey's candy bar. Further note that the question asks "How much of " the same whole, since the result of an addition problem of fractions must refer to the same whole.

14. Write a word problem for 1/2 + 1/3 Incorrect: Ashley has 1/2 a dozen of cookies. Chris gives her 1/3 of a dozen of cookies for her birthday. How many cookies does she have? This is incorrect because all quantities do NOT refer to the same whole. Note here that the "whole" for the 1/2 and for the 1/3 is the same whole ? a dozen cookies. However, the question is not phrased in terms of the same whole. It asks for the number of cookies when it should ask for "How much of a dozen cookies" does she have.

15. Write a word problem for 1/2 ? 1/3 Correct: Mary had 1/2 a cup of flour. She gave Joan 1/3 cup of flour. How much of a cup of flour does Mary have left? Note here that the "whole" for the1/2 and for the 1/3 is the same whole ? a cup of flour. Further note that the question asks "How much of" the same whole (i.e. how much of a cup of flour).

16. Write a word problem for 1/2 ? 1/3 Incorrect: There?s 1/2 a pizza on the table. John eats 1/3 of it. How much of a pizza is left? This is incorrect. You can see this by drawing the representations: The shaded area represents 1/2 a pizza. The part that has diagonal lines in it represents 1/3 of ?it?, i.e. of the 1/2 pizza. You can see from the representation then that 2 from 6 total parts are left in the shaded region once the eaten part is removed. This gives an answer of 1/3 of a pizza left. This clearly is NOT 1/2 - 1/3.

17. Incorrect Subtraction Problem continued Incorrect: There?s 1/2 a pizza on the table. John eats 1/3 of it. How much of a pizza is left? The trouble with the word problem as stated is that the fractions do not refer to the same whole. Here the "whole" for the 1/2 and the "whole" referred to in the question are the same ? one pizza. However, the "whole" referred to by the 1/3 is NOT the same whole of one pizza; the "whole" for the 1/3 is 1/2 a pizza.

18. Group Activity For each fraction expression below: a.) Write a word problem representing the expression. [Use different contexts than those used in class.] b.) Identify the "whole" for each fraction used in the problem and for the answer. c.) Solve the problem using area model representations. 1.) 2/3 ? 1/2 2.) 3/4 + 1/3


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