Chapter 6 Rational Number Operations and Properties Section

1
Chapter 6Rational 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 ½ ¼
• 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 ½.
• 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 gt 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
• Theres 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
• Theres 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