FRACTIONS

 Historical Development  Developmental Perspective  Equivalent Fractions  Fraction Addition  Fraction Multiplication and Division  Ratios and Proportions  Strategies

 Wrote in Unit Fractions  This is 1 over another number  Examples: ½, and ¼.  Only exception would be two thirds  Everything else was written in a form of a sum of unit fractions  Let’s Try 3/5 written out in the sum of unit fractions

 Used 60 as the Denominator  This leant its hand to a type of decimal system  Greeks followed Babylonians in their use of minutes and second in astronomy  To this day we still use minutes and seconds in astronomy and in GPS coordinates  Romans followed, but used base twelve

 We follow the ways of India (Brahmagupta)  They however omitted the fraction bar Example: 3 8 would be 3/8 Example: would be 11 and 2/7

 “Fair Share” Problems, this means they can split something among people  For Example 3 cookies among 2 people means each get 1 and a ½.  This would be taught in the primary grades, and students grasp the idea of ½ and the idea of ½ and ½ makes 1. killed-cookie-monster/

 Students build off of the Fair Share problems to come up with the concept of parts of a whole  If students have not gotten through the Fair Share problem, this may be a place to start  Students seem to treat as part of a whole, rather than treating as a separate number  The last thing in the book says students do not grasp the idea of adding Fractions, and go more by gut than by proven methods.  Decimals are soon introduced and students stay away from fractions

 May be written in form of an equation  a * x = b  a is the amount of parts in the whole  x is the fraction  b is the amount of parts of the whole being selected

 Example Show x and y are equal x=2/3 and y = 8/12 Set up in the Fraction Algorithm

 Divide until no remainder, than divide that into whole number  This will find the greatest common divisor which when divided into both will give the reduced form 456/732

 How would you add items with different units? Such as feet to yards, centimeters and kilometers.  You must transfer them into the same units.  If students understand it is just changing the units it may help Example: 5 yards + 4 feet

 The algorithm is as follows d*a + b* c d*b Does not matter if LCM or not

 Multiply across. a * c = a*c b d b*d What does this look like though?

 Same, Change, Flip Keep first factor the same, change the division to multiplication, and multiply a / c = a*c b d b*d a * d = a*d b c b*c Again, What does this look like?

 Comparing two quantities of different units  Example: Amount of Goals in a season of games  Word Problem: I can purchase 1 pound of candy for $1.20. How many pounds can I purchase for $1.80?

 Students believe that it is okay to treat fractions as two separate whole numbers, instead of a fraction itself.  Students often believe that it is okay to go along the top and bottom when adding or subtracting fractions.  Students believe that the multiplication rule relates to the rule of addition in that the denominator stay the same.

 Student change the sign for division, but fail to take the reciprocal of the number that is being divided.  Once able to do decimals, fractions will go away forever.

 Making Learning Visible  Fractions is one of those items without a shortcut, students must see why they work, and a proof does not do this.  Using Manipulatives  If you have ¼ and add ½, by showing on a ruler it is ¾ will reinforce this idea, students are stuck in there old way. SHOW THEM THE VOID IN THEIR KNOWLEDGE

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