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: 11 2 7 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. http://www.melecotte.com/2008/10/how-i- 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