Math for Parents
Tim Whiteford PhD

Fraction Concepts and Skills

1.      Common Fraction Models

  There are three different models that can be used with common fraction     
  experiences;
          a.      Region or area
         b.     Length or measurement
         c.      Set

  They are presented here in the order in which they should be introduced to children.

 2.    Common Fraction Value

  A fraction has no magnitude or value without knowledge of the size of the  whole, or "what the one is".

 3.    Common Fraction language.

  Care should be taken with the development of fraction language to avoid misconceptions and confusion. e.g. "third", "reduce" etc.

 4.    Common Fractional Parts

  A fractional part of a whole occurs when the whole is divided or fair shared into two or more equal parts.

   Each divided part must be the same size bit not necessarily the same shape.

 5.    Counting common fractional parts.

   Fractional parts can be counted in exactly the same way as can be horses, cars, chairs, megabytes and dollars.

 6.    Common Fraction symbols

 The top number shows how many parts. It is a counting number - or an adjective.

  The bottom number shows the size of each part. It is a naming (nominal) number – or a noun.

  The terms denominator and numerator can be introduced later when the need for specificity and differentiation arises.

  Develop the written notation (e.g. 1/4) as fractional parts are counted.

  7.  Mixed whole and common fraction numbers.

 Why are some fractions called improper? (or "vulgar" in the UK!!!!).What's wrong with them? We need them in order to conduct certain operations.

  We can count as many of a fractional part as we want.

 8. Equivalent Common fractions

 We can rename, trade, or regroup common fractional pieces in just the same  way as we can with whole numbers. This procedure can be developed conceptually by using models and appropriate language.  
 
  1/2 is the same size as 2/4 = 3/6 = 4/8 etc. Look for the pattern.

  9. Common Fraction sense. ideas associated with common   
We can develop an intuitive feel for fractions which will help us estimate and compute.

 Which is more 3/4 or 2/3, 2/5 or 3/8, 7/10 or 2/3? Try this conceptually, not procedurally.

 Name a fraction close to 1 but not more than 1. Now name another fraction in between this one and 1. Do this five times,  what do you notice.?

 Find 3/4 of something. Now find a 1/3 of this. What do you have? Which operation did you use?

 Find 1/4 of something. How many 1/8s of the same 1 can you get from this 1/4? Which operation did you use? 

 10. Addition and subtraction of fractions

     Informal relationships

     a. Working with related fractions in the same “family” such 
        as:

            ½,  ¼, 1/8   or ½, 1/3, 1/6, 1/12, or ½, 1/5, 1/10  

      b. Working with different denominators to develop the sense of a common 
          denominator such as:

             ½, 1/3, ¼, 1/6

             ¼ + 1/3 is the same as 3/12 + 4/12 (use the term renaming a fraction with 
              both denominators the same – fraction models can be used for this)

               It is important here for students to understand and  have facility with 
               common multiples when finding the common denominator.

11. Multiplication and division of fractions  11. Multiplication and division

 Really a middle school topic but the ideas can be developed informally through the same models used for whole numbers (repeated addition and  area concepts)  

 And, finally, here’s why you flip the second fraction and change the sign when  dividing fractions.

 How many quarters are there in the first half of a football game?

 1/2 ÷ 1/4

 = 1/2 ÷ 1/4 x 1   (multiplying by 1 does not change the problem)

= 1/2 ÷ 1/4 x (1/4 x 4/1)      (a number times its reciprocal is 1 so 1/4 x 4/1 is 1) can be rewritten as ¼ x 4/1)

= 1/2 (÷ 1/4 x  1/4) x 4/1     (simply move the placement of the ( ),s÷ ¼  x ¼ which is   also the same as 1)

= 1/2 x 1 x 4/1      (just remove the 1 as it makes no difference to the problem)

= 1/2 x 4/1 = 4/2 = 2 (there are two quarters in the first half of a football game)

1. Look for opportunities to include fractional parts of things such as half a glass of orange juice or half a chocolate chip cookie. Especially  with very young children.

2. There's no need to use fraction notation (e.g. 1/2 or 3/5) until children have developed a good understanding of fractions; especially the use of the language of fractions  in everyday conversation. 

3.Fractions are involved in telling the time so the analog clock-face is a good way of teaching half and quarter. 

4. Fractions occur most in measurements of some sort, especially with food.

5. Tenths appear on odometers (in a decimal form) and many other places. Use the faction language  to start with rather than the decimal language.

6. Remember to always identify the ONE when using fractions. (A 1/4 lb of ham, please" is better than "1/4 of ham" in the super-market if your child is listening. 

7. Visual fractions are more difficult to find but they do appear (gas prices, sign posts, prices  etc)

8. Mental operations with fractions are easy if you visualize the fractions. A fourth of 4/7 is the same as a fourth of four M&Ms.