Fractions

Fraction Insets

The children will have an idea of halves and quarters. When ‘shopping’ know half pound of tea, quarter pound of sweets, and so on. They have used fraction terms with regard to time: ‘It is half past five’, ‘It is a quarter to eleven’. They have used the fraction terms for sharing: ‘Half a bar of chocolate’. They have used the fraction terms as regards to money.

The presentation of fractions begins sensorially at about 4 years or earlier. In the beginning the child works with them in a sensorial way, making various designs. The names are taught quite simply.

Direct Aim

a) The early work with fractions using the circles, squares and triangles,helps the child to form the concept of a fraction.

b) The work with the equivalence of fractions makes the child thoroughly familiar with the relationships that exist between the various families, and is an essential preparation for the four operations.

c) To familiarize the child with the four operations using fractions.

Age

Approximately from 5 years.

Materials

  • Ten metal insets circular in shape. The first inset is a complete and undivided circle. The other circular insets are divided into 2, 3, 4, 5, 6, 7, 8, 9 and 10 equal parts.
  • Divided squares and divided triangles are also used. These are used for division.

Control of Error

There is no automatic control of error.

First Introduction

The children use the fraction insets sensorially. They make inset designs.

  1. ‘Here we have a whole circle – One circle’.
  2. ‘Here we have the whole circle divided into 2 halves – Each one is called a half’.
  3. ‘Here we have a whole divided into 3 parts – Each part is called one third of the whole circle’.
  4. Present the rest of the terms similarly: one quarter (one fourth, one fifth, one sixth, one seventh, one eight, one ninth, one tenth).
  5. The children still continue to make designs, but now encourage the children to use the terms: ‘Here we have a half. How many halves are there? Two. We shall call this the family of two. Here we have a third. How many thirds are there? Three. We shall call this the family of three’, and so on.

Exercise

  1. The child can take a sector from each ‘family’ and lay them out in a straight line.
  2. He will see the relationship of the sizes – of how, for example, there is a very little difference between the ninth and the tenth.
  3. He can transfer them to paper.
  4. The child begins to see the relationship of, say, 1/4 of something in relation to the whole thing.
  5. ‘What have we got here?’ – one third. ‘How many more do we need to make up the whole family?’ – 2 thirds.
  6. In the same way, 2 sevenths need 5 sevenths more.
  7. He can make mixed numbers, such as 39, using paper circles.

HE DOES NO WRITING AT THIS STAGE.

 

Wall Chart

This should have the names of the fractions written out in full. The child should also name his fractions in full: three fifths.

Direct Aim

The understanding of fractions, i.e. the relationship between the part and the whole.

 

Introduction of the Written Symbol

  1. Concentrate on one sector from each family at first.
  2. ‘Take one half. This one is from the family of 2. We write it like this 1/2′ (write in his book).
  3. ‘The lower number tells you what family you have taken it from, or how many equal parts the whole was divided into. The top number tells you how many numbers you have’.
  4. Repeat with the remaining fractions. Take name slips–1; 1/2, 1/2, 1/3, 1/3, 1/3,1/4, 1/4,1/4,1/4, etc.
  5. Taking the sectors from the insets, place a slip on each corresponding sector. All the slips can be introduced in one lesson.
  6. The child can go back to his early exercises, repeat them, but this time labelling his work. He uses the divided circles, squares and triangles.
  7. Ask him to get you 1/3, 1/4 – writing them down in his book – or get him to bring you some sectors and write down what he has brought.
  8. ‘Take 1/3 from the family of 3. How many are left? 2. Therefore 2/3’.
  9. ‘Take 1/4-how many are left? 3, 3/4’.
  10. ‘Take out 5/6’. Ask the child to write it down!
  11. ‘Take out all the sixths– 6/6=1.
  12. ‘Make mixed numbers and record them!
  13. Taking the eights, draw the whole and then fill in the eighths. Shade in, say, 3/8. Ask the child what fraction of the whole figure he has coloured, and how many did he leave plain? 5/8.
  14. Encourage the child to do much of this work writing on paper the fraction he had shaded in.
  15. Give blank stencilled copies (to the children) of divided circles, squares, triangles. ‘I want you to color in 16, /6, 3/6, 3/4, 2/8, and so on.

 

The Equivalence of Fractions

This is the most important part of the preparatory work. The child will go on for a long time.

Presentation

  1. Take one half.
  2. Invite the child to find 2 similar fractions that will make a half. He can use 2 quarters. Ask him to try using other fractions to make the half, viz. 3 sixths, 4 eighths, 5 tenths.
  3. Ask him to try making 1/3 in another way. e.g.1/3=2/6=3/9. His work is purely experimental at first.  

  4. ‘Make 2/3 in another way’, e.g. 4/6 and 6/9.

There is no writing yet.

Exercise

The children can make little booklets, e.g.

 

Later

Addition of Fractions

It is essential that the child should have worked with the equivalence of fractions very thoroughly before he starts the four operations.

Exercises

  1. ‘Let us add 1/2 + 1/4’.
  2. Take out the respective sectors. ‘How are we going to do it?
  3. They will all have to belong to the same family – so if we change 1/2 to 2/4, now 2/4 + 1/4 = 3/4.
  4. Let us add:

   a) 1/3 + 1/9     1/3=3/9 so 3/9+1/9= 4/9. 

   b) 1/4 + 2/8     2/8 + 2/8 = 4/8 = 1/2.

There is no writing of any sort done yet. When possible a larger fraction should take the place of smaller fractions, e.g. 4/8 = 1/2. The child will discover through his own activity that the material is limited, e.g. 1/3 + 1/7 cannot be added. Prepared slips with fractions will hinder his discovery.

Later

Recording:

a) 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2

b) 2/9 + 2/3 = 2/9 + 6/9 = 8/9.

Always encourage the child to estimate. Is the answer going to be more, or less?

Subtraction of Fractions

(Take-away aspect)

Exercises

  1. ‘Let us subtract 1/4 from 1/2. Take out the 1/2 fraction (inset). There are 2 quarters in one half, so change it into quarters.Now we can take away 1/4, and our answer is: 1/4′.
  2. 2/5 – 3/10     Change 2/5 for 4/10. Take away 3/10. Answer:1/10.
  3. What is the difference between 1/3 and 1/6. Answer /6.

As with addition, encourage the child to estimate.

Multiplication of Fractions

Estimation is of the utmost importance all the time. When multiplying with a fraction, the answer is smaller than the multiplicand wherever the multiplier is a (proper) fraction.

  1. Multiply a fraction by a whole number first, e.g. (a) 1/4 x 3 (b) 2/7 x 4. For (a), you are taking 1/4 three times, for (b),you are taking 2/7 four times.
  2. When the child has done several of these sums, letting him experiment for himself, introduce him to sums with the multiplier as a fraction, e.g. 1/4 x 1/2. Explain to the child that we are going to take 1/4 half times, and a half is less than one, so the answer will be less than 1/4 (the multiplicand). ‘What is the half of 1/4? 1/8. So change the 1/4 for 2/8 — we are taking it half times — so our answer is ‘1/8’ (or: look for 2 equal parts – ‘what 2 parts make up 1/4?’, ‘2/8’. So 1/4 taken 1/2 times = ‘1/8’).
  3. Another example: 1/2 x 2/5. 1/2 is taken 2/5 times. ‘What is 1/5 of 1/2?’ or ‘What 5 parts make up 1/2?’ ‘5/10’ Change the 1/2 for 5/10. We are taking it 2/5 times, so take away two of the ‘fifth’ parts which are tenths of the whole = 2/10 or 1/5 of the whole. Hence, 1/2 x 2/5 = 1/5.
  4. Another example: 2/3 x 3/4. ‘What is one fourth of 2/3?’ ‘1/6’. ‘We are taking it 3 times, but 3/6 = 1/2, so 3/6 = 1/2, so 2/3 x 3/4 = 1/2’.

Here are some more examples of fractions to do with material:

1) 1/2 x 1/4;1/2 x 1/3;1/2 x 1/5;1/3 x 1/3;1/2 x 3/4;

2) 1/2 x 2/3;1/2 x 2/5;1/2 x 3/5;1/2 x 4/5;1/2 x 1/4;

3) 2/3 x 3/4; 2/3 x 1/3; 3/4 x 1/3; 3/4 x 2/3; 4/5 x 5/8;

4) 3/5 x 2/3; 3/5 x 5/6; 5/7 x 2/5; 3/5 x 1/3.

Division of Fractions

Materials

Large Fraction Skittles
  • The fraction insets
  • Divided skittles: in 2 =halves (red inside)

                   in 3=thirds (orange inside)

                   in 4 = quarters (green inside)

  • one complete skittle

N.B. The divided skittles stand for the divisors

First Exercises (dividing fractions by whole numbers)

  1. At first, only have whole numbers for divisors.
  2. Encourage estimation.
  3. Example 1: 1/2 ÷ 2  ‘What equal parts make up one half? 2 quarters. Give one quarter to each skittle. In division our answer is always the share of one quarter. So, 1/2 ÷ 2 = 1/4
  4. Example 2: 1/3 ÷ 3   What 3 equal parts make up 1/3? 3/9 Answer to the share of 1: 1/9. So1/3÷3=1/9

Second Exercises (dividing fractions by fractions)

  1. Gradually introduce division by a fraction:1/2÷2/3
  2. Lay out the 2 ‘one third’ skittles
  3. They each get a quarter: However, our answer in division is always the answer to a whole, and if we looked at our skittles and said the answer was 1/4 we would only be giving one third of the answer, as 1/4 is the share of one third — so we take the remaining ‘third’ skittle, put it beside the others and give it a quarter as well.
  4. Put the three skittles together to make a whole and put their share underneath.
  5. So1/2 ÷ 2/3 = 3/4

By observing the quotient, when the divisor is a fraction, it will be noticed that the quotient is always greater than the dividend.

Fraction Insets (Circles)