**Fractions**

**Material Description:**

- Ten green frames each with a circular red inset – one inset is whole, the others are divided into equal parts, each part having a knob
- Printed slips – one set with symbols of fractions, another with labels for common denominators
- A set of questions for the operations
- Working Mat and felt mat

**Note:**

There are three levels of work for fractions;

- Manipulation: to build the foundation for a sensorial understanding of fractions and learn their names and symbols.
- The Concept of Equivalence
- Application of the Four Operations to numbers below the unit; The application of the operations is limited to those fractions with a common denominator and multiplications involve one whole number multiplicand and division to divisors with whole numbers.

These are offered after the first two operations of the Decimal System and the Small Insets.

**1. Manipulation**

Introduction:

Working with all the frames on the Working and Felt Mats present the fractions sensorially.

- The child removes one or more inset from it’s frame, places them at random on the mat and returns them.
- Remove a few insets from many different frames and ask the child to place specific ones in their frame
- This is essentially the Sensorial work done with the Small Insets

**Presentation:**

At a mat with the frames, plasticine and Golden Bead unit, introduce the first four frames

First frame – whole

- Isolate the inset, bringing it forward, and link the first frame to the unit of the Golden Bead material, saying, “This is also one”.
- Take a plasticine ball and press it to show the transformation of the Golden Bead into a circular ‘whole’.
- Replace the Golden Bead and plasticine

Second frame – halves

- Transfer the halves into the frame of the whole, then the thirds. Continue interchanging, introducing the other frames and handling them in the same way
- Isolate both frames bringing them forward, in line with the whole. Point around it’s circumference saying, “This is also a whole unit, but I can break it into equal parts. When we break a unit into equal parts we make fractions.”
- Hold the two halves together, show the child the breaking of the whole by ‘folding’ the two semi-circles, sensorially demonstrating equivalence
- Place them on the mat, slightly apart and ask how many there are, say, ‘This is the family of two’

Thirds and Fourths frame –

- As halves, showing the equivalence piece by piece and naming them

**Language (oral):**

- Give a Three Period Lesson on the names of the fractions with the frames of halves, thirds and fourths (beginning with the fourths if this lesson continues directly from the presentation)
- Or, ask the child how many pieces are in the halves frame and pointing to each say, “This is one half. Tell the child that each piece is also called a ‘fraction’ (a part of a whole). Replace. Remove the insets of thirds and do the same, then for quarters or fourths
- Place one inset from each frame on the mat and give the second and third parts of the lesson
- When they have been identified out of sequence put them into sequence and count up and down.
- Later introduce the next three frames and later still the final three. Counting from one whole during the first period and involving known names in the second and third periods

**Exercises with the Names:**

1. The child’s own work with the frames

2. Once the child knows the names and relates them to their shapes introduce the following;

At a mat isolate one frame at a time and remove one piece, ask the child to identify it (one-fifth) and then placing it next to the frame ask, “How many fifths are in the frame? (four-fifths).

Replace it and form fractions from other frames, including ones with more than one numerator.

Then reverse the process asking the child to identify what is in the frame and then what has been removed

3. Ask the child to form specific fractions from any frame, e.g. ‘two-thirds’, ‘three-quarters’

**Language (symbols):**

At a chowki, take three frames, blank paper and graphite pencil

The line

- Take the frame of halves and place it with the dividing line horizontal. Saying, “The line we write when we record fractions indicates that the whole (pointing to the circumference) has been divided into two parts.” Remove one inset and put it by the frame and point to the line saying, “This is half of the whole”.
- Now write the horizontal line, pointing to the written line say, “This line shows that the number represents a fraction”

The Denominator

- In the same presentation write the figure ‘2’ below the line, saying, “This number shows that the fraction belongs to the family of two”
- Next go to the frame of thirds, remove one inset, draw the line and, writing the denominator say, “This piece belongs to the family of three”
- Invite the child to write the denominator of other fractions, ask the child if she remembers the ‘family’ from the presentation

The Numerator

- After some time ask the child to write the numerator of a fraction and then ask her how many pieces are outside of the frame, write this number above the line. Ask her the name of the fraction and read what is written, e.g. ‘one-third’
- Repeat this procedure for other fractions.

The Printed Slips – Symbols

- Bring the fractions and sequenced symbols to the working mat.
- Invite the child to read the first symbol and place it on the mat, she then finds the fraction it represents, removes it from the frame and places it by the symbol
- After sufficient experience suggest that she mixes the symbols before placing them

The Printed Slips – Labels

- Bring the fractions and labels for each piece in the corresponding colour packets.
- Place the packets beneath the fraction they correspond to.
- Invite the child to open one of the packets, read the labels and place them directly onto the fraction insets in the frames.

**The Concept of Equivalence:**

**Introduction:**

- Ask the child to bring all the frames with fractions to the mat.
- Remove the inset of one-half and place it on the felt mat and ask the child to name it,
- Then explore which pieces are equivalent to it, beginning with the thirds and proceeding in sequence, replacing the insets which do not fit and putting those that do to one side.

**Presentation:**

- Bring the fractions and Equivalence Question Cards to the working mat.
- Invite the child to remove and read the first Question Card and place it on the mat, she then finds and makes the simplest fraction it represents placing it under the fraction card, before looking for equivalent pieces in the other frames and placing them adjacent

**Exercises:**

- The child’s own work with the frames
- Later find the equivalence with other insets
- Still later, give the paper fraction pieces, in respective colours, to enable the child to record her discoveries

**Application of the Four Operations:**

**Presentation:**

Addition**:**

Collect the fraction frames, blank and squared paper, a pencil, question cards and bring the material to the mat,

- Make two addends by isolating the respective insets, placing them on the mat.
- Tell the child, “We are doing an addition”, join the pieces together and count them to find the sum. Do not write anything at this stage
- Repeat the procedure several times using the question cards, until the child is familiar with the procedure, later ask the child to form the second addend and how to record.

**Exercises:**

- The child’s own work with the frames
- Later, give the fraction pieces cut into respective colours to enable the child to record her discoveries

Subtraction**:**

Collect the fraction trays, blank and squared paper, a pencil, question cards and bring the material to the mat,

- Make a minuend by isolating the respective insets, placing them on the mat.
- Tell the child, “We are doing a subtraction” and remove the subtrahend and count those that remain to find the difference.
- Repeat the procedure several times the question cards, showing her how to record.

**Exercises:**

- The child’s own work with the frames
- Later, give the fraction pieces cut into respective colours to enable the child to record her discoveries

Multiplication**:**

Collect the fraction trays, blank and squared paper, a pencil, question cards and bring the material to the mat,

- Make three multiplicands by isolating the respective insets, placing them on the mat.
- Tell the child, “We are doing a multiplication”, join the multiplicands together and count them to find the product.
- Repeat the procedure several times the question cards, showing the child how to record the answer

**Exercises:**

- The child’s own work with the frames
- Later, give the fraction pieces cut into respective colours to enable the child to record her discoveries

Division**:**

Collect the fraction trays, blank and squared paper, a pencil, question cards and skittles and bring the material to the mat,

- Make the dividend by isolate the respective insets, placing them on the mat.
- Tell the child, “We are doing a division”, place the skittles and distribute the insets equally amongst them. Count the insets belonging to one skittle to find the quotient.
- Repeat the procedure several times the question cards, showing the child how to record the answer.

**Exercises:**

- The child’s own work with the frames

**Direct Aim:**

To acquire:

- the concept and knowledge of fractions
- the concept of equivalence
- the understanding of how operations can be applied to fractions

**Age at Presentation:**

Five to Six years