Fractions · Lesson 1 of 9
Equal Shares and Parts of a Whole
“Understand fractions by sharing whole quantities fairly and identifying equal parts.”
• Recognise equal sharing and name the whole being shared. • Identify fractional units and compare the sizes of unit fractions. • Describe a part of a whole even when equal parts have different shapes. • Interpret familiar fraction words and everyday sharing situations.
A fair share needs equal parts
Imagine sharing one roti with a friend. If you each receive an equal amount, each share is half of the roti. Now imagine sharing that same roti equally among four children. Each child receives a quarter. Fractions give us a way to name these shares precisely, even though each share is smaller than one whole roti.
Before naming a fraction, decide what counts as one whole. Here the whole is one roti. In another situation it might be one kilogram of rice, one glass of juice, or one sheet of paper. A piece is called half only when it is one of two equal amounts of that whole. Cutting something into two pieces does not automatically make both pieces halves.
One of the equal parts obtained by dividing one whole unit into a chosen number of equal parts. It is also called a unit fraction.
We write one half as 1/2 and one quarter as 1/4. The bottom number tells us how many equal parts make one whole; the top number 1 says that we are taking one such part. The names one third, one fifth, and one sixth mean one of three, five, and six equal parts respectively. We can read 1/5 as “one fifth” or “one upon five”.
Problem
A merchant packs 1 kg of rice into four packets of equal weight. What does each packet weigh?
- 1.Treat 1 kg as the whole quantity.
- 2.Divide that quantity into four equal weights. Each packet is one of the four equal parts.
- 3.Each packet weighs 1/4 kg. Four such packets together still weigh 1 kg.
Problem
Three guavas together weigh 1 kg and are approximately the same size. What is the approximate weight of one guava?
- 1.The similar sizes suggest that their weights are approximately equal.
- 2.Sharing the total weight of 1 kg among three guavas gives a share of 1/3 kg.
- 3.Each guava weighs about 1/3 kg. This is approximate because real guavas may not weigh exactly the same.
Why more sharers get smaller shares
A larger bottom number does not make a unit fraction larger. Keep the whole unchanged and picture the number of children sharing it. If nine children share the same roti that five children could have shared, each of the nine must receive less. This reasoning helps us compare unit fractions without measuring or calculating their decimal values.
Problem
Which is greater: 1/5 of a roti or 1/9 of the same-sized roti?
- 1.Both situations begin with the same one whole roti.
- 2.Five equal shares are larger than nine equal shares of that roti.
- 3.Therefore 1/5 > 1/9. The symbol > means “is greater than”; equivalently 1/9 < 1/5.
For the same reason, 1/2 > 1/4 and 1/100 > 1/200. Each comparison is about equal parts of the same whole. Half of a very small roti need not be more food than a quarter of a much larger roti. When fractions describe physical amounts, check that their whole units are comparable.
Nine is greater than five, but 1/9 is smaller than 1/5. The bottom number counts the equal shares of one whole. More shares make each share smaller; it does not count how many shares you receive.
Equal amounts can have different shapes
A rectangular chikki can be divided into six equal strips. It can also be divided into equal triangles or other shapes. To decide whether each piece is one sixth, compare the amount of chikki in each piece, not simply its outline. For a flat chikki of uniform thickness, equal areas mean equal amounts.
The larger chikki fragment in a quarter-based picture contains three quarter-pieces. Its size is therefore three quarters of the whole, written 3/4. A whole can also be partitioned into thirds: one of three equal pieces is 1/3. To identify a piece, imagine repeating it or rearranging copies until they cover the original whole without gaps or overlaps.
Problem
Use the 24-square whole to identify the eight darker pieces in the diagram.
- 1.Piece a covers two small squares, so it is 2/24 = 1/12 of the whole. Piece d covers four squares, so it is 4/24 = 1/6.
- 2.Piece b is half of a rectangle covering twelve squares, so it has the amount of six squares: 6/24 = 1/4. Piece c is half of six squares, so it is 3/24 = 1/8.
- 3.Piece e is an L-shaped group of three squares, giving 3/24 = 1/8. Piece f has the same amount as four squares, giving 4/24 = 1/6.
- 4.Piece g covers one square, giving 1/24. Piece h contains two half-squares, so it also gives 1/24. Different outlines can represent equal amounts.
Problem
A large fish weighs 1/2 kg and a small fish weighs 1/4 kg. What is their combined weight?
- 1.Both weights use the same whole unit, 1 kg.
- 2.Half a kilogram contains two quarter-kilograms. Together with one more quarter-kilogram, there are three quarter-kilograms.
- 3.The fish weigh 3/4 kg together. This visual way of combining shares will later become a general addition method.
Problem
Four friends share three glasses of juice equally. How much does each drink?
- 1.Split each glass into four equal quarter-glass shares.
- 2.Give every friend one quarter from each of the three glasses.
- 3.Each friend receives three quarter-glass shares: 3/4 of a glass. No juice is left and everyone gets the same amount.
Fraction words in daily life
Words such as half, quarter, three quarters, and one and a half are part of everyday speech. They describe amounts of food, lengths, weights, and time. Recognising these names helps you connect symbols to quantities you already understand. A whole and one quarter means a quantity slightly larger than one; two and a half means two wholes and another half.
| Fraction name | Symbol | Size compared with one whole |
|---|---|---|
| Quarter | 1/4 | Less than one |
| Half | 1/2 | Less than one |
| Three quarters | 3/4 | Less than one |
| One and a quarter | 1 1/4 | Between one and two |
| One and a half | 1 1/2 | Between one and two |
| Two and a half | 2 1/2 | Between two and three |
The expression tri-pada for three quarters occurs in ancient Indian writing. Familiar names such as teen paav in Hindi and mukkaal in Tamil also express three quarters. Ask family members and classmates which words they use for halves, quarters, and amounts larger than one. Record the language as well as the meaning.
Quiz
One roti is divided into five equal shares. Each share is…
Which is larger when the whole is the same?
A chikki is cut into two unequal pieces. Which statement is correct?
Three quarter-pieces of one whole are written…
Which order goes from smallest to largest?
Two differently shaped pieces can each be one sixth of the same flat whole if…
Practice Problems
- Draw a whole and divide it into 2, 3, 4, and 6 equal parts in separate pictures. Label one part in each.
- Explain why 1/100 is greater than 1/200 when both use the same whole.
- Show how four friends can share three glasses of juice equally without leaving any juice.
- A rectangular chikki is arranged as 6 columns and 4 rows of equal small squares. What fraction is one square, two squares, three squares, four squares, or six squares? Draw both rectangular and triangular pieces with some of these areas.
- Draw two different partitions of the same rectangle into six equal areas. Explain why differently shaped pieces can still have the same fractional size.
- Arrange quarter, half, three quarters, one and a quarter, one and a half, and two and a half in increasing order. Ask someone for the names in another language.
- Explain why cutting a sheet into four pieces is insufficient to conclude that each piece is one quarter.
Key Takeaways
• A fraction describes an amount relative to a chosen whole. • A fractional unit is one of several equal parts of one whole. • For the same whole, a unit fraction becomes smaller as its denominator grows. • Equal fractional amounts may have different shapes. • Everyday fraction words connect symbols with familiar quantities.
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Building and Locating Fractions