Fair Share
Chapter at a Glance
This chapter introduces students to the concepts of fractions — specifically halves ($1/2$), quarters ($1/4$), and three-quarters ($3/4$) — through the sharing of food (parathas, chocolates, chikki sweets), paper folding, and marble games. Students learn that a fair share requires dividing a whole into equal parts. They explore fractions of continuous shapes (area) and discrete groups (sets of objects), investigate the relationship between halves and doubles on number lines, and learn to identify, shade, and assemble fractional cards to form a whole.
Key Definitions & Terminology
- Whole: The complete, undivided object or group.
- Half: One of $2$ equal parts when a whole is divided equally between two shares.
- Quarter: One of $4$ equal parts when a whole is divided equally among four shares.
- Three-Quarters: A portion made by combining $3$ quarter parts together (three out of four equal parts).
- Double: Twice the amount of a given quantity (e.g., $10$ is double of $5$).
- Half of a Group: Half the number of discrete items in a set (e.g., half of $8$ marbles is $4$ marbles).
Formulas, Rules & Properties
- Fractional Equivalence Rules:
- $2\text{ halves} = 1\text{ whole}$
- $4\text{ quarters} = 1\text{ whole}$
- $2\text{ quarters} = 1\text{ half}$
- Double-Half Inverse Rule:
- If a number $A$ is half of $B$, then $B$ is double of $A$.
- If $A = B \div 2$, then $B = 2 \times A$.
Core Concepts & Topics
- The Concept of Equal Sharing:
- Distributing items so that each part has the exact same size, shape, or quantity.
- Testing shape equality by folding paper sheets so that the edges overlap perfectly.
- Continuous Fractions (Symmetric Folding):
- Shading halves and quarters of circles, rectangles, squares, and triangles.
- Folding a rectangular or square sheet of paper in half in multiple different ways (vertical, horizontal, and diagonal folds).
- Completing incomplete drawings by mirroring a given half or drawing three missing quarters.
- Fractions of Discrete Groups:
- Sharing sets of items (like mangoes, birds, flowers, or marbles) equally.
- Shading fractions of collections:
- Shading half of $8$ birds $= 4$ birds.
- Shading a quarter of $8$ birds $= 2$ birds.
- Numerical Halves and Doubles:
- Locating doubles and halves of numbers on a number line.
- Solving distance problems on number lines using fractional relationships.
- Logical Fraction Puzzles:
- Finding a secret quantity using inequality clues based on halves and doubles.
Worked Examples
- Chikki Eating Fractions:
- Problem: Shabnam has a square piece of chikki.
- (a) She eats it from $3$ sides. How much is left? → Less than half.
- (b) Shade a chikki that has been eaten from $2$ sides such that exactly half is left. → Shade exactly half of the square.
- Amit and Bala's Unfair Sharing:
- Problem: An old man gives Amit and Bala a mango tree, a solar lamp, and a blanket. Amit suggests: Amit keeps the mango fruits, Bala keeps the tree; Amit keeps the lamp at night, Bala keeps it during the day; Amit keeps the blanket in winter, Bala keeps it in summer. Is this fair?
- Solution: No, this is highly unfair. Amit keeps all the utility: fruits grow on the tree; the lamp is only needed at night; the blanket is only needed in winter. A fair share would mean sharing the fruits equally, dividing the night hours of the lamp equally, and sharing winter blanket months equally.
- Half and Double Statements:
- $4\text{ marbles are half of } 8\text{ marbles}$.
- $8\text{ marbles are double of } 4\text{ marbles}$.
- $10\text{ marbles are double of } 5\text{ marbles}$.
- $5\text{ marbles are half of } 10\text{ marbles}$.
- $6\text{ marbles are half of } 12\text{ marbles}$.
- $12\text{ marbles are double of } 6\text{ marbles}$.
- Number Line Jumps:
- Question: How far is $13$ from the double of $7$ on a number line?
- Solution: Double of $7 = 14$. Distance $= 14 - 13 = 1$ step.
- Question: How far is $5$ from half of $14$ on a number line?
- Solution: Half of $14 = 7$. Distance $= 7 - 5 = 2$ steps.
- Guess Who am I? Marble Puzzles:
- Puzzle 1: I have less than the double of $3$ marbles ($6$), and more than the half of $8$ marbles ($4$). How many do I have? Options: ${4, 8, 5}$ → Answer: $5\text{ marbles}$ (since $4 < 5 < 6$).
- Puzzle 2: I have less than the double of $4$ marbles ($8$), and more than the half of $10$ marbles ($5$). How many do I have? Options: ${6, 3}$ → Answer: $6\text{ marbles}$ (since $5 < 6 < 8$).
- Birds & Flowers Group Fractions:
- Problem 1: In a group of $8$ birds:
- Shabnam colors half $\rightarrow 4\text{ birds}$.
- Mukta colors a quarter $\rightarrow 2\text{ birds}$.
- Shabnam colored double the number of birds Mukta colored.
- Problem 2: Lakshanya and Peehu have $16$ flowers each:
- Lakshanya ties half of her flowers $\rightarrow 8\text{ flowers}$.
- Peehu ties a quarter of her flowers $\rightarrow 4\text{ flowers}$.
- Lakshanya tied double the number of flowers Peehu tied.
Practical Activities & Experiments
- Equality Overlap Folding: Folding paper parathas and shapes in half to verify if the halves overlap perfectly (proving equality).
- Symmetric Paper Folding: Taking rectangular and square paper sheets and folding them to discover all the distinct ways to construct half-shapes (diagonal vs. straight folds).
- Halves and Quarters Drawing: Drawing the missing half of a symmetrical picture (butterfly, heart, house) or drawing three missing quarters to complete a circular grid.
- Fraction Shading grids: Dividing grids into equal sectors and shading $1$ quarter, $2$ quarters, $3$ quarters, or $4$ quarters.
- NCERT Cardboard Fraction Cards: Using perforated cards from the back of the NCERT textbook to assemble segments (half, quarter) together to construct a complete circle or square.