Learning Objective: Compare unit fractions and see how sharing among more people results in smaller individual shares.
As the denominator increases (sharing with more children), each fractional share becomes smaller.
Compare sharing with 2 children (1/2) versus 5 children (1/5) or 9 children (1/9).
1/9 is smaller than 1/5, and 1/2 is greater than 1/4.
Larger denominators in unit fractions represent smaller individual shares.
Learning Objective: Visualize proper fractions and mixed fractions as distances on the number line between whole integers.
Fractions are marked by dividing each unit interval into equal fractional parts.
Test improper fractions like 5/4 to see how they cross 1 into mixed number territory (1 and 1/4).
Every fraction has a precise point associated with it on the number line.
Number lines extend whole numbers to fractional quantities.
Learning Objective: Convert between improper fractions and mixed numbers seamlessly.
Breaking improper fractions (numerator > denominator) into whole units plus a fractional remainder.
Set numerator to 9 and denominator to 2 to get 4 and 1/2.
Mixed numbers contain a whole part and a fractional part less than 1.
Fractions greater than one can be expressed as mixed numbers.
Learning Objective: Explore equivalent fractions using stacked proportional strips (fraction wall).
Aligning fractional lengths shows that 1/2 = 2/4 = 3/6 = 4/8.
Check how many pieces of 1/6 make 1/2.
Equivalent fractions denote the same length or share using different fractional units.
Multiplying or dividing numerator and denominator by the same number yields equivalent fractions.
Learning Objective: Add and subtract fractions by finding a common denominator (Brahmagupta's method).
Converting fractions to a common denominator before combining numerators.
Add 1/4 and 1/3 to get 7/12.
Fractions with different denominators must be converted to common fractional units before adding or subtracting.
Brahmagupta's ancient method remains the standard procedure for fraction arithmetic today.
When a whole number of things are shared equally among some number of people, fractions tell us how much each share is. When one unit is divided into several equal parts, each part is called a fractional unit or unit fraction (e.g., 1/2, 1/3, 1/4, ..., 1/100).
If one roti is shared among more children, each child gets a smaller share. Thus, 1/2 is greater than 1/4, and 1/9 is less than 1/5 (1/9 < 1/5).
Q1. Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh __ kg.
Answer: 1/3 kg.
Q2. A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is __ kg.
Answer: 1/4 kg.
Q3. Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank __ glass of sugarcane juice.
Answer: 3/4 glass.
Q4. The big fish weighs 1/2 kg. The small one weighs 1/4 kg. Together they weigh __ kg.
Step-by-Step Explanation: 1/2 + 1/4 = 2/4 + 1/4 = 3/4.
Answer: 3/4 kg.
Q5. Arrange these fraction words in order of size from the smallest to the biggest in the empty box below: One and a half, three quarters, one and a quarter, half, quarter, two and a half.
Answer: Quarter (1/4), half (1/2), three quarters (3/4), one and a quarter (1 1/4), one and a half (1 1/2), two and a half (2 1/2).
Fractions represent parts of a whole, such as pieces of chikki divided into equal parts. By dividing a whole into 3 equal pieces, each piece is 1/3. A piece containing 3 smaller parts of 1/4 is equal to 3/4.
The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece? (a through h)
Answer: (a) 1/12 (b) 1/4 (c) 1/8 (d) 1/6 (e) 1/8 (f) 1/6 (g) 1/24 (h) 1/24.
Fractional quantities are measured by collecting fractional units (e.g., 5 times half = 5/2).
In a fraction like 5/6, 5 is the numerator (showing how many parts) and 6 is the denominator (showing the fractional unit 1/6).
Q1. Continue this table of 1/2 for 2 more steps.
Answer: 6 times 1/2 (= 6/2), 7 times 1/2 (= 7/2).
Q2. Can you create a similar table for 1/4?
Answer: 1-time quarter (1/4), 2 times quarter (2/4), 3 times quarter (3/4), 4 times quarter (4/4).
Q3. Make 1/3 using a paper strip. Can you use this to also make 1/6?
Answer: Yes, folding each third in half yields sixths (1/6).
Q4. Draw a picture and write an addition statement as above to show: (a) 5 times 1/4 of a roti (b) 9 times 1/4 of a roti.
Answer: (a) 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 5/4 (b) 1/4 + ... (9 times) = 9/4 = 2 + 1/4.
Q5. Match each fractional unit with the correct picture (1/3, 1/5, 1/8, 1/6).
Answer: Match each unit fraction to its corresponding partitioned circular/geometric sector diagram.
Every fraction has a point associated with it on the number line. Fractions greater than one have numerators larger than their denominators.
A mixed number or mixed fraction contains a whole number part and a fractional part less than 1 (e.g., 2 2/3).
Q1 & Q2 (Page 160). Number line fractional lengths for 1/10, 3/10, 4/5. How many fractions lie between 0 and 1? Lengths of blue/black lines.
Answer: Uncountable number of fractions lie between 0 and 1.
Q1 & Q2 (Page 162). How many whole units are there in 7/2, 4/3, 7/3? Can all fractions > 1 be written as mixed numbers?
Answer: 7/2 has 3 whole units; 4/3 has 1 whole unit; 7/3 has 2 whole units. No, fractions where numerator is a multiple of denominator (like 8/4 = 2) cannot be written as mixed numbers.
Q3 (Page 162). Write the following fractions as mixed fractions (9/2, 9/5, 21/19, 47/9, 12/11, 19/6).
Answer: 4 1/2, 1 4/5, 1 2/19, 5 2/9, 1 1/11, 3 1/6.
Write the following mixed numbers as fractions (Page 163): 3 1/4, 7 2/3, 9 4/9, 3 1/6, 2 3/11, 3 9/10.
Answer: 13/4, 23/3, 85/9, 19/6, 25/11, 39/10.
When two or more fractions represent the same share or length, they are called equivalent fractions (e.g., 1/2 = 2/4 = 3/6 = 4/8).
Fractions where shares are equal are equivalent fractions. Multiplying or dividing both numerator and denominator by the same non-zero number produces an equivalent fraction.
Q1 & Q2 (Page 164). Are lengths 1/2 and 3/6 equal? Are 2/3 and 4/6 equivalent? How many 1/6 make 1/2? How many 1/6 make 1/3?
Answer: Yes; Yes; 3 pieces of 1/6 make 1/2; 2 pieces of 1/6 make 1/3.
Q1, Q2 & Q3 (Page 165). Are 3/6, 4/8, 5/10 equivalent? Write two equivalent fractions for 2/6. Complete 4/6 = ...
Answer: Yes, verified on the fraction wall. Equivalent for 2/6 include 1/3, 3/9. 4/6 = 2/3 = 6/9 = 8/12 = 10/15...
Q1 & Q2 (Page 166). Three rotis shared by four children (division, addition, multiplication facts). Two rotis shared by four children. Anil's cake sharing.
Answer: Share = 3/4 (Division: 3 ÷ 4 = 3/4; Addition: 3 = 3/4 + 3/4 + 3/4 + 3/4; Multiplication: 3 = 4 × 3/4). For 2 rotis among 4, share is 1/2. Anil gets 2/5 cake.
Q1 (Page 168). Find missing numbers for equivalent fraction relations.
Answer: (a) 10 (b) 9 (c) 14, 10.
Adding & Subtracting Fractions (Brahmagupta's Method): Convert fractions to equivalent fractions with a common denominator (common fractional unit), then add or subtract their numerators while keeping the denominator the same.
Lowest Terms: A fraction is in its simplest form when numerator and denominator have no common factor other than 1.
History: The word bhinna means 'broken' in Sanskrit. Modern fraction notation originated in India (Bakshali manuscript, Brahmagupta) and spread globally.
Q1 (Page 179). Add fractions using Brahmagupta's method (a through m).
Answer: (a) 13/7 (b) 13/12 (c) 3/2 (d) 20/21 (e) 77/60 (f) 22/15 (g) 22/15 (h) 49/40 (i) 23/4 (j) 62/21 (k) 77/60 (l) 199/105 (m) 83/12.
Q2 & Q3 (Page 179). Rahim's paint mixing and Geeta/Shamim's lace length for a table cloth perimeter of 1 meter.
Answer: Total lace = 2/5 + 3/4 = 8/20 + 15/20 = 23/20 = 1 3/20 m. Yes, it is sufficient (1 3/20 m > 1 m).
Q1 & Q2 (Page 181-182). Subtract fractions using Brahmagupta's method and word problems (Jaya's school walk, Jeevika vs Namit park round time).
Answer: (a) 1/3 (b) 2/15 (c) 7/18 (d) 1/6. Subtractions: (a) 1/12 (b) 61/15 (c) 16/7. Word problems: Jaya walks 1/5 km daily; Namit takes less time than Jeevika by 1/12 minutes.
Expressing fractions in lowest terms (Page 173): 17/51, 64/144, 126/147, 525/112.
Answer: 1/3, 4/9, 6/7, 75/16.
Comparing fractions and sorting ascending/descending (Page 174).
Answer: Common denominator conversions successfully order all comparison sets correctly.