Wednesday, 4 March 2026

Mathematics

Ch 4: Mastering Mathematical Operations - Part II (Multiplication & Division) ✖️➗

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Chapter 4: Mastering Mathematical Operations - Part II (Multiplication & Division) ✖️➗

Welcome, PSTET Aspirants! 📚✨

After mastering addition and subtraction, we now move to the next big milestone in a child's mathematical journey—multiplication and division. These operations are fundamentally about efficiency and fairness. Multiplication helps us count large groups quickly, while division helps us share and compare fairly.

For the PSTET exam (Paper 1), you must understand not just how to perform these operations, but how to teach them to young learners using concrete examples, visual aids, and real-life contexts. This chapter is your complete guide, packed with colorful icons, comparison tables, and step-by-step methods to make your preparation thorough and effective.

Let's dive deep into the world of equal groups and fair shares! 🚀

4.1 Multiplication – The Concept of Repeated Addition 🧮✖️

Multiplication is introduced to children as a shortcut for repeated addition. Instead of saying "2 + 2 + 2 + 2," we can say "4 groups of 2" or "4 × 2." This shift from addition to multiplication is a giant leap in mathematical thinking.

🍎 Meaning of Multiplication Through Arrays and Equal Groups

The two most powerful visual models for teaching multiplication are equal groups and arrays.

ConceptWhat It IsReal-Life ExampleVisual RepresentationNumber Sentence
Equal Groups 🍪Combining multiple groups that have the same number of items."There are 3 plates. Each plate has 4 cookies. How many cookies in all?"[Plate 1: 🍪🍪🍪🍪] + [Plate 2: 🍪🍪🍪🍪] + [Plate 3: 🍪🍪🍪🍪]3 × 4 = 12
Arrays 📊Arranging objects in rows and columns. Rows are horizontal, columns are vertical."Arrange 12 chairs in 3 rows with 4 chairs in each row."Row 1: ⬤ ⬤ ⬤ ⬤
Row 2: ⬤ ⬤ ⬤ ⬤
Row 3: ⬤ ⬤ ⬤ ⬤		3 (rows) × 4 (columns) = 12
Repeated Addition ➕Adding the same number multiple times.4 + 4 + 4 = 12Number line: Jump 4 steps, three times, landing on 12.3 × 4 = 12

How to Teach Arrays:

  • Start with physical objects like buttons, cubes, or dots.

  • Explain that rows go across (like rows in a classroom) and columns go up and down.

  • An array with 3 rows and 4 columns represents 3 × 4.

  • Show children that turning the array (4 rows and 3 columns) gives the same total but a different multiplication fact (4 × 3). This visually introduces the commutative property.

Classroom Activity Idea: Give students 12 counters and ask them to make as many different arrays as possible (1×12, 2×6, 3×4, 4×3, 6×2, 12×1). This shows factor pairs!

📊 Tables 2 to 10 (and Beyond up to 20)

Memorizing multiplication tables is essential for fluency, but understanding comes first. Encourage students to see patterns.

TablePattern Tip 🔍Key Fact to Remember
Table of 2All answers are even numbers. It's just "doubling."2 × anything = double of anything.
Table of 3Digits often repeat patterns.3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Table of 4Double the 2 times table.4 × 6 = (2 × 6) × 2 = 12 × 2 = 24
Table of 5Answers end in 0 or 5.5 × even = ends in 0; 5 × odd = ends in 5.
Table of 6Often the trickiest; practice with songs.6 × 8 = 48 (think: 5 × 8 = 40, plus 8 = 48)
Table of 7Use patterns or skip counting.7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Table of 8Double the 4 times table.8 × 7 = (4 × 7) × 2 = 28 × 2 = 56
Table of 9Digit sum is always 9 (e.g., 18: 1+8=9; 27: 2+7=9).Also, 9 × n = (10 × n) - n. 9 × 7 = 70 - 7 = 63
Table of 10Answers end in 0.10 × n = n with a zero at the end.
Tables 11-20Use the distributive property. For 13 × 6, think (10 × 6) + (3 × 6) = 60 + 18 = 78.Break the number into tens and ones.

Trick for Table of 19: 19 × n = (20 × n) - n. Example: 19 × 7 = (20 × 7) - 7 = 140 - 7 = 133. 🤯

🏛️ Properties of Multiplication

Understanding the properties of multiplication helps in mental math and builds a foundation for algebra. These properties are explicitly tested in PSTET pedagogy questions.

PropertyRuleExampleKid-Friendly LanguageIcon
Order Property (Commutative) 🔄Changing the order of factors does not change the product.4 × 3 = 12
3 × 4 = 12		"It doesn't matter which group you count first—the total stays the same!"🔄
Grouping Property (Associative) 🤝Changing the grouping of factors does not change the product.(2 × 3) × 4 = 6 × 4 = 24
2 × (3 × 4) = 2 × 12 = 24		"When multiplying three numbers, we can multiply any two first and then the third."🤝
Property of Zero 0️⃣Any number multiplied by zero equals zero.7 × 0 = 0
0 × 9 = 0		"If you have zero groups of something, you have nothing at all!"0️⃣
Property of One (Identity) 1️⃣Any number multiplied by one stays the same.6 × 1 = 6
1 × 8 = 8		"One group of something is just that something."1️⃣
Distributive Property 📦Multiplying a sum (or difference) by a number is the same as multiplying each part separately and then adding (or subtracting).7 × 8 = 7 × (5 + 3) = (7 × 5) + (7 × 3) = 35 + 21 = 56
Also used for 12 × 9 = (12 × 10) - (12 × 1) = 120 - 12 = 108		"Break a tough fact into easier ones!"📦

🔢 Multiplying 2-Digit and 3-Digit Numbers

Once students know tables and place value, they can multiply larger numbers. The key is to keep place value clear.

A. Multiplying by 1-Digit Numbers (Without and With Carry)

Example 1: 23 × 4 (Without Carry)

StepOperationExplanation
Step 1Multiply the ones: 4 × 3 = 12Write 2 in ones place, carry over 1 ten.
Step 2Multiply the tens: 4 × 2 = 8Add the carried 1: 8 + 1 = 9. Write 9 in tens place.
Result23 × 4 = 92Done!

Example 2: 45 × 7 (With Carry)

StepOperationExplanation
Step 17 × 5 = 35Write 5 in ones place, carry over 3 to tens.
Step 27 × 4 = 28Add carry 3: 28 + 3 = 31. Write 31.
Result45 × 7 = 315Final answer.

Example 3: 3-Digit by 1-Digit (123 × 4)

text
   1 2 3
×     4
--------
   4 9 2
--------

Step-by-Step:

  1. Multiply ones: 4 × 3 = 12 (write 2, carry 1) 🔴

  2. Multiply tens: 4 × 2 = 8 + carry 1 = 9 (write 9) 🟢

  3. Multiply hundreds: 4 × 1 = 4 (write 4) 🔵

  4. Answer: 492

B. Multiplying by 2-Digit Numbers (The Column Method)

When multiplying by a 2-digit number, we multiply by the ones digit first, then by the tens digit (adding a zero placeholder), and finally add both results.

Example: 24 × 13

text
     2 4
   ×  1 3
   -------
     7 2   (24 × 3)  ← Partial Product 1
 + 2 4 0   (24 × 10) ← Partial Product 2 (notice the zero)
   -------
   3 1 2
   -------

Step-by-Step Explanation:

  1. Step 1 (Ones): Multiply 24 by the ones digit (3): 24 × 3 = 72. ✏️

  2. Step 2 (Tens): Multiply 24 by the tens digit (1): 24 × 10 = 240. We write it as "24" shifted one place left (or with a zero at the end). ✏️

  3. Step 3 (Add): Add the two partial products: 72 + 240 = 312. ➕

Example 2: 3-Digit by 2-Digit (123 × 45)

text
     1 2 3
   ×   4 5
   -------
     6 1 5   (123 × 5)
 + 4 9 2 0   (123 × 40)
   -------
   5 5 3 5
   -------

Common Error Alert: Students often forget the zero placeholder when multiplying by the tens digit. Emphasize that multiplying by 40 is the same as multiplying by 4 and then by 10, so we shift left.

4.2 Division – The Concept of Equal Sharing and Grouping 🧮➗

Division is the inverse of multiplication. It answers two types of questions: "How many in each group?" (sharing) and "How many groups?" (grouping).

🤲 Meaning of Division Through Fair Share and Repeated Subtraction

ConceptQuestionStory ExampleActionNumber Sentence
Equal Sharing (Partitive) 🍬"How many does each one get?""12 candies are shared equally among 3 friends. How many candies does each friend get?"Distribute one by one until all are gone.12 ÷ 3 = 4
Equal Grouping (Quotative) 📦"How many groups can we make?""12 candies are packed into bags of 3 each. How many bags are needed?"Count how many groups of 3 fit into 12.12 ÷ 3 = 4
Repeated Subtraction 🔄"How many times can we subtract?"Start with 12, subtract 3 repeatedly: 12 - 3 = 9, 9 - 3 = 6, 6 - 3 = 3, 3 - 3 = 0. We subtracted 4 times.12 - 3 - 3 - 3 - 3 = 012 ÷ 3 = 4

Classroom Activity:

  • Sharing Activity: Give 15 beads to 3 students. Ask them to share equally. They will distribute one by one until all beads are gone. Each gets 5. This is 15 ÷ 3 = 5.

  • Grouping Activity: Give 15 beads and ask them to make groups of 3. They will make 5 groups. This is also 15 ÷ 3 = 5!

This shows that division can mean two different things, but the mathematical operation is the same.

🔗 Relationship Between Multiplication and Division (Inverse Operation)

Just like addition and subtraction, multiplication and division are opposites. This forms the basis of fact families.

The Fact Family Triangle:

text
     12
    /  \
   ×    ÷
  /      \
 3        4

For the numbers 3, 4, and 12:

  • Multiplication Facts:

    • 3 × 4 = 12

    • 4 × 3 = 12

  • Division Facts:

    • 12 ÷ 3 = 4

    • 12 ÷ 4 = 3

Teaching Tip: If a child knows 7 × 8 = 56, they automatically know 56 ÷ 7 = 8 and 56 ÷ 8 = 7. This builds confidence and fluency.

Missing Number Problems:

  • 5 × ? = 35 → Think: 35 ÷ 5 = 7 ✅

  • ? ÷ 6 = 4 → Think: 4 × 6 = 24 ✅

📐 Division as: Dividend ÷ Divisor = Quotient (+ Remainder)

Every division problem has specific names for its parts.

TermMeaningExample: 17 ÷ 5 = 3 Remainder 2
DividendThe total number being divided.17
DivisorThe number you are dividing by (the size of each group or number of groups).5
QuotientThe answer (the number of groups or the number in each group).3
RemainderThe amount left over that cannot be divided equally.2

Rule: The remainder must always be less than the divisor. If the remainder is greater than or equal to the divisor, you can make another group!

Checking Division: (Quotient × Divisor) + Remainder = Dividend

  • Check: (3 × 5) + 2 = 15 + 2 = 17 ✅

🔢 Long Division Method for 2-Digit and 3-Digit Numbers by 1-Digit Numbers

Long division is a systematic way to divide larger numbers. The mnemonic "Does McDonald's Sell Burgers?" helps remember the steps: Divide, Multiply, Subtract, Bring Down.

A. 2-Digit by 1-Digit (with Remainder)

Example: 57 ÷ 4

StepActionCalculationVisual
1. DivideHow many 4s in 5?1 (since 1 × 4 = 4)Write 1 above the 5.
2. Multiply1 × 4 = 4Write 4 below the 5.
3. Subtract5 - 4 = 1Write 1 below.
4. Bring DownBring down the 7Now we have 17.↓
5. Divide AgainHow many 4s in 17?4 (since 4 × 4 = 16)Write 4 above the 7.
6. Multiply4 × 4 = 16Write 16 below 17.
7. Subtract17 - 16 = 1This is the remainder.
Result14 R1Check: (14 × 4) + 1 = 56 + 1 = 57 ✅

B. 3-Digit by 1-Digit (with Remainder)

Example: 157 ÷ 4

StepActionCalculationVisual
1. Divide4 into 1? No. Move to 15. How many 4s in 15?3 (3 × 4 = 12)Write 3 above the 5.
2. Multiply & Subtract3 × 4 = 12. 15 - 12 = 3.
3. Bring DownBring down the 7.New number: 37.↓
4. Divide AgainHow many 4s in 37?9 (9 × 4 = 36)Write 9 above the 7.
5. Multiply & Subtract9 × 4 = 36. 37 - 36 = 1.Remainder 1.
Result39 R1Check: (39 × 4) + 1 = 156 + 1 = 157 ✅

C. Special Case: Zero in the Quotient

Example: 2146 ÷ 7

text
     3 0 6 R4
   ---------
7 | 2 1 4 6
     2 1
     ---
       0 4  (Bring down 4)
         0  (7 into 4? 0 times)
        ---
         4 6 (Bring down 6)
         4 2 (7 × 6 = 42)
         ---
           4 (Remainder)

Explanation:

  1. 7 into 21 = 3 (write 3) ✅

  2. Bring down 4 → 7 into 4? 0 times! Write 0 in the quotient. ✅

  3. Bring down 6 → Now we have 46. 7 into 46 = 6 (remainder 4) ✅

  4. Answer: 306 R4

Key Lesson: When a digit is smaller than the divisor, the quotient gets a zero in that place.

4.3 Solving Multi-Step Word Problems 🧩📝

In real life, problems rarely involve just one operation. PSTET expects you to be comfortable with problems requiring two or more operations.

The R.U.D.E. Method (Enhanced for Multi-Step)

StepActionExample Problem
"Riya bought 3 packets of biscuits. Each packet had 8 biscuits. She ate 5 biscuits. How many biscuits are left?"
R - Read and Visualize 🎬Read the entire problem. Visualize the scene. Identify what is given and what is asked.Riya has packets. Packets contain biscuits. She eats some. We need what's left.
U - Unpack and Identify Steps 🧩Break the problem into smaller, logical steps. Decide which operation is needed for each step.Step 1: Find total biscuits (Multiplication).
Step 2: Subtract the ones she ate (Subtraction).
D - Decide and Draw 📐Write the number sentence for each step. Use bar models if helpful.Step 1: 3 × 8 = 24
Step 2: 24 - 5 = 19
E - Execute and Evaluate ✅Calculate carefully. Check if the answer makes sense. Is it reasonable?Answer: 19 biscuits. Check: 24 total, eat 5, leaves 19. Reasonable.

Bar Model Method for Multi-Step Problems 📊

Bar models are powerful visual tools for PSTET.

Problem: "A pen costs ₹15. A notebook costs ₹25. Aman buys 2 pens and 1 notebook. He gives ₹100. How much change does he get?"

Step 1 (Total Cost Bar):

text
|← 2 Pens (2 × 15 = 30) →|← Notebook (25) →|
|←───────── Total Cost ? ─────────→|

Step 2 (Change Bar):

text
|←─────────── 100 ───────────→|
|←── Total Cost (55) ──→|← Change ? →|

Solution:

  1. Cost of pens = 2 × 15 = ₹30

  2. Total cost = 30 + 25 = ₹55

  3. Change = 100 - 55 = ₹45

Common Multi-Step Problem Types

Problem TypeOperations InvolvedExampleSolution Pathway
Shopping Scenarios 🛒Multiplication + Addition / Subtraction"A pen costs ₹15. A notebook costs ₹25. Aman buys 2 pens and 1 notebook. He gives ₹100. How much change does he get?"1. Cost of pens: 2 × 15 = 30
2. Total cost: 30 + 25 = 55
3. Change: 100 - 55 = 45
Classroom Arrangement 🪑Multiplication + Division"There are 48 students. They sit in 6 rows with equal students in each row. How many students are in 4 such rows?"1. Students per row: 48 ÷ 6 = 8
2. Students in 4 rows: 4 × 8 = 32
Fair Distribution 🍫Division + Subtraction"A box of 60 chocolates is shared equally among 8 children. Each child gets the maximum whole number. How many chocolates are left?"1. 60 ÷ 8 = 7 R4
2. Remainder = 4 chocolates left.
Comparison Problems ⚖️Multiplication + Subtraction"Raj's age is 9 years. His father's age is 4 times his age. How much older is the father?"1. Father's age: 4 × 9 = 36
2. Difference: 36 - 9 = 27 years
Combined Operations 🔄All four operations"A shop has 5 shelves. Each shelf has 12 packets of biscuits. If 15 packets are sold, how many are left? Later, the remaining are packed into boxes of 5 each. How many boxes are needed?"1. Total: 5 × 12 = 60
2. After sale: 60 - 15 = 45
3. Boxes: 45 ÷ 5 = 9 boxes

Keywords Chart for Word Problems

OperationKeywordsExample Phrase
Multiplication ✖️each, every, times, product, multiplied by, total (in groups)"Each packet has 8 biscuits" → Multiply
Division ➗shared equally, divided by, per, each group, how many groups, split"Shared among 4 friends" → Divide
Addition ➕altogether, total, in all, sum, combined, both"How many in total?" → Add
Subtraction ➖left, remaining, difference, how many more, fewer, less than"How many are left?" → Subtract

Chapter 4 Exercises: Test Your Mastery 🧪📝

A. Concept Check (Fill in the Blanks) ✍️

  1. An arrangement of objects into rows and columns is called an ________.

  2. The property that states 6 × 0 = 0 is called the ________ property.

  3. In 45 ÷ 6 = 7 R3, the number 45 is called the ________, 6 is the divisor, 7 is the ________, and 3 is the ________.

  4. 7 × 8 = 56 implies that 56 ÷ 7 = ________ and 56 ÷ 8 = ________.

  5. The mnemonic for long division steps is "Does ________ ________ ________ ________" which stands for Divide, Multiply, Subtract, Bring Down.

  6. In a division problem, the remainder must always be ________ than the divisor.

B. Match the Following (Properties Edition) 🔗

Column A (Equation)Column B (Property)
1. 5 × 9 = 9 × 5A. Identity Property
2. (2 × 3) × 4 = 2 × (3 × 4)B. Commutative Property
3. 12 × 1 = 12C. Distributive Property
4. 7 × (10 - 2) = (7 × 10) - (7 × 2)D. Associative Property

C. Table Practice (Fill the Grid) 🔢

×6789
424
535
648
763

D. Solve the Following (Algorithms) 🔢

  1. 34 × 6 = ?

  2. 52 × 14 = ? (Use column method)

  3. 89 ÷ 3 = ? (with remainder)

  4. 428 ÷ 5 = ? (Use long division)

  5. 205 × 8 = ?

  6. 637 ÷ 7 = ?

E. Multi-Step Word Problems 📖

  1. The School Trip: A school hired 5 buses for a trip. Each bus can carry 40 children. If 185 children go on the trip, how many empty seats are there? 🚌
    (Clues: Total seats = multiplication; Empty seats = subtraction)

  2. The Bakery: A baker makes 24 cupcakes. He puts them into boxes. Each box holds 6 cupcakes. He sells each box for ₹50. How much money does he get if he sells all the boxes? 🧁
    (Clues: Number of boxes = division; Total money = multiplication)

  3. The Garden: There are 48 marigold plants and 32 rose plants in a garden. The gardener wants to plant them in rows such that each row has only one type of plant and 8 plants in each row. How many rows are needed in total? 🌻🌹
    (Clues: Rows for marigolds = division; Rows for roses = division; Total rows = addition)

  4. The Library: A library has 7 shelves. Each shelf holds 25 books. If 60 books are borrowed, how many books are left? Later, the remaining books are arranged equally on 5 new shelves. How many books on each new shelf? 📚
    (Clues: Total books = multiplication; Left books = subtraction; New arrangement = division)

  5. The Farmer: A farmer has 96 apples. He packs them in boxes of 8. Each box sells for ₹120. How much money will he get if he sells all boxes? 🍎
    (Clues: Number of boxes = division; Total money = multiplication)

F. Find the Missing Number 🕵️

  1. 6 × ? = 54

  2. ? × 9 = 72

  3. 48 ÷ ? = 6

  4. ? ÷ 7 = 9

  5. 8 × ? = 0

  6. ? ÷ 5 = 5

Answer Key 🔑

A. Concept Check

  1. Array

  2. Zero

  3. Dividend, Quotient, Remainder

  4. 8, 7

  5. McDonald's Sell Burgers (or any variation)

  6. less

B. Match the Following

1-B, 2-D, 3-A, 4-C

C. Table Practice

×6789
424283236
530354045
636424854
742495663

D. Algorithm Solutions

  1. 34 × 6 = 204

  2. 52 × 14 = (52 × 4) + (52 × 10) = 208 + 520 = 728

  3. 89 ÷ 3 = 29 R2 (Check: 29 × 3 = 87, +2 = 89)

  4. 428 ÷ 5 = 85 R3 (Long division: 5 into 42 = 8 (40), remainder 2; bring down 8 = 28; 5 into 28 = 5 (25), remainder 3)

  5. 205 × 8 = 1640

  6. 637 ÷ 7 = 91 (7 × 91 = 637)

E. Word Problems

  1. Step 1: Total seats = 5 × 40 = 200 seats.
    Step 2: Empty seats = 200 - 185 = 15 empty seats.

  2. Step 1: Number of boxes = 24 ÷ 6 = 4 boxes.
    Step 2: Total money = 4 × 50 = ₹200.

  3. Step 1: Marigold rows = 48 ÷ 8 = 6 rows.
    Step 2: Rose rows = 32 ÷ 8 = 4 rows.
    Step 3: Total rows = 6 + 4 = 10 rows.

  4. Step 1: Total books = 7 × 25 = 175 books.
    Step 2: Books left = 175 - 60 = 115 books.
    Step 3: Books per new shelf = 115 ÷ 5 = 23 books.

  5. Step 1: Number of boxes = 96 ÷ 8 = 12 boxes.
    Step 2: Total money = 12 × 120 = ₹1440.

F. Find the Missing Number

  1. 9

  2. 8

  3. 8

  4. 63

  5. 0

  6. 25

Chapter Summary: Quick Revision Notes 📝

ConceptKey PointExample
MultiplicationRepeated addition of equal groups3 × 4 = 4 + 4 + 4
ArraysRows × Columns3 rows of 4 = 12
PropertiesCommutative, Associative, Zero, Identity, Distributive5 × 9 = 9 × 5
DivisionEqual sharing or grouping12 ÷ 3 = 4
Inverse OperationMultiplication "undoes" division4 × 3 = 12, so 12 ÷ 3 = 4
Long Division StepsDivide, Multiply, Subtract, Bring Down57 ÷ 4 = 14 R1
RemainderAlways less than divisor17 ÷ 5 = 3 R2
Multi-Step ProblemsBreak into steps, identify keywordsTotal - Used = Left

PSTET Success Tip: For the exam, focus on the pedagogical aspects—how to introduce concepts, common errors students make (like mixing up rows and columns, forgetting the zero placeholder in multiplication, or not checking that the remainder is smaller than the divisor), and remedial teaching strategies. Practice solving word problems quickly, as time management is crucial in a 150-minute, 150-question paper.

Common Student Errors to Watch For:

  1. Multiplication: Forgetting to add the carry-over number. 😵

  2. Multiplication by 2-digit: Forgetting the zero placeholder when multiplying by tens. 0️⃣

  3. Division: Stopping too early when there's a remainder. ⛔

  4. Division: Writing the remainder larger than the divisor. 🔄

  5. Word Problems: Adding when they should multiply, or vice versa. 🤔

Remedial Strategies:

  • Use manipulatives (counters, blocks) for concrete understanding.

  • Use grid paper to keep digits aligned in columns.

  • Practice estimation before calculation to check if the answer is reasonable.

  • Create fact family triangles for fluency.

You've now mastered the four fundamental operations. Remember, multiplication and division are not just procedures—they are ways of thinking about the world. Every time you share food, pack items, or calculate costs, you're using these operations!

Keep practicing, and remember: Every great mathematician started with these basics! 🌟

Happy Studying, Future Teachers! 📚🍎

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