Grade 6 · Math

Build the foundation for HCF without naming it yet.

1. 3 minWhat a factor is
   A factor divides a number exactly, with nothing left over. Factors of 6: 1, 2, 3, 6.
2. 6 minList them
   Every factor of 12. Then of 18. Work systematically from 1 upward so none is missed.
3. 3 minThe overlap
   Circle the factors that appear in both lists: 1, 2, 3, 6. Do not name it HCF yet — just let them see the overlap exists.

Tutor: Self-study this week. RD Sharma Class 6 has the practice sets.

The mirror of factors, and the foundation for LCM.

1. 3 minWhat a multiple is
   A multiple is what you get by multiplying: multiples of 4 are 4, 8, 12, 16. The list never ends — that is the difference from factors.
2. 6 minList and compare
   First eight multiples of 4. First eight of 6. Circle any that appear in both.
3. 3 minFactor or multiple
   You call a pair of numbers, they say whether the first is a factor or a multiple of the second. The two get confused constantly.

The classification CBSE tests directly and uses everywhere else.

1. 3 minThe rule
   A prime has exactly two factors: 1 and itself. A composite has more. 1 is neither — that is a favourite exam question.
2. 6 minSieve
   Numbers 1 to 30. Cross out composites, circle primes. They should end with 10 primes.
3. 3 minRecall
   Primes under 20, from memory: 2, 3, 5, 7, 11, 13, 17, 19.

The method that makes HCF and LCM mechanical instead of guesswork.

1. 3 minFactor trees
   Break 36 down: 36 = 4 × 9 = 2 × 2 × 3 × 3. Keep splitting until everything is prime.
2. 6 minPractice
   Factor trees for 24, 40 and 60. Write each answer as a product of primes.
3. 3 minCheck
   Multiply the primes back to confirm the original number. Self-checking is a habit worth building now.

Give the week's work its name and its method.

1. 3 minName it
   The biggest number in both factor lists is the Highest Common Factor. They already found it on Monday without knowing the term.
2. 6 minTwo methods
   Method 1: list factors, take the largest common one. Method 2: prime factorise both, multiply the shared primes. Do 24 and 36 both ways and get 12 twice.
3. 3 minWeek test
   HCF of 18 and 27. Then of 45 and 60. Either method, but show the working — CBSE marks working, not just answers.

Tutor: This is the 🔴 gap in the requirements doc. If Friday is shaky two weeks running, book the maths tutor.

Teach LCM by listing multiples — the most intuitive method before formal algorithms.

1. 3 minReview: multiples and common multiples
   Multiples of 3: 3, 6, 9, 12, 15, 18… Multiples of 4: 4, 8, 12, 16, 20… Circle numbers that appear in both lists — those are common multiples.
2. 6 minFind the smallest common one
   List the first seven multiples of 5 and the first five of 7 — the smallest they share is 35, and it is the seventh multiple of 5, so a short list misses it. Identify the smallest common multiple — that is the LCM. Then try 6 and 8: multiples of 6 are 6,12,18,24…; multiples of 8 are 8,16,24… LCM is 24.
3. 3 minWhy it matters
   Explain: LCM helps add fractions like 1/6 + 1/8 — you need a common denominator, and the smallest one is the LCM of 6 and 8.

Compute LCM using prime factorisation without listing multiples.

1. 2 minReview: prime factorisation
   Recall: 12 = 2 × 2 × 3, 18 = 2 × 3 × 3. Every composite number breaks into primes uniquely.
2. 4 minPrime factorise two numbers
   Do 24 and 36: 24 = 2 × 2 × 2 × 3, 36 = 2 × 2 × 3 × 3. Write each prime with its highest power: 2³ and 3².
3. 4 minMultiply highest powers
   LCM = 2³ × 3² = 8 × 9 = 72. Verify: 72 ÷ 24 = 3, 72 ÷ 36 = 2 — no remainder.
4. 2 minWhy this works
   LCM must contain every prime from both numbers at least as many times as it appears in either. 24 needs three 2s; 36 needs two 3s — so we take both.

Tutor: RD Sharma Class 6, Exercise 2.5 — problems 1 to 8.

Verify and internalize the identity HCF × LCM = product of two numbers using concrete examples.

1. 2 minReview: HCF from week 1, LCM from days 6-7
   Recall: HCF of 12 and 18 is 6; LCM of 12 and 18 is 36. Confirm they remember both methods (prime factorisation and division).
2. 4 minCompute product and compare
   For 12 and 18: compute 12 × 18 = 216. Then compute HCF × LCM = 6 × 36 = 216. Show they match.
3. 4 minTry another pair
   Use 15 and 20. HCF = 5, LCM = 60. Verify: 15 × 20 = 300; 5 × 60 = 300. Then try 9 and 14 (coprime): HCF = 1, LCM = 126; 9 × 14 = 126.
4. 2 minState the rule
   For any two whole numbers a and b: HCF(a, b) × LCM(a, b) = a × b. Emphasise it only holds for two numbers.

Tutor: Use RD Sharma Class 6, Exercise 2.7 — problems 1–5 for drill.

Teach how to decide whether a word problem needs HCF or LCM.

1. 3 minReview: HCF and LCM meaning
   HCF is the largest number that divides two or more numbers exactly. LCM is the smallest number that appears in all their multiplication tables.
2. 6 minClue words and situations
   HCF problems: splitting into equal parts, cutting rods/ropes into same length pieces, arranging things in rows/columns with no remainder. LCM problems: events repeating at different intervals, finding when things happen together again, smallest number divisible by several numbers.
3. 3 minPractice sorting
   Two word problems: (1) Two ropes 12 m and 18 m long — cut into equal pieces with nothing left. What is the longest possible length? (2) Buses leave every 15 minutes and 20 minutes. After how many minutes will they leave together again? Identify which needs HCF and which needs LCM.

Tutor: Focus on the clue words — 'largest', 'longest', 'equal parts' → HCF; 'together again', 'smallest', 'after how long' → LCM.

Final review and assessment of everything so far: factors, multiples, prime/composite, prime factorisation, HCF (two methods), LCM (two methods), and the identity HCF × LCM = product of two numbers.

1. 3 minReview day 9 word problems
   Recall yesterday’s HCF vs LCM word problems: when to use which. Example: 'Two ribbons 18 cm and 24 cm long are cut into equal pieces — longest possible piece?' (HCF). 'Two lights blink every 4 and 6 seconds — when next together?' (LCM).
2. 5 minQuick-fire calculations
   HCF of 24 and 36 by listing common factors. HCF of 48 and 72 by prime factorisation (48 = 2⁴ × 3, 72 = 2³ × 3² → take the lower power of each shared prime: 2³ × 3 = 24). LCM of 15 and 20 by listing multiples (15,30,45,60; 20,40,60). LCM of 18 and 24 by prime factorisation (2 × 3², 2³ × 3 → 2³ × 3² = 72).
3. 4 minWritten test
   Five questions: (1) List all factors of 28. (2) First five multiples of 9. (3) Is 49 prime or composite? Why? (4) HCF of 30 and 45 by any method. (5) LCM of 12 and 18 by prime factorisation.

Tutor: Week-1 summary sheet provided. Next week starts Fractions — no new work needed this evening.

Introduce negative numbers as opposites and locate them on the number line.

1. 3 minReview HCF vs LCM word problems
   Recall yesterday’s word problem: A 24 cm and 36 cm ribbon cut into equal pieces — find the longest possible length. Answer: HCF(24,36) = 12 cm. Another: Bells ring every 4 and 6 minutes — next time both ring together? Answer: LCM(4,6) = 12 minutes.
2. 5 minNegative numbers as opposites
   If +5 means 5 steps forward, −5 means 5 steps backward. Temperature: +10°C is 10 above zero; −4°C is 4 below zero. Opposite of +7 is −7, and vice versa.
3. 4 minNumber line with negatives
   Draw a horizontal line, mark 0 in the middle. To the right: 1, 2, 3,… To the left: −1, −2, −3,… Show that −2 is two steps left of 0; +3 is three steps right. Locate −4 and +5 on the same line.

Introduce integers on the number line and compare them using position.

1. 3 minReview: negatives on the number line
   Recall day 11: +5 means five steps forward, −5 five steps back; −4°C is four below zero. The opposite of +7 is −7. Mark −4 and +3 on a number line with 0 in the middle.
2. 4 minIntegers on the number line
   Draw a horizontal line with 0 in the middle. Mark −3, −2, −1, 1, 2, 3. Explain negative numbers are left of zero, positive right. Show −2 is to the left of 1, so −2 < 1.
3. 5 minCompare and order
   Place these on the number line: −4, 0, 3, −1. Then order them from smallest to largest: −4, −1, 0, 3. Do three more sets: (−5, 2, −2), (0, −3, 4), (1, −6, −1).

Add integers using number line and absolute value rules, including same-sign and opposite-sign cases.

1. 3 minReview: ordering integers
   Recall from day 12 that −5 < −2 < 0 < 3 < 7. On the number line, moving right adds a positive integer; moving left adds a negative integer.
2. 5 minSame sign: add absolute values
   −4 + (−3): same signs, add |−4| and |−3| → 7, keep the sign → −7. Do 5 + 2 = 7 and (−5) + (−2) = −7.
3. 4 minOpposite sign: subtract absolute values
   −6 + 4: opposite signs, |−6| − |4| = 2, sign of larger absolute value (−6) → −2. Do 7 + (−3) = 4 and (−9) + 5 = −4.

Tutor: Use RD Sharma Class 6, Exercise 5.2 for extra practice.

Subtract integers using the number line and rule-based strategies.

1. 3 minReview addition of integers
   Recall: adding a positive moves right; adding a negative moves left. Example: 5 + (–3) = 2, because from 5 move 3 steps left.
2. 6 minSubtraction as adding the opposite
   Subtracting a number is the same as adding its opposite. So 4 – (–2) = 4 + 2 = 6, and 4 – 7 = 4 + (–7) = –3. Do three examples: 6 – 9, –5 – 2, –3 – (–4).
3. 3 minNumber line check
   Verify each subtraction on the number line: start at the first number, face left (because subtracting), then move |second number| steps. Confirm result matches the opposite-addition method.

Tutor: RD Sharma Class 6, Chapter Integers, Exercise 5.3 — problems 1–10.

Assess mastery of integer addition and subtraction through targeted practice and a short test.

1. 2 minReview subtraction of integers
   Subtracting a positive integer moves left on the number line; subtracting a negative integer moves right. Example: 5 − (+3) = 2, 5 − (−3) = 8.
2. 4 minMixed practice
   Solve: (−7) + (+4), (+9) − (−2), (−6) − (+3), (+10) + (−10). Check each answer by reasoning on the number line.
3. 6 minTimed test
   Five problems: (−4) + (+7), (+8) − (+12), (−5) − (−9), (+3) + (-8), (−10) − (−4). 5 minutes. Score: 4/5 required to pass.

Tutor: This is the final assessment for integers this week. If passed, they may proceed to Week 4 material.

Introduce the three fraction types with precise definitions and examples, building directly on integer operations.

1. 3 minReview integer operations
   Recall day 15: −7 + 4 = −3, 5 − (−2) = 7, (−5) − (−9) = 4. Confirm the two sign rules — same signs add and keep the sign, opposite signs subtract and take the sign of the larger.
2. 4 minProper fractions
   A proper fraction has numerator smaller than denominator: 2/5, 3/7, 1/4. Value is always less than 1.
3. 5 minImproper and mixed
   An improper fraction has numerator ≥ denominator: 5/3, 7/4, 9/9. A mixed number is a whole plus a proper fraction: 1 2/3, 2 1/4. Show 5/3 = 1 2/3 using a circle divided into 3 equal parts.

Tutor: RD Sharma Class 6, Exercise 7.1 — only questions 1–5.

Connect equivalent fractions to multiplication and division, then use HCF to reduce.

1. 2 minReview: types of fractions
   Recall proper (numerator < denominator), improper (numerator ≥ denominator), and mixed numbers (whole + fraction). Example: 3/4, 7/5, 2 1/3.
2. 4 minWhat equivalent means
   Multiply numerator and denominator by the same number: 1/2 = 2/4 = 3/6. Show with 2/5 × 3/3 = 6/15.
3. 6 minReduce using HCF
   To write a fraction in lowest terms, divide numerator and denominator by their HCF. Example: 12/18 → HCF is 6 → 12÷6=2, 18÷6=3 → 2/3.

Introduce comparing fractions with unlike denominators using common denominators derived from LCM.

1. 2 minReview equivalent fractions
   Recall: multiplying numerator and denominator by the same number gives an equivalent fraction. Example: 2/3 = 4/6.
2. 7 minCompare 2/3 and 3/5
   Find LCM of 3 and 5 (15). Convert: 2/3 = 10/15, 3/5 = 9/15. Since 10 > 9, 2/3 > 3/5.
3. 3 minDo it yourself
   Compare 5/6 and 7/9. LCM of 6 and 9 is 18. Convert: 5/6 = 15/18, 7/9 = 14/18. So 5/6 > 7/9.

Introduce adding and subtracting fractions with unlike denominators using LCM of denominators.

1. 3 minReview: Comparing fractions
   Recall how to compare 2/3 and 3/5: find common denominator (LCM of 3 and 5 = 15), convert to 10/15 and 9/15, then compare numerators.
2. 6 minAdd and subtract with unlike denominators
   Step 1: Find LCM of denominators (e.g., for 1/4 + 2/3, LCM of 4 and 3 is 12). Step 2: Convert each fraction (1/4 = 3/12, 2/3 = 8/12). Step 3: Add or subtract numerators (3/12 + 8/12 = 11/12). Do same for subtraction: 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
3. 3 minOne full example
   Compute 5/6 − 1/4: LCM of 6 and 4 is 12. 5/6 = 10/12, 1/4 = 3/12. 10/12 − 3/12 = 7/12. Ensure answer is in lowest terms (it is).

Tutor: Practice from RS Aggarwal Class 6, Chapter on Fractions — exercises on addition and subtraction with unlike denominators.

Consolidate and assess all topics covered this month through a structured written test.

1. 2 minReview: Adding and subtracting fractions with unlike denominators
   Recall the steps: find LCM of denominators, convert to equivalent fractions with that denominator, then add or subtract numerators. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
2. 8 minTimed written test
   Five problems: (1) HCF of 24 and 36 by prime factorisation; (2) LCM of 8 and 12 by prime factorisation; (3) Verify HCF × LCM = product for 15 and 20; (4) Integer subtraction: −7 − (−3); (5) Fraction addition: 2/5 + 3/10. Time limit: 7 minutes.
3. 2 minSelf-check using answer key
   Reveal answers: (1) 12; (2) 24; (3) HCF=5, LCM=60, 15×20=300 and 5×60=300; (4) −4; (5) 7/10. Child checks own work and notes any errors.

Tutor: This is the final assessment for the month. No new content — only consolidation and evaluation.