In this guide
The week before CTET is not the time to learn new topics. It is the time to make sure nothing you already know slips away under pressure. This sheet collects the content facts and the pedagogy principles that maths questions keep returning to, so you can revise both halves of the section in one sitting.
In Paper I, mathematics has 30 questions: roughly half content at the NCERT Classes 1–5 level and half pedagogy. In Paper II, the Mathematics and Science section has 30 questions each for maths and science: 20 content and 10 pedagogy per subject. Check the latest information bulletin before the exam, since the pattern can change.
Numbers
| Fact | Detail |
|---|---|
| Place value and face value | In 4,725 the face value of 7 is 7; its place value is 700 |
| Indian and international grouping | 1,00,000 = one lakh = 100,000 = one hundred thousand; 1,00,00,000 = one crore = ten million |
| 0 and 1 | 0 is the smallest whole number; 1 is the smallest natural number; 1 is neither prime nor composite |
| Primes | 2 is the smallest and only even prime; primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 |
| Divisibility | By 3: digit sum divisible by 3. By 4: last two digits divisible by 4. By 6: by 2 and 3. By 9: digit sum divisible by 9. By 11: difference of alternate digit sums is 0 or a multiple of 11 |
| HCF × LCM | For two numbers, HCF × LCM = product of the numbers |
| Order of operations | Brackets, then division and multiplication (left to right), then addition and subtraction (left to right) |
Roman numerals: I 1, V 5, X 10, L 50, C 100, D 500, M 1,000. A symbol is not repeated more than three times in a row; V, L and D are never repeated. A smaller symbol before a larger one is subtracted, in six standard pairs: IV 4, IX 9, XL 40, XC 90, CD 400, CM 900. So 49 is XLIX, not IL, and 1,994 is MCMXCIV.
Quick check: HCF of 12 and 18 is 6, LCM is 36, and 6 × 36 = 216 = 12 × 18. For more, see numbers and operations.
Fractions, decimals and percentages
- A fraction names equal parts of a whole or of a collection.
- Equivalent fractions: multiply or divide numerator and denominator by the same number (2/3 = 8/12).
- Compare by making denominators equal: 3/4 = 9/12 and 2/3 = 8/12, so 3/4 is larger.
- Add unlike fractions through a common denominator: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
- Decimals: 0.3 = 0.30, so 0.3 > 0.25. Tenths × tenths gives hundredths: 0.3 × 0.2 = 0.06.
- Percentage means "per hundred": 25% = 25/100 = 1/4.
Geometry
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Cube and cuboid | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square pyramid | 5 | 8 | 5 |
| Triangular pyramid (tetrahedron) | 4 | 6 | 4 |
Euler's formula for polyhedra: F + V − E = 2. For a cube, 6 + 8 − 12 = 2.
- Angle sum of a triangle is 180°; of a quadrilateral, 360°.
- Right angle 90°; straight angle 180°; complete turn 360°.
- Lines of symmetry: square 4, rectangle 2, equilateral triangle 3, isosceles triangle 1, scalene triangle 0, regular hexagon 6, circle infinitely many. A parallelogram that is not a rectangle or rhombus has none.
Mensuration and measurement
| Figure | Perimeter | Area or volume |
|---|---|---|
| Rectangle (l, b) | 2(l + b) | l × b |
| Square (side a) | 4a | a² |
| Triangle | Sum of sides | ½ × base × height |
| Circle (radius r) | 2πr | πr² |
| Cube (edge a) | - | Volume a³ |
| Cuboid (l, b, h) | - | Volume l × b × h |
Conversions: 1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm; 1 kg = 1,000 g; 1 L = 1,000 mL; 1 L = 1,000 cm³; 1 hectare = 10,000 m²; 1 hour = 60 minutes; 1 day = 24 hours.
Measurement pedagogy: non-standard units (handspans, footsteps, cups) come before standard units, because they show why we need a common unit.
Data and algebra
For the data 3, 7, 7, 8, 10: mean = 35 ÷ 5 = 7; median (middle value in order) = 7; mode (most frequent) = 7; range = 10 − 3 = 7. For an even number of values, the median is the mean of the two middle values: for 2, 4, 6, 8 it is (4 + 6) ÷ 2 = 5.
Pictographs need a key ("one symbol = 5 children"). Bar graphs need equal widths and a uniform scale.
Algebra: a variable stands for any number in a pattern ("the number of matchsticks for n squares in a row is 3n + 1"). Solve equations by keeping both sides balanced: if 2x + 3 = 11, then 2x = 8 and x = 4. Check: 2 × 4 + 3 = 11.
Pedagogy on one page
| Idea | What to remember |
|---|---|
| Aim of school maths (NCF 2005 position paper) | Mathematisation of the child's thinking; narrow aim (numeracy) and higher aim (thinking and reasoning mathematically) |
| Nature of maths | Abstract, logical, precise, structured, about patterns; not the same as computation |
| Reasoning | Inductive: examples to rule. Deductive: general statements to particular conclusions, used in proof |
| Sequence | Concrete → pictorial → abstract (Bruner: enactive, iconic, symbolic) |
| Van Hiele levels | 0 visualisation, 1 analysis, 2 informal deduction, 3 deduction, 4 rigour (some books number 1–5) |
| Children's strategies | Counting all, counting on, making ten, doubles, compensation; value them |
| Problem solving | Pólya: understand, plan, carry out, look back; open-ended tasks and problem posing |
| Errors | Windows into thinking; systematic errors reveal a wrong rule; diagnose before remedial teaching |
| Remedial teaching | Targets a diagnosed gap, uses a different approach and concrete material, re-assesses |
| Aids | Dienes blocks and bundles for place value (proportional), abacus later (non-proportional), geoboard for area and perimeter, tangram (7 pieces) for spatial sense, fraction strips for equivalence |
| Assessment | Continuous, formative, focused on reasoning; oral questions, observation, portfolios, open-ended tasks |
| Problems of teaching | Maths anxiety, rote learning, language of word problems, school-life gap |
| Community maths | Maths of shops, farms, crafts, rangoli and calendars in the classroom |
The detailed guides: nature of maths, evaluation, errors and remedial teaching.
Quick self-test
- The place value of 6 in 36,452 is: (a) 6 (b) 600 (c) 6,000 (d) 60,000
- 49 in Roman numerals is: (a) IL (b) LXI (c) XXXXIX (d) XLIX
- A solid has 5 faces and 5 vertices. Its number of edges is: (a) 6 (b) 8 (c) 9 (d) 10
- The median of 9, 2, 5, 7, 4 is: (a) 4 (b) 5 (c) 5.4 (d) 7
- A child says, "It is a rectangle because it looks like a door." The child is at van Hiele level: (a) visualisation (b) analysis (c) informal deduction (d) rigour
- According to the NCF 2005 position paper, the main goal of maths education is: (a) speed in calculation (b) memorising formulas (c) mathematisation of the child's thinking (d) preparing for board exams
- The best first aid for teaching regrouping in addition is: (a) abacus (b) calculator (c) tangram (d) bundles of ten sticks or Dienes blocks
- A square and a rectangle have the same perimeter of 16 cm; the rectangle is 6 cm long. The square's area exceeds the rectangle's by: (a) 2 cm² (b) 4 cm² (c) 6 cm² (d) 0
Answers
- (c). 6 is in the thousands place.
- (d). 40 is XL and 9 is IX; IL is not a valid form.
- (b). F + V − E = 2 gives 5 + 5 − E = 2, so E = 8 (a square pyramid).
- (b). In order: 2, 4, 5, 7, 9; the middle value is 5.
- (a). Judging by overall appearance is level 0.
- (c). Mathematisation is the stated main goal.
- (d). Proportional materials come before the abacus.
- (b). Square side 4 cm, area 16 cm². Rectangle breadth = 8 − 6 = 2 cm, area 12 cm². Difference 4 cm².
The last week
- Day 1: numbers, Roman numerals, divisibility rules; redo this self-test.
- Day 2: fractions, decimals, geometry tables; draw each solid and count.
- Day 3: mensuration and conversions; five mixed problems.
- Day 4: the pedagogy table; explain each row aloud in one sentence.
- Day 5: one full previous-year maths section, timed.
- Day 6: review every wrong answer and name its reason.
- Day 7: light revision of this sheet only.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Central Board of Secondary Education website .
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