Mathematics
गणित · Class 8 · Code BEE-8
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chapters done
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The syllabus asks for
- review of sets and set notation
- universal set and complement
- disjoint and overlapping sets
- subset, proper subset and number of subsets
- union, intersection and difference; Venn diagrams
- word problems with two sets
Key points
- A set is a well-defined collection; it is written by the listing method or the set-builder method.
- n(A) is the number of members; ∈ and notin test membership.
- The universal set U is the largest set in the problem, and A' is everything in U that is not in A; n(U)=n(A)+n(A').
- Disjoint sets have no common member, so n(A∩ B)=0; overlapping sets share at least one.
- n(A∪ B)=n(A)+n(B)-n(A∩ B), and n(A∩ B)=n(A)+n(B)-n(A∪ B).
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The syllabus asks for
- review of the decimal number system and place value
- binary number system: conversion to and from decimal
- quinary and other bases
- addition, subtraction and multiplication in base 2 and base 5
Key points
- A number in base b uses only the digits 0 to b-1, and each place is worth b times the place on its right.
- To convert to decimal, expand by place values and add.
- To convert from decimal, divide repeatedly by the base and read the remainders from the bottom upwards.
- To convert between two other bases, go through decimal in two steps.
- To compare numbers in different bases, change both to decimal first.
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The syllabus asks for
- rational numbers and decimal expansion
- irrational numbers and surds
- scientific notation of numbers
- simplification of numbers in scientific notation
Key points
- A rational number is (p)/(q) with q≠0; as a decimal it terminates or recurs.
- A recurring decimal becomes a fraction by multiplying by 10, 100 or 1000 for one, two or three repeating digits, subtracting, and solving.
- An irrational number never repeats and cannot be written as a fraction; √n is irrational unless n is a perfect square.
- A surd is an exact irrational root; simplify by taking out the largest perfect-square factor, and add only like surds.
- Rationalise a denominator by multiplying the top and bottom by the surd.
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The syllabus asks for
- ratio and its simplification
- proportion: means and extremes
- continued proportion
- direct and inverse proportion
- word problems of ratio and proportion
Key points
- A ratio compares two quantities of the same kind and has no unit; write both in the same unit and simplify by the HCF.
- To divide a quantity in the ratio a:b:c, add the terms, find one part, then multiply.
- Four quantities are in proportion if a:b=c:d, which holds exactly when ad=bc.
- Three quantities are in continued proportion if a:b=b:c, and then the mean proportional is b=√ac.
- Direct variation: x∝ y, x=ky, (x1)/(y1)=(x2)/(y2); both quantities rise together.
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The syllabus asks for
- cost price, selling price, profit and loss
- profit percent and loss percent
- marked price and discount
- commission
Key points
- profit=SP-CP when SP exceeds CP; loss=CP-SP when CP exceeds SP.
- Profit percent and loss percent are measured against the cost price.
- SP=CP(1±(%)/(100)); CP=(100timesSP)/(100±%).
- discount=MP-SP, and discount percent is measured against the marked price.
- SP=MP(1-(discount%)/(100)).
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The syllabus asks for
- the unitary method
- application of direct variation
- application of inverse variation
- mixed problems; time and work, time and distance
Key points
- In direct variation, find the value of one unit by division and multiply.
- With three quantities, compare the unknown with each one separately (D: new/old, I: old/new) and multiply.
- In inverse variation, find the constant total (man-days, man-hours, speed × time) and divide by the new value.
- total work=workerstimesdays; with hours, work=workerstimeshours per daytimesdays.
- For taps or pipes, work in "fraction of the tank per hour" and add the fractions; the time is the reciprocal of the sum.
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The syllabus asks for
- principal, rate, time and simple interest
- amount and the interest formula
- finding rate or time when the others are known
- simple interest in daily life: saving and borrowing
Key points
- I=(PTR)/(100) and A=P+I, with T always in years.
- Rearranged: P=(100I)/(TR), R=(100I)/(PT), T=(100I)/(PR).
- When the amount is given, use A=Pleft(1+(TR)/(100)right) and solve for P.
- Simple interest is the same every year, so it grows in a straight line.
- Income tax on interest is a percent of the interest only; amount received =P+I-tax.
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The syllabus asks for
- area of triangles
- area of quadrilaterals (parallelogram, rhombus, trapezium)
- area of a circle and its parts
- surface area and volume of a cylinder
- area and volume problems of the SEE type
Key points
- Triangle: frac12 bh; equilateral frac√34a^2; isosceles frac b4√4a^2-b^2.
- Any quadrilateral with diagonal d and offsets h1,h2: frac12d(h1+h2).
- Shaded area = whole figure - unshaded part.
- Rectangle lb; square a^2; parallelogram bh; rhombus and kite frac12 d1d2; trapezium frac12(a+b)h.
- Circle: circumference 2π r, area π r^2; sector area (θ)/(360°)π r^2; arc length (θ)/(360°)2π r.
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The syllabus asks for
- laws of indices
- simplification using the laws
- negative and zero indices
- exponential equations
Key points
- a^m× a^n=a^m+n; a^m÷ a^n=a^m-n; (a^m)^n=a^mn; (ab)^n=a^nb^n; left(frac abright)^n=(a^n)/(b^n).
- a^0=1 for every non-zero a, and a^-n=(1)/(a^n).
- Simplify inside brackets first, then apply powers, then multiply or divide like bases by adding or subtracting indices.
- An exponential equation is solved by writing both sides as powers of the same base and equating the indices.
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The syllabus asks for
- review of factorisation
- highest common factor of algebraic expressions
- lowest common multiple
- factorisation by grouping and by using identities
Key points
- Terms, coefficient, constant and degree describe the parts of an expression; only like terms can be combined.
- HCF: every factor common to all the expressions, to its lowest power.
- LCM: every factor appearing in any expression, to its highest power.
- For two expressions, HCFtimesLCM= product of the expressions.
- ab+ac=a(b+c); group four terms in pairs when no single factor is common to all of them.
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The syllabus asks for
- simplification of algebraic fractions
- addition and subtraction of algebraic fractions
- multiplication and division of algebraic fractions
- equations involving algebraic fractions
Key points
- Factorise the top and bottom before cancelling anything; only factors cancel.
- The LCM of the denominators is the common denominator; multiply top and bottom by whatever each fraction is missing.
- When subtracting, put the second numerator in brackets and distribute the minus sign.
- Multiply fractions straight across; divide by inverting the second fraction.
- Solve a fractional equation by multiplying every term by the LCM, and state the values the variable may not take.
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The syllabus asks for
- linear equations in one variable
- simultaneous linear equations
- inequality and its graph on the number line
- graph of a linear equation
Key points
- Two simultaneous equations are solved graphically by plotting both lines; the point where they cross gives x and y.
- A quadratic equation ax^2+bx+c=0 is solved by factorising and putting each factor equal to zero; roots p, q give (x-p)(x-q)=0.
- A linear equation is solved by clearing fractions, expanding brackets, collecting like terms and dividing by the coefficient.
- Simultaneous equations are solved by elimination (make one coefficient match, then add or subtract) or substitution (make one variable the subject).
- A linear equation has infinitely many solutions, and its graph is a straight line; two points fix the line and a third checks it.
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The syllabus asks for
- straight lines, rays and angles
- pairs of angles: complementary, supplementary, adjacent, vertically opposite
- parallel lines and a transversal
- corresponding, alternate and co-interior angles
Key points
- Complementary angles add to 90°; supplementary angles add to 180°; angles round a point add to 360°.
- Vertically opposite angles are equal; angles in a linear pair are supplementary.
- When a transversal cuts two parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°.
- Two lines are parallel if any one of those three conditions holds.
- Remember the shapes: F for corresponding, Z for alternate, C for co-interior.
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The syllabus asks for
- triangles: classification and properties
- quadrilaterals and their properties
- construction of triangles and quadrilaterals
- polygons: interior and exterior angles
- the circle: radius, chord, arc, sector
Key points
- The angles of a triangle add to 180°; an exterior angle equals the sum of the two interior opposite angles.
- Equilateral, isosceles and scalene are classified by sides; acute, right and obtuse by angles.
- The angles of a quadrilateral add to 360°; a parallelogram has equal opposite angles and supplementary adjacent angles, and its diagonals bisect each other.
- For a polygon of n sides: interior angles sum to (n-2)×180°; exterior angles always sum to 360°.
- The perpendicular from the centre to a chord bisects the chord; the angle in a semicircle is 90°.
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The syllabus asks for
- congruent figures and congruent triangles
- tests of congruence (SSS, SAS, ASA, RHS)
- similar figures and similar triangles
- ratio of areas of similar triangles
Key points
- Congruent figures have the same shape and size; similar figures have the same shape at different sizes.
- The tests of congruence are SSS, SAS, ASA, AAS and RHS; there is no SSA test.
- The order of the letters in △ ABCcong△ PQR gives the correspondence, and corresponding parts of congruent triangles are equal.
- Two triangles are similar if two angles match (AA), or if the three pairs of sides are in the same ratio (SSS similarity), or if two pairs of sides are in the same ratio with equal included angles (SAS similarity).
- In similar triangles, corresponding sides are in the same ratio k and areas are in the ratio k^2.
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The syllabus asks for
- prisms and pyramids: faces, edges and vertices
- nets of solids
- surface area of prisms and pyramids
- volume of prisms and pyramids
Key points
- A prism has two identical parallel ends and rectangular sides; a pyramid has one base and triangular sides meeting at an apex.
- F+V-E=2 for every such solid.
- A net is a flat pattern that folds into the solid; a cube has eleven nets.
- Cube surface area 6a^2, volume a^3; cuboid surface area 2(lb+bh+hl), volume lbh.
- Volume of a prism = base area × length; volume of a pyramid =frac13× base area × height.
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The syllabus asks for
- the coordinate plane and plotting points
- distance between two points on the axes and in the plane
- simple shapes drawn from coordinates
Key points
- A point is written (x,y), x first; the axes divide the plane into four quadrants, and the signs of the coordinates decide which.
- On a horizontal line the distance is the difference of the x-coordinates; on a vertical line it is the difference of the y-coordinates.
- For any two points, d=√(x2-x1)^2+(y2-y1)^2, which is Pythagoras' theorem.
- Joining points and comparing the side lengths and the angles identifies the shape; equal sides alone give a rhombus, not a square.
- A triangle is right-angled exactly when the squares of two sides add to the square of the third.
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The syllabus asks for
- tessellation and its use in design
- regular tessellation
- semi-regular tessellation
- symmetry in tessellation
Key points
- A tessellation covers a plane with no gaps and no overlaps; at every vertex the angles add to exactly 360°.
- Exactly three regular polygons tessellate alone: the equilateral triangle (6×60°), the square (4×90°) and the regular hexagon (3×120°).
- A polygon tessellates alone only if its interior angle divides 360° exactly.
- Regular pentagons and octagons cannot tessellate alone; an octagon can be used with a square (135+135+90=360).
- In semi-regular tessellations the same arrangement of two or more regular polygons occurs at every vertex, written as a list such as 4.8.8.
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The syllabus asks for
- translation
- reflection and axes of reflection
- rotation about a point
- enlargement and scale factor
Key points
- Translation slides a figure: (x,y)→(x+a,y+b); the image is congruent and faces the same way.
- Reflection flips it in a mirror line: in the x-axis y changes sign, in the y-axis x changes sign, in y=x the coordinates swap.
- Rotation turns it about a centre: about the origin, 90° anticlockwise is (-y,x), 180° is (-x,-y) and 270° anticlockwise (or 90° clockwise) is (y,-x).
- Enlargement multiplies every coordinate by the scale factor k; lengths become k times and areas k^2 times as large.
- Translation, reflection and rotation give congruent images; enlargement gives a similar image.
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The syllabus asks for
- bearing and its measurement
- drawing and reading a scale drawing
- distance and direction problems using bearing
Key points
- A bearing is measured clockwise from north and written in three figures, from 000° to 360°.
- North 000°, east 090°, south 180°, west 270°.
- The back bearing is the bearing ± 180°: add 180° if the bearing is under 180°, subtract 180° if it is over.
- scale=(drawing length)/(real length), and one unit must be used throughout.
- In a scale drawing a fresh north line is drawn at every point before the next bearing is measured.
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The syllabus asks for
- collection and tabulation of data
- bar graph and pie chart
- mean, median and mode of individual series
- interpretation of statistical data
Key points
- Data are tallied, tabulated with a total, and then presented as a table, a bar graph or a pie chart.
- On a bar graph the height of the bar reads the frequency; bars have equal width and equal gaps.
- On a pie chart angle=(frequency)/(total)×360°, and the angles must add to exactly 360°.
- mean=(Σ x)/(n); the median is the middle value of the ordered list; the mode is the most frequent value.
- With an even number of values the median is the mean of the two middle ones.
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The paper
Basic Education Examination (BEE), grade 8. The CDC basic-level assessment scheme gives continuous (internal) assessment 50 % weight and the external written paper 50 %; the local level (palika) conducts and marks it. The practice sets here are written to a 50-mark, 2-hour written paper unless your palika's grid says otherwise.