Chapter 3 Lines, angles and shape Vivian IGCSE Chapter 3 Lines, angles and shape Vivian

Learning target Use the correct terms for geometrical figures Classify, measure and construct angles Calculate unknown angles using angle relationships Talk about the properties of triangles, quadrilaterals, circles and polygons. Use a ruler and a pair of compasses to bisect lines and angles Use instruments to construct triangles and other geometrical figures Calculate unknown angles in irregular polygons

Reviewing 3.1 Lines and angles Terms of geometrical figures: point, line, parallel, intersecting line, angle, perpendicular, perpendicular bisector Terms of angles: acute angle, right angle, obtuse angle, straight angle, revolution , reflex angle

Reviewing 3.1 Lines and angles cont… Angle relationships 1. complementary angles 2. supplementary angles 3. angles round a point 4. vertically opposite angles

Reviewing 3.1 Lines and angles cont… Using angle relationships to find unknown angles (homework: P50, Exercise 3.3, 3) Angles and parallel lines 1. corresponding angles (F-shape) 2. alternate angles (Z-shape) 3. co-interior angles (C-shape)

Reviewing 3.1 Lines and angles cont… 1. corresponding angles (F-shape) 2. alternate angles (Z-shape) 3. co-interior angles (C-shape)

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Do you know the shapes?

3.2 Triangles Definition: A triangle is a plane shape with three sides and three angles Classification: lengths of the three sides and sizes of the angles: scalene triangle, isosceles triangle, equilateral triangle; acute-angle triangle, right-angle triangle, obtuse triangle

3.2 Triangles cont… Angle properties of triangles

3.2 Triangles cont… Angle properties of triangles cont… 1. Angles in a triangle add up to 180° 2. The exterior angle is equal to the sum of the opposite interior angles

3.2 Triangles cont…

3.2 Triangles cont…

3.2 Triangles cont…

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Do you know the shapes?

3.3 Quadrilaterals Definition: quadrilaterals are plane shapes with four sides and four interior angles Classification: parallelogram, rectangle, square, rhombus, trapezium, kite Could you tell me some properties of the quadrilaterals? Such as the positional relations or the length relations of opposite sides; the size of opposite angles; and the relation between the two diagonals.

3.3 Quadrilaterals cont… The angle sum of quadrilateral

3.3 Quadrilaterals cont…

3.4 Polygons Definition: a polygon is a plane shape with three or more straight sides. Classification: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon

3.4 Polygons Do you know the angle sum of a polygon? The angle sum of a polygon is (n-2) ×180°, where n is the number of sides.

3.4 Polygons concave polygon and convex polygon

3.4 Polygons The sum of exterior angles of a convex polygon ? If I=sum of the interior angles, E=sum of the exterior angles and n=number of sides of the polygon Since the angle sum of a polygon is (n-2) ×180°, where n is the number of sides. I+E=180n E=180n-I I=(n-2) ×180° SO, E =180n-(n-2) ×180°=360°

Homework---- on your notebook! Page 59 Exercise 3.6, question2 Page62 Exercise 3.7, question5

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After the third chapter, test your self by this! Calculating unknown angles: On a line or round a point Vertically opposite angles Using angle relationships associated with parallel lines Angle properties of polygons

Fractions

Chapter5: Fractions Equivalent fraction: if multiply or divide both the numerator and the denominator by the same number at the same time, the new fraction still represents the same amount of the whole as the original fraction. The new fraction is known as an equivalent fraction. eg. 2/3=(2×5)/(3×5)=10/15 or 36/28=(36÷4)/(28÷4)=9/7

Chapter5: Fractions Simplest form (or lowest terms): the fraction whose numerator and denominator has no common factor other than 1, the fraction cannot be divided any further. That is, to divide the numerator and denominator by the Highest Common Factor of both.

Chapter5: Fractions Exercise: express each of the following in the simplest form possible. a) 3/15 b) 16/24 c) 21/28 d)5/8

Chapter5: Fractions Exercise: which three of 5/6, 20/25, 15/18, 35/42, 44/56 are equivalent fractions?

Chapter5: Fractions Operations on fractions: multiplying fractions: simply multiply the numerators and then the denominators; and sometimes to simplify the answer. Or it can be faster to cancel the fractions before multiplication. For mixed numbers, to change the mixed number to an improper fraction first, then multiply. Adding and subtracting fractions: fractions can be added or subtracted only when they have the same denominator. Thus, using lowest common multiple to obtain a common denominator before addition or subtraction. Dividing fractions: to divide one fraction by another fraction, simply multiply the first by the reciprocal of the second.

Chapter5: Fractions Excise: (2/3)×(5/9)= (1/4) ×(8/9)= (2/7) × = × = (2/7) × = × = (1/2)+(1/4)= (3/4)+(5/6)= - = (3/4) ÷(1/2)= ÷ =

Chapter5: Fractions For fractions with decimals in either numerators or denominators, or both: Convert the decimals to integers Simplify the fraction to be the simplest eg. 0.1/3 36/0.12 1.3/2.4

https://www.youtube.com/watch?v=-6AhN38OR14 https://www.youtube.com/watch?v=wL4hICyMLKU

Chapter5: Fractions----percentages A percentage is a fraction whose denominator is 100 To convert a fraction into a percentage, we can multiply 100 to both denominator and numerator and then cancel. eg. 8/15=

Chapter5: Fractions----percentages Practice: fraction decimal 2.5 percentage       0.833 0.075 0.32 2.5 0.333 83.3% 250% 32% 250%

Chapter5: Fractions----percentages Percentages of a given quantity 1. change the percentage into a fraction 2. multiply the fraction by the given number and simplify A is 60% of B. B is 5. Solve A.    

Chapter5: Fractions----percentages Eg1. 5% of 600  

Chapter5: Fractions----percentages Finding a number as a percentage of another: First, to write the first number as a fraction of the second number; Then multiplied by 100. Then add a percentage sign. eg. Express 16 as a percentage of 48.     33.3% = =

Chapter5: Fractions----percentages 40% Try by yourself: Express 14 as a percentage of 35 Express 3.5 as a percentage of 14 Express 1.3 as a percentage of 5.2 36 people live in a block of flats. 28 of these jog around the park each morning. What percentage of people living in the block of flats go jogging around the park? 25% 25% 77.8%

Chapter5: Fractions----percentages Now come to some examples harder 1. A can of fruit has a mass of 530g. The fruit has a mass of 500g. Find the mass of the fruit as a percentage of the total mass.  

Chapter5: Fractions----percentages 2. During the first week of October, a bookshop sold 880 books. In the second week, it sold 15% fewer books. How many books did it sell in the second week? 880×(1-15%)=748

Chapter5: Fractions----percentages  

Chapter5: Fractions----percentages Example 1. A shop keeper buys an article for $500 and sells it for $600. What is the percentage profit?  

Chapter5: Fractions ----percentages Example A person buys a car for $16000 and sells it for $12000. Calculate the percentage loss.  

Try: A pair of shoes were sold for $63. The pair of shoes were discounted by 30%. What was the original price of shoes before the sale?

5. you can call anyone in target group to solve your question Group work For percentages section, in groups, select 10 questions for students in other groups. Evaluation: 1. the base point of each group is 10. 2. if your answer is right, +1 3. if the student cannot answer your question, +1 4. group star for sun, sun for heart, heart for star 5. you can call anyone in target group to solve your question 6. each student answer at least one question

HW: WORKSHEET

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Chapter5: Fractions ----standard Form Standard form is used to express very small and very large numbers in a compact and efficient way. In standard form, numbers are written as a number multiplied by 10 raised to a given power. 10,000,000,000 0.11111111111 1×1010 1.1×10-10

Chapter5: Fractions ----standard Form  

Chapter5: Fractions ----Estimation To estimate, roundthe numbers before calculation. Although you can use any accuracy, usually the numbers in the calculation are rounded to one significant figure. Example: 3.9×2.1≈

A review of significants Count the significants: 345000 345.90 0.345000 0.000345 345.000 3.45006 30.0045 All non-zero numbers are significants All zeros between non-zeros are significants The zeros at the beginning of a number are never significant. Zeros at the end of a number are sometimes significants. If the decimal is visible, the zeros are significant. If the decimal is invisible, the zeros are not significant.

Try: 1. Estimate: 2. Estimate: 3. Estimate: 4. Estimate:

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Chapter6 Equations and transforming formulae Expansion If there is a “minus” before the bracket eg. -3(x+4) 4(y-7)-5(3y+5) 8(p+4)-10(9p-6)

Chapter6 Equations and transforming formulae Linear equation: there is no variable with a power greater than one Solving an equation with one variable means to find the value of the variable When solving equations you must make sure that you always do the same to both sides.

Chapter6 Equations and transforming formulae Solve the equations: 5x-2=3x+6 5x+12=20-11x 2(y-4)+4(y+2)=30 6/7p=10

Chapter6 Equations and transforming formulae Factorizing algebraic expressions 15x+12y 18mn-30m 36p2q-24pq2 15(x-2)-20(x-2)3

Chapter6 Equations and transforming formulae Transformation of a formula Subject eg. S=ut+5at2 F=ma x=(-b+b-2)/2a

Chapter6 Equations and transforming formulae Make the variable shown in brackets the subject of the formula in each case: x+y=c (y) X1/2+y=z (x) (a-b)/c=d (b)

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