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Maths

Maths. YEAR 10. Powers and roots. Exponentiation 求幂 : base (a) , power (n) a^n

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Maths

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  1. Maths YEAR 10 www.themegallery.com

  2. Powers and roots • Exponentiation求幂: base (a) , power (n) a^n • How to read it? Read as: a raised to the n-th power, araised to the power ofn, or possibly araised to the exponent ofn, or more briefly as a to the n. Some exponents have their own pronunciation: for example, a^2 is usually read as a squared and a^3 as a cubed. www.themegallery.com

  3. Example: 5^2=25 (5 raised to the power of 2 equal to 25) 25 is a power of 5, 5 is a root of the power ( square rootorperfect square) • 4^3=64 ( 4 raised to the powerof 3 or 4 cubed) 64 is a cube number. 4 is said to be the cube root of 64 www.themegallery.com

  4. Directed number (integers) • Positive Zero Negative • Number line can be either horizontal （水平） or vertical （垂直） • The rules： • ++ makes + • -+ makes – • - - makes + • Sea level (P16) www.themegallery.com

  5. Fractions • b/a numerator / denominator • Equivalent fractions (equal in value) • Simplest form: numerator and denominator have no common factors • 90/120=? SIMPLEST FORM www.themegallery.com

  6. Operations on fractions • Adding or subtracting fractions:find the LCM of the denominators • What is LCM? • Lowest common multiple • How to find LCM? 2 WAYS • 1) list the multiples of each number and then pick out • 2) express each of the numbers as a product of prime factors and then work out www.themegallery.com

  7. Continue • Dividing fractions • reciprocal (multiplicative inverse) • The product of reciprocals is always 1 www.themegallery.com

  8. Decimals • 5.268 • Decimal point: the union end and fractions begin. • Changing (transform) fractions to decimals www.themegallery.com

  9. Recurring decimals (Repeating) • Dot • Recurring decimals are rational numbers (why) www.themegallery.com

  10. Rounding numbers • Rounding to the nearest ten • Rounding to the nearest unit • Rounding to decimal places (e.g. 1 decimal place. 2 decimal places) • Rounding to significant figures • Work out the answer to one more place than you need. If the extra number is 5 or more, add 1 to the number before it. If the extra number is less than 5, leave the number before it as it is. www.themegallery.com

  11. Significant figures • 1,2,3,4,5,6,7,8,9 All non-zero digits are counted as significant figures • ZERO (0) ??? Significant or not significant? • Zero appearing anywhere between 2 non-zero digits are significant e.g. 100002 • Leading zeros are not significant e.g. 0.00052 • A zero to the left of a decimal point is significant if there is a non-zero digit to its right. e.g. 10000.02 • A zero to the right of a decimal point is significant if there is a non-zero digit to its left. E.g. 120.2300 • Zeros in a number not containing a decimal point can be ambiguous e.g. 12000 www.themegallery.com

  12. Ratio and proportion • Ratio: a comparision between two or more amounts. Usually expressed as "a to b" or a:b • a:b a being the antecedent [ˌæntɪˈsi:dnt] and b being the consequent • Hotdogs and pies are sold in a ratio of 3:4 at a local football match. If 840 pies and hotdogs were sold. How many of them were pies? • Solution: • ①Determine the ratio ( the ratio is 3:4); ②Determine the total number of parts (3+4=7) ③Calculate the value of each part (each part:840/7=120) ④Determine the number of parts needed (Number of pies sold=4×120=480) www.themegallery.com

  13. Map scale • The map of Deutschland [ˈdɔitʃlənd] • Map scale is 1:1 500 000 • This means that 1 unit of measurement on the map must be multiplied by 1 500 000 to get the distance in real life. • Express these map scales in the form 1:n • 5cm to 2km • To do Unit conversion www.themegallery.com

  14. Proportion • Show the relationship of two variables whose ratio is constant • a and b are proportional if the ratio a/b is constant • E.g. Ivan-make a cake. The proportions of sugar or egg is always given • Direct proportion • Inverse proportion: density Vs. Volume ( certain mass: m=ρ ×V) www.themegallery.com

  15. Rate • Rate: compare 2 different quantities that are measure in different units. E.g. velocity =s/t • Rate Vs. Proportion • Rate: the most common type of rate is "per unit time" which can be expressed as a percentage ( %) • Proportion: a:b www.themegallery.com

  16. Percentages • Percentage is a fraction that has a denominator of 100 • X%=X/100 • Percentages of a given quantity • E.g. 14% VAT on $100 (in China VAT is 17% or 13%) means if you buy a computer, the price tag says $100, but the store charged you for $114. $14 is VAT( value added tax) VAT=100×(14/100)=14 www.themegallery.com

  17. Profit and loss • Make money: the difference between the purchase price and the costs of bringing to market • GOAL for businessmen is profit maximization (利润最大化） • E.g. the box you made in your economic lesson • COST: glue+paper+time=￥1 =$ 0.15 • Price：$ 5 • Robber！！！ www.themegallery.com

  18. Continue • Profit=Price-Cost • Loss=Cost-Price You buy an air ticket cost $ 1000 • Percentage profit • Percentage loss www.themegallery.com

  19. Discount • Reduced price • Pay=original price ×（1-discount） www.themegallery.com

  20. Money • Foreign currency：Dollar Pound Euro Yen • Exchange e.g. ＄A AUD ＄100=￥680.98 -> ＄1=￥6.8098 Cappuccino＄5 =？RMB • The influence of RMB appreciation www.themegallery.com

  21. Time • 12-hour and 24-hour system • ante meridiem [ˈænti meˈridiem] (a.m., "before midday") and post meridiem (p.m., "after midday") www.themegallery.com

  22. New words in questions • Find the value of • Evaluate • Work out • Express fractions in its simplest form • The product of (X) www.themegallery.com

  23. Thank You ! www.themegallery.com

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