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1. 9 NUMBER 1/2 3 3.14 Year 9 ¾ 0.001 1.25 5 2345

2. Note 1: Place Value • Each digit in a figure has a place value. Each place value is ten times larger than the place value on the right. • A decimal point shows where the ‘whole number’ part ends and the ‘fraction’ part begins. Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths e.g. 294 = 2 hundreds plus 9 tens plus 4 ones = 2 × 100 + 9 × 10 + 4 × 1

3. Place Value

4. Note 1: Place Value Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths e.g. 4.83 = 4 ones plus 8 tenths plus 3 hundredths = 4 × 1 + 8 × 1/10 + 3 × 1/100 e.g 503.08 = 5 hundreds plus 3 ones plus 8 thousandths = 5 × 100 + 3 × 1 + 8 × 1/1000

5. What are these fractions as decimals? = 0.1 1 100 = 0.01 1 1000 = 0.001 IWBPg 28 Ex 2.01

6. Note 2: Multiples • The counting numbers (or natural numbers) can be written as: • {1, 2, 3, 4, 5, 6, …………} • Multiples of a number are obtained by multiplying each of the counting numbers by that number e.g. Multiples of 7 are: 7 x { 1, 2, 3, 4, 5, 6,……} = { 7, 14, 21, 28, 35, 42, ……}

7. Note 2: Multiples e.g. Multiples of 4 are: 4 x { 1, 2, 3, 4, 5, 6,……} = { 4, 8, 12, 16, 20, 24, ……} IWBPg 99 Ex. 4.01

8. Note 3: Factors The factors of a number are all the counting numbers that divide into it exactly. e.g. The factors of 18 are: { 1, 2, 3, 6, 9, 18} e.g. The factors of 21 are: { 1, 3, 7, 21}

9. IWBPg 105-106 Ex. 4.04

10. Note 4: Prime and Composite numbers Prime numbers are counting numbers that have exactly two factors: the number itself and 1. Examples: { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…} Composite numbers are counting numbers that have more than 2 factors. Examples: {8, 14, 21, 30, 35, 100, …}

11. Note 4: Prime and Composite numbers What is the only even prime number? 2 Composite numbers are composed of prime numbers e.g. 16 = 2 x 2 x 2 x 2 IWBPg 114 Ex. 4.07

12. Starter Place the numbers 1 to 6 in the circles so that each side adds to 12 6 1 2 5 3 4

13. Note 5 : Ordering Decimals A decimal number line shows us how to compare decimals Decide whether one number is: greater than > less than < e.g.Which number is greater? 5.09 ___ 5.12 2.8 ____ 2.79 > < e.g.Put the following numbers in order from smallest to largest 0.51, 0.052, 0.52, 0.0052, 0.512 0.0052, 0.052, 0.51, 0.512, 0.52

14. Note 5 : Adding / subtracting Decimals Decimals are added/subtracted in the same way as whole numbers are. To add/subtract you must have the decimal points lined up vertically. Zeros are added so that all numbers have the same number of digits after the decimal point. e.g. 3.3 + 6.8 5.4 + 1.07 3.3 5.40 + 6.8+ 1.07 10.1 6.47 IWB Pg 34 Ex 2.02 #2.. Pg 37 Ex 2.03 #2 Pg 38 Ex 2.04

15. Note 6: Rounding How many places would I like to round to? Look to the right of that digit If it is 5 or greater, round up If it is 4 or less, leave unchanged (round down) Examples: (round to 1dp) 5.139 28.55 0.092 = 28.6 = 5.1 = 0.1 Examples: (round to 3dp) 7.813499 0.05661 IWB Pg 63Ex 2.20 = 0.057 = 7.813

16. Note 7: Estimating Answers • When estimating, round the numbers to something that is easy to calculate in your head. IWB Pg 65 Ex 2.21 Examples: 384 + 116 41.2 x 11.2 50.6 ÷ 7 = 40 x 10 = 49 ÷ 7 = 400 + 100 = 400 = 7 = 500 3 x 30 = \$ 90 George goes to buy 3 CD’s for \$29.95 each. How much will it cost him approximately? Calculator Check 3 x 29.95 = \$ 89.95

17. Note 8: Powers exponent, power • We use powers to show repeated multiplying • 32 is said as ‘3 squared’ • 53 is said as ‘5 cubed’ an base Ex 5.01 pg 124 Ex 5.02 pg 125 Ex 5.03 pg 126 Examples: 95 = 9 x 9 x 9 x 9 x 9 = 59049 43 = 64 34 = 81 28 = 256

18. Note 9: Square Roots • The opposite to squaring a number (i.e 42) is to take the square root of that number. • If you multiply a number by itself, you get a second number. The first number is the square root (√) of the second number. Examples: √81 = 9, because 9 x 9 = 81 Ex 5.08 pg 138 Ex 5.09 pg 141 PUZZLE pg 143 √16 = 4 √49 = 7 √225 = 15 4 x 4 = 16 7 x 7 = 49

19. Note 10: Integers • Integers are whole numbers, that can be either positive or negative.

20. Comparing Integers • We can compare integers using the following symbols: < means ‘less than’ e.g. 6 < 13 > means ‘greater than e.g. 43 > 41 Fill in the appropriate symbol: □ < > < □ 8 -3 □ 5 -2 □ -8 Pg 70-71 Ex 3.01 Pg 73-74 Ex 3.02

21. Note 11: Adding Integers • To add integers, start from where the first integer is. • Move right down the number to add a positive number • Move left down the number line to add a negative integer. Ex 3.03 pg 78 Ex 3.04 pg 79 -3 + 5 = 2 4 + -3 = 1

22. Note 12: Subtracting Integers • Notice: 5 − 4 = 1 Also, 5 + −4 = 1 We can simplify subtraction problems by adding the opposite of the second integer to the first integer. e.g. 1 − +3 5 − −3 −1 − −3 = 1 + −3 = 5 + +3 = −1 + +3 = −2 = +8 = +2 Ex 3.06 pg 81 Ex 3.07 pg 82

23. Note 13: Multiplying Integers • Rules for multiplying pairs of integers:

24. Remember BEDMAS! Multiplying integers 1.) 2 x 4 2.) 2 x −4 3.) −4 x −4 = = = 8 −8 16 3.) 9 x −7 4.) −6 x −4 6.) −5 x +2 = = = −63 24 −10 7.) 3(2 x −5 +1) 8.) (−6 −−4) x (−10 + 2 x 3) = = (−6 + 4) x (−10 + 6) 3(−10 + 1) =3(−9) = (−2) x(−4) = 8 = −27

25. Multiplying/dividing Integers + − − + Ex 3.15 pg 93 Ex 3.16 pg 95 Ex 3.17 pg 96 e.g.−12 ÷ −2 = 6

26. Note 14: Order of OperationsBEDMAS B rackets Exponents Division Multiplication Addition Subtraction When there is only x and ÷ work left to right When there is only + and − work left to right

27. BEDMAS Examples 5 + 3 x 4 6 x ( 3 + 2 ) = 5 + 12 = 6 x 5 = 17 = 30 13 − 4(3−1) 6(13−10) + 5(20÷4) = 13 − 4(2) = 6(3) + 5(5) = 13 − 8 = 18 + 25 = 5 = 43 IWB Pg 146 Ex 6.01 Pg 148 Ex 6.02

28. Note 15: BEDMAS with integers and exponents • We can combine many of the skills from this unit to solve higher level questions. Solve using BEDMAS 4[2 −(12−6)2 ÷ 3] (3 + 2)2 − 2 x 5 = 4[2 −(6)2 ÷ 3] = (5)2 − 2 x 5 = 4(2 − 36 ÷ 3) = 4(2 − 12) = 25 − 2 x 5 = 4(−10) = 25 − 10 = −40 Ex 6.02 pg 148 = 15

29. Number Strategies • Multiplying and dividing by multiples of 10 Multiply 27 by: a.) 10b.) 100 c.) 1000 thousands 7 0 2

30. Number Strategies • Multiplying and dividing by multiples of 10 Multiply 27 by: a.) 10b.) 100c.) 1000 thousands 2 7 0 0

31. Number Strategies • Multiplying and dividing by multiples of 10 Multiply 27 by: a.) 10 b.) 100c.) 1000 thousands 2 7 0 0 0

32. Number Strategies • Multiplying and dividing by multiples of 10 Divide 56000 by: a.) 10b.) 100 c.) 1000 thousands 0 0 5 6

33. Number Strategies • Multiplying and dividing by multiples of 10 Divide 56000 by: a.) 10b.) 100c.) 1000 thousands 5 6 0

34. Note 16: x and ÷ by multiples of 10 • Multiplying by multiples of 10 cause the place value digits to glide to the LEFT the same number of place values as there are zeros in the multiplier. • e.g. 24 x 100 = 2400 • Dividing by multiples of 10 cause the place value digits to glide to the RIGHT the same number of place values as there are zeros in the divisor. • e.g. 94 000 ÷ 1000 = 94

35. Carry out these calculations a.) 79 x 10 b.) 533 x 100 c.) 21 x 1000 790 53000 21 000 d.) 230 ÷ 10 e.) 9500 ÷ 100 f.) 2300 ÷ 1000 23 95 2.3 g.) 142 x 20 h.) 60 x 300 i.) 25 x 3000 2840 18 000 75 000 j.) 25 000 ÷ 5000 k.) 480 000 ÷ 20 000 5 24 l.) Loati’s friend sell 2000 daffodils on Daffodil Day to raise funds for the local hospice. They raised \$ 14 000. What was the selling price for each daffodil? \$ 14 000 ÷ 2000 = \$ 7.00