Numbers of Different Size: Notation and Presentation

Numbers of Different Size: Notation and Presentation Content Powers and roots, index notation Multiples and factors Prime factorisation H.C.F./L.C.M. Standard form What is it for why do we do this? These are organising mechanisms. They allow us to simplify the way numbers can be written and help us to look for relationships between them. A large part of mathematics is about categorising objects in terms of their relationships. Hence we need to find better ways of organising them. Example : Srinivasa Ramanujan and G. H. Hardy 729 is not dull: No, it is a very interesting number, it is the smallest number expressible as a sum of two cubes in two different ways. (Can you find out which ones? This is a good spreadsheet task; the answers are ³ + 2³ and 0³ + 9³). Illustration: 25 can be expressed as a sum of two squares. (Hence it is a Pythagorean triple). Example 2: Grains of rice on a chessboard. on the first square, 2 on the second, 4 on the third, 8 on the fourth, doubling up as we go. How many on the 64 th square? Use a scientific calculator and the power button. See what happens as you go up (using 2). Soon the number gets sufficiently big that you need a new notation. See what the notation is doing. Finally interpret 2 63. (Why 63?). What are the key algorithms that need to be taught? Prime factorisation giving the method for lcm and hcf. Index notation (including fractional and negative indices) and rules of indices. Converting to and from standard form and doing calculations on numbers in standard form. Additional techniques and facts that need to be practised Meaning and finding prime, multiple, factor, root, square, cube. Structure of Delivery. Multiples/Factors Prime factorisation HCF/LCM 2. Index notation rules of indices standard form convert calculate 3. Surds rational/irrational

Module Scheme Lesson Topic Starter Activity Main Activity Multiples/Factors Multiple, factors bingo Divisibility tests 2 Prime factorisation QQ Powers and roots Prime factorisation 3 HCF/LCM I QQ Divisibility List method 4 HCF/LCM II QQ Prime factorisation Prime factors method 5 Index notation Chessboard Index notation 6 Rules of indices QQ Indices Rules of indices 7 Standard form Large + small numbers Standard form 8 Standard form convert Graphical calculators Practice and formalise 'How?' 9 Standard form calculate Graphical calculators Practice and formalise 'How?' 0 Surds Fractional indices Simplifying surds Rational/irrational QQ Surds Spot the irrational 2 Review QTest Summary

Lesson : Multiples and Factors Objectives Understand the purpose of the unit Remember the meanings of multiple, factor, prime, square and cube Know the divisibility tests for numbers up to Starter Pair based bingo. Each pair fill in their first bingo card. (One person is writing the other is checking) Teacher reads out clues in random order. Complete a line of 4 in any order for a point (call it out?). Winner is complete card or most lines. Second card available if time is. Introduction to the module Introductory talk from: What is it for why do we do this? and Structure of Delivery on page of this document. Activity Investigate divisions to look for divisibility tests. (Worksheet A) Exposition Divisibility tests up to : 2 even 3 sum of digits is a multiple of 3 4 half it and it's even 5 ends in 0 or 5 6 even + sum of digits is a multiple of 3 7 no test 8 halve it, halve it again, it's even 9 sum of digits is a multiple of 9 0 ends in 0 sum of even digits sum of odd digits is a multiple of. Follow up with quick practice. Homework Make a table like the first one on the worksheet. Fill it in for all of the numbers from 30 to 50 inclusive. Materials Calculator. Worksheets S and A.

Lesson 2: Prime Factorisation Objective Find the prime decomposition of any number using index notation for repeated factors Starter Quick questions: 'powers and roots' Say the words and write the symbols on the board. Pupils write reminder and answer into their books. Read/Write Reminder and answer 2² ("two squared") 2² = 2 2 = 4 3² (etc.) etc. 5² 0² 2³ ("two cubed") 3³ 4³ 0³ 2 5 ("two to the power of 5") 2 5 = 2 2 2 2 2 = 32 5 4 etc. Activity Short activity. 'Split it up'. Game in pairs. First person chooses a number. Second person makes the first split. If you get a number you can split up take turns to keep going till you cannot split it up anymore. Winner makes the most splits. 30 = 5 6 = 3 2 48 = 4 2 2 2 4 3 2 2 Discussion points: What kind of number can we not split any more? (Prime numbers underline them to show this). What name to we give to numbers we can split a number into (Factors).

Exposition 30 = 5 6 2 3 30 = 5 2 3 (It is good style to put the factors in order). 30 = 2 3 5 (Emphasise detail: '=' signs are in a line). 24 = 2 2 3 4 2 2 24 = 2³ 3 Exercise and homework Worksheet A2 Materials Worksheet A2 only.

Lesson 3: HCF/LCM I Objective Find the highest common factor and least common multiple by the list method. Starter For each number: say what it is divisible by (from 2, 3, 4, 5, 6, 8,9 0, ). Write up as a table to list answers, insist on (quick) explanation. Number Answer 23 3 345 5 22 2 6 2, 4 52 3, 9 35 3, 5, 9 620 2, 3, 4, 5, 6, 9, 0 353 3, 54835 5, 592 2, 4, 8, Exposition Take two numbers. Write out a list of all factors. Put them in order. 24:, 2, 3, 4, 6, 8, 2, 24 30:, 5, 6, 30 The highest common factor of 24 and 30 is 6. Now write out multiples alternating between the numbers. Stop when you find the same multiple in each list. 24: 48, 72, 96, 20 30: 60, 90, 20 The least common multiple of 24 and 30 is 20. Activity Worksheet A3 Materials Worksheet A3 only. (Calculators should not be allowed).

Lesson 4: HCF/LCM II Objective Find the highest common factor and least common multiple by the prime factor method. Starter Reminder method on the board (but do not go through). Complete a prime decomposition for each of these: 2, 28, 45, 68, 308 Very quick work through. Emphasis layout and style. 24 = 2 2 3 4 2 2 24 = 2³ 3 Activity Worksheet A4 Exposition Example : 5 = 3 5 0 = 2 5 LCM = 3 2 5 = 30 HCF = 5 Example 2: 24 = 2 2 2 3 30 = 2 3 5 LCM = 2 2 2 3 5 = 20 HCF = 2 3 = 6 Emphasise layout and method. Notes: (for students) The find the least common multiple, cross out any repeated factors. Multiply together all of the remaining factors. To find the highest common factor, multiply together all repeated factors. Exercise/Practice/Homework Find the LCM and HCF of:. 2 and 4 2. 20 and 30 3. 68 and 80 4. 98 and 63 5. 24 and 80 6. 24 and 368 Materials Worksheet A4 only. (Calculators should not be allowed).

Lesson 5: Index Notation You will need: Chessboard, Rice Objective Understand how to use index notation to show numbers raised to different powers. Use positive and negative indices. Starter Tell the chessboard story. Ask for guesses. Get volunteers to demonstrate with a real chessboard and rice. Use calculators to work it out. (Note it's 2 63.). You will necessarily bump into standard form notation when the calculation is done. This can be quickly described e.g. that's 9 with 8 noughts after it. How would you say that? We need a notation for such long calculations. We write 2 2 2 (63 times) as 2 63. Activity Worksheet A5. Pupils should work in pairs. Exposition Emphasise the 2 and 2 relationship in the table. 2 2 2 = 2³ = 8 2 2 = 2² = 4 2 = 2 = 2 2 0 = = 2 - = ½ 2 = 2-2 = ¼ 2 2 Emphasis correct terminology: Two cubed, two squared, two to the power of negative one etc. Exercise/Practice/Homework Lots of quick practice, first on items from the table, then increasingly including powers of other numbers e.g. 3, 4, 5 and 0. Materials Worksheet A5, Chessboard, Rice.

Lesson 6: Rules of Indices Objective Understand how to use to combine numbers given in index notation. Use index notation for fractional powers. Starter Fill in the gaps: Index Value Words 2 2 two squared 3 3 3 3 6 0 5 0 2? 4 5 5 5 3-4 25 0 three to the power of negative 2 Worksheet S6 is a copy of this. It may be best used as an OHT. Activity Worksheet A6. Questions 8 to 4 in the exercise are important. They could be attempted for homework and commented on in marking. The ideas will be rehearsed at the start of lesson 0. Exposition Emphasise and ensure notes for the three rules of indices. (The conclusion of each of the three sections on page of worksheet A6). Exercise/Practice/Homework Worksheet A6 has a good range of examples. Enough to include homework. However as above Questions 8 to 4 need emphasis. Materials Worksheet S6 and A6. Calculators must not be allowed.

Lesson 7: Standard Form Objective Write large and small numbers in standard form. Starter Have available a good reference source e.g. CD Rom encyclopaedia, dictionary of mathematics, physics reference, astronomy reference, internet access. Brainstorm: Give examples of really large amounts and really small amounts. Include: speed of light, distance to the sun, distance to the next nearest star, diameter of an electron, etc., What do you call them how do you say that amount? (Thousand, million, billion, trillion, zillion (?) what happens then? thousandth, billionth,. (?)) How do we avoid messy decimals or lots of zeros in measurements? We use different units for increasingly big and small things, notably in the decimal system we have kilo-, mega-, centi-, milli-, pico-,.. e.g. 6 kilometres = 6 000 metres, 3 centimetres = 3 metre etc. 00 What is the largest number? (Infinity is not a number!) A google is and a googleplex is ). Activity Worksheet A7 Exposition Summarise with the boxed section at the top of the worksheet A7. Homework The last question on the worksheet could generate good display work. Materials Worksheet A7. Reference materials.

Lesson 8: Standard Form Conversion Objective Convert between numbers written in standard form and without standard form. Starter Use a scientific calculator to find out how to convert between normal form and standard form. Use numbers less than 8 digits long or exponents between 8 and 8. On a Casio fx-83w you can change between mode9: Norm and mode8: Sci to convert. For example: To convert normal to standard form, set to mod8. Type a number which is not in standard form and press '='. To convert standard form to normal, set to mode9. Type a number (e.g. 3.6 0 5 is entered as 3.6 EXP (-) 5), then press '='. Use the table on worksheet S8 as an OHT or given out. Activity Worksheet A8 Exposition Summarise and emphasise the example at the top of worksheet A8. Use the arrow method shown to help support pupils who have not firmly generated their own strategies. Materials Worksheet S8 and A8. Scientific calculators.

Lesson 9: Standard Form Calculations Objective Calculate with numbers written in standard form. Starter Use a scientific or graphical calculator to do calculations with numbers already in standard form. For example how long does it take light to travel from the sun? Speed of light = 2.998 0 8 m/s Distance to the sun = 50 0 9 m Time taken = distance speed = 50 0 9 m 2.998 0 8 = m Now practice with simpler examples. Draw conclusions. Use worksheet S9 to help organise the work. Activity Worksheet A9. Calculators must not be used. Exposition Work through the examples in the box at the top of worksheet A9. You may find it necessary to work through the conversions to correct standard form at the start of the activity. The full examples are best left to a summary. Exercise/Practice/Homework The exercise is long enough to allow for homework practice. Materials Worksheets S9 and A9. Scientific calculators for the starter (but not for the activity).

Lesson 0: Surds Objective Be able to write numbers in surd form. Simplify the surd form and be clear about where this is possible. Starter Worksheet S0 is a reminder for fractional (and negative) indices. It can be used as a cut out the cards and match activity in pairs or used as an OHT. Activity Worksheet A0 Exposition Work through the examples at the top of worksheet A0 Materials Worksheet S0 and A0

Lesson : Rational and Irrational Objective Identify rational and irrational numbers. Starter Start the lesson with quick practice simplifying surds in the form of those in the exercise from worksheet A0 (and others) Activity If you have access to algebra software let pupils look at the values of π, e and 2 to a large number of decimal places to try to find a pattern. Worksheet A continues the practice of simplifying surds to find irrational numbers. Exposition Tell the story and follow the examples from worksheet A. Exercise/Practice/Homework Pupils should be encouraged to find other irrational quantities e.g. by an internet search. Materials Worksheet A. Algebra software if available.

Lesson 2: Review Objective Recall and apply the skills of the module. Activity Pupils should work in small groups to produce a report on one of the aspects of the work covered in this module (ensure that all aspects are covered by at least one group): Prime factorisation Finding HCF and LCM Index notation and the rules of indices Standard form Surds Rational and irrational numbers There report should include: Essential vocabulary with meanings Examples to demonstrate methods. Pupils should be ready to (briefly) present their work by the end of the lesson. Homework Each group member should take one part of their report to complete for homework, so that the group as a whole contributes a complete piece of work e.g. for display or publishing to share for the whole class.

Worksheet S: Types of Number Bingo This are is your first bingo card. In each square write a different number between and 40 This are is your second bingo card. In each square write a different number between and 40 Remember: A multiple of a number is a number made by multiplying the number by a whole number. The multiples of 6 are 6, 2, 8, 24, A factor of a number is a number which divides exactly into that number. The factors of 2 are, 2, 3, 4, 6, 2 A prime number is a number with exactly two different factors. 7 is a prime number. It can only be divided exactly by and 7. A square number is made by multiplying a number by itself. 25 is a square number it is 5 5. A cubic number is made by multiplying a number by itself, then by itself again. 27 is a cubic number. It is 3 3 3.

Example Clues: Smallest factor of 2 2 The first prime 3 The second prime 4 The first square 5 6 7 8 9 A square between 5 and 5. 0 The second multiple of 5 2 The second largest factor of 24 3 The next prime after 4 5 6 A square between 0 and 20. 7 8 9 The closest prime to 20. 20 2 22 23 2 less than the square of 5. 24 25 A square between 20 and 30. 26 27 The smallest number with both 7 and 4 28 as factors. 29 30 The second multiple of 5. 3 32 33 34 35 36 A square between 30 and 40. 37 38 39 The third multiple of 3. 40 The fourth multiple of 0.

Worksheet A: Divisibility Tests You will need a calculator. Take a number bigger than 0. Which numbers can you divide it exactly by..?.. Make a table of your results. 2 3 4 5 6 7 8 9 0 8 30 Write a list of all of the numbers that you can divide by 2. Can you see a rule that would help you decide from the number whether or not you could divide it by 2..?.. Write the rule into this table. Do the same for numbers that you can divide by 3. Then for those you can divide by 4 and so on up to. Divide by 7 Test

Worksheet A2: Prime Decomposition Example: Exercise:. Copy and complete: 8 = 2 6.. 300 = 2 50 3 50 2 50 5 5 24 = 2 3 2 5 5 24 = 2² 3 5² 8 =... 2. Copy and complete: 20 =..... 20 =... 20 =.². 3. Follow the working out from questions and 2. Complete a prime decomposition for these numbers: (a) 5 (b) 22 (c) 4 (d) 2 (e) 40 (f) 6 (g) 36 (h) 52 (i) 63 (j) 40 (k) 252 (l) 220 (m) 254 (n) 3234

Worksheet A3: HCF and LCM with Lists HCF is the highest common factor LCM is the least common multiple Exercise:. Copy and complete: List the factors - 4:. 2:. HCF = List the multiples in turn until you get one the same: 4:. 2:. LCM =.. 2. Copy and complete: List the factors - 48:. 72:. HCF = 3. List the multiples in turn until you get one the same: 6:. 4:. LCM =.. 4. Copy the method above to find the HCF of: (a) 8 and 24 (b) 48 and 60 (c) 88 and 32 5. Copy the method above to find the LCM of:

(a) 2 and 6 (b) 20 and 24 (c) 48 and 54 Worksheet A4: HCF and LCM by the Factors Method Exercise:. Copy and complete: List the factors - 7:. 4:. HCF = Complete the factorisation: 7 = 4 = List the multiples in turn until you get one the same - 7:. 4:. LCM =.. Look at the factorisation. Look at the HCF and LCM. Explain how you can use the factors to work out the HCF and LCM. 2. Copy and complete: List the factors - 5:. 0:. HCF = Complete the factorisation: 5 = List the multiples in turn until you get one the same - 5:. 0:. LCM =.. 0 = Look at the factorisation. Look at the HCF and LCM. Explain how you can use the factors to work out the HCF and LCM. 3. Copy and complete: List the factors - 24:. 30:. HCF = Complete the factorisation: 24 = 30 = List the multiples in turn until you get one the same - 24:. 30:. LCM =..

Look at the factorisation. Look at the HCF and LCM. Explain how you can use the factors to work out the HCF and LCM.

Worksheet A5: Index Notation Copy and complete the table: Hint: start in the middle work up to the top then down to the bottom.. = 2 6 =.. =. =.. = 2 4 = 6 2 2 2 = 2³ = 8 2 2 2 =. = 4 2 =. = 2 2 2 =. =. 2 = 2 =. =. 2 2 = 2 2 = 2-2 = 4 2 2 2 2 =. =. =..=. =. =..=. =. =. Look at your table. Work out the value of these: (a) 3³ (b) 0 5 (c) 4 - (d) 3-2 (e) 5 0 (f) 3-3 (g) 0 - (h) 57 0 (i) 5-3

Worksheet S6: Indices Practice Fill in the table: Index Value Words 2 2 two squared 3 3 3 3 6 0 5 0 2..?.. 4 5 5 5 3-4 25 0 three to the power of negative 2

Worksheet A6: Rules of Indices Copy and complete: 3 4 3 5 = (3 3 3 3) (3 3 3 3 3) = 3 3 3 3 3 3 3 3 3 = 3..?.. 5² 5³ = 5 5 5 5 5 = 5..?.. 7 5 7 6 = 7..?.. x a x b = x..?.. 2 7 2 4 = (2 2 2 2 2 2 2) (2 2 2 2) = 2 2 2 2 2 2 2 2 2 2 2 = 2 2 2 = 2..?.. 4 9 4 3 = 4 4 4 4 4 4 4 4 4 4 4 4 = 4..?.. 8 7 8 5 = 8..?.. x a x b = x..?.. (Now write it in good algebra: x x a..?.. b = x ) (7³)² = (7 7 7)² = (7 7 7) (7 7 7) = 7 7 7 7 7 7 = 7..?.. (9³) 5 = (9 9 9) (9 9 9) (9 9 9) (9 9 9) (9 9 9) = 9..?.. (2 4 ) 5 = 2..?..

(x a ) b = x..?.. Exercise Write each one with a single index. For example: (4 3 ) 6 = 4 8. (a) 3 7 3 4 (b) 5 9 5 2 (c) 3 6 3 7 2. (a) 3 8 3 5 (b) 2 9 2 7 (c) 7 7 3 3. (a) (6 3 ) 2 (b) (9 6 ) 5 (c) ( 7 ) 4 4. (a) 6 7 7-2 (b) 8 5 8 0 (c) 2-3 2 4 3 x 5. (a) 4 x 5 x (b) 2 x 0 x (c) 3 x 6. (a) (0 4 ) - (b) (2 5 ) 0 (c) (7-2 ) -2 7. (a) 3 5 3-5 (b) (20 a ) a (c) (0 4b ) 2c (d) 5-3 5-2 (e) 4a x 2a x (f) x 3a x 5a 8. If 6? 6? = 6 =6. What must? be? 9. If 6 ½ 6 ½ = 6. What must 6 ½ be? What symbol do we use to show this? 0. What is: (a) 25 ½ (b) 2 ½ (c) 49 ½. 8 / 3 8 / 3 8 / 3 = 8 = 8. What must 8 / 3 be? What symbol do we use to show this? 2. What is: (a) 27 / 3 (b) 25 / 3 (c) / 3 3. Copy and complete: 27 2 / 3 = (27 / 3 )² = (..)² =.. 4. Copy and complete: 6-3 / 2 = (6 / 2 ) -3 = (.) -3 = =

Worksheet A7: Standard Form Standard Form is an agreed way of writing large and small numbers. It is often called Scientific Notation (especially on calculators). To be in standard from a number must be written like this: A number from up to (but not including) 0 0 a power For example: 3.65 0 7, 2 0 8, 00.3 0 3 are all in standard form. 0.45 0 5, 3.6 2 4, 8 0 7 are not in standard form. Exercise. Write these numbers as powers of 0. (e.g. 000 = 0 4 ) (a) 00 (b) 0000 (c) 0000000 (d) 0 (e) (f) 0000000000000 2. Write these numbers as powers of 0. (e.g. 0.0 = 0 2 ) (a) 0. (b) 0.00 (c) 0.000 (d) 0.0000 (e) 0.0000000 (f) 0.00000000000 3. Copy and complete: (e.g.) 300 = 3 00 = 3 0² (a) 5000 = 5 000 = 5 0.?. (b) 700 = 7. = 7. (c) 600000 = 6.. =.. (d) 0.03 = 3 0.0 = 3. (e) 0.4 = 4. =.. (f) 0.0006 =.. =.. 4. Copy and complete: (e.g. 500 =.5 000 =.5 0 3 ) (a) 230 = 2.3 00 = 2.3. (b) 47000 = 4.7. = 4.7. (c) 4350 = 4.35. =.. (d) 0.34 = 3.4 0. =.. (e) 0.045 = 4.5. =.. (f) 0.00562 = 5.62. =..

5. Search on the internet and in reference books or CD Roms for examples of standard form (also called scientific notation). To convert normal to standard form, set to mod8. Type a number which is not in standard form and press '='. To convert standard form to normal, set to mode9. Type a number (e.g. 3.6 0 5 is entered as 3.6 EXP (-) 5), then press '='. Worksheet S8: Standard Form Investigation Use these tables to keep your results: I typed this normal number This is what I get in standard form 4560000 4.56 0 6 I typed this number in standard form This is the normal number I got 6.079 0 3 0.006079

Write a paragraph, with examples, to explain how to convert between normal numbers and standard form. Worksheet A8: Converting Standard Form Remember: Standard form has A number from up to (but not including) 0 0 a power Examples 2 3 4 5 6 7 8 567000000 = 5.67 0 8 2 3 4 5 6 0.000007823 = 7.283 0 6 2 3 4 5 6 7 3.45 0 7 = 34500000 2 3 4 5 2.9 0 5 = 0.000029 Exercise. Convert these into standard form: (a) 2000 (b) 500 (c) 700000 (d) 5700 (e) 450 (f) 73000 (g) 56200 (h) 65000 (i) 67320 2. Convert these into standard form: (a) 0.3 (b) 0.004 (c) 0.00006 (d) 0.45 (e) 0.067 (f) 0.00045 (g) 0.582 (h) 0.00672 (i) 0.08724 3. Remove the standard form (write them as ordinary numbers):

(a) 5 0 3 (b) 7 0 4 (c) 8 0 7 (d) 2.6 0 2 (e) 4.8 0 3 (f) 2.9 0 6 (g) 6.92 0 (h) 7.04 0 3 (i) 2.93 0 8 4. Remove the standard form (write them as ordinary numbers): (a) 3 0 3 (b) 7 0 2 (c) 0 5 (d) 3.6 0 (e) 4.5 0 3 (f) 5.9 0 4 (g) 4.89 0 4 (h) 5.82 0 2 (i) 6.82 0 3 Worksheet S9: Calculating with Standard Form Investigation Make sure that you have set the mode to Scientific Notation (Standard Form) Use these tables to keep your results: Not correct standard form Correct standard form 34 0 7 3.4 0 8 0.78 0 6 7.8 0 7 My calculation My result (8 0 9 ) (4 0 3 ) 3.2 0 3

My calculation My result (8 0 9 ) (4 0 3 ) 2 0 6 Write a paragraph, with examples, to explain how to calculate with numbers in standard form. Worksheet A9: Calculating with Standard Form Examples Conversion 24 0 8 = (24 0) (0 8 0) = 2.4 0 7 0.63 0 5 = (0.63 0) (0 5 0) = 6.3 0 6 Calculation (8 0 9 ) (4 0 3 ) = (8 4) (0 9 0 3 ) = 32 0 2 = 3.2 0 3 (8 0 9 ) (4 0 3 ) = (8 4) (0 9 0 3 ) = 2 0 6 (3 0 4 ) (8 0 7 ) = (3 8) (0 4 0 7 ) Exercise

. Copy and complete: (a) 57 0 9 = (57 0) (0 9 0) =.. (b) 68 0 8 = (. 0) (. 0) =.. (c) 89 0 6 = (. 0) (. 0) =.. 2. Copy and complete: (a) 0.8 0 8 = (0.8 0) (0 8 0) = 8. (b) 0.25 0 5 = (. 0) (. 0) =.. (c) 0.85 0 7 = (. 0) (. 0) =.. 3. Convert these into correct standard form: (a) 3.4 0 7 (b) 5.82 0 9 (c) 7.9 0 6 (d) 0.8 0 5 (e) 0.782 0 8 (f) 0.75 0 6

4. Copy and complete: (6 0 7 ) (7 0 4 ) = ( ) (0 7 0 4 ) =.. =.. 5. Copy and complete: (9 0 2 ) (3 0 7 ) = (..) (0 2.) =.. 6. Copy and complete: (3 0 6 ) (4 0 4 ) = ( ) (0 6 0 4 ) =.. =.. 7. Calculate: (a) 7 0 8 3 0 9 (b) 6 0 7 3 0 4 (c) 0 8 4 0 3 8. Calculate: (a) 3.4 0 5 2.6 0 (b) 9 0 5 6 0 3 (c) 5 0 6 8 0 3 9. A printer makes 3.6 0 7 sheets of paper. Each sheet is 5 0 4 metres thick. How tall will the paper be if all of the sheets are stacked in a pile? 0. The speed of light is approximately 2.998 0 8 m/s. If the distance to a certain star is 4.6 0 24 m. How long does it take light to reach us from there?

Worksheet S0: Fractional Indices Match up the cards in sets of 3. One from each column. 8 2 3 2 4 6 2 6 4 4 25 2 9 ( ) 2 3 8 2 3 27 4 ( 3 8) 2 3 2 9 2 6 4 6 27 ( ) 2 3 27 3 64 5 64 8 2 3 4 25 4 2 8 ( ) 3 9 2 64 4 3 64

Worksheet A0: Surds A surd contains the square (or other) root of a prime number. Remember: The prime numbers are 2, 3, 5, 7,, 3 etc.,,, are all surds. is not a surd, because 0 is not a prime number. We can use factors to turn a non-surd into surd form. Examples: 8 = 9 2 = 9 2 = 3 2 = 3 2 24 = 4 6 = 4 3 2 = 4 3 2 = 2 3 2 = 2 3 2 75 + 50 = 25 3 + 50 2 = 25 3 + 50 = 5 = 5 3 + 5 2 ( 3 + 2) 2 You must break down the number into prime factors and factors which are square numbers. This is called simplifying surds. Exercise. Copy and complete: 48 = 6... = 6... (a) =...... =...... (b) 20 =... 5 =...... =...... =......

2. Copy and complete: 35 =... 5 (a) = =... 5............ (b) 30 = =.................. =......... =......... =......... 3. Copy and complete: 8 + 45 = 9... + 9... =...... +...... =... =...... +...... (... +...) 4. Simplify the following surds: (a) 8 (b) 50 (c) 63 (d) 48 (e) 35 (f) 42 (g) 2 + 20 (h) 32 + 48

Worksheet A: Rational and irrational number Pythagoras of Samos is best remembered for the theorem about right angled triangles named after him. However, his work as a mathematician was mostly concerned with the properties of numbers. Sometimes these ideas had mystic aspects. Numbers had a certain perfection and purity. However, members of Pythagoras' college found that the square root of 2 had a very messy property. It is a decimal number which never ends. Worse than this, the decimal part never has any pattern. So upset were they, that all members of the college were sworn to secrecy. It is said that one member who revealed the secret outside the college was stoned to death for his crime. Numbers which have a decimal part which never ends and has no pattern are called irrational numbers. The square root of any prime number is irrational. π = 3.45926535 8979323846 2643383279 50288497 693993750 5820974944 592307864 0628620899 8628034825 34270679 82480865 3282306647.. is irrational. Any number which contains the square root of a prime number (a surd) or π is an irrational number. Examples 75 = 25 3 = 25 3 = 5 3 is irrational 27 75 9 3 3 3 3 = = = is not irrational. (It is called rational). 25 3 5 3 5 Exercise In each case simplify the expression to show whether it is rational or irrational.. 49 2. 2 3. 2 36 4. 8 2 5. 9 π 6. 64 2 7. 2 3 8. 8 + 2 2 9. 3 π 6

0. 50 + 5 25