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It is important to be able to identify the different types of whole numbers and recognize their properties so that we can apply the correct strategies needed when completing calculations. 4 6 4 Blaise Pascal developed this triangle by simply adding two whole 5 0 0 5 Pascal's Triangle numbers together each time 6 5 0 5 6 7 5 5 7 8 8 56 70 56 8 8 9 6 84 6 6 84 6 9 Give this a go! Q The ant nest below has a tunnel system that leads down to a main chamber. After one ant enters the tunnel from the top, how many different ways can it get to the main chamber if it only travels downwards the entire way? Main Chamber Work through the book for a great way to solve this P Learning H SERIES TOPIC
How does it it work? Place values 4 Numbers can be separated into columns that represent different multiples of 0. The column where a number is found determines the place value of that number. 000 000 00 000 0 000 000 Millions Hundreds of thousands Tens of thousands Thousands Hundreds Tens Ones N U M B E R S 00 0 Remember: When multiplying by multiples of 0, just add the same number of zeros to the end # 00 00 5 # 00000 500000 # 0000 0000 What is the place value of 4 in the number 4 50? Method : Multiplying the number by the multiple of 0 matching its position in the number 4 50 4 is in the thousands position Identify the position of the 4 in the number ` 4 # 000 Multiply 4 by the place value ` place value of 4 is 4000 Method : The place value of a number can also be found by changing all the other numbers to a 0 4 50 04 000 Change all the other numbers to a zero ` place value of 4 is 4000 Ignore all zeros in front of the 4 H SERIES TOPIC P Learning
Multiplying the number by the multiple of 0 matching its position in the number. (i) Write 6 405 using words (ii) Write 6 405 in expanded form Hundreds of thousands Tens of thousands Thousands Hundreds Tens Ones 6 4 0 5 00 000 0 000 000 00 0 (i) Using words: Six hundred and thirty one thousand, four hundred and five Name using groups of three (ii) Expanded form: ^6 # 00000h+ ^# 0 000h+ ^# 000h+ ^4 # 00h+ ^0 # 0h+ ^5# h Multiply each number by the place value and add together Here is another example. (i) Write 07 8 using words (ii) Write 07 8 in expanded form 000 000 00 000 0 000 000 Millions Hundreds of thousands Tens of thousands Thousands Hundreds Tens Ones 0 7 8 00 0 (i) Using words: One million, seventy two thousand, one hundred and thirty eight Name using groups of three (ii) Expanded form: ^# 000000h+ ^0 # 00 000h+ ^7 # 0 000h+ ^ # 000h+ ^# 00h+ ^# 0h+ ^8 # h Multiply each number by the place value and add together P Learning H SERIES TOPIC
Your Turn Place value Write down the place values for each of these numbers a 46 b 4 60 Place value of : Place value of : Place value of : Place value of 4: c 560 4 d 7 80 6 Place value of 5: Place value of 7: Place value of 6: Place value of 8: Write each of these ordinary numbers in: (i) worded form (ii) expanded form a 560 (i) Two thousand, five hundred and sixty (ii) ( # 000) + (5 # 00) + (6 # 0) b 06 (i) (ii) c 89 06 (i) (ii) 4 H SERIES TOPIC P Learning
WHOLE NUMBERS * WHOLE NUMBERS * How does it work? Your Turn Place value d 708 00 (i) (ii) Place Value.../.../0... e 9 0 060 (i) (ii) Write the ordinary number for each of these: a Four hundred and thirty nine thousand, two hundred and six b ^4 # 000000h+ ^ # 00000h+ ^0 # 0000h+ ^# 000h+ ^0 # 00h+ ^# 0h+ ^0 # h c Eighty one thousand and five d ^9 # 0000h+ ^8 # 000h+ ^9 # 00h+ ^9 # 0h+ ^8 # h e Any number whose place values for 4, 5 and are 4000, 5 and 00 f Three million, thirty thousand and thirty P Learning H SERIES TOPIC 5
Adding and subtracting large numbers When adding or subtracting large numbers, make sure the place values are lined up correctly. Here are some addition examples to refresh your memory. Calculate 89 + 47 89 + 47 8 9 + 4 7 Ensure matching place values are aligned 8 9 + 4 7 Carry over the 'tens' value 8 0 0 ` 89 + 47 800 You can check your answer by simply entering the sum into your calculator. Always check that your answer makes sense if using a calculator. It is easy to accidentally press a wrong button when entering numbers and operations into a calculator. Calculate 7 9 + 0 0 + 0 56 7 9 + 7 9 + 0 09 + 0 56 0 0 9 0 5 6 Ensure matching place values are aligned 7 9 + 0 0 9 0 5 6 4 9 7 4 0 Carry over the 'tens' value ` 79 + 009 + 056 49740 7 9 + 0 0 9 + 0 5 6 49 740 6 H SERIES TOPIC P Learning
There are many different accurate ways to subtract large numbers. You should always use the method that you were taught or know best. Here is an example using one way. Calculate 765 489 Step : 7 6 5 4 8 9 Line up matching place values Step : 7 6 5 4 8 9 Since 5 9 we cannot subtract, ` place a between 5 and The is in front of the 5 (in the tens position), making it 5 is traded and carried to to make it for using the to make 5 Step : 7 6 5 4 8 9 0 6 5 9 equals 6 and equals 0 Step 4: 7 6 5 4 8 9 0 6 Since 6 8we cannot subtract, ` place a between 4 and 6 The is in front of the 6 (in the tens position), making it 6 is traded and carried to 4 to make it 5 for using the to make 6 Step 5: 7 6 5 4 8 9 8 0 6 6 8 equals 8 and 7 5 equals ` 765-489 806 7 6 5 4 8 9 806 P Learning H SERIES TOPIC 7
Here is another example for subtraction. Calculate 8 4 576 Step : 8 4 5 7 6 Line up matching place values Step : 8 4 5 7 6 Since 4 6 we cannot subtract, ` place a between 7 and 4 The is in front of the 4 (in the tens position), making it 4 is traded and carried to 7 to make it 8 for using the to make 4 Step : 8 4 5 7 6 8 4 6 equals 8 Step 4: 8 4 5 7 6 8 Since 8 we cannot subtract, ` place a between and 5 The is in front of the 6 (in the tens position), making it is traded and carried to 5 to make it 6 for using the to make Step 5: 8 4 5 7 6 5 8 8 equals 5 Step 6: 8 4 5 7 6 5 8 Since 6 we cannot subtract, ` place a between and The is in front of the (in the tens position), making it is traded and carried to to make it for using the to make Step 7: 8 4 5 7 6 6 6 5 8 6 equals 6, 8 equals 6, equals Do all the subtractions since there are no more columns with the top bottom ` 84-576 6658 8 4 5 7 6 6 658 8 H SERIES TOPIC P Learning
Your Turn Adding and subtracting large numbers Calculate each of these addition questions showing all working. a 5 6 0 + 8 8 5 0 6 4 b 7 4 0 0 + 0 8 0 9 4 0 c 9 9 4 8 6 4 + d 8 4 9 5 6 0 4 6 + Large Whole Numbers.../.../0... e 4 6 9 9 9 5 7 6 0 9 + f 8 7 6 8 9 0 5 4 + Combo Time! Calculate the sum (+ ) of three hundred and forty five thousand, two hundred and nine and eighteen thousand, seven hundred and ninety six. P Learning H SERIES TOPIC 9
Your Turn Adding and subtracting large numbers Calculate each of these substraction questions showing all working. a 5 6 8 5 b 5 7 5 6 4 0 c 6 5 6 8 0 4 8 d 5 4 4 5 e 0 0 4 0 0 0 f 7 0 0 0 0 6 7 8 9 4 Combo Time! Calculate the difference ( ) between: five hundred and seventy thousand, two hundred and seventeen and ninety eight thousand, four hundred and twenty one 0 H SERIES TOPIC P Learning
Long multiplication As you did when adding and subtracting, keep your place value columns lined up neatly. You need to be aware of the place value of the number you are multiplying by. Calculate 49 # 4 9 + 8 4 8 4 9 8 5 8 + + 6 7 0 # # Line up matching place values Multiply the 49 by Carry over any tens values after multiplying For 49 # tens, put a 0 in the ones column and multiply by Carry over any tens values after multiplying 4 9 4 8 5 8 7 8 0 4 5 7 8 # + Add the two new numbers together ` 49 # 4578 Here is another example. Be careful to line up the columns correctly. Calculate 4 # 506 4 5 0 6 + + 4 8 # Line up matching place values Multiply the 4 by 6 Carry over any tens values after multiplying 4 5 0 6 0 0 5 0 0 8 0 # For 4 # 0 tens, put a 0 in the ones column first and multiply by 0 4 5 0 6 0 0 5 0 0 8 0 + + 0 0 5 0 0 # For 4 # 5 hundreds, put a 0 in the ones and tens columns and multiply by 5 Carry over the tens value after multiplying 4 # 5 0 6 5 8 + 0 0 0 0 0 + 5 0 0 4 0 8 Add the two new numbers together ` 4 # 506 408 P Learning H SERIES TOPIC
Your Turn Long multiplication WHOLE NUMBERS * WHOLE NUMBERS * Calculate each of these multiplication questions showing all working. Check your answers on the calculator. a 0 6 # b 5 8 9 # Long Multiplication.../.../0... c 9 5 7 0 6 # d 8 7 6 4 5 # e 0 7 # f 0 0 5 # H SERIES TOPIC P Learning
Your Turn Long multiplication Calculate each of these multiplication questions showing all working. Check your answers on the calculator. a b 5 8 4 0 5 c 9 0 8 0 9 d 8 6 4 4 5 e 5 4 7 f 6 4 8 5 P Learning H SERIES TOPIC
Short and long division Short and long division are only different due to the size of the number you are dividing by. Here is a short division question. Calculate 75 408 ' 6 Step : Step : Step : Step 4: Step 5: 6 7 5408 g 6 7 5 4 08 g 5 4 6 7 5 4 08 g 5 6 4 4 6 7 5 4 0 8 g 5 6 8 4 4 6 7 5 4 0 8 g Divide 7 by 6 Put the whole number answer () above the 7 Make the remainder () the tens digit for the next number Divide 5 by 6 Put the whole number answer () above the 5 Make the remainder () the tens digit for the next number Divide 4 by 6 Put the whole number answer (5) above the 4 Make the remainder (4) the tens digit for the next number Divide 40 by 6 Put the whole number answer (6) above the 0 Make the remainder (4) the tens digit for the next number Divide 48 by 6 Put the answer (8) above the 8 ` 75408 ' 6 568 If there is a remainder at the end, always write it as a fraction. remainder fraction the amount left over the divisor Calculate 58 ' Step : 5 g 8 Divide 5 by Put the whole number answer () above the 5 Make the remainder () the tens digit for the next number Step : 7 5 8 g Divide by Put the whole number answer (7) above the There is no remainder this time Step : divisor g 7 g5 8 amount left over Divide 8 by Put the whole number answer () above the 8 Write the remainder as a fraction ( ) to the right ` 58 ' 7 4 H SERIES TOPIC P Learning
Here is a long division question. Calculate 659 ' 4 Step : 4 4 g 6 5 9 5 6 Divide 6 by 4 Put the whole number answer (4) above the Multiply 4 by the answer (4) and write this underneath the 6 Step : 4 4 g 6 5 9 5 6 6 5 Subtract 56 from 6 Drop the 5 down next to the answer Step : 4 4 4 g 6 5 9 5 6 6 5 5 6 Divide 65 by 4 Put the whole number answer (4) above the 5 Multiply 4 by the answer (4) and write this underneath the 65 Step 4: 4 4 4 g 6 5 9 5 6 6 5 5 6 9 9 Subtract 56 from 65 Drop the 9 down next to the answer Step 5: 4 4 7 4 g 6 5 9 5 6 6 5 5 6 9 9 9 8 Divide 99 by 4 Put the whole number answer (7) above the 9 Multiply 4 by the answer (7) and write this underneath the 99 Step 6: 4 4 7 4 g 6 5 9 5 6 6 5 5 6 9 9 9 8 Subtract 98 from 99 Write the remainder as a fraction ` 659 ' 4 447 4 P Learning H SERIES TOPIC 5
Your Turn Short and long division WHOLE NUMBERS * WHOLE NUMBERS * Calculate each of these short division questions showing all working. Check your answers on the calculator. a 4767 ' b 680 ' 5 Short & Long Division.../.../0... g g c 69 ' 4 d 054 ' 6 g g Calculate each of these short division questions showing all working. (psst: remember to write any remainders as a simplified fraction) Check your answers on the calculator. a 8965 ' 7 b 879 ' g g c 96 ' 8 d 580' 6 g g 6 H SERIES TOPIC P Learning
Your Turn Short and long division Calculate each of these long division questions showing all working. Check your answers on the calculator. g a 5 8 5 5 b 8 9 4 7 g g c 4 5 8 5 d 7 5 7 8 g P Learning H SERIES TOPIC 7
Arithmetic laws There are three laws of arithmetic that can be powerful tools to help with calculations.. Commutative laws The order that we add or multiply numbers can be switched without changing the answer. Addition a + b b + a + + 4 and 5 5 and 4 4 + 5 5 + 4 9 Multiplication a # b b # a 4 lots of 5 5 lots of 4 4 # 5 5 # 4 0 The commutative law for multiplication can be shown also in terms of rows and columns. rows # columns columns # rows 4 rows of 5 5 columns of 4 rows 4 rows 4 columns 4 5 5 columns 4 # 5 5 # 4 0 The commutative law does not work for subtraction or division as the order of the terms is important. 4-5! 5-4 4 5! 5 4 4-5 - and 5-4 4 5 0.8 and 5 4.5 8 H SERIES TOPIC P Learning
Your Turn Commutative laws Shade groups of boxes to match these descriptions and write the calculation they represent. a Two rows of four b Four columns of two c Four rows of two d Two columns of four # # # # e Four columns of one f Three rows of six g Two rows of seven h Four rows of one # # # # Fill in the mathematical sentences for each of these diagrams showing the commutative law. a + + + + b # # c # # d P Learning H SERIES TOPIC 9
Your Turn Commutative laws Write the expression for the commutative law represented by this diagram. 4 Draw two different diagrams below to demonstrate that # 6 6 #. 5 Earn yourself an awesome stamp for this one. Draw all four different pairs of diagrams that represent the commutative law for multiplication with an answer of 4. 0 H SERIES TOPIC P Learning
. Associative laws The numbers we group together when adding or multiplying can change without changing the answer. Addition (a + b) + c a + (b + c) + + + ( + 4) + 5 7 + 5 + + + + (4 + 5) + 9 Multiplication (a # b) # c a # (b # c) # lots of 4 # 5 ( # 4) # 5 lots of 5 # 5 60 # # 4 lots of 5 # (4 # 5) lots of 0 # 0 60 The associative law can make adding/multiplying terms easier. + 5 + 9 ( + 9) + 5 40 + 5 65 # 5 # 4 # (5 # 4) # 00 00 What about subtraction or division? There are only a few very special cases where the associative law works for subtraction and division. When a 0 is involved like this: When a is involved like this: (6 - ) - 0 6 - ( - 0) (8 ' ) ' 8 ' ( ' ) (0 ' 7) ' 5 0 ' (7 ' 5) In all other cases, the associative law does not work for subtraction and division. ( - 4) - 5! - (4-5) ( ' 4) ' 5! ' (4 ' 5) ( - 4) - 5-6 and - (4-5) 4 ( ' 4) ' 5 0.5 and ' (4 ' 5).75 P Learning H SERIES TOPIC
Your Turn Associative laws Complete these equations for the associative law of addition shown in each diagram. a + + + 4 + 6 + 6 + 4 6 + 6 8 + 4 b + + + + + + + + c + + + + + + + + d + + + + + + + + H SERIES TOPIC P Learning
Your Turn Associative laws The associative law lets us choose what we want to do first to make calculations easier and quicker. For example: 4 + 9 + 6 is made easier by adding the 4 and 6 together first to make it 9 + 0 9 Pairing up terms that make nicer whole numbers before adding to other terms is a great trick. Use the associative law for addition to simplify these calculations: a 5 + 9 + 75 + + 9 + 9 b 8 + 5 + 8 + + 8 + 8 c + 6 + 7 + + + d 0 + 4 + 5 + + + e 7 + 4 + 56 + f + 80 + 9 + 45 P Learning H SERIES TOPIC
Your Turn Associative laws Complete these equations for the associative law of multiplication shown in each diagram. a # # 6 # # # # # # 6 b # # 40 # # # # # # 40 c # # 68 # # # # # # 68 d # # # # # # # # 4 H SERIES TOPIC P Learning
Your Turn Associative laws 4 Shade groups of squares in these grids to match the associative law of multiplication underneath. a # # # ( # ) ( # ) # b # # 4 # ( # 6) (4 # ) # 6 c # # 4 # ( # ) (4 # ) # 4 d # # # (5 # ) ( # 5) # 0 5 Use the associative law for multiplication to simplify these calculations: a 5 # 8 # 0 # # 8 b # 50 # 7 # # 7 # 8 # 7 c 4 # 9 # 75 # # d 5 # # # # # # P Learning H SERIES TOPIC 5
Your Turn Associative laws 6 Tick the box true or false to show which of these subtraction or division statements are special cases where the associative law does work. a c e g i ( ' 4) ' ' (4 ' ) (0-4) - 0 - (4 - ) (0 - ) - 0 0 - ( - 0) (0 ' ) ' 0 ' ( ' ) ( - ) - - ( - ) True False b (5 ' 0) ' 5 ' (0 ' ) True False True False d (0 ' 6) ' 0 ' (6 ' ) True False True False f (5 ' ) ' 5 5 ' ( ' 5) True False True False h (8-0) - 8 - (0 - ) True False True False j (0 - ) - 0 0 - ( - 0) True False 7 Larger strings of numbers can also have the associative law applied to them. Group terms together in these expressions that make the calculation much easier. a + 4 + 8 + 4 + Explain why you grouped the numbers you did. b # # 5 # c Use the associative law to rearrange into three pairs to make the calculation easier. + + 7 + 4 + 6 + 9 8 Write an expression that represents these dot diagrams and use the associative law to simplify. a # b # 6 H SERIES TOPIC P Learning
. Distributive law This law allows you to spread a multiplication out (or expand) into smaller parts to make it simpler. Using the order of operations and calculating the brackets first: # + # # (4 + 5) # (9) 7 Distributing the multiplication to each term within the brackets: # # + # + # + # (4 + 5) # 4 + # 5 + 5 7 Also works if a subtraction is inside the brackets. # # - # - # - # (4 - ) # 4 - # 8-6 So the distributive law is: a # (b + c) a # b + a # c These signs must always match a # (b - c) a # b - a # c These signs must always match Using this law in reverse makes the multiplication of large numbers easy to calculate mentally. For example, we can use the distributive law to calculate 7 # 4 using either of these: 0 # 4 + 7 # 4 or 0 # 4 # 4 7 # 4 (0 + 7) # 4 0 # 4 + 7 # 4 480 + 8 508 7 # 4 (0 ) # 4 0 # 4 # 4 50 508 You could have split 7 up into any sum/subtraction you found easiest to use. P Learning H SERIES TOPIC 7
Your Turn Distributive law Fill in the missing values for each of these examples of the distributive law. a # 5 + # + # 6 b # 8-4 7 # 8-7 # c 6 # + 6 # 9 # + d # 7 - # 0 5 # 7 - e # 9 + 45 + 5 # f # - 6 - a Use the distributive law to expand these multiplications: a) 5 # 8 + 5 # + 5 # b) + 5 # 4 + 6 5 # + 5 # + b Explain why both have the same value even though the terms in the brackets are different. c Write another similar expression to those in a that will also give the same answer. Use the distributive law to simplify and evaluate these multiplications: a 6 # 5 6 # + b 8 # 98 8 # - 6 # + 6 # 8 # - 8 # + - c # # + d 5 # 9 5 # - # + # 5 # - 5 # + - 8 H SERIES TOPIC P Learning
Your Turn Distributive law 4 You can apply the distributive law more than once to simplify a calculation: For example: 5 # 59 5 # (60 - ) 5 # 60-5 # 5 # 60-5 OR 5 # 60-5 (0 + 5) # 60-5 0 # 60 + 5 # 60-5 00 + 00-5 475 5 # (0 + 40) - 5 5 # 0 + 5 # 40-5 500 + 000-5 475 Apply the distributive law twice to simplify and calculate these: a 4 # 7 b 45 # 8 c # 75 d 5 # e 8 # 5 f 0 # 08 P Learning H SERIES TOPIC 9
Divisibility tests Divisibility tests are used to see if a small whole number will be a factor of a larger composite number. A number is always divisible by if the last digit is an even number (i.e. 0,, 4, 6 or 8) 4 is divisible by as the last digit (4) is even A number is always divisible by if the sum (+) of all its digits is divisible by 4 is divisible by because + + 4 9 (which is divisible by ) A number is always divisible by 4 if the number formed by the last two digits is divisible by 4 4 is divisible by 4 because the last two digits form the number 4 (which is divisible by 4) A number is always divisible by 5 if the last digit of the number is a 0 or 5 65 is divisible by 5 because the last digit is a 5 A number is always divisible by 6 if it is divisible by both and 4 is divisible by 6 because it is even (so divisible by ) and + + 4 9 (which is divisible by ) A number is always divisible by 8 if the number formed by the last three digits is divisible by 8 8 is divisible by 8 because the last three digits form the number 8 (which is divisible by 8) A number is always divisible by 9 if the sum (+) of all its digits is divisible by 9 4 is divisible by 9 because + + 4 9 (which is divisible by 9) A number is always divisible by 0 if the last digit of the number is a 0 840 is divisible by 0 because the last digit is 0 Investigate the divisibility tests for 7 and. They are a little more involved but interesting! 0 H SERIES TOPIC P Learning
* DIVISIBILITY TESTS FOR NUMBERS How does it work? Your Turn Divisibility tests Use the divisibility tests to determine whether each of these numbers are divisible by the numbers listed on the right hand side. Draw a line to all the numbers each one is divisible by. The first number is completed for you. 60 6 96 49 4 45 5 07 5 6 588 8 78 00 00 9 756 0 8640 87 600.../.../0... P Learning H SERIES TOPIC
Exponent notation for numbers Exponent notation uses a small number called a power or exponent to show how many times a number is multiplied by itself. Simplify these products by using exponent notation and then calculate: (i) 4 # 4 4 4 4 # 6 Two 4s in the multiplication, so the exponent is We say 4 squared When a number is multiplied by itself once, this is called squaring the number (ii) # # # # Three s in the multiplication, so the exponent is 8 We say cubed When a number is multiplied by itself twice, this is called cubing the number. The same pattern continues for any number of multiplications (iii) # # # # # # # # # # 6 79 Six s in the multiplication, so the exponent is 6 We say to the power of 6 A mixture of numbers multiplied together can also be simplified using exponent notation (iv) 4 # 5# 5# 5# 4 # 5 4 # 5# 5# 5# 4 # 5 4 # 4# 5 # 5# 5# 5 4 4 # 5 6 # 65 0 000 Group identical numbers We say 4 squared times 5 to the power of 4 Doing the reverse to simplifying is called expanding. Write these in expanded form: (i) 7 4 7 4 7 # 7 # 7 # 7 The exponent is 4, so four 7s multiplied together (ii) 9 7 7 9 9 # 9 # 9 # 9 # 9 # 9 # 9 The exponent is 7, so seven 9s multiplied together 4 Be careful: A lot of people make this mistake: 7 7 # 4, which is NOT true. 4 7! 7 # 4 Make sure you can see the difference. H SERIES TOPIC P Learning
EEXPONENT NOTATION * How does it work? Your Turn Exponent notation for numbers Write each of these products using exponent notation. a 5 # 5 b 4 # 4 # 4.../.../0... XPONENT NOTATION * c # # # # d # # # e 7 # 7 # 7 # 7 # 7 # 7 f # # # # # # Write each of the mixed products using exponent notation and then calculate. a # # # # b 5 # 5 # 4 # 4 c 6 # 6 # 6 # 6 # 7 # 7 # 7 d # # # # e # 8 # 8 # # 8 # 8 # 8 f 4 # # # 4 # # # # Change each of these to expanded form. a b 8 4 c 6 5 d 7 e 5 # 7 f # 4 g 7 5 # 4 h # # 5 4 P Learning H SERIES TOPIC
Your Turn Puzzle time The area of a square can be written using exponent notation: units units Area # units 4 units 6 Using each of the different grey squares below twice and the black square only once, form a rectangle on the grid above. You can do this by shading in the squares using a pencil or cut some similar-sized squares out of another sheet of paper and try to complete like a jigsaw. The top left-hand corner of the rectangle is already completed for you, so only one more 6 grey square can be used. use twice 4 5 6 use once use twice use twice use twice When finished, have a go at writing two different expressions for the total area of the rectangle using exponent notation. Hint: For one expression multiply the side lengths together. Area expression Area expression 4 H SERIES TOPIC P Learning
Square roots and cube roots Finding the square root or cube root of a number is the opposite operation to squaring or cubing. The radical symbol ( ) is used for roots, with used for square root and for cube root. The square root sign is asking: What number multiplied by itself once will get the number inside me? Calculate the square root of these whole numbers (i) 9 9 9 # 9 written as a product of its prime factors Because # 9 or (ii) 6 6 6 6 6 # 6 6 written as a product of its prime factors Because 6 # 6 6 6 6 6 We look closely at prime factors next The cube root sign is asking: What number multiplied by itself twice will get the number inside me? Calculate the cube root of these whole numbers (i) 8 8 or 8 # # 8 written as a product of its prime factors Because # # 8 The little root number indicates how many times the same number appears in the multiplication. (ii) 4 4 7 or 4 7 # 7 # 7 4 written as a product of its prime factors Because 7 # 7 # 7 7 4 7 7 You could be asked to write a value using square or cube root notation. Rewrite these numbers (i) 4 as a square root (ii) as cube root 4 4 # 4 6 ` 4 6 # # 7 ` 7 P Learning H SERIES TOPIC 5
WHOLE NUMBERS * WHOLE NUMBERS * How does it work? Your Turn Square roots and cube roots Calculate each of these square roots a 4 b 6 Square roots & Cube roots.../.../0... c 5 d 49 e 8 f Calculate each of these cube roots a 7 b 64 c 6 d 5 Write each of these values using square root notation a b 8 c 6 d 4 Write each of these values using cube root notation a b c 5 d 7 6 H SERIES TOPIC P Learning
Where does it it work? Factor trees Composite numbers can be divided exactly (with no remainder), by other smaller or equal whole numbers called factors. Composite numbers: 5 9 4 4 Factors:,, 5, 5,, 9,,, 4, 6,,, 4,,, 4, 6, 8,, 4 Prime numbers only have and themself as factors. Prime numbers: 7 Factors:,,, 7,, All composite numbers can be written as the product (#) of prime factors (all the prime numbers that divide exactly into them). Let s see how. Express is a another way of saying write in Mathematics. Express 8 as a product of its prime factors 8 Split 8 into two smaller factors 6 Solid circle around prime numbers to stop that branch Split 6 into two smaller factors Solid circle around prime numbers to stop that branch Once every branch has reached a prime number, multiply all the prime numbers together ` 8 # # # Simplify answer ALWAYS at the prime number. Don t ever do this because is NOT a prime number Remember: A prime number has two factors, itself and P Learning H SERIES TOPIC 7
Where does it work? Here are some more examples. Express 8 as a product of its prime factors 8 Split 8 into two smaller factors 9 Solid circle around prime numbers to stop that branch Once every branch has reached a prime number, multiply all the prime numbers together ` 8 9 # There is often more than one way to create a factor tree for numbers with a lot of factors. Express 48 as a product of its prime factors 48 Split 48 into two smaller factors 8 6 Split 6 and 8 into two smaller factors 4 Solid circle around prime numbers to stop that branch Split 4 into two smaller factors Solid circle around prime numbers to stop that branch Once every branch has reached a prime number, multiply all the prime numbers together ` 48 # # # # 4 # Simplify answer 8 H SERIES TOPIC P Learning
Where does it work? Your Turn Factor trees Fill in the missing values on the following factor trees and write the number as a product of its primes. a b 8 4 ` ` 8 c d 56 4 4 4 ` 56 ` e 84 f 8.../.../0... PRIME FACTOR TREES ** PRIME FACTOR TREES 4 ` 84 ` 8 P Learning H SERIES TOPIC 9
Where does it work? Your Turn Factor trees Complete a factor tree for each number below and express them as a product of their prime factors. a 8 b 0 ` 8 ` 0 c 4 d 60 ` 4 ` 60 e 96 f 44 ` 96 ` 44 40 H SERIES TOPIC P Learning
Where does it work? Greatest common factor (GCF) The GCF is the largest number that divides exactly into two or more composite numbers. Write all the factors of each number then circle the largest one which appears in both lists. Find the greatest common factor for these pairs of numbers (i) 6 and 8 Factors of 6:,,, 6 Factors of 8:,, 4, 8 List all the factors for each number Circle the largest number common to both lists ` The GCF for 6 and 8 is: (ii) 8 and Factors of 8:,,, 6, 9, 8 Factors of :,,, 4, 6, List all the factors for each number Circle the largest number common to both lists ` The GCF for 8 and is: 6 We can use the list of prime factors for larger numbers to find the GCF. Find the GCF for these pairs of larger numbers (i) 7 and 96 Factors of 7:,,,, List all the prime factors for each number Factors of 96:,,,,, ` The GCF for 7 and 96 is: 4 # # # (ii) 58 and 64 Factors of 58:,,,,, List all the prime factors for each number Factors of 64:,,,,, ` The GCF for 58 and 64 is: 48 # # # # P Learning H SERIES TOPIC 4
Where does it work? Your Turn Greatest common factor (GCF) Find the greatest common factor for these pairs of numbers. a 8 and b 6 and 5 GCFs.../.../0... * GREATEST COMMON FACTORS * GREATEST COMMON FACTORS c 0 and 8 d 8 and 4 e 4 and 8 f 6 and 6 Use the prime factors to find the GCF for these larger numbers. a 4 and 84 b 9 and 7 c 80 and 490 d 56 and 640 4 H SERIES TOPIC P Learning
Where does it work? Lowest common multiple (LCM) The LCM is the smallest number that is common to the multiplication tables of two or more numbers. Write down the multiples of the numbers and stop once you find the lowest common multiple. Find the lowest common multiple for these pairs of numbers (i) and 5 # 4 # 6 # Multiples of :, 4, 6, 8, 0,, 4,... List some multiples of the first number # # 5 # 7 # # 5 Multiples of 5: 5, 0,... List the multiples of the second number until there is a match # 5 ` The LCM for and 5 is: 0 (ii) 6 and 8 Multiples of 6: 6,, 8, 4, 0,... Multiples of 8: 8, 6, 4,... List some multiples of the first number List the multiples of the second number until there is a match ` The LCM for 6 and 8 is: 4 We can use the list of prime factors for larger numbers to find the LCM by looking at the differences. Find the LCM for these pairs of larger numbers (i) 0 and 00 Prime factors of 0:,, 5 Prime factors of 00:,, 5, 5 ` The LCM for 0 and 00 is: 00 # 00 List all the prime factors for both numbers Circle all the different factors in the smaller number Multiply the larger number by the different factor (ii) 4 and 88 Prime factors of 4:,,, Prime factors of 88:,, 97 ` The LCM for 5 and 88 is: 88 # # 8 List all the prime factors for both numbers Circle all the different factors in the smaller number Multiply the larger number by the different factors P Learning H SERIES TOPIC 4
** LOWEST COMMON MULTIPLE Where does it work? Your Turn Lowest common multiple (LCM) Find the lowest common multiple for these pairs of numbers. a and 9 b 5 and 0 LCMs.../.../0... LOWEST COMMON MULTIPLE c 4 and 6 d 5 and 6 e 6 and 7 f and 6 Use the prime factors to find the LCM for these larger numbers. a 60 and 08 b 4 and 50 c 68 and 80 d 0 and 85 44 H SERIES TOPIC P Learning
What else can you do? Pascal s triangle This amazing triangle developed in 65 by French mathematician Blaise Pascal uses the addition of two whole numbers to create it. The number pattern forms the shape of a triangle and contains many mathematical applications. To create Pascal s triangle, each number on the line is obtained by adding the two numbers above it. The first seven lines of Pascal s Triangle 0 + + 0 0 + + + 0 0 + + + + 0 0 + + + + + 0 4 6 4 0 + + 4 4 + 6 6 + 4 4 + + 0 5 0 0 5 0 + + 5 5 + 0 0 + 0 0 + 5 5 + + 0 6 5 0 5 6 The pattern continues in the same fashion for each added row of numbers The second diagonal of Pascal s triangle contains all the counting numbers Counting numbers Counting numbers 4 6 4 5 0 0 5 6 5 0 5 6 P Learning H SERIES TOPIC 45
What else can you do? Here are some more patterns found within Pascal s triangle. The third diagonal of Pascal s triangle contains triangular numbers Triangular numbers Triangular numbers 4 6 4 5 0 0 5 6 5 0 5 6 Triangular numbers are formed by creating equilateral triangles using dot diagrams starting from dot,,,,,... 6 0 5 A very well known number pattern which occurs frequently in nature is the Fibonacci Sequence. The Fibonacci sequence is also within Pascal s triangle and is found by adding terms along the lines shown + + + + 5 + 4 + 8 + 6 + 5 + 4 6 4 5 0 0 5 6 5 0 5 6 Sunflowers contain a Fibonacci sequence Each number in a Fibonacci Sequence is found mathematically by adding the two numbers before it,,,, 5, 8,,,... 0 + + + + + 5 5 + 8 8 + 46 H SERIES TOPIC P Learning
What else can you do? Your Turn Pascal s triangle Another special pattern is called the Sierpinkski Triangle. This is a special fractal pattern made using triangles. Each dark equilateral triangle is split into four smaller equilateral triangles at every step. This pattern can be reproduced using Pascal s triangle by simply separating the odd and even numbers. In Pascal s triangle below, colour in all the odd numbered hexagons to see this pattern emerge! 4 6 4 5 0 0 5.../.../0... * PASCAL S TRIANGLE * PASCAL S TRIANGLE 6 5 0 5 6 7 5 5 7 8 8 56 70 56 8 8 9 6 84 6 6 84 6 9 0 45 0 0 5 0 0 45 0 55 65 0 46 46 0 65 55 40 66 0 495 79 94 79 495 0 66 78 86 75 87 76 76 87 75 86 78 4 9 64 00 00 00 4 00 00 00 64 9 4 5 05 455 65 00 5005 645 645 5005 00 65 455 05 5 P Learning H SERIES TOPIC 47
What else can you do? Applications of Pascal s triangle Pascal s triangle can often be useful when solving problems like the ones shown here. Each number in Pascal s triangle represents the number of paths that can be taken to get to that point. Show all the different downward paths that can be taken to get to the circled number in the triangle The number circled is, so there are different downward paths leading to this point Path Path Path The total number of different paths to the bottom of a Pascal triangle is found by adding the numbers across. For this four-line Pascal triangle: (i) How many different paths can be taken to reach the bottom of the triangle below? The total number of different paths + + + 8 (ii) How many paths to reach the bottom if one more line was added? 4 6 4 The total number of different paths + 4+ 6+ 4+ 6 48 H SERIES TOPIC P Learning
What else can you do? Your Turn Applications of Pascal's triangle Write down how many different downward paths there are to each of the points circled on this triangle. A B C D E F Number of downward pathways to: A B C D E F Show the six different downward paths that lead to the circled point on this triangle from the top. Start 6 Start Start Start Path Path Path Start Start Start Path 4 Path 5 Path 6 P Learning H SERIES TOPIC 49
What else can you do? Your Turn Applications of Pascal's triangle The ant nest below has a tunnel system that leads down to a main chamber. After one ant enters the tunnel from the top, how many different ways can it get to the main chamber if it only travels downwards the entire way? Hint: Fill in Pascal s triangle values. Remember me? Main Chamber 50 H SERIES TOPIC P Learning
Cheat Sheet Here is a summary of the important things to remember for whole numbers. Place value Writing numbers using words, name using groups of three digits. To write in expanded form, multiply each number by the place value and add together. The place value of a numeral in a large number is found by multiplying the numeral by the matching position value. Adding and subtracting large numbers When adding or subtracting large numbers, make sure the place values are lined up correctly first. 000 000 00 000 0 000 000 Millions Hundreds of thousands Tens of thousands Thousands Hundreds Tens Ones N U M B E R S Long multiplication Make sure the place values are lined up correctly first. Add zeros on each line to match the place value of the number you are multiplying by. Add together the new numbers formed after multiplying. Short and long division Keep all place values lined up neatly. Be careful and methodical with each step. Always write the remainder as a fraction. Exponent notation for numbers Exponent notation is used to show how many times a number is multiplied by itself. # # # # 5 Square and cube roots The square root or cube root of a number is the opposite operation to squaring or cubing. The symbols used are for square root and for cube root. 9 because # 9 and 7 because 7 # # Factor trees These are used to write any composite number as the product of prime number factors only. Greatest common factor (GCF) The GCF is the largest number that divides exactly into two or more composite numbers. Lowest common multiple (LCM) The LCM is the smallest number that is common to the multiplication tables of two or more numbers. Pascal s triangle Each number in Pascal s triangle is the sum of the two numbers above it. Each number is the number of different downward paths that can be taken to get to that point. The number of different downward paths to the bottom of a Pascal s triangle is found by adding together all the values across the bottom. 00 0 P Learning H SERIES TOPIC 5
Notes 5 H SERIES TOPIC P Learning
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