Mathematics. Name: Class: Transforming Life chances

Transcription

1 Mathematics Name: Class: Transforming Life chances

2 Children first- Aspire- Challenge- Achieve Aspire: To be the best I can be in everything that I try to do. To use the adults and resources available both at school and at home, to aspire for personal excellence and professional competence. Challenge: To aim high, to push my limits to be able to strive for the highest possible achievements. To make every minute count to by maximising all learning opportunities both at school and at home. To seek challenge and to use my thinking tools to develop my thinking and push myself forward. To be responsible and in control of my own destiny. To be a skilled, independent, reflective learner. Achieve: To demonstrate the highest levels of thinking and habits. To question, to challenge, to think independently and interdependently to achieve my personal academic aims. To be proud of who I am and what I achieve. You are who you choose to be! T6 T T T T Target Term Term Term Term Target Arithmetic Arithmetic Arithmetic Arithmetic Arithmetic Paper Paper Paper Paper Paper Paper Paper Paper Paper Paper Total Total Total Total Total

3 Number and Place value Ma: Can read, write and order whole numbers up to 0,000,000 Place value is the number system that we use to describe the position of each digit within a number. Whole numbers are numbers that do NOT include fractions and decimals. Millions Hundreds of Tens of Thousands Hundreds Tens Ones Thousands Thousands M HTh TTh T H T Ø We can use this model of place value to help us read, write and order whole numbers up to 0,000,000. Read 68 T H T Ø = thousand, 6 hundred and eighty-three five thousand, six hundred and eighty-three 6 8 Read 679- to help read the number count back three digits from the right and place a comma. 67,9 HTh TTh T H T Ø hundred and thousand, hundred and =00, ,000+ 7, Four hundred and sixty-seven thousand, two hundred and ninety-three. Ordering Whole Numbers- Note: you should be able to order whole numbers with up to digits. Let s look at an example with digit numbers. Put these numbers in ascending order:,,,, Ascending means smallest to biggest. Let s look at the thousands column. There are two numbers with thousands: and. We now need to look at the hundreds column. We can see that has hundreds but has only hundred so it is smaller. The next smallest number is. We then have numbers with in the thousands column. and. We can see that is bigger as it has hundreds. So, our numbers in ascending order are,,,, Hint: When answering questions such as this, make sure your final list has all the numbers and none have been left out. T H T Ø Look at numbers as a whole What is the biggest number? In this case it is a thousand number- therefore you need to think about place value up to Thousands. Look at numbers in place value order- Thousands-Hundreds-Tens-Ones

4 Mb: Can read, write and order numbers up to decimal places Remember: When working with decimals, it is very important that you know and understand the place value of numbers. Thousan ds (000) Hundred s (00) Tens (0) Ones () Tenth s ( 0 ) Hundredth s ( 00 ) Thousandth s ( 000 ) For example, we can use this chart to see that the number.7 is made up of: tens, ones, tenths (or /0 or 0.), hundredths (or /00 or 0.0) and 7 thousandths (or 7/000 or 0.007).. 7 This number is read or written in words as thirty two point four five seven. Read.098 -This number is read or written in words as twenty one point zero nine eight. We are sometimes given decimal numbers written in words and we have to read, understand the number and write it in figures. Let s look at some examples: Write forty nine point zero nine four in figures This is written in figures as 9.09 Write thirty point two zero eight in figures This is written in figures as 0.08 We will sometimes be asked questions to show that we understand place value. Let s look at an example: Write this total as a decimal = We can see from our place value chart that this number will be. Write a number in the box to make this correct: 7.6 = Ones. 0 Tenths 00 Hundredths 000 Thousandths We know that the number 7.6 is 7 ones, 6 tenths, hundredths and thousandths or 7.6 = Therefore the missing number is 0.0.

5 Ordering Decimals Remember: When working with decimals it is very important that you know and understand the place value of numbers. Thousan ds (000) Hundred s (00) Tens (0) One s () Tenth s ( 0 ) Hundredth s ( 00 ) Thousandth s ( 000 ) For example, we can use this chart to see that the number.7 is made up of:. 7 tens, ones, tenths (or /0 ), hundredths (or /00) and 7 thousandths (or 7/000). We can use this understanding of place value to help us order decimal numbers. Put these numbers in order from smallest to largest:.,.7, 7.,.7,.77 To answer this question, let s start by looking at the ones. There are two numbers which have the smallest number of ones:.7 and.77 Let s first look at.7: ones 0 00 ones Look at the whole number first- the smallest whole number is They both have 0.7 tenths We then look at the hundredths column, one has 0.0 and the other therefore.7 is smaller. The nest smallest large number is 7. We then look at the tenths column. One has 0.7 and one has 0.- therefore. is smaller. The final number is The decimal numbers in order from smallest to largest will be:.7,.77,.,.7, 7.

6 Let s now try to order decimal numbers which include thousandths digits. Put the numbers below in order from largest to smallest: Add in 0s so that each number has the same digits To order these decimals, let s first put them all in the correct column in the place value table: Writing them in the correct columns will help us compare the size of the numbers. Starting with comparing the ones, it is clear that will be the smallest number as that is the only number which hasn t got 7 ones. We know therefore that will come last on our list. (Remember, the question this time asks us to order from largest to smallest. Sometimes, it is easier to start with looking for the smallest numbers and then writing the numbers in reverse.) To find the largest of the remaining numbers, let s first look down the tenths column. The top three numbers have 6 in the tenths column but 7.6 only has in the tenths column. Therefore, 7.6 is smaller and will be the next smallest number on our list. We now need to compare the remaining numbers: 7.600, 7.67 and We have already seen that the three numbers have 7 ones and 6 tenths. We therefore now need to look at the hundredths column and 7.67 both have 7 in the hundredths column but 7.6 has nothing in the hundredths column. In other words, 7.6 has 0 hundredths and is smaller than 7.67 and Lastly, if we compare 7.67 and 7.67, we can see that 7.67 is larger because 7.67 has thousandths but 7.67 has 0 thousandths. Another method is to write each of the numbers out with the same number of decimal places to show that there is nothing in these place value columns: 7.67, 7.670, 7.600, 7.6, 6.776

7 Mc: Can round any whole number to the nearest 0, 00 and 000 Round to the nearest 0 Round to the nearest 00 Round to the nearest 000 What is,6 rounded to the nearest 0? What is,6 rounded to the nearest 00? What is,6 rounded to the nearest,000? When rounding to the nearest 0, we need to look at the digit in the ones column. 6 is less than so we round down.,6 rounded to the nearest 0 is,60. Note: Multiples of 0 end in at least one zero. This is a good way of checking quickly that you have rounded to the correct multiple. When rounding to the nearest 00, we need to look at the digits in the tens column. 6 0 is less than 0 so we round down.,6 rounded to the nearest 00 is,600. Note: Multiples of 00 end in at least two zeros. This is a good way of checking quickly that you have rounded to the correct multiple. When rounding to the nearest,000, we look at the digits in the hundreds column is more than 00 so we round up.,6 rounded to the nearest 000 is Note: Multiples of 000 end in at least three zeros. This is a good way of checking quickly that you have rounded to the correct multiple Using number lines can help to visualise where the number is and how close it is to the next 0,00 or 000 What is 7,00 rounded to the nearest,000? Note: When a number lies exactly half way between, we round up. In this example 7,00 is exactly half way between 7,000 and 8,000 so we round up. 7,00 rounded to the nearest,000 is 8,000.

8 Md: Can round decimals to the nearest whole number and to one or two decimal places Round to the nearest whole number When rounding to the nearest whole number, we look at the tenths column. Round to the nearest 0 Round to the nearest 00 When rounding to the nearest tenth, we look at the digit in the hundredths column. When rounding to the nearest hundredth, we look at the digit in the thousandths column. What is 9.7 rounded to the nearest whole number? 7 tenths is more than tenths so we round up. 9.7 rounded to the nearest whole number is 0 What is.68 rounded to the nearest whole number? 6 tenths is more than tenths so we round up..68 rounded to the nearest whole number is 6 What is 8.9 rounded to the nearest whole number? tenths is less than tenths so we round down. 8.9 rounded to the nearest whole number is 8 What is.8 rounded to the nearest tenth? 8 hundredths is more than hundredths so we round up..8 rounded to the nearest tenth is. The question may be phrased differently. For example: What is.8 rounded to one decimal place? We still need to round to the nearest tenth. We, therefore, look at the digit in the hundredths column. tenths is less than tenths so we round down..8 rounded to one decimal place is.8 What is 6.9 rounded to the nearest hundredth? thousandths is less than thousandths so we round down. 6.9 rounded to the nearest hundredth is 6.9 The question may be phrased differently. For example: What is. rounded to two decimal places? We still need to round to the nearest hundredth. We, therefore, look at the digit in the thousandths column. thousandths is exactly halfway so we round up.. rounded to two decimal places is

9 Me: Can use place value to multiply whole numbers by 0, 00 or 000 When you multiply by 0 the original number gets 0 times bigger. Hundreds Tens Ones x 0 = 90 The 9 moves one place value column to the left Thousands Hundreds Tens Ones x 0 = 70 All digits move one place value column to the left When we multiply by 0, each digit moves one place to the left: The Ones digit moves to the Tens column, Tens moves to the Hundreds column etc. The space in the column is filled with a 0, which is called a place holder. Multiplying by 00 Multiplying by 000 When we multiply by 00, each digit moves two places to the left and the spaces in the columns are filled with a 0, the place holder. When we multiply by,000, each digit moves three places to the left and the spaces in the columns are filled with a 0, the place holder Thousands Hundreds Tens Ones 0 0 x 00 =,00 7 x 000 =7,000 HTh TTh Th H T Ø x 00 = 60,00 6 x 000 = 6,000 HTH TTH Th H T Ø M HTh TTh Th H T Ø

10 Mg: Can use place value to divide whole numbers by 0, 00 or 000 When you divide by 0 the original number gets 0 times smaller. Hundreds Tens Ones = 9 The 9 moves one place value column to the right H T Ø / = 7. All digits move one place value column to the right When we divide by 0, we are making the number smaller so each digit moves place to the right. Thousands move to the hundreds column, hundreds move to the tens column, tens move to the Ones column, Ones move to the tenths column. THE DECIMAL POINT DOES NOT MOVE. Dividing by 00 Dividing by 000 When we divide by 00, each digit moves two places to the RIGHT When we divide by,000, each digit moves three places to the RIGHT 6 00 = H T Ø /0 / =7,000 TTh Th H T Ø /0 /00 / = 60,00 TTh Th H T Ø /0 / = 6,000 TTh Th H T Ø /0 /00 /

11 Mh: Can use place value to divide decimal numbers by 0, 00 or 000 Thousands Hundreds Tens Ones. Tenths Hundredths Thousandths When we divide by 0, we are making the number smaller so each digit moves place to the right. Thousands become Hundreds, Hundreds become Tens, Tens become Ones. When we are dividing by 00, we are making the number 00 times smaller so each digit moves places to the right. When we are dividing by 000, we are making the number 000 times smaller so each digit moves places to the right. Note: Move to the right to reduce the value. The same rules apply to decimal numbers. There will be occasions where you will need to add a 0 as a place holder for the answer to the calculation. When we divide a decimal number by 0, each digit moves one place to the right so 7. 0 =.7 When we divide a decimal number by 00, each digit moves two places to the right so = 0.7 When we divide a decimal number by 000, each digit moves three places to the left so = 0.07 T Ø. /0 /00 /000 / Note: Dividing by 00 is the same as dividing by 0 and 0 again. Dividing by 000 is the same as dividing by 0, 0 and 0 again or 00 and 0.

12 Mi: Can use negative numbers in context and calculate intervals across zero Remember: Negative numbers are numbers smaller than zero. Positive numbers are numbers which are greater than zero. This number line shows the numbers from -0 to. Remember on a horizontal line, that as we move to the left, the numbers get smaller. We can see on the number line that -9 is smaller than -6. We can use number lines to help us add and subtract with negative numbers. Let s look at some examples: Using the number line below, work out the difference between - and - To find the difference between - and -, we need to count how many steps it is from - to - (shown by the blue arrows) We can see that we have gone up in steps. Therefore: The difference between - and - is.

13 It is important to remember that number lines are not always horizontal. You will sometimes see vertical number lines, especially when we are talking about temperatures. Think of them looking like thermometres. Let s look at some examples During the day the temperature reached C but at night the temperature dropped to C. By how much did the temperature drop? Start at + Count to 0 + to 0 is Count how many steps there are from + to -. Draw a vertical numberline Place 0 on the numberline Count from the positive number to 0 Start at 0 and count to - 0 Count from the negative number to 0 Add the two totals together Check the difference Finish at 0 to - is Using this vertical number line, we can see that if the temperature dropped from C to - C, it dropped by 9 C.

14 Number: Addition and Subtraction Ma: Can use mental methods of computation for addition Remember: when faced with any calculation we should look at the numbers involved and ask ourselves, can I do the calculation mentally or in my head? Remember: solving a calculation mentally or in my head does not mean that we cannot jot things down to help us such as the numbers involved or a number line. Number line ) (8 = ) Draw an empty number line and write the largest number on it. Add the multiples of 0 from the second number (8) Add on the ones from the second number Partitioning ) = = = Partition the numbers into their place value groups i.e. Tens and Ones Add the multiples of 0 together Add the ones together Total the two sums to calculate the final answer Rounding and adjusting ) = Look at the numbers: 9 is very close to 0- adding multiples of is easier than adding 9. 0 = 9 + therefore (6 + 0) = the answer 6 =

15 Number line ) = Adding decimal numbers Draw the number line Write the largest number on the line Add the whole number Add the decimal Partitioning ) = + = = Note: when adding the whole number the decimal stays the same Partition the numbers into their place value groups i.e. Ones and tenths Add the ones together Add the tenths together Total the two sums to calculate the final answer Rounding and adjusting 8 +. = 9. ) = Look at the numbers:.8 is very close to - adding whole numbers is easier than adding = therefore (.6 + ) 0. = the answer = 9. When adding more than two numbers it is important to look for numbers that make number bonds to ensure the calculation is accurate and quick = 0+(0+0)= 80 (7+)+= 80+=9

16 Mb: Can use mental methods of computation for subtraction Remember: when faced with any calculation we should look at the numbers involved and ask ourselves, can I do the calculation mentally or in my head? Remember: solving a calculation mentally or in my head does not mean that we cannot jot things down to help us such as the numbers involved or a number line. Number line Partitioning ) 9 - = Rounding and adjusting ) - 9 = (0+) Counting on (the difference between the two numbers) We can also use a number line to help us do this calculation by working backwards. ) 87 9 = = 80 = 87 9 = ( 87 0 ) + So, 9 = = 66 or 9 0 = 8 8 = 8 So, 9 = 8 Place the smallest number on the number line Place the largest number on the number line Count on in multiples of 0 Count on in ones Total the jumps to calculate the different by counting on (0+) Starting with, first subtract which will makes 0. Then subtract 0 which makes 0 lastly subtract to get to 9. Altogether, to get from to 9 we have subtracted, 0 and. Therefore, 9 = Partition and Subtract multiples of 0 Subtract ones Total the numbers or Leave the largest number Subtract multiples of 0 Subtract ones gives the answer 9 + = 0 We can use this to subtract 0 and then adjust by adding the back on after. This makes calculating the answer easier!

17 Number line ) = Subtracting decimal numbers Count on to the nearest whole number Count on in whole numbers Add on the final decimal Total the jumps to calculate the answer Counting on (the difference between the two numbers) We can also use a number line to help us do this calculation by working backwards Partitioning ) 9..8= 9. = =.6 So, 9..8 = Start with the largest number and count back Subtract the decimal to give a whole number Count back in whole numbers Count back the remaining decimal Subtract the whole number Subtract the decimal Rounding and adjusting ) We will use the rounding and adjusting strategy here. We now need to adjust this because we have subtracted 0. too much. We therefore need to add that back on. Round one of the numbers up Subtract the whole number from the decimal Adjust =..8 rounds to = =.6

18 Mc: Can use efficient written methods of addition including column addition with more than digits Remember when we look at a calculation we should always ask: Can I do it in my head? With/without jottings? Do I need a written method? Sometimes numbers are too large or there are too many numbers to calculate in our head. We need a reliable written method to help us. Partitioning Expanded Method Short addition Place value: What is the value of each digit here? For example: = 78 = = = 7 so = Add the ones Add the tens Add the hundreds Total the numbers = = ten (0) and ones Place the ten in the tens column = is 00 Larger numbers Decimal numbers Expanded Method Short addition Expanded Method Short addition

19 Md: Can use efficient written methods of subtraction including column subtraction Subtraction can sometimes be calculated more easily by finding the The counting on Method Let s do the same calculation more efficiently: difference between the numbers. This can be illustrated on a number line. 7 =? = 80 To find the difference between these two numbers we can count up from 7 to on a number line = = Now add the steps together: = 69 so 7 = (80) + 0 (00) + () 69 so 7 = 69 Use base 0 apparatus to help with this strategy = The expanded Method 8 subtract 7 = 0 subtract 0 = hundred subtract 00 = = the difference between 78 and =69 So, 7 = subtract gives a negative number so we need to take a 0 from the tens column. 0 subtract 70 gives a negative number so we need to take a 00 from the Hundreds column.

20 Expanded Method Short subtraction Expanded Method Short Subtraction = Short subtraction- further examples Me: Can add with decimals to two places (including money) Expanded Method Short addition Expanded Method Short addition T Ø. /0 / T Ø. /0 / T Ø. /0 / T Ø. /0 /

21 Adding decimals (including money) is just the same as whole numbers. Just make sure that you line up the decimal points and fill in the numbers in the correct place value columns. Put 0s as place value holders to stop you getting confused easy as that! Mf: Can subtract with decimals to two places (including money) = (Counting on method) (.9) () +.6 (8.6).79 Counting on to the nearest tenth Counting on to the nearest unit (Expanded) (Short subtraction) = (). 0 (7.0). 6 So, =.6 Subtraction with money (counting on) To get from. to we need to add 0.76 To get from to 7.0 we need to add.0 In total, to get from. to 7.0, we have added.6 Short Subtraction with money T Ø. /0 / With money, line up the decimal points and place the numbers into the correct columns. The methods for subtraction work for all numbers (decimals including money!)

22 Number: Multiplication and Division Mb: Can use tables and place value with multiples of 0 We can use multiplication facts to help us understand other times tables questions: E.g. if we know that 6 x = we also know that 6 x 0 = 0 and that 60 x 0 = is ten times larger than 6. Our answer will also be ten times larger. H T U. 0 th 00 th is ten times larger than, so x 60 = 0 H T U. 0 th 00 th is ten times larger than 0 so 0 X 60 = 00 Th H T U. 0 th 00 th The same principles apply with division 6 x 8 = = = 60 If the question is how many 8s in 80 you can use your knowledge of 6 x 8 = 8 to help you figure it out. 8 is 0 x smaller than is 0 x larger than = = 60 7 x = 0 7 =0 0 = 70 If the question is how many 7s in 0 you can use your knowledge of x7- to help is 0x smaller than 0 0 is 0 x larger than,00 7 = We know that 7 =. 00 is 00 times larger than. So the answer is:,00 7 = 00 6,00 90 = We know that 6 9 = is 00 times larger than 6. So the answer is: 6,00 90 = = We know that 9 7 = is 00 times larger than 90. So the answer is: 90 7 =

23 Mc: Can use mental methods of computation for multiplication To double 96 we can partition and multiply by : 96 = x = 80 6 x = 80 + = 9 so double 96 is 9 Doubling Double a digit number: 7 = x = x = 0 x = = 8 so double 7 is = p 8.00 x = p x = = 7.80 so double 8.90 is 7.80 Multiply by 6. = x 6 = 80 x 6 = = 0 so x 6 is 0 Partitioning What is x 6? = x 6 = x 6 = 80 x 6 = = 80 so x 6 is 80 Multiply.8 by 7.8 = x 7 = x 7 = =.6 so.8 x 7 is.6 Using easier known facts x- double and double again x we can x0 and then ½ ( ) x0 times by 00 then ½ x times by 00 then ¼ ( ) x x x 0 x 8 = x = 0 x = 8 x 0 = 0 0 = 60 x 00 = = x 00 = = 0 so x is 96 so x = 60 so x 0 = 600 so x 8 = 0

24 Md: Can use mental methods of computation for division Halving a digit number Halving a digit number Halve a decimal = = p 78 We know that halving is the same as dividing by. To halve 78 we can partition and divide by : 78 = = 8 = + = 9 So half of 78 is 9 00 = 0 0 = 8 = = 79 So half of 8 is =.0 60p = 0p.0 + 0p =.80 So half of 7.60 is = = 0 = = = = Using easier known facts (divide by ) 8 (divide by 8) 6 8 What is 8 7? Half of 6 is Half of is 66 Half of 66 is so 6 8 = We know that 0 multiplied by 7 is 70. This fact helps us here. 8 can be partitioned into 70 and. 8 = = 0 7 = What is 96 6? 96 = = = = 6 So 96 6 = 6

25 + = 6 So 6 = = So 8 7 = Me: Can use efficient written methods of multiplication including short and long multiplication Multiplication is easier when you a) know your times tables facts and b) can set your work out correctly! Expanded Multiplication Short multiplication Th H T Ø 6 8 X = x 8 SET your calculation our correctly to avoid errors DON T forget the value of the digits when multiplying. 68= Th H T Ø 6 8 X = x = x 00 You MUST make sure that you multiple each digit in the top row (68) by the number in the bottom row () x 8 = Put in the Ts column Put the in the Øs column x 60 = 0 Put the in the H column The in the Ts column The 0 in the Ø column x 00= =00

26 Expanded Multiplication Expanded Multiplication Expanded Multiplication HTh TTh Th H T Ø HTh TTh Th H T Ø HTh TTh Th H T Ø X 7 X x Short Multiplication Short Multiplication Short Multiplication HTh TTh Th H T Ø HTh TTh Th H T Ø 6 6 HTh TTh Th H T Ø X 7 X x

27 Mf: Can use efficient written of division including short and long division When we are dividing large numbers by a single digit we can use short division. 8 6 = = =6 r 6 r 7 9 ) We say, how many 6s in 8? The answer is so we put the above the 8 because there is only group of 6 in 8 with a remainder of. ) x 6 = 6 so we have a remainder of. We put this in front of the to make. ) We say, how many 6s in? The answer is. We put this above the next to the. ) is our answer. ) So, 8 6 = ) We say, how many s in 0? The answer is 0 so we put the 8 above the 8 ) We move the next to the 6 to make 6. ) We say, how many s in 6? The answer is 6. We put this above the 6 next to the 0. There is a remainder of ) We put the next to the 0 to make 0 ) We say, how many s in 0? The answer is. We put the above the - next to the 6 6) 6 is our answer. 7) So, 60 = 6 ) How many 7s in 9? The answer is remainder. We put above the 9 and a in front of the. ) How many 7s in? The answer is remainder. We put above the and a in front of the. ) How many 7s in? The answer is 6 remainder. We put the 6 above the and the remainder next to it. ) Our answer is 6 r. ) So 9 7 = 6 r. Remainders as fractions: 9 7 = 6 r. We can write the remainder as a fraction. In this example it would be: 6 and because 7 is our divisor and we only have out of a possible group of 7. 7

28 So 9 7 = 6 7 Chunking: The Chunking Method as a step towards understanding long division NOTE: To understand the chunking method, we must understand division as grouping as well as sharing. We can illustrate this using a simple division fact that we already know. Let s look at an example = x = 0 x = 0 We have subtracted 7 groups of so = 7 Note: We can physically use cubes, alongside the number line, to illustrate the process of subtracting groups of from. To help us divide larger numbers, we can chunk or group the divisor Using the chunking method: 7 = 0 7 We have subtracted 0 groups and then 8 groups of the divisor. This totals 8. Therefore, 7 = 8 7 = (0 x ) - (8 x ) 0

29 We need to use our multiplication and division facts and be able to multiply by 0 and 00. We also need to subtract. We have to use a lot of skills to divide efficiently! 68 = (00x) 68-0 (0x) 8-8 (6x) =6 Therefore, 68 = 6 You want to subtract the largest amount you can each time, using number facts you can easily work out. It is easiest to work with multiples of 0 and 00. NOTE: keep the divisor lined up and the multiples lined up This will help with efficient calculations at the end Chunking 768 = (0x) 8-8 (x) = Therefore, 768 = Division is complicated. It is always a good idea to estimate first. Here we could partition 768 into 700 and 68. There are roughly groups of in 00, so we would have 7 lots of in 700. This is plus an extra groups from the 68. We should expect an answer of roughly. 6 = We then divide 7 by which is 7 with a remainder. To calculate the remainder, we subtract 7 groups of from 7. Long division. We say divided into 6 which is not an easy division so we say divided into divided by. The answer is with a remainder. To calculate the remainder we need to work out what is left if you subtract groups of from 6.. This time we put the above the and write two groups of beneath the 6 so 8 is written beneath 6. We then subtract 8 from 6. The remainder of 7 is written below the 8. 7 r 6 x 0 = 0 x = 0 x 6 = x 7 = We also bring down the from the calculation above. 8 So 6 = 7 r

30 We put the 7 at the top next to the and write the 7 groups of beneath the 7. We then subtract 68 from 7. The remainder of is written next to the answer of Chunking or long division???? Mg: Can multiply a simple decimal by a single digit Grid method is a good way to first begin multiplying decimals by a whole number.6 x X x 0=0 x=0 x 0.6= = 6.6 x = 6.9 x = ( =) lots of 0.6 Th H T Ø x. 0. = x = x 0. = x 0 6. H T Ø x = x = x 0.9. = x 9 0 = x Leading to Leading to

31 .6 x = 6 Th H T Ø x x = H T Ø x Mi: Can identify factors and common factors Let us see if we can find two numbers which multiply together to make 6: We know that multiplied by makes 6. Therefore, we say that and are FACTORS of 6. But x 6 = 6 as well so and 6 are also FACTORS of 6 It is best if we work systematically to find factor pairs. Factors of and and and 8 Factors of 8 and 8 and 9 and 6 and 6 Therefore the factors of are:,,,, 6, 8, and Therefore the factors of 8 are:,,, 6, 9 and 8 Common Factors of both and 8 are:,, and 6 Lowest common factor (LCM) is (we discount ) Highest common factor (HCF) is 6

32 Mj: Can recognise and describe square numbers We say that is a SQUARE number because both factors of are the same (x). If you multiply any number by itself, the answer you get will be a square number. Let s look at some more examples of square numbers: x x x x The first square number is. We know this because x = The first 0 square numbers are: x = therefore, we know that is a square number x = 9 therefore, we know that 9 is a square number x = 6 therefore, we know that 6 is a square number You need to learn and recognise these numbers ² ² ² ² ² 6² 7² 8² 9² 0² It is also important to know the notation used when working with square numbers. We say that squared is because x =. In mathematics we write this as =. Mk: Can recognise and identify prime numbers Remember: A prime number is a number that has only factors ( and itself). For example is a prime number as the only factors of are and.

33 Prime factor: a factor of a number that also happens to be a prime number. For example 7 is a prime factor of because 7 is a factor of and 7 is a prime number. Its only factors are and 7. We should know and recognise all the prime numbers up to 0. These are: (the only even prime number) Number: Fractions, Decimals and percentages Ma: Can identify, name and write equivalent fractions of a given fraction represented visually The numerator (the number at the top) tells us how many pieces we have. The denominator (the number at the bottom) tells us how many equal parts the whole (shape, number, quantity) has been divided into. You should be familiar with the following fractions: / / and / are shown on this number line. 0 You should also be familiar with thirds and sixths

34 0 6 or 6 6 or or 6 6 or 6 6 or You should also be familiar with eighths (and quarters) 0 8 or 8 8 or 8 or 8 or or 8 8 or You should also be familiar with fifths and tenths 0 0 or or 0 or 6 0 or or or or

35 You will see that some fractions are equivalent to each other. This means that their value is the same. For example is equivalent to 6 as we could see on the number line 8 above. Let s look at these equivalent fractions represented in a different visual way. Imagine the drawing below is a chocolate bar. We can see here that is equivalent to and These fractions are all equivalent to and These fractions are all equivalent to 8 6 and These fractions are all equivalent to 8 Hint: knowing your multiplication and division facts will help you find equivalent fractions quickly and efficiently! Mb: Can use common factors to simplify fractions

36 When fractions are equivalent we know that the equivalent fractions have the same value. For example if I had of a chocolate bar or of the same 8 chocolate bar, I would have exactly the same amount of chocolate. What do equivalent fractions have in common? We know that fractions are closely linked to division. tells us that something has been divided into equal parts and we have out of equal parts. tells us that something has been divided into 8 equal parts and we have out of 8 equal parts (equal 8 to ½). Remember: A factor is a number that divides into another number with no remainders. For example is a factor of. A common factor is a factor that two larger numbers have in common. For example is a factor of and 6 because divides into and also divides into 6. To simplify a fraction, we must find a factor that the numerator and denominator have in common. Let s look at an example:. Simplify 8 The numerator is and the denominator is 8. To simplify this fraction we must find a common factor. Both and 8 have the factor in common. We must now divide both the numerator and the denominator by. 8 = Both fractions are equivalent. (See fraction wall) 8 can be simplified to. The fraction is now in its simplest form. A fraction can be put in its simplest form and be simplified. To simplify a fraction in one step, it is necessary to find the largest common factor.. Simplify 0 0 = 0 can be simplified to Both fractions are equivalent. ) Look for the largest common factor of and 0. ) is the largest common factor of and 0. ) Divide both and 0 by. Simplify = 8 6 can be simplified to ) Look for the largest common factor of 8 and 6. ) 9 is the largest common factor of 8 and 6.

37 Both fractions are equivalent. (See fraction wall) ) Divide both 8 and 6 by 9. Hint: knowing your multiplication and division facts will help you simplify fractions quickly and efficiently! Mc: Can compare and order fractions When fractions have a common denominator, it is straightforward to compare them and put them in order. Put the following fractions in order from smallest to largest: Clearly, the correct order for these fractions is: Sometimes the fractions do not have a common denominator but the fractions are familiar. For example we know that is smaller than and that is bigger than etc. These fractions are familiar and can be compared easily on a number line. 6 Which is the smallest fraction? or? =? =? Before we can compare these fractions, we must make sure they both have a common denominator. To do this, we need to find the lowest common multiple of all the denominators. The lowest common multiple is the lowest number that both numbers go into. In this case, we need to find the lowest common multiple of and. What is the lowest number that these numbers go into? The answer is. We now need to write each fraction with as a denominator.

38 =? To find an equivalent fraction, we need to look at what we have multiplied the denominator by to get. Whatever we have multiplied the denominator by, we need to multiply the numerator by =? = 9 Order the following fractions from smallest to largest: 6 8 Before we can compare these fractions, we must make sure they all have a common denominator. To do this, we need to find the lowest common multiple of all the denominators. The lowest common multiple is the lowest number that all the other numbers go into. In this case, we need to find the lowest common multiple of 6, 8,, and. What is the lowest number that all of these numbers go into? The answer is 6. We now need to write each fraction with 6 as a denominator. (The first fraction already has a denominator of 6.) 8 =? 6 =? 6 =? 6 When finding equivalent fractions, we multiply or divide the numerator and denominator by the same number. =? 6 x 8 =? 6 = 8 6 = 6 = 6 x 8 = 6 = 8 6 = 6 Now that they have a common denominator, we can now compare the fractions and put them in order: = 6

39 Don t forget to give the original fractions in your answer: The fractions in order from smallest to largest are: 8 Md: Can add and subtract fractions When fractions have a common denominator, it is straightforward to compare them and also to add and subtract them. For example, 6 We can see this represented visually too: Addition + = 6 6 can also be written as + Simplify the fraction We can see this represented visually too: Subtraction = 0 0 = 7 out of 0 pieces is equal to 7 0 if we subtract of those pieces we are left with 0 Sometimes fractions don t have a common denominator. If this is the case we must first convert the fractions so that they do have a common denominator. We CANNOT add fractions which do not have the same denominator. X X X = = 9 = 8 8

40 X X X Before we can add these fractions, we must make sure they have a common denominator. To do this, we need to find the lowest common multiple of the denominators. The lowest common multiple is the lowest number that both numbers go into. Me: Can multiply fractions by whole numbers (fractions of quantities) We can think of fractions in two ways: as a number and as an operator. When we place fractions on a number line and think of them as part of our number system we are thinking of fractions as numbers. But fractions can also be used as operators. This requires us to use the fraction to carry out a calculation. Let s look at the example below: We can find of a set of objects. Look at the visual example below: In this example, the fraction is being used as an operator not a number because we have to find of a set of objects. We have to carry out an operation of is 9 To multiply a whole number by a fraction, you need to use the fraction as an operator. This means reading the multiplication sign as a fraction of a number. In the example left, of is the same as x How would you read x? There are a number of ways we could think of x lots of one quarter or one quarter times ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

41 x = of = x = of = 6 x = of = 9 x = of = Remember: Finding of something is the same as dividing by As a number sentence. For example, grapes are shared equally between friends. Each child will get: x = = grapes x = Using the bar model First draw your bar The calculation is x, so the length of the bar is. Next you need to find of the bar, so you need to divide it into equal pieces Remember: Finding of something is the same as dividing by So x = To find one piece you need to divide the whole bar into As there are pieces, each piece is worth ( x = ) What about x 0? of 0 is 8 Using the Bar Method: 0 x 0

42 To find you need to divide the bar into pieces. 0 We can see that dividing the bar into means each section is 0 =. If each section represents, then two sections is x = 8. Therefore, x 0 = 8 8 Mf: Can multiply pairs of fractions, writing the answer in its simplest form Remember: Fractions can be used as numbers or operators. When multiplying two fractions, use one of the fractions as an operator. This means you are finding a fraction of a fraction Remember: Finding of something is the same as dividing by. x This can be read as of a quarter, using one of the fractions as an operator. Draw a shape and divide it into quarters four equal parts. We need to find of one of the quarters. Finding of something is the same as dividing by To finish finding out the answer to of a quarter, we need to make all the parts in the diagram equal. This means we need to divide all the quarters into half. This shows one of the quarters divided into, or of a quarter. The part that is cross-shaded is ½ of a quarter. There are eight parts in total.

43 of a quarter is one eighth. x = 8 x This can be read as / of a third. First draw a diagram to show We know that finding is the same as dividing by. So next we need to divide the third into. x = This method can still be used when the numerator is greater than one. x This can be read as of three quarters First draw a diagram to show divided by

44 x = 8 Written method If you understand how to visually multiply fractions together, you have probably worked out the method for multiplying fractions together. To multiply fractions the two numerators are multiplied together and the two denominators are multiplied together. x x = x 8 x x = 0 x x = = 6 Mg: Can divide fractions by whole numbers Remember: Dividing by is the same as finding Dividing a fraction by a whole number is the same as multiplying two fractions together. This is because when you multiply two fractions together, one is being used as an operator. For example, x can be read as a half of a quarter. We know that finding a half of a number (in this case ¼) is the same as multiplying by ½. Therefore, x = Remember that when we are multiplying two numbers, we can work it out in any order. So, x = x is one eighth. This can also be written as: x = Remember that finding ¼ of a number is the same as dividing the number by = =

45 Written method We can divide a fraction by a whole number using our knowledge of multiplying fractions. Let s look at an example: Therefore, we can say that = x = x = 8 Remember that dividing by is the same as multiplying by ¼ 6 6 = Equal to x 6 x = 6 0 x = x 6 = = Equal to x 7 x = 7 x = x 7 = Mh: Can convert mixed numbers to improper fractions Remember that fractions can be larger than is called a mixed number. We read this as one and one quarter. The word and here is important because = + 0 = =

46 is called an improper fraction or top-heavy fraction. An improper fraction is a fraction where the numerator (top number) is larger than the denominator (bottom number). is equal to. Mixed fraction or = = Improper fraction = How do we calculate without pictures? = 8 = 6 + so the calculation is 6 + = 8 Write as an improper fraction. 7 Start by writing as sevenths whole one = 7 sevenths whole ones = sevenths whole ones = sevenths Therefore, = 7 If = 7, then 7 = = 7 7 = 7 Mj: Can read and write decimal numbers as fractions Remember place value In order to read and write decimal numbers as fractions, it is vital that you remember and understand the value of digits in numbers. thousands hundreds tens ones. tenths 0 hundredths 00 thousandths 000

47 The numbers get smaller as the chart moves to the right, so thousands are the biggest numbers and thousandths are the smallest numbers on this chart. The chart mirrors itself in terms of the names of the columns with tens on the left of the decimal point and tenths on the right, then hundreds and hundredths etc. Just as ten ones are ten and 0 tens are hundred, so 0 tenths are one and 0 hundredths are tenth. If I want to read 0., I can put the number into the chart. thousands hundreds tens ones. tenths 0 0. hundredths 00 thousandths 000 So, 0. = 0 0. = four tenths = 0. = five tenths = 0.6 = six tenths = 6 0 = 0 = 0 = 0.7 = seven tenths = 7/0 0.8 = eight tenths = 8/0 = / 0.9 = nine tenths = 9/0 What about 0.0 thousands hundreds tens ones. tenths hundredths thousandths 0. 0 thousands hundreds tens ones. tenths hundredths thousandths = three hundredths = = four hundredths = 00 = In the chart, we can see that this number can be read as one hundredth. This can be written as / = 00 This is two hundredths or = 00 = 0

48 0.0 = five hundredths = 00 = 0 What is 0.7 as a fraction? If we use the place value chart again, we can see the 0.7 is tenths and 7 hundredths. 0.7 = We know that 0 = 0 00 Therefore, 0.7 = = 7 00 What is 0. as a fraction? Here, 0. = 00 In this example though, is not a fraction in its simplest form because and have a common factor of. = Therefore, 0. = 00 = 9 0 Divide both and 00 by We can keep counting up in hundredths. There are some well known decimal hundredths that convert to fractions. 0.0 = ten hundredths = 0 00 = 0 0. = 00 = (one quarter) 0.7 = 7 = (three quarters) 00 What about 0.00? thousands hundreds tens ones. tenths hundredths thousandths Once written in the chart, this can easily be read as one thousandth or 0.0 = thousands hundreds tens ones. tenths hundredths thousandths = two thousandths = 000 = 00 What is 0.8 as a fraction? 0.8 = = =

49 0. is one hundred and twenty five thousandths or 0. is approximately is approximately two thirds or This can be reduced to 00 or 0 or 8. Mk: Can recognise approximate proportions of a whole number using percentages Remember: A percentage represents the number of parts out of 00. For example, 76% means 76 out of 00 and 00% means 00 out of 00 (or a whole). Percentages can also be written as fractions and decimals. Equivalence 0% = % = 7% = = = = = 0. = 0. = 0.7 Number line 0 % 0% 7% 00% 00 = 0 00 = 7 00 = Pictorial 0% = 0 00 = 0 = 0. We can see in this diagram that 0 squares are shaded out of a total of 00 squares (0/00 = 0%). We can also see that row is shaded out of a total of 0 rows (/0 = 0%)

50 out of 0 blocks are shaded = 0 = 0% % = 0 = % = 00 = 0% = 0 00 = 7% = 7 00 = out of 0 blocks are shaded. = 0 = 0% 0 00 Using known equivalent facts to help you calculate percentages. You must learn the following facts: 0% = Once you know these percentages 0 and their fraction equivalents, you We know that = 0% can find others. We therefore can work out that: For example, 0% = 0 From this we can work out that: 0% = 0 = 0% = 60% = 80% Count how many squares there are in total- this is your denominator Count how many squares are shaded- this is your numerator Look at the fraction- is there an equivalent that would support your calculation? Convert to a decimal or percentage- what is the question asking you? Often we are asked to find percentages of certain numbers. Sometimes, the easiest way to do that is convert the percentage to a fraction and then work it out. Let s look at some examples: % of We know that % = of = Therefore, % of = 0% of or 60% = 6 0 We know that 0% = of = 6 Calculating percentages of an amount using known facts: % of 8= % =0% + % First we need to find 0% by dividing the number by =.8 0% =.8 % of = % = 0% + 0% + 0% + % first we need to find 0% by dividing the number by 0 0 =. 0% =. 0% =. x = or =

51 If we know 0% we can now find % as half of 0 is we can half 0% % =.9 % = 0% =.8 + 0% =. 8 % =. 9 % =. 7 Half of =. Half of 0.9= 0. 0% = 7. If we know 0% we can now find % as half of 0 is we can half 0% %= 6. 0% =. 0% = 7. 0 % = 6. % =. 7 % = Half of = 6 + Half of 0.= 0. Ml: Can recognise simple equivalence between fractions, decimals and percentages To recognise simple equivalences between fraction, decimals and percentages we must first remember that equivalent means equal to. Fractions, decimals and percentages are all ways of representing parts of a whole. Fractions Decimals Percentages Fractions Decimals Percentages 0.0 0% % % % 0. % % 0.7 7% % 0.0 0% 0.0 0% % 0.0 0% % 0. 0% 0

52 % % 0