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Maths methods taught in Key Stage 2 Learning Together for Life Mathematics – LKS2 No ratio required in LKS2 Written division moved to UKS2 No calculator skills included Carroll / Venn diagrams no longer required Y3: Formal written methods for + & — Y3: Compare, order & + & — easy fractions Y3: Vocabulary of angles & lines Y3: Time including 24h clock & Roman numerals Y4: Recognise equivalent fractions/decimals Y4: Solve fractions & decimals problems Y4: Perimeter/area of compound shapes Y4: Know multiplication tables to 12 x 12 Mathematics – UKS2 No calculator skills included No probability included Data handling greatly reduced content Y5: Use decimals to 3dp, including problems Y5: Use standard multiplication & division methods Y5: Add/subtract fractions with same denominator Y5: Multiply fractions by whole numbers Y6: Long division Y6: Calculate decimal equivalent of fractions Y6: Use formula for area & volume of shapes Y6: Calculate area of triangles & parallelograms Y6: Introductory algebra & equation-solving Addition Learning Together for Life Split the numbers up! The art of this is to split the numbers up into units, tens, hundreds etc. Add each “bit” separately then join them together at the end. If you are asked to find the “sum” of some numbers, add them. An example of this: Calculate 34 + 55 Split it up into tens and units: Tens: 30 + 50 = 80 Units: 4 + 5 = 9 Final answer: 89 Another example of this: Calculate 187 + 264 Split it up into tens and units: Hundreds: 100 + 200 = 300 Tens: 80 + 60 = 140 Units: 7 + 4 = 11 Final answer: 451 Try these: 1. 26 + 53 = 79 2. 57 + 35 = 92 3. 98 + 57 = 155 4. 365 + 478 = 843 Using a standard method. Column Addition 386+542 + Now you try… • Use a method you haven’t used before – a number line, expanded method or standard column addition. 587 + 348 = When you have your answer, compare your answer and method with the person next to you. Do you have the same answer? Did you find it in a different way? Subtraction Learning Together for Life The key to subtracting is… …to estimate an answer first. Perform the subtraction in stages. If you are asked to find the “difference” between two numbers, subtract them. What does this mean? Calculate 72 – 38 Make it easier: 72 – 30 Answer: 42 42 - 8 Answer: 34 Firstly take subtract 30 Then subtract 8 A different method: Calculate 72 – 38 Make it easier: 72 – 40 Answer: 32 Actual answer: 34 +2 to the number you’re subtracting +2 to your answer Another example: Calculate 91 – 23 Make it easier: 91 – 20 Answer: 71 71 - 3 Subtract 20 Subtract 3 Actual answer: 68 Try these: 1. 85 - 36 = 49 2. 52 - 25 = 27 3. 96 - 58 = 38 4. 124 - 89 = 35 Column method - Exchanging & cancelling. 832–279 Multiplication Learning Together for Life In Year 3 • Understand how to use partioning and arrays to solve TU x U eg 13 x 3 10 x 3 = 30 3x3=9 Calculate 23 × 8 Split the large number up: This is 2 × 8 with a zero on the end 20 × 8 = 160 3 × 8 = 24 Add the answers together as you have a total of 23 eights. Answer: 184 Have a go at these: 1. 17 × 6 = 102 2. 26 × 4 = 104 3. 43 × 7 = 301 4. 64 × 9 = 576 In Year 3 x 10 3 3 30 9 30 + 9 = 39 Grid Method 255 x 5 = ? 200 50 5 200 x 5 50 x 5 5x5 x 5 1000 250 25 Finally, add the three numbers together to get your answer. 1000 + 250 So 255 x 5 = 1 275 + 25 = 1275 Grid Method – part 2 255 x 25 = ? x 200 50 5 20 4000 1000 100 5 1000 250 25 Add up each column, then add the resulting numbers together. 4000 + 1000 + 100 1000 + 250 + 25 = = 5100 1275 6375 Over to you! Have a go at solving these multiplications using the grid method. 65 x 8 74 x 45 92 x 53 Traditional method 21 x 13 = ? x 2 1 1 3 3 x 1 = 3 write down the 3. 6 3 2 1 0 2 7 3 3 x 2 = 6 write down the 6 10 x 1 = 10 write down the 10 1 x 2 = 2 write down the 2 Add the numbers Traditional method 45 x 34 = ? 1 2 x 4 5 3 4 4 x 5 = 20 write down the 0, carry the 2. 1 8 0 1 3 5 0 1 5 3 0 4 x 4 = 16, add 2 write down the 18 30 x 5 = 150 write down the 50, carry the one 3 x 4 =12, add the 1, write down the 13 Add the numbers Lattice method – part 1 25 x 5 = ? 5 x 5 = 25 2x5= 10 2 5 1 2 0 5 5 1. Make the lattice (grid) as shown 2. Multiply each number above a column by the numbers in every row 3. Write the answers in the lattice. Making sure you have only 1 digit in each triangle Lattice method – part 2 25 x 5 = ? 2 5 1 2 0 1 2 2+0=2 5 5 5 Add along the diagonal Lattice method – part 1 36 x 8 = ? 6 x 8 = 48 3x8 =24 3 6 2 4 4 8 8 1. Make the lattice (grid) as shown 2. Multiply each number above a column by the numbers in every row 3. Write the answers in the lattice. Making sure you have only 1 digit in each triangle Lattice method – part 2 36 x 8 = ? 3 6 2 4 4 2 8 4+4=8 8 8 8 Add along the diagonal line Lattice method – part 1 36 x 13 = ? 1x6=6 3 1x3=3 6 0 0 3 3x3=9 0 6 1 9 8 1 1. Make the lattice (grid) as shown 3 2. Multiply each number above a column by the numbers in every row 3 x 6 = 18 3. Write the answers in the lattice. Making sure you have only 1 digit in each triangle Lattice method – part 2 36 x 13 = ? 3 0 6 3 0 1 4 0 1 9 6 6 + 1 + 9 = 16 6 8 8 1 3 Add along the diagonal line Have a go at calculating 21 × 32 Do it in as many ways possible… Which of these did you use to calculate 21 × 32? 30 2 20 1 600 30 40 2 Answer: 672 32 × 21 32 + 640 Answer: 672 2 0 0 0 6 1 6 4 7 0 0 3 3 2 2 2 Answer: 672 Are there any other methods? Choose a method to calculate 43 × 17? 10 7 40 3 400 30 280 21 Answer: 731 43 × 17 301 + 430 Answer: 731 4 0 0 2 7 3 4 8 13 0 2 3 1 1 7 1 Answer: 731 Are there any other methods? Multiply these using whichever method you like (no calculators!): 1. 26 × 14 = 364 2. 74 × 39 = 2886 3. 124 × 16 = 1984 4. 249 × 179 = 44571 643 x 27 = 17,361 6 1 4 0 3 0 2 1 2 6 8 2 2 14 7 2 8 1 1 7 6 1 3 Using decimals 23.6 x 3.2 = 75.52 2 0 3 0 6 1 3 6 8 9 0 1 10 2 4 6 2 1 7 5 2 5 Real Life Problem (6a) 1. Amy bought 48 teddy bears at £9.55 each. Work out total amount she paid. £458.40 Real Life Problem (6a) 2. Nick takes 26 boxes out of his van. The weight of each box is 32.9kg. Work out the total weight of the 26 boxes. 855.4 kg Division Learning Together for Life Starter For each number in the table, put a tick if it is divisible by 2, 3, 4, 5, or 6. How can you work these out without actually working out the division? Number 26 120 975 12,528 Divisible by 2? Divisible by 3? Divisible by 4? Divisible by 5? Divisible by 6? Multiples Investigation Do you know any of the rules for checking divisibility? A number can be divided by 2 if: It ends in a 0, 2, 4, 6 or 8 A number can be divided by 3 if: The sum of its digits is a multiple of 3 A number can be divided by 4 if: The number made by the last 2 digits is a multiple of 4 A number can be divided by 5 if: It ends in a 0 or 5 A number can be divided by 6 if: It can be divided by both 2 and 3 (ends in an even number and is a multiple of 3) Divisibility Rules (8) • A number is divisible by 8 if • the number made by the last three digits will be divisible by 8 Divisibility Rules (9) • A number is divisible by 9 if • the sum of all the digits will add to 9 Multiplication is the inverse of division. Multiplication and division are inverse operations; this means they are the opposites of each other. By knowing the answer to one problem you can work out all the others. Example 20 ÷ 5 ÷ × 4 We can use our tables – how? 300 ÷ 6 is easy because we already know 30 ÷ 6. 30 ÷ 6 = 5 So 300 ÷ 6 =50 2800 ÷ 4 is easy because we know 28 ÷ 4. 28 ÷ 4 = 7 So 2800 ÷ 4 = 700. Have a go at these: 1. 2000 ÷ 5 = 400 2. 320 ÷ 4 = 80 3. 33,000 ÷ 3 = 11,000 4. 42,000,000 ÷ 7 = 6,000,000 Division: Learning to show the remainder 58 ÷ 4 = 14 r2 14 1 4 58 r 2 Division: Learning to show the remainder For each question show the three ways of showing the remainder. 1 - 28 ÷ 5 5 - 73 ÷ 2 2 - 37 ÷ 2 6 - 58 ÷ 4 3 - 49 ÷ 4 7 - 87 ÷ 4 4 - 39 ÷ 5 8 - 67 ÷ 5 The “bus stop” method You do need to know your tables! The most common method is called “the bus stop”, because it looks like the numbers are waiting in a bus stop. The divisor goes outside the “bus stop”. You can also add multiples of your divisor, but this will take you longer! Calculate 135 ÷ 3 Bus Stop Draw the bus stop: Divisor 0 3 1 4 1 3 5 1 5 Divide each value by 3, not forgetting to carry any remainders. The answer is on top! Answer: 45 Multiples We need to do the following calculations: 10 × 3 = 30 Total so far: 30 10 × 3 = 30 Total so far: 60 10 × 3 = 30 Total so far: 90 10 × 3 = 30 Total so far: 120 5 × 3 = 15 Total so far: 135 How many 3s in total? Answer: 45 What about 322 ÷ 14 Bus Stop Draw the bus stop: Divisor 0 14 3 2 3 2 3 4 2 Divide each value by 14, not forgetting to carry any remainders. The answer is on top! Answer: 23 Multiples We need to do the following calculations: 10 × 14 = 140 Total so far: 140 10 × 14 = 140 Total so far: 280 3 × 14 = 42 Total so far: 322 How many 14s in total? Answer: 23 Calculate these, without a calculator! 1. 168 ÷ 7 = 24 2. 222 ÷ 6 = 37 3. 384 ÷ 12 = 32 4. 952 ÷ 17 = 56 Long Division Methods Method 1 We are going to try to solve 839 ÷ 27 Method 1 839 -270 839 ÷ 27 (27 x 10 = 270) Method 1 839 -270 569 839 ÷ 27 (27 x 10 = 270) Method 1 839 -270 569 -270 299 839 ÷ 27 (27 x 10 = 270) (27 x 10 = 270) Method 1 839 -270 569 -270 299 -270 29 839 ÷ 27 (27 x 10 = 270) (27 x 10 = 270) (27 x 10 = 270) 839 ÷ 27 Method 1 839 -270 569 -270 299 -270 29 -27 2 (27 x 10 = 270) (27 x 10 = 270) (27 x 10 = 270) (27 x 1 = 27) 839 ÷ 27 Method 1 839 -270 569 -270 299 -270 29 -27 2 (27 x 10 = 270) (27 x 10 = 270) (27 x 10 = 270) (27 x 1 = 27) 10 + 10 + 10 + 1 = 31 Method 1 839 ÷ 27 =31 r 2 Or 31 2 27 Long Division Methods Method 2 We are going to try to solve 839 ÷ 27 Method 2 839 ÷ 27 becomes 27 ) 8 3 9 Method 2 839 ÷ 27 27 ) 8 3 9 27 x 10 = 270 Miles off! Method 2 839 ÷ 27 27 ) 8 3 9 27 x 10 = 270 27 x 20 = 540 Getting better! Method 2 839 ÷ 27 27 ) 8 3 9 27 x 10 = 270 27 x 20 = 540 27 x 30 = 810 Much better! Method 2 27 x 30 = 810 839 ÷ 27 Put the 810 underneath. 27 ) 8 3 9 -810 29 27 x 30 = 810 Method 2 839 ÷ 27 27 ) 8 3 9 -810 29 -27 2 27 x 30 = 810 27 x 1 = 27 Method 2 839 ÷ 27 31 27 ) 8 3 9 -810 29 -27 2 27 x 30 = 810 27 x 1 = 27 30 + 1 = 31 Method 2 839 ÷ 27 31 27 ) 8 3 9 -810 29 -27 2 r 2 Don’t forget the remainder! Method 2 839 ÷ 27 =31 r 2 Or 31 2 27 Long Division Methods Method 3 We are going to try to solve 839 ÷ 27 Method 3 839 ÷ 27 becomes 27 ) 8 3 9 Method 3 839 ÷ 27 Calculate 8 ÷ 27 27 ) 8 3 9 Method 3 839 ÷ 27 We can’t do it, so we write the answer 0 here 0 27 ) 8 3 9 Method 3 839 ÷ 27 So we next look at 83 ÷ 27 0 27 ) 8 3 9 Use you repeated subtraction here if this helps Method 3 2 x 27 = 54 839 ÷ 27 03 27 ) 8 3 9 3 x 27 = 81 Method 3 3 x 27 = 81 839 ÷ 27 03 27 ) 8 3 9 -81 2 We need to take off 81 from the 83 to get the remainder Method 3 839 ÷ 27 03 27 ) 8 3 9 -81 29 Drop the next digit next to the 2 Method 3 839 ÷ 27 Now we are going to do 29 ÷ 27 and put the answer here 031 27 ) 8 3 9 -81 29 Method 3 839 ÷ 27 Now we are going to do 29 - 27 to get the remainder 031 27 ) 8 3 9 -81 29 27 Method 3 839 ÷ 27 031 27 ) 8 3 9 -81 29 27 2 r 2 Method 3 839 ÷ 27 or = 31 31 2 27 r 2 Again use the method that gives you the correct answer !! Long division Question : 2987 23 12 9 23 times table 23 46 69 92 115 138 161 184 207 230 29 68 23 6 22 2987 20 Answer : 129 r 20 227 Now try 1254 17 and check on your calculator – Why is the remainder different? How would this be calculated.... 2 468 • How many 2s in 4........ • How many 2s in 6........ • How many 2s in 8......... 1)Work out the following:- 2742 ÷ 3 0 9 1 4 3 2 27 4 12 (a) Answer = 914 7364 ÷ 7 1 0 5 2 7 7 3 36 14 (b) Answer = 1052 3231 ÷ 9 0 3 5 9 9 3 32 53 81 (c) Answer = 359 Fractions Review Level 1 2 Fractions Recognise & use 1⁄2 & 1⁄4 Find and write simple fractions. Understand equivalence of e.g. 2/4 = 1/2 recognise, find, name and write fractions 1/3 ¼ 2/4 and ¾ of a length, shape, set of objects or quantity. 3 Use & count in tenths. Recognise, find & write fractions. Recognise some equivalent fractions for ¼½¾ Add/subtract fractions with the same denominator within one whole. Order fractions with common denominator 4 Identify equivalent fractions. Add & subtract fractions with common denominators. Recognise common equivalent fractions including decimal equivalents for ¼ ½ ¾ and tenths and hundredths. 5 6 Compare & order fractions. Compare & simplify fractions. Add & subtract fractions with different denominators, with mixed numbers. Use equivalents to add fractions. Multiply and divide fractions by units. Divide fractions by whole numbers. Write decimals as fractions. Multiply and divide fractions with different denominators, writing the answer in its simplest form. Link percentages to fractions & decimals Multiply simple fractions. Multiply fractions and mixed numbers Today we are learning I am starting the lesson on level _____________________ By the end of this lesson I want to be able to _____________________ Fractions A number in the form Numerator Denominator Or N D Fractions The denominator can never be equal to 0. 12 0 Does not = exist! Fractions A fraction with a numerator of 0 equals 0. 0 4 = 0 0 156 = 0 Fractions • If the numerator is larger than the denominator, it is called an improper fraction. Find the improper fraction 7 10 4 56 10 11 8 9 73 12 Maths with Fractions Four basic functions • Multiply • Divide • Add • Subtract Multiplication • Multiply the numerators and put in the numerator of the result • Multiply the denominators and put in the denominator of the result 7 8 x 4 9 = 7x4 8x9 = 28 72 Multiplication - Let’s Try It! 7 9 7 5 x x 1 2 1 3 = = 7 4 18 7 7 30 15 4 x x 9 11 7 14 = = 36 77 210 56 210 56 These numbers get pretty big! What if we needed to multiply again? Let’s make the fraction more simple, so it will be easier to use in the future. Simplification • Divide by the Greatest Common Factor 28 72 But what is a Common Factor? Factors • A factor is a number that can be divided into another number with no remainder – 8’s factors are: • • • • 1 (8/1 =8) 2 (8/2 = 4) 4 (8/4 = 2) 8 (8/8 = 1) – 3 is NOT a factor of 8, because 8 is not evenly divisible by 3 (8/3 = 2 with R=3) Common Factors • A common factor is a factor that two numbers have in common – For example, 7 is a factor of both 21 and 105, so it is a common factor of the two. – The greatest common factor is the largest factor that the two number share So let’s go back to our simplification problem from before… Simplification • Divide both numerator and denominator by the Greatest Common Factor 28 72 Factors are 1, 2, 4, 7, 14, 28 Factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 48, 72 Greatest Common Factor is 4 28 ÷ 4 = 7 72 ÷ 4 = 18 28 7 So = 72 18 Simplification - Let’s Try It! 3 1 = 9 3 7 1 = 21 3 18 2 = 63 7 24 2 = 84 7 6 2 = 15 5 78 13 = 114 19 Division Just like multiplication with one more step:• Turn the ÷ into a x symbol. • Invert / flip the second fraction and multiply 3 8 ÷ 1 2 = 3 8 x 2 1 = 6 8 Division - Let’s Try It! 7 1 ÷ 9 2 7 5 ÷ 1 3 14 = 9 = 4 9 44 ÷ = 7 11 63 21 20 5 4 ÷ 7 10 = 50 7 Addition • To add two fractions, you must make sure they have a Common Denominator 3 8 + 5 16 What is a Common Denominator? Common Denominator • A common denominator is a number with which both of the denominators share at least one factor that is not the number 1 – For example, if the denominators are 4 and 7, then a common denominator is 28. – 28 shares the factors 1, 2 and 4 with the number 4, and the factors 1 and 7 with the number 7. So let’s go back to our simplification problem from before… Addition • To add two fractions, you must make sure they have a Common Denominator • What can you multiply each fraction by to give the smallest common denominator? 3 5 + 8 16 8 goes into 16 two times 3 8 x The smallest number that has both of these as factors is 16 Once you have a common denominator, add the numerators. 2 2 = 6 16 16 goes into 16 one time 5 16 x 1 1 = 6 5 11 + = 16 16 16 5 16 Addition - Let’s Try It! 1 1 + 4 2 6 8 + 2 3 = = 3 4 4 2 1 + = 16 8 2 17 13 12 16 + 3 4 = 25 16 Subtraction • • To subtract two fractions, they also must have a Common Denominator What can you multiply each fraction by to give the smallest common denominator? 3 - 5 8 16 8 goes into 16 two times 3 2 6 8 x 2 = 16 The smallest number that has both of these as factors is 16 Once you have a common denominator, subtract the numerators. 16 goes into 16 one time 5 1 5 16 x 1 = 16 6 5 1 = 16 16 16 Subtraction - Let’s Try It! 7 8 6 8 - - 1 2 1 2 = = 3 9 8 16 1 5 4 4 - - 3 8 7 16 = = 3 16 13 16 Review • A fraction has a numerator and a denominator • The denominator can never be 0 • You can multiply, divide, add and subtract fractions • A common factor is a number that both denominators are evenly divisible by • A common denominator is a number that both denominators share a factor with