# SOLVING PROBLEMS. As outcomes, Year 4 pupils should, for example: Pupils should be taught to:

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1 SOLVING PROBLEMS Pupils should be taught to: Choose and use appropriate number operations and appropriate ways of calculating (mental, mental with jottings, written methods, calculator) to solve problems As outcomes, Year 4 pupils should, for example: Use, read and write: operation, sign, symbol, number sentence, equation Make and justify decisions: choose the appropriate operation(s) to solve word problems and number puzzles; decide whether calculations can be done mentally or with pencil and paper; explain and record how the problem was solved. For examples of problems see sections on: puzzles (page 78), real life (page 82), money (page 84), measures (page 86) and time (pages 88 and 100). Make up number stories to reflect statements like: = = = = 31 Four portions of fries at 90p each cost 360p or 3.60 altogether. Recognise the operation represented by the in examples such as: = = = = 14 Look at a set of subtractions of different pairs of numbers. Discuss which is easiest/hardest to do and justify why. 74

2 Making decisions As outcomes, Year 5 pupils should, for example: As outcomes, Year 6 pupils should, for example: Use, read and write, spelling correctly: operation, sign, symbol, number sentence, equation Use, read and write, spelling correctly: operation, sign, symbol, number sentence, equation... Make and justify decisions: choose the appropriate operation(s) to solve word problems and number puzzles; decide whether calculations can be done mentally or with pencil and paper or a calculator; explain and record how the problem was solved. Make decisions: choose the appropriate operation(s) to solve word problems and number puzzles; decide whether calculations can be done mentally or with pencil and paper or a calculator; explain and record how the problem was solved. For examples of problems see sections on: puzzles (page 79), real life (page 83), money (page 85), measures (page 87) and time (pages 89 and 101). For examples of problems see sections on: puzzles (page 79), real life (page 83), money (page 85), measures (page 87) and time (pages 89 and 101). Make up number stories to reflect statements like: = = = = 70.5 If 8 equal pieces are cut from 564 mm of string, each piece is 70.5 mm long. Make up number stories to reflect statements like: = = = = compact discs at 6.83 each will cost Recognise the operation represented by the in examples such as: = = = = 6 Recognise the operation represented by the in examples such as: = = = = Look at multiplications of different pairs of numbers. Discuss which is easiest/hardest to do and justify why. Look at divisions of different pairs of numbers. Discuss which is easiest/hardest to do and justify why. 75

3 SOLVING PROBLEMS Pupils should be taught to: Explain methods and reasoning about numbers orally and in writing As outcomes, Year 4 pupils should, for example: Explain calculations that have wholly or partly been done mentally, beginning to use conventional notation and vocabulary to record the explanation Add 17 and 3 to get 20, then 20 more to get = 32, so is the same, so is is = = = = 16 and = 90, so is 90, 100, is 35, but this takes away 2 too many, so add back 2 to make There are three hundreds and 90 and 7, that s Take away 200, that s 300, add 20 is Half of 80 is 40, and half of 7 is 3.5, so it s Extend to calculations that cannot entirely be done mentally = = = 612 See also pencil and paper procedures for: addition (page 48), subtraction (page 50), multiplication (page 66) and division (page 68). 76

5 SOLVING PROBLEMS Pupils should be taught to: Solve mathematical problems or puzzles, recognise and explain patterns and relationships, generalise and predict. Suggest extensions by asking What if...? As outcomes, Year 4 pupils should, for example: Solve puzzles and problems such as: Find three consecutive numbers which add up to 39. What other numbers up to 50 can you make by adding three consecutive numbers? Find a pair of numbers with: a sum of 11 and a product of 24; a sum of 40 and a product of 400; a sum of 15 and a product of cubes can be arranged to make a cuboid. What other cuboids can you make with 72 cubes? You can make 6 by using each of the digits 1, 2, 3 and 4 once, and any operation: for example, 6 = (21 + 3) 4 or 6 = (3 4) (1 2) Use each of the digits 1, 2, 3 and 4 and any operation to make each number from 1 to 40. Can you go further? Arrange the numbers 1, 2, 3... to 9 in the circles so that each side of the square adds up to 12. Draw three rings. Use each of the numbers from 1 to 9. Write them in the rings so that each ring has a total of 15. Find different ways to do it. Use a computer program to solve number puzzles: for example, to fill a given number of carriages on a train with given numbers of people. Each represents a missing digit. a. Choose three digits from this set: 1, 3, 4, 8. Replace each to make this statement true: = 38 b. Find the missing digits = = = 120 c. Find different ways of completing: = 252 Count all the triangles in this diagram (11). Start with a rectangular sheet of paper. By folding, making one straight cut, and then unfolding, make this hexagon. 78

6 Reasoning about numbers and shapes As outcomes, Year 5 pupils should, for example: Solve puzzles and problems such as: Find: two consecutive numbers with a product of 182; three consecutive numbers with a total of 333. As outcomes, Year 6 pupils should, for example: Solve puzzles and problems such as: Find: two consecutive numbers with a product of 1332; two numbers with a product of 899. Choose any four numbers from the grid. Add them up. Find as many ways as possible of making Complete this multiplication table Each represents one of the digits 1 to 6. Use each of the digits 1 to 6 once. Replace each to make a correct product. Use a binary tree computer program to sort a set of numbers according to their properties. Write a number in each circle so that the number in each square box equals the sum of the two numbers on either side of it. 13 = A number sequence is made from counters. There are 7 counters in the third number. (1) (2) (3) How many counters in the 6th number? the 20th...? Write a formula for the number of counters in the nth number in the sequence. With 12 squares you can make 3 different rectangles. Find how many squares can be rearranged to make exactly 5 different rectangles. A two-digit number is an odd multiple of 9. When its digits are multiplied, the result is also a multiple of 9. What is the number? Find ways to complete: + + = 1 Each represents a missing digit. Use your calculator to solve: 6 = 6272 A pentomino is a shape made from five identical squares touching edge to edge. Divide this shape into two pentominoes. Do it in four different ways. For how many three-digit numbers does the sum of the digits equal 25? Each letter from A to G is a code for one of these digits: 1, 3, 4, 5, 6, 8, 9. Crack the code. A + A = B A A = DF A + C = DE C + C = DB C C = BD A C = EF Use a computer program to investigate and generalise a number relationship: for example, the number of times a bouncing ball will touch the sides of a billiard table. Use a calculator to solve these. a. Each represents a missing digit. Solve: 2 = b. One whole number divided by another gives What are the two numbers? This is half a shape. Count all the rectangles in this diagram (26). Sketch some of the different whole shapes the original could have been. Mark any lines of symmetry of these shapes. Using straight cuts, divide a square into 6 smaller squares. 79

7 SOLVING PROBLEMS Pupils should be taught to: Make and investigate a general statement about familiar numbers or shapes by finding examples that satisfy it As outcomes, Year 4 pupils should, for example: Find examples that match a general statement. For example, explain and start to make general statements like: The sum of three odd numbers is odd. Examples: = = 633 If 14 < < 17, then any number between 14 and 17 can go in the box. Examples: 16, 14.5, Half way between any two multiples of 10 is a multiple of 5. Examples: 90 and 120 are both multiples of 10; half way between them is 105, which is a multiple of 5. Multiples of 4 end in 0, 2, 4, 6 or 8. Examples: 12, 64, 96, 108, 6760 Any odd number is double a number add 1. Example: 63 = If I multiply a whole number by 10, every digit moves one place to the left. Examples: = = = 3660 The perimeter of a rectangle is twice the length plus twice the breadth. Example: The perimeter of a 5 cm 3 cm rectangle is: 5 cm + 3 cm + 5 cm + 3 cm = 16 cm. This is the same as 5 cm 2 add 3 cm 2. The number of lines of reflective symmetry in a regular polygon is equal to the number of sides of the polygon. Example: a regular hexagon has 6 sides and 6 lines of symmetry. Start to express a relationship orally in words. Explain how to find the number of days in any number of weeks. Explain how to find the change from 1 after buying two first class stamps. Describe a short way to work out the perimeter of a rectangle. The rule is add 4. Start with 0. Explain how to find the first five numbers in the sequence. What would the 10th number be? A sequence starts 1, 4, 7, 10, Explain in words the rule for the sequence. 80

9 SOLVING PROBLEMS Pupils should be taught to: Use all four operations to solve word problems involving numbers in real life As outcomes, Year 4 pupils should, for example: Solve story problems about numbers in real life, choosing the appropriate operation and method of calculation. Explain and record using numbers, signs and symbols how the problem was solved. Examples of problems Single-step operations I think of a number, then subtract 18. The answer is 26. What was my number? A beetle has 6 legs. How many legs have 9 beetles? How many legs have 15 beetles? Kate has 38 toy cars. John has half as many. How many toy cars has John? A box holds 70 biscuits. How many biscuits are left if you eat 17 biscuits? How many people can have 5 biscuits each? How many biscuits are there in 6 boxes? How many boxes are needed to hold 200 biscuits? To cook rice, you need 5 cups of water for every cup of rice. You cook 3 cups of rice. How many cups of water do you need? Multi-step operations There are 129 books on the top shelf. There are 87 books on the bottom shelf. I remove 60 of the books. How many books are left on the shelves? There are 4 stacks of plates. 3 stacks have 15 plates each. 1 stack has 5 plates. How many plates altogether? I think of a number, add 2, then multiply by 3. The answer is 15. What was my number? There are 36 children in the class. Half of them have flavoured crisps. One third of them have plain crisps. How many children have crisps? See also problems involving money (page 84), measures (page 86), time (pages 88 and 100), and puzzles (page 78). 82

11 SOLVING PROBLEMS Pupils should be taught to: Use all four operations to solve word problems involving money As outcomes, Year 4 pupils should, for example: Use, read and write: money, coin, pound,, pence, note, price, cost, cheaper, more expensive, pay, change, total, value, amount... Solve problems involving money, choosing the appropriate operation. Explain and record how the problem was solved. Shopping problems What is the total cost of a 4.70 book and a 6.10 game? It costs 80p for a child to swim. How much does it cost for 6 children to swim? A jigsaw costs 65p. How many can you buy for 2? How much change do you get? A CD costs 4. Parveen saves 40p a week. How many weeks must she save to buy the CD? Lauren has three 50p coins and three 20p coins. She pays 90p for a Big Dipper ride. How much does she have left? Dad bought 3 tins of paint at 5.68 each. What was his change from 20? Peter offered two silver coins to pay for a 14p pencil. Investigate how much change he got. A chocolate bar costs 19p. How many bars can be bought for 5? For her party Asmat spent: 2.88 on apples 3.38 on bananas 3.76 on oranges Will a 10 note cover the cost? Explain your reasoning. Converting pounds to pence and vice versa How many pence is: ? Write in pounds: 356p p p... Calculating fractions Harry spent one quarter of his savings on a book. What did the book cost if he saved: ? Gran gave me 8 of my 10 birthday money. What fraction of my birthday money did Gran give me? 84

12 Problems involving money As outcomes, Year 5 pupils should, for example: Use, read and write, spelling correctly, the vocabulary of the previous year, and extend to: discount Solve problems involving money, choosing the appropriate operation. Explain and record how the problem was solved. Shopping problems Find the total of: 9.63, and 3.72; 66p, 98p, 48p and How much does one of each cost? 4 for for for 3.24 What change do you get from 20 for 13.68? Kobi saved 15p a week for one year. How many pounds did he save? Four people paid 72 for football tickets. What was the cost of each ticket? How much change did they get from 100? Petrol costs 64.2p per litre. What do you pay to fill a 5 litre can? Which amounts up to 1 cannot be paid exactly with fewer than six coins? You have four 35p and four 25p stamps. Find all the different amounts you could stick on a parcel. Take 1 worth of coins: four 1p, three 2p, four 5p, three 10p and two 20p. Find all the different ways of using the coins to pay 50p exactly. As outcomes, Year 6 pupils should, for example: Use, read and write, spelling correctly, the vocabulary of the previous year. Solve problems involving money, choosing the appropriate operation. Explain and record how the problem was solved. Shopping problems What is the total of , 3.43 and 11.07? How much does one of each cost? 10 for for for 1.55 Find the cost of 145 bottles of lemonade at 21p each. What change do you get from 50? Things at half price now cost: What was the original price of each item? Three people won on the lottery to be shared equally between them. How much does each one get? Costs of rides are: Galaxy 1.65 Laser 2.80 Big wheel 1.45 Spaceship 2.70 Amy went on two rides. She had 5.65 change from 10. Which two rides did she go on? Use a calculator or a written method people go to a football match. Each ticket costs What is the total cost of all the tickets? Converting foreign currency Exchange rates for 1 are: 1.6 US dollars 8.7 French francs 220 Spanish pesetas How many dollars, francs, pesetas do you get for 5? Converting to European or foreign currency There are 1.43 euros to 1. What is the price in pounds of a car costing euros? Use a calculator or a written method. There are 2560 lira to 1. Find the price in lira of a house costing Calculating fractions and percentages The deposit on a 230 chair is 50%. How much is the deposit? There is 25% off prices in a sale. How much do you get off ? Calculating fractions and percentages The agent s fee for selling a house is 5%. Calculate the fee on a house sold for Use a calculator or a written method. There is a 15% discount in a sale. How much is the discount on ? 85

13 SOLVING PROBLEMS Pupils should be taught to: Use all four operations to solve word problems involving length, mass or capacity As outcomes, Year 4 pupils should, for example: Solve story problems involving: kilometres, metres, centimetres, millimetres... kilograms, half kilograms, grams litres, half litres, millilitres... miles, pints... and explain and record how the problem was solved. Measure the lengths of these lines to the nearest mm. Two shelves are 75 cm and 87 cm long. What is their total length in metres? What is the difference in their lengths in centimetres? A family sets off to drive 524 miles. After 267 miles, how much further do they have to go? A potato weighs about 250 g. Roughly how much do 10 potatoes weigh? How many times heavier is a 1 kg potato? A bottle of salad dressing holds 300 millilitres. A tablespoon holds 15 millilitres. How many tablespoons of dressing are in the bottle? A full jug holds 2 litres. A full glass holds 1 4 of a litre. How many glasses full of water will the jug fill? Change this recipe for ginger nuts for 6 people to a recipe for 12 people for a party. 125 g flour 50 g fat 75 g sugar 30 ml treacle 1 teaspoon ground ginger Each side of a regular hexagon is 14 centimetres long. How long is its perimeter? See also problems involving numbers in real life (page 82), money (page 84), time (pages 88 and 100), and puzzles (page 78). 86

15 SOLVING PROBLEMS Pupils should be taught to: Use all four operations to solve word problems involving time As outcomes, Year 4 pupils should, for example: Solve story problems involving units of time, and explain and record how the problem was solved. Raiza got into the pool at 2:26. She swam until 3 o clock. How long did she swim? The cake went in the oven at 1:20. It cooked for 75 minutes. What time did it come out? Lunch takes 40 minutes. It ends at 1:10 pm. What time does it start? Mary got up at 7:35. She left for school 45 minutes later. Her journey took 15 minutes. What time did she arrive at school? The football team kicked off at 1:30 pm. They played 45 minutes each way. They had a 10 minute break at half time. At what time did the game finish? Jan went swimming on Wednesday, 14 January. She went swimming again 4 weeks later. On what date did she go swimming the second time? The swimming pool shut for repairs on Friday, 20 March. It opened again on Friday, 10 April. For how many weeks was the swimming pool shut? See also using timetables (page 100), problems involving numbers in real life (page 82), money (page 84), measures (page 86), and puzzles (page 78). 88

16 Problems involving time As outcomes, Year 5 pupils should, for example: Solve story problems involving units of time, and explain and record how the problem was solved. The car race began at 08:45 and finished at 14:35. How long did the race last? The sun sets at 19:30 and rises again at 06:30. How many hours of darkness? Of daylight? A train leaves at 09:45 h and arrives at 15:46 h. How long does the journey last? These are the start and stop times on a video cassette recorder. START 14:45 STOP 17:25 As outcomes, Year 6 pupils should, for example: Solve story problems involving units of time, and explain and record how the problem was solved. Lamb must be cooked for 60 minutes for every kg. Chicken must be cooked for 50 minutes for every kg. Complete this table of cooking times. kilograms Cooking time in minutes (lamb) Cooking time in minutes (chicken) For how long was the video recording? Four children in a relay team swim in a race. Here are their times for each lap. LAP 1 Craig 92.4 seconds LAP 2 Fiona 86.3 seconds LAP 3 Harun 85.1 seconds LAP 4 Jenny 91.8 seconds What is their total time for the four laps? See also using timetables (page 101), problems involving numbers in real life (page 83), money (page 85), measures (page 87), and puzzles (page 79). See also using timetables (page 101), problems involving numbers in real life (page 83), money (page 85), measures (page 87), and puzzles (page 79). 89

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