# Decimals and Percentages

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1 Decimals and Percentages Specimen Worksheets for Selected Aspects Paul Harling

2 b recognise the number relationship between coordinates in the first quadrant of related points Key Stage 2 (AT2) on a line or in a shape e.g. the vertices of a rectangle, a graph of the multiples of 3: Number: Programme of Study c consolidate knowledge of addition and Subtraction facts to 20;know the multiplication Facts 10X10; develop a range of mental methods for finding quickly from known facts those that they cannot recall; use some 1. Pupils should be given opportunities to: properties of numbers, including multiples, a develop flexible and effective methods of factors and squares, extending to primes, cubes computation and recording, and use them with and square roots; understanding; d develop a variety of mental methods of b use calculators, computers and a range of computation with whole numbers up to 100, and other resources as tools for exploring number explain patterns used; extend mental methods to structure and to enable work with realistic develop a range of non-calculator methods of data; computation that involve addition and c develop the skills needed for accurate and subtraction of whole numbers. progressing to appropriate use of equipment methods for multiplication and division of up to three-digit by two-digit whole numbers: 2. Developing an understanding of place value e understand multiplication as repeated addition, and extending the number system and division as sharing and repeated Pupils should be taught to: subtraction; use associated language and recognise situations to which the operations a read, write and order whole numbers, apply; understanding that the position of a digit f understand and use the relationships between signifies its value; use their understanding of the four operations, including inverses; place value to develop methods of computation, g extend methods of computation to include to approximate numbers to the nearest 10 or addition and subtraction with negative numbers, 100, and to multiply and divide by powers of 10 all four operations with decimals, and when there are whole-number answers; calculating fractions and percentages of b extend their understanding of the number quantities. using a calculator where appropriate: system to negative numbers in context, h understand and use the features of a basic decimals with no more than two decimal places calculator. interpreting the display in the context in the context of measurement and money: of the problem, including rounding and c understand and use, in context, fractions and percentages to estimate. describe and compare proportions of a whole. remainders. 4. Solving numerical problems Pupils should be taught to: 3. Understanding relationships between a develop their use of the four operations to numbers and developing methods of computation Pupils should be taught to: a explore number sequences, e.g. counting in different sizes of step, doubling and halving, using a multiplication square, explaining patterns and using simple relationships; progress to interpreting, generalising and using simple mappings, e.g. C=15n for the cost of n articles at 15p, relating to numerical, spatial or practical situations, expressed initially in words and then using letters as symbols; solve problems, including those involving money and measures, using a calculator where appropriate: b choose sequences of methods of computation appropriate to a problem, adapt them and apply them accurately; c check results by different methods, including repeating the operations in a different order or using inverse operations; gain a sense of the size of a solution, and estimate and approximate solutions to problems.

3 Level Descriptions with the four operations. including mental recall Level 1 of multiplication facts up to 10x 10. Pupils count, order, add and subtract numbers They add and subtract decimals to two places. when solving problems involving up to 10 In solving problems with or without a calculator. objects. pupils check the reasonableness of their results They read and write the numbers involved, by reference to their knowledge of the context or Pupils recognise and make repeating patterns, to the size of the numbers. counting the number of each object in each They recognise approximate proportions of a repeat. whole and use simple fractions and percentages to describe these. Level 2 Pupils explore and describe number patterns, Pupils count sets of objects reliably, and use and relationships including multiple. factor and mental recall of addition and subtraction facts to square. They have begun to use simple formulae 10. expressed in words. They have begun to understand the place value Pupils use and interpret co-ordinates in the first of each digit in a number and use this to order quadrant. numbers up to 100. They choose the appropriate operation when Level 5 solving addition and subtraction problems. Pupils use their understanding of place value to They identify and use halves and quarters, such multiply and divide whole numbers and as half of a rectangle or a quarter of eight decimals by and objects. They order. add and subtract negative numbers They recognise sequences of numbers, including in context. odd and even numbers. They use all four operations with decimals to two places. Level 3 They calculate fractional or percentage parts of Pupils show understanding of place value in quantities and measurements, using a calculator numbers up to 1000 and use this to make where appropriate. approximations. They have begun to use decimal Pupils understand and use an appropriate nonnotation and to recognise negative numbers, in calculator method for solving problems that contexts such as money, temperature and involve multiplying and dividing any three-digit calculator displays, by an two-digit number. Pupils use mental recall of addition and They check their solutions by applying inverse subtraction facts to 20 in solving problems operations or estimating using approximations. involving larger numbers. They construct, express in symbolic form, and They use mental recall of the 2, 5 and 10 use simple formulae involving one or two multiplication tables, and others up to 5x5, in operations. solving whole-number problems involving multiplication or division, including those that give rise to remainders. Pupils use calculator methods where numbers include several digits. They have begun to develop mental strategies, and use them to find methods for adding and subtracting numbers with at least two digits. Level 4 Pupils use their understanding of place value to multiply and divide whole numbers by 10 or In solving number problems. pupils use a range of mental and written methods of computation Level 6 Pupils order and approximate decimals when solving numerical problems and equations such as x 2 = 20. using trial-and improvement methods Pupils are aware of which number to consider as 100 percent or a whole. In problems involving comparisons and use this to evaluate one number as a fraction or percentage of another. They understand and use the equivalences between fractions. decimals and percentages, and calculate using ratios in appropriate situations. When exploring number patterns, pupils find and describe in words the rule for the next term or nth term of a sequence where the rule is linear They formulate and solve linear equations with whole number coefficients They represent mappings expressed algebraically, interpreting general features and using graphical representation in four quadrants where appropriate

4 One point to you! Matching Decimals

5 First to one Square up

6 On the right line Rounding decimals

8 Shortcuts

9 Copyright: Paul Harling (Maths Plus series: Decimals and Percentages)(WLE) Decimal move.. You need a counter for each person who is playing and 2 minis from a set of M.A.B. Instructions 2 to 4 people may play this game. Take 2 minis from your M.A.B. set. Mark the decimals 0.1 to 0.6 in blue on one of them. Mark the decimals 0.7 to 1.2 in red on the other, Each player takes it in turns to throw both dice. Subtract the decimal on the blue die from the decimal on the red die. Your answer will tell you how far to move on the chart. The first player to reach 10 is the winner.

10 How to handle hundredths

11 Copyright: Paul Harling (Maths) Plus series: Decimals and Percentages)(WLE) Race to the line!. Play this game with a friend. You both need a calculator. Most of the numbers on the grid below are sums of two different numbers from this set. For example, 44 9 is the sum of 3 4 and But watch out! There are two hidden bug numbers that cannot be found by adding listed numbers. To play the game you have to find which numbers from the list are needed to make all five numbers in any 1 row, column or diagonal on the grid. You race your friend to see who can work out a whole line first. So look at the grid carefully and choose which line you are going to work out. Then write down the five calculations as quickly as you can. Record like this = l 5 The first to finish wins, but your partner can challenge your answers!

12 Recurring decimals

13 Percentages You need a dictionary, centimetre squared paper, scissors and glue I Use the dictionary to find the meaning of each of these words. century centenary centurion centenarian 2 On the squared paper draw a square 10 cm by 10 cm. This large square is made up of 100 small squares. Copy this pattern exactly onto your square. Then cut it out and stick it in your book. Now copy these sentences underneath the pattern. 4 squares out of 100 are black. 4 / 100 is black. 4 per cent is black. 20 squares out of 100 are grey. 20 / 100 is grey. 20% is grey. 12 squares out of 100 are blue. 12 / 100 is blue. 12%is blue. 3 Look back to your pattern. a How many squares are not coloured? b What fraction is this? c What percentage is not coloured? 4 Write each of these percentages in decimal form. a 4% b 20% c 12%

14 Working percentages Collect advertisements that include percentages. They will have phrases and sentences like these

15 Pocket-money percentages Carefully trace this percentage wheel. Each sector (part) is 10% of the full circle. The full circle is 100%. These wheels show how six teenagers use their pocket money. Each gets 1 5O a day. I For each person, use the percentage wheel to work out:

16 a what percentage of the 1 50 is spent on each item. b how much money is spent on each item. 2 If each person s pocket money went up to 2 a day, what would happen to: a the percentage spent on each item? b the amount of money spent on each item? 3 Imagine you have 5. How would you use it? Draw and label a diagram to show what you would buy, and the percentage and amount of money you would spend on each item. ercentage puzzle Play this game with a friend. You each need a calculator with a % key, and a set of counters. (The two sets of counters should be different in colour.) Rules Take turns to: Choose a percentage and a number from these sets. Work out the chosen percentage of the chosen number. (Look back to page 18 if you need help.) Find the result of the calculation on the grid below and cover it with a counter. The first player to get a line of four counters wins.

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