Number and Place Value

Transcription

1 Maths Revision

2 Number and Place Value Use numbers from -000 to (The curriculum is not specific about how far to use negative numbers) Counting Count forwards and backwards in, 6, 7,, 9,, 0, steps of powers of 0 (0, 00, 000, ) 7,,,,,, 6, 600, 7, 0,, 7, 67, 77, 7, Find 0, 00 or 000 more or less than a given number. What is 00 less than 90? What is 000 more than 9? Count forwards and backwards through zero 6,,,,,, 0, -, -, - Negative numbers Use negative numbers in context and calculate intervals across zero. The temperature inside is 9 C and outside is - C. What is the difference in temperature between inside and outside? Place Value Recognise the place value of each digit in up to four-digit numbers. hundred thousands ten thousands thousands hundreds tens ones Compare and Order Numbers Compare using <, > or = 9 < 0 > 0 Smallest Greatest Page of 6

3 Identify, Represent and Estimate Use models and representations of numbers. 0 can be represented by Rounding Round numbers to the nearest 0, 00, 000, or and any whole number. Remember rounds up. 00 rounded to the nearest 000 is 000 (the 00 rounds up). 0 to the nearest is (the 0 rounds down). 7 rounded to the nearest 0 is 0 (7 is nearer to 0 than 00). Read and Write Numbers in Numerals and Words is three hundred and forty-four thousand, two hundred and eighty-five. Roman Numerals Use the following Roman numerals to represent numbers to 00: Roman Numeral I V X 0 L 0 C 00 D 00 M 000 CCXIX = 9 DCXXVI = 66 CMXLVIII = 9 MDCCCLXXI = 7 Solve Problems Here are years written in Roman Numerals. Order the years from earliest to latest: MMIX MCMXCIX MMXV MCMCXIX (999) MMIX (009) MMXV (0) Page of 6

4 Addition and Subtraction Add and Subtract Mentally Add and subtract three-digit numbers and ones, tens and hundreds = = = 76 Mental Methods Add and subtract numbers mentally with larger numbers = 7 Estimate, Round, Levels of Accuracy and Inverse Estimate by rounding to check accuracy: Inverse: check 79 7 = 7, by = 79 Multiplication Tables Multiplication and division facts to x Page of 6

5 Multiplying and Dividing Use place value and known facts: 00 = 000, 60 7 = 90 Multiply by 0 and and divide by : =, 0 = 0, = Multiplying and dividing whole numbers and decimals by 0, 00 and 000: When multiplying or dividing a number by 0, 00 or 000, keep the digits in the number together. When multiplying the number gets larger and when dividing the number gets smaller. The numbers will move in place value by the number of 0 s. 0 = = = = = = 0. Common Multiples, Factor Pairs, Common Factors and Commutativity is a common multiple of and 6, because is a multiple of and a multiple of 6. All the factor pairs of 6 are and 6, and, and, and 7. Use this to solve: 6 pencils are shared between tables. How many pencils does each table receive? The common factors of and 6 are,, and because they are factors of both and 6. Commutativity means changing the order of the numbers in a calculation does not change the answer: 9 = 9 = 0 9 = 90 Prime Numbers Prime numbers only have and itself as factors. Prime factors are factors of a number that are prime numbers: the prime factors of are and 7, the prime factors of are and. 6 Composite numbers are non-prime numbers: is a composite number because is a factor. Recall the prime numbers to 9:,,, 7,,, 7 and 9 Square and Cube Numbers The square numbers are,, 9, 6,, 6, 9, 6,, 00,,, 69, 96,, e.g. = = 9 7 = 7 7 = 9 The cube numbers are,, 7, 6,, e.g. = = = = Page of 6

6 Order of Operations BODMAS is a way of remembering the order in which operations are carried out. Brackets first: ( + ) = 9 = 7 Order - square or cube: + = + 9 = Division and Multiplication: + = + 6 = 0 Addition and Subtraction: (as in examples above) Division and multiplication are carried out in the order they are in the expression. Addition and subtraction are carried out in the order they are in the expression. Formal Methods becomes Answer: becomes 9-7 Answer: becomes Answer: Use formal methods to multiply up to digit numbers by digit numbers. 7 becomes 7 becomes 7 6 becomes 7 0 Answer: Answer: Answer: 6 Use the formal long multiplication method for multiplying digit numbers by digit numbers. 7 becomes 7 becomes Answer: Answer: 7 Page of 6

7 Use short division for up to digit numbers divided by one-digit numbers. 7 becomes Answer: 9 7 becomes r Answer: 97 r Use long division for up to digit numbers divided by two-digit numbers. Express remainders as whole numbers, fractions or decimals. 6 becomes r Answer: r 6 6 becomes = 6 Answer: 0 6 becomes Answer:. Solve Problems Multi-step problems people visit a cinema on one day. There are two films showing. 9 adults and 96 children see an adventure film, 6 adults and a number of children see an animation. How many more children see the animation than the adventure film? = adults = 69 children = 69 children see the animation = 77 more children see the animation than the adventure film Page 6 of 6

8 Using knowledge of factors, prime numbers, square and cube numbers, explain why the numbers to 0 are placed in this Venn Diagram in this way. Prime Numbers Square Numbers Composite Numbers Missing number problems: = or 6 = Word Problems A teacher has four new boxes of pencils, each with pencils, and a tray with 7 pencils. The teacher shares equally all the pencils between tables. How many pencils does each table receive? = new pencils + 7 = pencils = 7 pencils per table Scaling Problems with Simple Fractions pizzas are cut into quarters. How many quarters of pizza will there be altogether? Correspondence Problems Jenna has t-shirts and pairs of shorts. How many different combinations of the t-shirts and shorts does Jenna have? 0 pencils are shared equally between classes. How many pencils will they each receive? Using the Distributive Law 9 7 = = = 7 Page 7 of 6

9 Tenths Fractions Counting:,,,,... Dividing into 0 equal parts or by Counting:,,,,... Dividing into 00 equal parts or by 00 or tenths by 0 There are 0 thousandths in one hundredth and 00 thousandths in one tenth. Fraction of a Set of Marbles Find of these marbles = = 0 Equivalent Fractions = 6 = 6 Page of 6

10 Write fractions that are equivalent to Use common factors to simplify fractions 9 = 9 and have as a common factor Expressing Fractions with the Same Denominator Use common multiples and 0 is the smallest common multiple of and becomes 0 becomes 0 Page 9 of 6

11 Mixed Numbers and Improper Fractions Mixed fraction (whole number and fraction) = Improper fraction (numerator is larger than the denominator) Add and Subtract Fractions with the Same Denominator and with Denominators that are Multiples, and with Different Denominators and Mixed Numbers Add or subtract the numerator, keeping the denominator the same. The answer can be expressed as an equivalent fraction. + = - = = = If the denominators are different, convert the fractions to equivalent fractions with the same denominator before adding or subtracting. + = + = = + = = 7 0 Compare and Order Unit fractions smallest greatest 6 Fractions with the same denominator < Fractions with denominators that are multiples > Page 0 of 6

12 Multiply Fractions Multiply proper fractions and mixed numbers by whole numbers. Proper fractions - multiply the numerator by the whole number: = = 0 Mixed numbers multiply the whole numbers and add the product of the fraction and whole number: = 6 + = 6 + = 6 Divide Fractions Divide proper fractions by whole numbers multiply the denominator by the whole number: = Decimal Equivalents Tenths and hundredths: 7 0 = = 0. = 0. = 0. = 0.7 Write decimals as a fraction: 0.67 = Calculate decimal fraction equivalents: = 0.7 because = 0.7 Decimal Place Value tenth hundredth thousandth 0.9 Page of 6

13 Multiplication and Division By 0, 00 and 000: 0. 0 = 00 = = 000 = 0.0 Multiply one-digit numbers up to decimal places by whole numbers = = 9 Rounding Decimals To the nearest whole number: 0. rounds to because the rounds up. rounds to because the rounds down (ignoring the ) To one decimal place: 0.0 rounds to 0. because the rounds up. Read, Write, Order and Compare Decimals 0. is zero point four five (not zero point forty-five) 0. < > 0.6 Percentages % means out of % = = % = 00 Solve Problems Fractions Adil divides his marbles into tenths. He wants to give two friends a number of the tenths of his marbles and keep the rest himself. Write ways that he could share the marbles. 0 0 eg Page of 6

14 Measure and Money Problems Ellie buys a new shirt for.7 and a pair of trousers for.0 in a sale. She pays with a 0 note. What change will she receive? A bag of potatoes weigh.kg. How much will bags weigh? Decimal Problems to Decimal Places A packet of sugar weighs.kg. kg is used to bake some cakes. How much will the packet weigh now?.kg 0.7kg = 0.9kg Knowing Percentage and Decimal Equivalents of,,,,,, Order the following from smallest to largest: %,, 0. * 0 * Page of 6

15 Ratio and Proportion Use Multiplication and Division Facts children share 6 pizzas. If more children join the group and each child is to have the same amount of pizza, how many more pizzas are needed? more pizzas Percentages Which is greater? % of litres 0% of 00ml Scaled Shapes The length and width of rectangle A are increased by a scale factor of to make rectangle B. What are the new dimensions of rectangle B? cm A cm B Use Fractions and Multiples A child has read 0 pages of a book and has How many pages are there left to read? of the book has been read which is 0 pages of the book is pages to read. of the book is 7 pages. There are 7 pages left to read. Page of 6

16 Algebra Formulae s + = t, if s =, what is t? t = + = Formulae are used in mathematics and science: Area of a rectangle: a = lw Perimeter of a rectangle: p = (l + w) (a = area, l = length and w = width) (p = perimeter) cm cm What is the area and perimeter of this rectangle? Express missing number problems algebraically. A number (g) is more than a number (h) so g = h + or h = g A locksmith charges callout and 0 per hour for any work. What formulae would calculate his charge for h number of hours? 0h + Sequences Linear number sequences:, 7,, The sequence starts with and each step is so the st term is =, the nd term is = 7, the rd is = therefore the nth term is n Equations Find possible pairs of numbers for a and b in a + b =. Some possible answers: a =, b = 9; a =, b = 6; a =, b = Variables The total of two numbers is. Both numbers are between and 0. Find all the possible combinations =, 7 + = Page of 6

17 Measurement Estimate, Measure, Compare, Add and Subtract In all cases, be able to estimate with some accuracy prior to measuring. Lengths (mm/cm/m) Measure and draw lines using a ruler in centimetres (cm) or millimetres (mm). This line is 9.cm or 9mm long. Mass (g/kg) Measure the mass of objects using different scales. apples weigh g. One is eaten, and the remaining apples weigh g. What is the mass of the eaten apple? g g = 0g Capacity (ml/l) Which jug has more water? 7ml 90ml Convert between units Length: km = 000m, m = 00cm or 000mm. cm = 0mm Mass: kg = 000g Capacity/ Volume: l = 000ml Time: year = 6 days (leap year 66 days), week = 7 days, 0 days hath September, April, June and November. All the rest have, Excepting February alone Which only has but days clear And 9 in each leap year. day = hours, hour = 60 minutes, minute = 60 seconds Page 6 of 6

18 Convert between metric and imperial units inch.cm miles km kg.lb (pounds) litre.7 pints A road sign says Sheffield miles. How many kilometres is it to Sheffield? 7km Perimeter, Area and Volume The perimeter is the measurement around the edge of a shape. The sides of this rectangle are cm and cm, so the perimeter is cm. Measure and calculate the perimeter of rectilinear shapes (including squares). cm cm Perimeter = cm 6cm cm Page 7 of 6

19 Area Area of rectilinear shapes by counting squares Area = 66cm Calculate the area of rectangles: multiply the length of two adjacent sides. cm Area = cm cm = cm cm Estimate the area of irregular shapes: Count the whole squares and the squares with more than half included in the shape. Page of 6

20 Shapes with the same area can have different perimeters. cm 6cm A cm B cm Find the area and perimeter of these rectangles. What do you notice? The area of both is cm, but the perimeter of A is cm and B is 6cm, so they are different. The area of a triangle is based on it being half of a rectangle that includes the triangle. h b The area of a triangle is half of the base (b) the height (h) or bh A similar idea is used to find the area of a parallelogram. Cut a triangle of one end and move to the other and the parallelogram becomes a rectangle. The area of a parallelogram is the base (b) the height (h) or bh. Page 9 of 6

21 Money Add and subtract giving change. Jude buys an apple and an orange costing p and p. How much change from 0p? Time Analogue clocks and / hour time. These clocks show quarter to nine: Record time in hours, minutes and seconds. The maths lesson lasted hour and minutes. The art lesson was one hour and twenty minutes. The art lesson was longer than the maths lesson. Morning is am, afternoon is pm A film lasts 6 minutes. How long is the film in hours and minutes? hours and 6 minutes Solve Problems equal bottles of water contain 00ml of drink. How many litres will 7 bottles hold? bottles hold 00ml, bottle will hold 0ml = 0.l 7 bottles will hold 0.l 7 =.7l A 6.kg bag of soil is divided into 0 pots equally. Each pot needs 0.kg. How much more does each pot need? 6. 0 = = 0.7kg is needed by each pot Page 0 of 6

22 Geometry Shape D Shapes Main shapes: circle, triangle, quadrilateral, square, rectangle, rhombus, parallelogram, pentagon, hexagon, octagon, decagon circle triangle quadrilateral square rectangle rhombus parallelogram pentagon hexagon octagon decagon Draw D shapes using given dimensions and angles. Draw a square with sides cm. Draw an isosceles triangle with one side of cm and sides of 7cm. Page of 6

23 Compare and classify shapes: Has at least one right angle Has sides The shapes are classified in this Venn diagram. Triangles Equilateral (all sides and angles equal) Isosceles ( sides and angles equal) Scalene (no sides and angles equal) Right-angled triangle (one angle a right angle) Page of 6

24 D Shapes Main shapes: sphere, cylinder, cube, cuboid, tetrahedron, square-based pyramid, triangular prism, pentagonal prism, hexagonal prism sphere cylinder cube cuboid tetrahedron square-based pyramid triangular prism pentagonal prism hexagonal prism Recognise, describe and build simple D shapes, including making nets. What shape is made from this net? Angles An angle measures a turn An acute angle is less than a right angle (90 ) A right angle is the corner of a square An obtuse angle is between a right angle and a straight line right angles make a straight line Page of 6

25 Draw and Measure Angles One of the lines must be on the 0. Read the scale on the other line round from 0. The answer is 7 The angles at a point and whole turn total 60 (four right angles). 0 0 Angles at a point on a line total One right angle = 90 Two right angles = 0 Three right angles = 70 Page of 6

26 Angles in a triangle add up to 0. 0 What is the unknown angle? Angles in a quadrilateral add up to What are the unknown angles in this parallelogram? Lines Horizontal Vertical Parallel Lines Perpendicular lines (at a right angle) Page of 6

27 Symmetry Identify lines of symmetry: Complete a symmetrical figure: Page 6 of 6

28 Regular and Irregular Polygons Regular polygons have equal sides and equal angles. square regular hexagon Irregular polygons do not have equal sides and angles. They may have equal angles or equal sides but not both. A rhombus has equal sides and a rectangle has equal angles but they are not regular (unless they are a square). rhombus rectangle irregular quadrilateral Angles in other regular polygons are: 7 60 pentagon hexagon octagon Circles The main parts of a circle are: radius circumference diameter The circumference is the distance around the perimeter of the circle. The radius is the distance from the centre to the circumference. The diameter is the distance from the circumference to the circumference on the other side through the centre of the circle. The diameter is double the radius. Page 7 of 6

29 Geometry Position and Direction Coordinates Coordinates in all four quadrants. y 7 6 A The coordinates are: A (, 7), B (-6, ) C (-, -7) B x C -7 - Some coordinates grids are drawn without squares. Work out the coordinates of points a and b. Compare the coordinates of the triangles to find the answer. y a a = (-, 7) b (0,) b = (-, ) (-,-) x (-6,-) (-,-) * not to scale Page of 6

30 Translation Translate shapes on a coordinates grid. Translate this triangle so point A translates to point B. y 7 B x - A Translations can also be on blank grids as in the coordinates section above. Page 9 of 6

31 Reflection Reflect shapes on a coordinates grid. Reflect this triangle about the y-axis x Reflections can also be on blank grids as in the coordinates section above. Page 0 of 6

32 Statistics Present data in these graphs and tables and solve problems: Pictograms Favourite Colour How many children chose their favourite colour? Bar Charts How many more children chose cheese and onion as their favourite crisps than ready salted? 0 Page of 6

33 Continuous data can have any value usually a measurement. The Height of Children How many children are shorter than m? (Add the first bars) Tables Here is a table of the chocolate bars sold to customers in a shop over days. Monday Tuesday Wednesday Thursday Saturn Twin 0 Stars 0 Cluster Treasure 0 Tiger 6 Plimmy Which chocolate bar is the most popular? Tiger Page of 6

34 Time Graphs Time graphs show the changing of data over time. These often take the form of line graphs but can also be a bar chart. Number of Children Who Have a School Meal How many school meals were served during the week? Line Graphs Length of a Shadow In which hour was the largest change in the length of the shadow? Page of 6

35 Timetables Train timetable from London to Newcastle Destination Journey A Journey B Journey C London 0:0 :0 6:0 Derby :0 :00 Sheffield :0 :0 :0 Hull :0 : 9: Newcastle : :0 Which train takes the least time to get from London to Hull? Pie Charts Pie charts show data by dividing a circle to represent the different proportions of the data. A class of children chose their favourite flavour of crisps. Here is a pie chart of the results. Salt and Vinegar Cheese and Onion Ready Salted In questions about pie charts children have to use the proportion of the pie to work out answers. In this pie chart, 0 children are asked how they travel to school. Estimate how many children travelled by bus. % of the children travelled by bus and bicycle, which is children. Estimate that bicycle represents child and bus represents. Mean The mean of a set of data is equivalent to sharing the data out equally. If test scores are,, 6,, the mean is found by adding the data ( = ) and then sharing between the scores by dividing by ( = 6). What is the mean of, 7, 0,,? Page of 6

36 Important Vocabulary This list is not exhaustive. Some vocabulary is described above. Vocabulary D shapes D shapes Algebra Analogue Area Calculation Capacity Commutativity Composite Number Meaning Flat shapes with no thickness. In theory a D shape cannot be picked up, but in practice shapes made of paper are counted as D. (A list of shapes is included in the section on shape.) A shape with dimensions that can be picked up. (A list of shapes is included in the section on shape.) Letters or symbols used in the place of numbers. A clock face with hands. The amount of space taken up by a shape. The working out of an answer using addition, subtraction, multiplication or division. How much a container holds. The answer is the same no matter which way the calculation is completed: e.g. + = + or =. A number that has more than factors. ( is not a composite number because it only has factor.) Cube number The result of multiplying a whole number by itself twice: e.g. = Denominator The bottom part of a fraction. Digit A single symbol used to make a numeral: 7 (All numbers are made from the ten digits 0,,,,,, 6, 7,, 9, 0.) Digital Discrete Equation A clock using digits to tell the time. A whole number of a set of objects. A statement where the value of each mathematical expression is equal: e.g. + = 7 Equivalent fraction A fraction which has the same value but is divided into a different number of parts: e.g. = Factor Factor pairs Fraction A factor of a number is a number into which the number can be divided with no remainders: e.g. the factors of are,,, and. Factor pairs are factors that are multiplied together to make the number: e.g. the factor pairs of are and, and. A number expressed as the number of parts into which the whole has been divided: e.g represents parts out of. Page of 6

37 Improper fraction A fraction where the numerator is larger than the denominator: e.g Integer A whole number with no parts: e.g.,, 09. A whole number with no fraction or decimal part: e.g. 6 or 7. Inverse Mass An inverse operation is the opposite or reverse of an operation: e.g. the inverse of 6 = is + = 6 or the inverse of 6 = is = 6. Often known as weight how much matter is in an object. Mixed number A whole number and a proper fraction: e.g. Numeral Numerator Perimeter Place value Polygon A symbol, symbols, word or words that stand for a number: 7 or thirty-seven. The top part of a fraction. The measurement around an object. The value of each digit in any number: In 7 the represents tens. A D shape with any number of sides. Prime factor A factor which is a prime number: e.g. is a prime factor of. Prime Number Proper fraction Quadrant Quadrilateral Rectilinear A number that only has factors: and itself. ( is not a prime number because it only has factor.) A fraction where the numerator is smaller than the denominator: e.g A quarter of the space represented by coordinates, bordered by the x and y axes. Any four sided shape. A shape with all angles as right angles (the right angle can be inside or outside the shape). 9 Scale The mathematical relationship between different measurements or number of objects. Square number The result of multiplying a whole number by itself: e.g. = The Distributive Law Translation Volume Weight Multiplying numbers by a number and adding, gives the same answer as multiplying the sum of the numbers by the other number: e.g. ( + ) = +. The movement of a shape without rotation or reflection. The amount of space taken up by an object. Mass is measured by how much something weighs, but this can change in different locations. Page 6 of 6