ࡱ>  Tjy bbjbj 4{{. . < p$h&&&&).)) )++++++$v(4Oq4')|)44O&&;6664&&)64)66I&wl60 \6\\p)j[-p6/17)))OO6)))4444\))))))))). N:  Problem Solving Solve problems with or without a calculator Level 4Interpret a calculator display of 4.5 as 4.50 in context of money Use a calculator and inverse operations to find missing numbers, including decimals as for example: 6.5 9.8 = % 4.8 % = 0.96 1/8 of % = 40 Use inverses to check results, for example, 703/19 = 37 appears to about right because 36 x 20 = 720 Carry out simple calculations involving negative numbers in contextWhat would 0.6 mean on a calculator display if the units were s, metres, hours, cars? What is the important information in this problem? Show me a problem that you would use a calculator to work out the answer. Show me a problem that you wouldnt use a calculator? How do you decide? Is it always quicker to use a calculator? What key words tell you that you need to add, subtract, multiply or divide? How would you use a calculator to solve this problem? Choose a number to put into a calculator. Add 472 (or multiply by 26) what single operation will get you back to your starting number? Will this be the same for different starting numbers? How do you know?  Understand and use an appropriate non-calculator method for Level 5 solving problems that involve multiplying and dividing any three digit number by any two-digit numberShow how you could work these out without a calculator: 348 27 309 44 19 423 Explain your choice of method for each calculation. Find the answer to each of the following, using a non-calculator method. 207 ( 23 976 ( 61 872 ( 55 317 people are going on a school coach trip. Each coach will hold 28 passengers. How many coaches are needed? 611 is the product of two prime numbers. One of the numbers is 13. What is the other one? Give pupils some examples of multiplications and divisions with mistakes in them. Ask them to identify the mistakes and talk through what is wrong and how they should be corrected. Ask pupils to carry out multiplications using the grid method and a compact standard method. Ask them to describe the advantages and disadvantages of each method. How do you go about estimating the answer to a division? Solve simple problems involving ordering, adding, subtracting negative numbers in contextImmediately before Sharon was paid, her bank balance was shown as -104.38; the minus sign showed that her account was overdrawn. Immediately after she was paid, her balance was 1312.86. How much was she paid? The temperatures in three towns on January 1st were: Apton -5C Barntown 2C Camtown -1C Which town was the coldest? Which town was the warmest? What was the difference in temperature between the warmest and coldest towns? The lowest winter temperature in a city in Canada was -15C. The highest summer temperature was 42C higher. What was the difference in temperature between the minimum and the maximum temperature? Addition makes numbers bigger. When is this statement true and when is it false? Subtraction makes numbers smaller. When is this statement true and when is it false? The answer is -7. Can you make up some addition/subtraction calculations with the same answer. The answer on your calculator is -144. What keys could you have pressed to get this answer? How does a number line help when adding and subtracting positive and negative numbers? Talk me through how you found the answer to this question. Apply inverse operations and approximate to check answers Level 5 to problems are of the correct magnitudeDiscuss questions such as: Will the answer to 75 0.9 be smaller or larger than 75? Check by doing the inverse operation, for example: Use a calculator to check : 43.2 x 26.5 = 1144.8 with 1144.8 ( 43.2  EMBED Equation.3  of 320 = 192 with 192 x 5 ( 3 3 ( 7 = 0.4285714with 7 x 0.4285714 Looking at a range of problems or calculations, ask: Roughly what answer do you expect to get? How did you come to that estimate? Do you think your estimate is higher or lower than the real answer? Explain your answers. How could you use inverse operations to check that a calculation is correct? Show me some examples. Calculate percentages and find the outcome of a given Level 6 percentage increase or decreaseUse written methods, e.g. Using an equivalent fraction: 13% of 48; 13/100 48 = 624/100 = 6.24 Using an equivalent decimal: 13% of 48; 0.13 48 = 6.24 Using a unitary method: 13% of 48; 1% of 48 = 0.48 so 13% of 48 = 0.48 13 = 6.24 Find the outcome of a given percentage increase or decrease. e.g. an increase of 15% on an original cost of 12 gives a new price of 12 1.15 = 13.80, or 15% of 12 = 1.80 12 + 1.80 = 13.80 Talk me through how you would increase/decrease 12 by, for example 15%. Can you do it in a different way? How would you find the multiplier for different percentage increases/decreases? The answer to a percentage increase question is 10. Make up an easy question. Make up a difficult question. Make and justify estimates and approximations of calculations; Level 7 estimate calculations by rounding numbers to one significant figure and multiplying and dividing mentallyExamples of what pupils should know and be able to doProbing questionsEstimate answers to: 5.16 x 3.14 0.0721 x 0.036 (186.3 x 88.6)/(27.2 x 22.8) Talk me through the steps you would take to find an estimate for the answer to this calculation? Would you expect your estimated answer to be greater or less than the exact answer? How can you tell? Can you make up an example for which it would be difficult to decide? Show me examples of multiplication and division calculations using decimals that approximate to 60. Why is 6 2 a better approximation for 6.59 2.47 than 7 2? Why it is useful to be able to estimate the answer to complex calculations? Use fractions or percentages to solve problems involving Level 8 repeated proportional changes or the calculation of the original quantity given the result of a proportional changeSolve problems involving, for example compound interest and population growth using multiplicative methods. Use a spreadsheet to solve problems such as: How long would it take to double your investment with an interest rate of 4% per annum? A ball bounces to of its previous height each bounce. It is dropped from 8m. How many bounces will there be before it bounces to approximately 1m above the ground? Solve problems in other contexts, for example: Each side of a square is increased by 10%. By what percentage is the area increased? The length of a rectangle is increased by 15%. The width is decreased by 5%. By what percentage is the area changed? Talk me through why this calculation will give the solution to this repeated proportional change problem. How would the calculation be different if the proportional change was? What do you look for in a problem to decide the product that will give the correct answer? How is compound interest different from simple interest? Give pupils a set of problems involving repeated proportional changes and a set of calculations. Ask pupils to match the problems to the calculations.  Solve problems involving calculating with powers, roots and numbers expressed in standard form, checking for correct order of magnitude and using a calculator as appropriateUse laws of indices in multiplication and division, for example to calculate:  What is the value of c in the following question, 48 X 56 = 3 x 7 x 2c Understand index notation with fractional powers, for example knowing that n1/2 = "n and n1/3 = 3"n for any positive number n. Convert numbers between ordinary and standard form. For example: 734.6 = 7.346 x 102 0.0063 = 6.3 x 10-3 Use standard form expressed in conventional notation and on a calculator display. Know how to enter numbers on a calculator in standard form. Use standard form to make sensible estimates for calculations involving multiplication and division. Solve problems involving standard form, such as: Given the following dimensions Diameter of the eye of a fly: 8 10-4 m Height of a tall skyscraper: 4 102 m Height of a mountain 8 103 m How many times taller is the mountain than the skyscraper? How high is the skyscraper in km? Convince me that 37 x 32 = 39 37 3-2 = 39 37 x 3-2 = 35 When working on multiplications and divisions involving indices, ask: Which of these are easy to do? Which are difficult? What makes them difficult? How would you go about making up a different question that has the same answer? What does the index of represent? What are the key conventions when using standard form? How do you go about expressing a very small number in standard form? Are the following always, sometimes or never true: Cubing a number makes it bigger The square of a number is always positive You can find the square root of any number You can find the cube root of any number If sometimes true, precisely when is it true and when is it false? Which of the following statements are true? 163/2 = 82 Length of an A4 piece of paper is 2.97 x 10-5km 8-3 =  272 = 36 3 "7 x 2"7 = 5"7  Co-ordinates and Graphs Use and interpret coordinates in the first quadrant Level 4Given the coordinates of three vertices of a rectangle drawn in the first quadrant, find the fourth What are the important conventions when describing a point using a coordinate? Im thinking of a co-ordinate that I want you to plot. I can only answer yes and no. Ask me some questions so you can plot the coordinate. How do you use the scale on the axes to help you to read a co-ordinate that has been plotted? How do you use the scale on the axes to help you to plot a co-ordinate accurately? If these three points are the three vertices of a rectangle how will you find the coordinates of the fourth point? Use and interpret coordinates in all four quadrants Level 5Plot the graphs of simple linear functions. Generate and plot pairs of co-ordinates for y = x + 1, y = 2x Plot graphs such as: y = x, y = 2x Plot and interpret graphs such as y = x, y = 2x , y = x + 1, y = x - 1 Given the coordinates of three points on a straight line parallel to the y axis, find the equation of the line. Given the coordinates of three points on a straight line such as y = 2x, find three more points in a given quadrant.If I wanted to plot the graph y = 2x how should I start? How do you know the point (3, 6) is not on the line y = x + 2? Can you give me the equations of some graphs that pass through (0, 1)? What about? How would you go about finding coordinates for this straight line graph that are in this quadrant? Plot the graphs of linear functions, where y is given explicitly in terms of x; Level 6 recognise that equations of the form y = mx+c correspond to straight-line graphsPlot the graphs of simple linear functions using all four quadrants by generating co-ordinate pairs or a table of values. e.g. y = 2x - 3 y = 5 - 4x Understand the gradient and intercept in y = mx + c, describe similarities and differences of given straight line graphs. e.g. y = 2x + 4 y = 2x - 3 Without drawing the graphs, compare and contrast features of graphs such as: y = 3x y = 3x + 4 y = x + 4 y = x 2 y = 3x 2 y = -3x + 4 How do you go about finding a set of coordinates for a straight line graph, for example y = 2x + 4? How do you decide on the range of numbers to put on the x and y axes? How do you decide on the scale you are going to use? If you increase/decrease the value of m, what effect does this have on the graph? What about changes to c? What have you noticed about the graphs of functions of the form y = mx + c? What are the similarities and differences? Construct functions arising from real-life problems and plot their corresponding graphs; interpret graphs arising from real situationsThe graph below shows information about a race between two animals the hare (red) and the tortoise (blue)  Who was ahead after 2 minutes? What happened at 3 minutes? At what time did the tortoise draw level with the hare? Between what times was the tortoise travelling fastest? By how much distance did the tortoise win the race? What do the axes represent? In the context of this problem does every point on the line have a meaning? Why? What does this part of the graph represent? What does this point on the graph represent? What sort of questions could you use your graph to answer? For real-life problems that lead to linear functions: How does the gradient relate to the original problem? Do the intermediate points have any practical meaning? Whats the relevance of the intercept in relation to the original problem? Plot graphs of simple quadratic and cubic functions Level 7 Construct tables of values, including negative values of x, and plot the graphs of these functions. y = x y = 3x + 4 y = 2x2 x + 1 y = x How can you identify a quadratic function from its equation? What about a cubic function? How do you find an appropriate set of coordinates for a given quadratic function? Convince me that there are no coordinates on the graph of y=3x+4 which lie below the x-axis. Why does a quadratic graph have line symmetry? Why doesnt a cubic function have line symmetry? How would you describe the symmetry of a cubic function? Understand the effect on a graph of addition of (or multiplication by) a constant Level 8 Given the graph of y=x2, use it to help sketch the graphs of y=3x2 and y=x2+3 Explore what happens to the graphs of the functions, for example: y = ax2 + b for different values of a and b y = ax3 + b for different values of a and b y = (x a)(x b) for different values of a and b Show me an example of an equation of a graph which moves (translates) the graph of y=x vertically upwards (in the positive y-direction) What is the same/different about: y=x, y=3x, y=3x+4 and 1/3x Is the following statement always, sometimes, never true: As 'a' increases the graph of y=ax becomes steeper Convince me that the graph of y=2x is a reflection of the graph of y=-2x in the x-axis Sketch, identify and interpret graphs of linear, quadratic, cubic and reciprocal functions, and graphs that model real situations Match the shape of graphs to given relationships, for example: the circumference of a circle plotted against its diameter the area of a square plotted against its side length the height of fluid over time being poured into various shaped flasks Interpret a range of graphs matching shapes to situations such as: The distance travelled by a car moving at constant speed, plotted against time; The number of litres left in a fuel tank of a car moving at constant speed, plotted against time; The number of dollars you can purchase for a given amount in pounds sterling; The temperature of a cup of tea left to cool to room temperature, plotted against time. Identify how y will vary with x if a balance is arranged so that 3kg is placed at 4 units from the pivot on the left hand side and balanced on the right hand side by y kg placed x units from the pivot Show me an example of an equation of a quadratic curve which does not touch the x-axis. How do you know? Show me an example of an equation of a parabola (quadratic curve) which has line symmetry about the y-axis does not have line symmetry about the y-axis How do you know? What can you tell about a graph from looking at its function? Show me an example of a function that has a graph that is not continuous, i.e. cannot be drawn without taking your pencil off the paper. Why is it not continuous? How would you go about finding the y value for a given x value? An x value for a given y value? Constructing and Using Formula Begin to use formulae expressed in words Level 4 Explain the meaning of and substitute integers into formulae expressed in words, or partly in words, such as the following: number of days = 7 times the number of weeks cost = price of one item x number of items age in years = age in months 12 pence = number of pounds 100 area of rectangle = length times width cost of petrol for a journey = cost per litre number of litres used Use formulae expressed in words, for example for a phone bill based on a standing charge and an amount per unit Recognise that a formula expressed in words requires an equals symbol, for example, 'I think of a number and double it', is different from 'I think of a number and double it. The answer is 12'. Give pupils two sets of cards, one with formulae in words and the other with a range of calculations that match the different formulae (more than one for each formula in words). Ask them to sort the cards. Show me an example of a formula expressed in words How can you change Cost of Plumbers bill = 40 per hour to include a 20 call-out fee. I think of a number and add twelve do you know what my number is? Why or why not? 'I think of a number and add 12. The answer is 17.' Do you know what my number is? Why? How do you know this calculation is for this rule/formula)? Why is it possible for more than one calculation to match with the same rule? Could there be others? Whats the same and whats different about the calculations for the same rule/formula? Construct, express in symbolic form, and use simple formulae Level 5 involving one or two operationsUse letter symbols to represent unknowns and variables. Understand that letter symbols used in algebra stand for unknown numbers or variables and not labels, e.g. 5a cannot mean 5 apples Know and use the order of operations and understand that algebraic operations follow the same conventions as arithmetic operations Recognise that in the expression 2 + 5a the multiplication is to be performed first Understand the difference between expressions such as: 2n and n +2 3(c + 5) and 3c + 5 n and 2n 2n and (2n) Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket. Simplify these expressions: 3a + 2b + 2a b 4x + 7 + 3x 3 x 3(x + 5) 12 (n 3) m(n p) 4(a + 2b) 2(2a + b) Substitute integers into simple formulae, e.g. Find the value of these expressions when a = 4. 3a2 + 4 2a3 Find the value of y when x = 3 y = 2x + 3 y = x - 1 x x +1 Simplify p=x+x+y+y Write p = 2(x+y) as p=2x+2y Give pupils three sets of cards: the first with formulae in words, the second with the same formulae but expressed algebraically, the third with a range of calculations that match the formulae (more than one for each). Ask them to sort the cards. Formulae should involve up to two operations, with some including brackets.How do you know if a letter symbol represents an unknown or a variable? What are the important steps when substituting values into this expression/formula? What would you do first? Why? How would you continue to find the answer? How are these two expressions different? Give pupils examples of multiplying out a bracket with errors. Ask them to identify and talk through the errors and how they should be corrected, e.g. 4(b +2) = 4b + 2 3(p - 4) = 3p - 7 -2 (5 - b) = > 10 -2b 12  (n  3) = 9  n Similarly for simplifying an expression. Can you write an expression that would simplify to, e.g.: 6m  3n, 8(3x + 6)? Are there others? Can you give me an expression that is equivalent to, e.g. 4p + 3q - 2? Are there others? What do you look for when you have an expression to simplify? What are the important stages? What hints and tips would you give to someone about simplifying expressions?removing a bracket from an expression? When you substitute a = 2 and b = 7 into the formula t = ab + 2a you get 18. Can you make up some more formulae that also give t = 18 when a = 2 and b = 7 are substituted? How do you go about linking a formula expressed in words to a formulae expressed algebraically? Could this formulae be expressed in a different way, but still be the same? Use formulae from mathematics and other subjects; Level 7 substitute numbers into expressions and formulae; derive a formula and, in simple cases, change its subjectFind the value of these expressions 3x2 + 4 4x3 2x When x = -3, and when x = 0.1 Make l or w the subject of the formula P = 2(l + w) Make C the subject of the formula F = 9C/5 + 32 Make r the subject of the formula A = r2Given a list of formulae ask: If you are substituting a negative value for the variable, which of these might be tricky? Why? Talk me through the steps involved in this formula. How do you know you do & before & when substituting values into this formula? What are the similarities and differences between rearranging a formula and solving an equation? Derive and use more complex formulae and change Level 8 the subject of a formulaChange the subject of a formula, including cases where the subject occurs twice such as: y a = 2(a +7). By formulating the area of the shape below in different ways, show that the expressions a2 - b2 and (a - b)(a + b) are equivalent.  Derive formulae such as How do you decide on the steps you need to take to rearrange a formula? What are the important conventions? What strategies do you use to rearrange a formula where the subject occurs twice? Talk me through how you went about deriving this formula.  Evaluate algebraic formulae, substituting fractions, decimals and negative numbersSubstitute integers and fractions into formulae, including formulae with brackets, powers and roots. For example: p = s + t + (5"(s2 + t2))/3 Work out the value of p when s = 1.7 and t = 0.9 Given a list of formulae ask: If you are substituting a negative value for the variable, which of these might be tricky? Why? Talk me through the steps involved in evaluating this formula. What tells you the order of the steps? From a given set of algebraic formulae, select the examples that you typically find easy/difficult. What makes them easy/difficult? Manipulate algebraic formulae, equations and expressions, finding common factors and multiplying two linear expressionsSolve linear equations involving compound algebraic fractions with positive integer denominators, e.g. (2x 6) (7 x) = -6 4 2 Simplify: (2x 3)(x 2) 10 (15 x) (3m 2)2 - (1 - 3m)2 Cancel common factors in a rational expression such as EMBED Equation.3  Expand the following, giving your answer in the simplest form possible: (2b - 3)2Give pupils examples of the steps towards the solution of equations with typical mistakes in them. Ask them to pinpoint the mistakes and explain how to correct. Talk me through the steps involved in simplifying this expression. What tells you the order of the steps? How do you go about finding common factors in algebraic fractions? Give me three examples of algebraic fractions that can be cancelled and three that cannot be cancelled. How did you do it? How is the product of two linear expressions of the form (2a b) different from (a b)?  Solving Inequalities Solve inequalities in one variable and represent Level 7 the solution set on a number lineAn integer n satisfies -8 < 2n d" 10. List all possible values of n. Solve these inequalities marking the solution set on a number line 3n+4 < 17 and n >2 2(x  5) d" 0 and x >-2 How do you go about finding the solution set for an inequality? What are the important conventions when representing the solution set on a number line? Why does the inequality sign change when you multiply or divide the inequality by a negative number? Solve inequalities in two variables and find the solution set Level 8 What are the similarities and differences between solving a pair of simultaneous equations and solving inequalities in two variables? Convince me that you need a minimum of three linear inequalities to describe a closed region. How do you check if a point lies inside the region outside the region on the boundary of the region?  Manipulating Expressions Construct, express in symbolic form, and use simple Level 5 formulae involving one or two operationsUse letter symbols to represent unknowns and variables. Understand that letter symbols used in algebra stand for unknown numbers or variables and not labels, e.g. 5a cannot mean 5 apples Know and use the order of operations and understand that algebraic operations follow the same conventions as arithmetic operations Recognise that in the expression 2 + 5a the multiplication is to be performed first Understand the difference between expressions such as: 2n and n +2 3(c + 5) and 3c + 5 n and 2n 2n and (2n) Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket. Simplify these expressions: 3a + 2b + 2a b 4x + 7 + 3x 3 x 3(x + 5) 12 (n 3) m(n p) 4(a + 2b) 2(2a + b) Substitute integers into simple formulae, e.g. Find the value of these expressions when a = 4. 3a2 + 4 2a3 Find the value of y when x = 3 y = 2x + 3 y = x - 1 x x +1 Simplify p=x+x+y+y Write p = 2(x+y) as p=2x+2y Give pupils three sets of cards: the first with formulae in words, the second with the same formulae but expressed algebraically, the third with a range of calculations that match the formulae (more than one for each). Ask them to sort the cards. Formulae should involve up to two operations, with some including brackets.How do you know if a letter symbol represents an unknown or a variable? What are the important steps when substituting values into this expression/formula? What would you do first? Why? How would you continue to find the answer? How are these two expressions different? Give pupils examples of multiplying out a bracket with errors. Ask them to identify and talk through the errors and how they should be corrected, e.g. 4(b +2) = 4b + 2 3(p - 4) = 3p - 7 -2 (5 - b) = > 10 -2b 12  (n  3) = 9  n Similarly for simplifying an expression. Can you write an expression that would simplify to, e.g.: 6m  3n, 8(3x + 6)? Are there others? Can you give me an expression that is equivalent to, e.g. 4p + 3q - 2? Are there others? What do you look for when you have an expression to simplify? What are the important stages? What hints and tips would you give to someone about simplifying expressions?removing a bracket from an expression? When you substitute a = 2 and b = 7 into the formula t = ab + 2a you get 18. Can you make up some more formulae that also give t = 18 when a = 2 and b = 7 are substituted? How do you go about linking a formula expressed in words to a formulae expressed algebraically? Could this formulae be expressed in a different way, but still be the same? Square a linear expression, and expand and simplify Level 7 the product of two linear expressions of the form (x n) and simplify the corresponding quadratic expressionMultiply out these brackets and simplify the result: (x + 4)(x - 3) (a + b)2 (p - q)2 (3x + 2)2 (a + b)(a - b) What is special about the two linear expressions that, when expanded, have: a positive x coefficient? a negative x coefficient? no x coefficient? How did you multiply out the brackets? Show me an expression in the form (x + a)(x + b) which when expanded has: (i) the x coefficient equal to the constant term (ii) the x coefficient greater than the constant term. What does the sign of the constant term tell you about the original expression? Factorise quadratic expressions including the difference of two squares Level 8Factorise x2 + 5x + 6 x2 + x - 6 x2 + 5x Recognise that x2 9 = (x + 3)(x 3)When reading a quadratic expression that you need to factorise, what information is critical for working out the two linear factors? What difference does it make if the constant term is negative? What difference does it make if the constant term is zero? Talk me through the steps you take when factorising a quadratic expression. Show me an expression which can be written as the difference of two squares. How can you tell? Why must 1000 x 998 give the same result as 9992 -1? Manipulate algebraic formulae, equations and expressions, finding common factors and multiplying two linear expressionsSolve linear equations involving compound algebraic fractions with positive integer denominators, e.g. (2x 6) (7 x) = -6 4 2 Simplify: (2x 3)(x 2) 10 (15 x) (3m 2)2 - (1 - 3m)2 Cancel common factors in a rational expression such as EMBED Equation.3  Expand the following, giving your answer in the simplest form possible: (2b - 3)2Give pupils examples of the steps towards the solution of equations with typical mistakes in them. Ask them to pinpoint the mistakes and explain how to correct. Talk me through the steps involved in simplifying this expression. What tells you the order of the steps? How do you go about finding common factors in algebraic fractions? Give me three examples of algebraic fractions that can be cancelled and three that cannot be cancelled. How did you do it? How is the product of two linear expressions of the form (2a b) different from (a b)?  Sequences Recognise and describe number patterns Level 4Describe sequences in words, for example: 8, 16, 24, 32, 40, 5, 13, 21, 29, 37, 89, 80, 71, 62, 53, Continue sequences including those involving decimals. Find missing numbers in sequences.What do you notice about this sequence of numbers? Can you describe a number sequence to me so, without showing it to me, I could write down the first ten numbers? How do you go about finding missing numbers in a sequence?Recognise and describe number relationships including multiple, factor and square Use the multiples of 4 to work out the multiples of 8. Identify factors of two-digit numbers Know simple tests for divisibility for 2, 3, 4, 5, 6, 8, 9 Find the factors of a number by checking for divisibility by primes. For example, to find the factors of 123 check mentally or otherwise, if the number divides by 2, then 3,5,7,11 Which numbers less than 100 have exactly three factors? What number up to 100 has the most factors? The sum of four even numbers is a multiple of four. When is this statement true? When is it false? Can a prime number be a multiple of 4? Why? Can you give me an example of a number greater than 500 that is divisible by 3? How do you know? How do you know if a number is divisible by 6? etc Can you give me an example of a number greater that 100 that is divisible by 5 and also by 3? How do you know? Is there a quick way to check if a number is divisible by 25? Recognise and use number patterns and relationships Level 5Find: A prime number greater than 100 The largest cube smaller than 1000 Two prime numbers that add to 98 Give reasons why none of the following are prime numbers: 4094, 1235, 5121 Use factors, when appropriate, to calculate mentally, e.g. 35 12 = 35 2 6 Continue these sequences: 8, 15, 22, 29, 6, 2, -2, -6, & 1, 1D 2, 1D 4, 1D 8, 1, -2, 4, -8 1, 0.5, 0.25 1, 1, 2, 3, 5, 8Talk me through an easy way to do this multiplication/division mentally. Why is knowledge of factors important for this? How do you go about identifying the factors of a number greater than 100? What is the same / different about these sequences: 4.3, 4.6, 4.9, 5.2, 16.8, 17.1, 17.4, 17.7, 9.4, 9.1, 8.8, 8.5, ... Ive got a number sequence in my head. How many questions would you need to ask me to be sure you know my number sequence? What are the questions? Solving Equations Use systematic trial and improvement methods and ICT tools Level 6 to find approximate solutions to equations such as x3 + x = 20Use systematic trial and improvement methods to find approximate solutions to equations. For example: a3+ a = 20 y(y + 2) = 67.89 Use trial and improvement for equivalent problems, e.g. A number plus its cube is 20, whats the number? The length of a rectangle is 2cm longer than the width. The area is 67.89cm2. Whats the width? Pupils should have opportunities to use a spreadsheet for trial and improvement methods How do you go about choosing a value (of x) to start? How do you use the previous outcomes to decide what to try next? How do you know when to stop? How would you improve the accuracy of your solution? Is your solution exact? Can this equation be solved using any other method? Why?Construct and solve linear equations with integer coefficients, using an appropriate methodSolve, e.g. 3c 7 = -13 4(z + 5) = 84 4(b 1) 5(b + 1) = 0 12 / (x+1) = 21 / (x + 4) Construct linear equations. For example: The length of a rectangle is three times its width. Its perimeter is 24cm. Find its area. How do you decide where to start when solving a linear equation? Given a list of linear equations ask: Which of these are easy to solve? Which are difficult and why? What strategies are important with the difficult ones? 6 = 2p 8. How many solutions does this equation have? Give me other equations with the same solution? Why do they have the same solution? How do you know? How do you go about constructing equations from information given in a problem? How do you check whether the equation works? Use algebraic and graphical methods to solve simultaneous Level 7 linear equations in two variablesGiven that x and y satisfy the equation 5x + y = 49 and y= 2x, find the value of x and y using an algebraic method Solve these simultaneous equations using an algebraic method: 3a + 2b = 16, 5a - b=18 Solve graphically the simultaneous equations x + 3y = 11 and 5x - 2y = 4 What methods do you use when solving a pair of simultaneous linear equations? What helps you to decide which method to use? Why might you use more than one method? Talk me through a couple of examples with your reasons for your chosen method. Is it possible for a pair of simultaneous equations to have two different pairs of solutions or to have no solution? How do you know? How does a graphical representation help to know more about the number of solutions?Manipulate algebraic formulae, equations and expressions, Level 8 finding common factors and multiplying two linear expressionsSolve linear equations involving compound algebraic fractions with positive integer denominators, e.g. (2x 6) (7 x) = -6 4 2 Simplify: (2x 3)(x 2) 10 (15 x) (3m 2)2 - (1 - 3m)2 Cancel common factors in a rational expression such as EMBED Equation.3  Expand the following, giving your answer in the simplest form possible: (2b - 3)2Give pupils examples of the steps towards the solution of equations with typical mistakes in them. Ask them to pinpoint the mistakes and explain how to correct. Talk me through the steps involved in simplifying this expression. What tells you the order of the steps? How do you go about finding common factors in algebraic fractions? Give me three examples of algebraic fractions that can be cancelled and three that cannot be cancelled. How did you do it? How is the product of two linear expressions of the form (2a b) different from (a b)?  Averages Understand and use the mode and range to describe sets of data Level 4Use mode and range to describe data relating to, for example, shoe sizes in the pupils class and begin to compare their data with data from another class. Discuss questions such as: How can we find out if this is true? What information should we collect? How shall we organise it? What does the mode tell us? What does the range tell us? List a small set of data that has a mode of 5. How did you do it? List a small set of data that has a mode of 5 and a range of 10. How did you work this out? Can you find two different small sets of data that have the same mode and range? How did you do it?Understand and use the mean of discrete data and compare two simple Level 5 distributions, using the range and one of mode, median or mean Describe and compare two sets of football results, by using the range and mode In your class girls are taller than boys. True or false? Solve problems such as, Find 5 numbers where the mode is 6 and the range is 8 How long do pupils take to travel to school? Compare the median and range of the times taken to travel to school for two groups of pupils such as those who travel by bus and those who travel by car. Which newspaper is easiest to read? In a newspaper survey of the numbers of letters in 100-word samples the mean and the range were compared: Tabloid: mean 4.3 and range 10, Broadsheet: mean 4.4 and range 14The mean height of a class is 150cm. What does this tell you about the tallest and shortest pupil? Tell me how you know. Find 5 numbers that have a mean of 6 and a range of 8. How did you do it? What if the median was 6 and the range 8? What if the mode was 6 and the range 8? Two distributions both have the same range but the first one has a median of 6 and the second has a mode of 6. Explain how these two distributions may differ.  Estimate the mean, median and range of a set of grouped data and Level 7 determine the modal class, selecting the statistic most appropriate to the line of enquiry  Estimate the median and range from a grouped frequency table. Calculate an estimate of the mean from a large set of grouped data. Appreciate a distinction between 'estimating the mean of...' and 'calculating an estimate of the mean of...'. Produce sets of grouped data with: an estimated range of 35 an estimated median of 22.5 an estimated median of 22.5 and an estimated range of 35 an estimated mean of 7.4 (to 1dp) .. .Why is it only possible to estimate the mean (median, range) from grouped data? Why is the mid-point of the class interval used to calculate an estimated mean? Why not the end of the class interval? Talk me through the steps you take to estimate the median from grouped data. Why is it important to use the lowest class value for the first class and the highest class value for the last class to estimate range? Why not the mid-point? Can you construct a spreadsheet of grouped data to calculate an estimated mean? When is the estimated mean not the most appropriate average, i.e. it is misleading as a representative value.Compare two or more distributions and make inferences, using the shape of the distributions and measures of average and rangeUse frequency diagrams to compare two distributions. Calculate the mean, median and mode for the same data. Use shape of distribution and measures of mean and range to, for example: Compare athletic performances such as long jump in Year 7 and Year 9 pupils Compare the ages of the populations of the UK and Brazil Make some statements from the shape of this distribution. What are the key similarities and differences between these two distributions? How would you decide whether to, for example buy brand A instead of brand B.  Collecting and Recording Data Collect and record discrete data. Level 4Record discrete data using a frequency table Structure data into a frequency table for enquiries such as: the number of goals scored during one season by a hockey team How did you decide on how to structure your table of results? Why did you choose these items? Might there be others? How did you go about collecting the data for this enquiry? What made the information easy or difficult to record?Group data, where appropriate, in equal class intervals Decide on a suitable class interval when collecting or representing data, for example: pupils time per week spent watching television using intervals of one hour, how long pupils take to travel to school - using intervals of 5 minutes Why did you choose this group size for organising the data? What would happen if you chose a different group size? How do you know these are equal class intervals? Why is it important to use equal class intervals? Ask questions, plan how to answer them and Level 5 collect the data requiredRespond to given problems by asking related questions. For example: Problem A neighbour tells you that the local bus service is not as good as it used to be. How could you find out if this is true? Related questions How can good be defined? Frequency of service, cost of journey, time taken, factors relating to comfort, access? How does the frequency of the bus service vary throughout the day/week? Decide which data would be relevant to the enquiry and possible sources. Relevant data might be obtained from: a survey of a sample of people; an experiment involving observation, counting or measuring; secondary sources such as tables, charts or graphs, from reference books, newspapers, websites, CD-ROMs and so on. Examples of questions that pupils might explore: How do pupils travel to school? Do different types of newspaper use words (or sentences) of different lengths? What was important in the way that you chose to collect data? How do know that you will not need to collect any more data? How will you make sense of the data you have collected? What options do you have in organising the data? What other questions could you ask of the data? How will you make use of the data you have collected? Design a survey or experiment to capture the necessary Level 6 data from one or more sources; design, trial and if necessary refine data collection sheets; construct tables for large discrete and continuous sets of raw data, choosing suitable class intervals; design and use two-way tablesInvestigate jumping or throwing distances: Check that the data collection sheet is designed to record all factors that may have a bearing on the distance jumped or thrown, such as age or height. Decide the degree of accuracy needed for each factor. Recognise that collecting too much information will slow down the experiment; too little may limit its scope. Study the distribution of grass: Use a quadrat or points frame to estimate the number of grass and non-grass plants growing in equal areas at regular intervals from a north-facing building. Repeat next to a south-facing building. Increase accuracy by taking two or more independent measurements. What was important in the design of your data collection sheet? What changes did you make after trialling the data collection sheet and why? Why did you choose that size of sample? What decisions have you made about the degree of accuracy in the data you are collecting? How will you make sense of the data you have collected? What options do you have in organising the data, including the use of two-way tables? How did you go about choosing your class intervals? Would different class intervals make a difference to the findings? How? Suggest a problem to explore using statistical methods, Level 7 frame questions and raise conjectures; identify possible sources of bias and plan how to minimise itDecide on questions to explore in a given context and raise conjectures, e.g. in PSHE questions posed might be: How available are fairly-traded goods in local shops? Who buys fairly-traded goods? A conjecture might be: People with experience or links with LEDCs are more likely to be aware of and to buy fairly-traded goods. Be aware of bias and have strategies to reduce bias, for example due to selection, non-response or timing. .What convinced you that this question could usefully be explored using statistical methods? Convince me that it is valid to explore this situation / hypothesis. How can you tell if data is biased? Can you give any examples of where data may be biased? How would you go about reducing bias? Any general guidelines for checking for bias? Data is biased. Always, never or sometimes true?  Probability In probability, select methods based on equally likely Level 5 outcomes and experimental evidence, as appropriate Find and justify probabilities based on equally likely outcomes in simple contexts. For example: The letters of the word REINDEER are written on 8 cards, and a card is chosen at random. What is the probability that the chosen letter is an E? On a fair die what is the probability of rolling a prime number? Estimate probabilities from experimental data. For example: Test a dice or spinner and calculate probabilities based on the relative frequency of each score. Decide if a probability can be calculated or if it can only be estimated from the results of an experiment. Can you give me an example of an event for which the probability can only be calculated through an experiment? Can you give me an example of what is meant by equally likely outcomes? Understand and use the probability scale from 0 to 1What words would you use to describe an event with a probability of 90%? What about a probability of 0.2? Sketch a probability scale, and mark these probabilities on it. See pages 278 and 280 of the Framework supplement of examplesWhat is the same / different with a probability scale marked with: fractions decimals percentages words? Give examples of probabilities (as percentages) for events that could be described using the following words: Impossible Almost (but not quite) certain Fairly likely An even chance Make up examples of any situation with equally likely outcomes with given probabilities of: 0.5, 1/6, 0.2 etc. Justify your answers. Understand that different outcomes may result from repeating an experimentUnderstand that if an experiment is repeated there may be, and usually will be, different outcomes. For example: Compare estimated probabilities obtained by testing a piece of apparatus (such as a dice, spinner or coin) with those obtained by other groups. Understand that increasing the number of times an experiment is repeated generally leads to better estimates of probability. For example: Confirm that the experimental probabilities for the scores on an ordinary dice approach the theoretical values as the number of trials increase. 'When you spin a coin, the probability of getting a head is 0.5. So if you spin a coin ten times you would get exactly 5 heads'. Is this statement true or false? Why? What is wrong with this coin!? What do you think? You spin a coin a hundred times and count the number of times you get a head. A robot is programmed to spin a coin a thousand times. Who is most likely to be closer to getting an equal number of heads and tails? Why?Find and record all possible mutually exclusive outcomes Level 6 for single events and two successive events in a systematic way Use a sample space diagram to show all outcomes when two dice are thrown together, and the scores added. One red and one white dice are both numbered 1 to 6. Both dice are thrown and the scores added. Use a sample space to show all possible outcomes. Give me examples of mutually exclusive events. How do you go about identifying all the mutually exclusive outcomes for an experiment? What strategies do you use to make sure you have found all possible mutually exclusive outcomes for two successive events, for example rolling two dice? How do you know you have recorded all the possible outcomes?Know that the sum of probabilities of all mutually exclusive outcomes is 1 and use this when solving problemsTwo coins are thrown at the same time. There are four possible outcomes: HH, HT, TH, and TT. The probability of throwing two heads is . What is the probability of not throwing two heads? Extend this to: Three coins Four coins Five coins.  Why is it helpful to know that the sum of probabilities of all mutually exclusive events is 1? Give me an example of how you have used this when working on probability problems.Understand relative frequency as an estimate of probability Level 7 and use this to compare outcomes of an experiment Recognise that repeated trials result in experimental probability tending to a limit, and that this limit may be the only way to estimate probability. Describe situations where the use of experimental data to estimate a probability would be necessary. What might be different about using theoretical probability to find the probability of obtaining a 6 when you roll a dice, and using experimental probability for the same purpose? True or false Experimental probability is more reliable than theoretical probability; Experimental probability gets closer to the true probability as more trials are carried out; Relative frequency finds the true probability.Know when to add or multiply two probabilities Level 8Recognise when probabilities can be associated with independent events or mutually exclusive. Understand that when A and B are mutually exclusive, then the probability of A or B occurring is P(A) + P(B), whereas if A and B are independent events, the probability of A and B occurring is P(A) P(B) Solve problems such as: A pack of blue cards are numbered 1 to 10. What is the probability of picking a multiple of three or a multiple of 5? There is also an identical pack of red cards. What is the probability of picking a red 5 and a blue 7? Show me an example of: A problem which could be solved by adding probabilities A problem which could be solved by multiplying probabilities What are all the mutually exclusive events for this situation? How do you know they are mutually exclusive? Why do you add the probabilities to find the probability either this event or this event occurring? If I throw a coin and roll a dice the probability of a 5 and a head is 1/12. This is not 1/2 + 1/6. Why not? What are the mutually exclusive events in this problem? How would you use these to find the probability?Use tree diagrams to calculate probabilities of combinations of independent eventsConstruct tree diagrams for a sequence of events using given probabilities or theoretical probabilities. Use the tree diagram to calculate probabilities of combinations of independent events. The probability that it will rain on Tuesday is 1/5. The probability that it will rain on Wednesday is 1/3. What is the probability that it will rain on just one of the days? The probability that Nora fails her driving theory test on the first attempt is 0.1. The probability that she passes her practical test on the first attempt is 0.6. Complete a tree diagram based on this information and use it to find the probability that she passes both tests on the first attempt.How can you distinguish between mutually exclusive and independent events from a tree diagram? Why do the probabilities on each pair of branches have to sum to 1? How can you tell from a completed tree diagram whether the question specified with or without replacement? What strategies do you use to check the probabilities on your tree diagram are correct? Talk me through the steps you took to construct this tree diagram and use it to find the probability of this event. Representing and Interpreting Data Continue to use Venn and Carroll diagrams to record Level 4 their sorting and classifying of informationUsing this Carroll diagram for numbers, write a number less than 100 in each space even not even a square number not a square number Use a Venn diagram to sort by two criteria, e.g. sorting numbers using the properties multiples of 8 and multiples of 6Give me an example of a Venn diagram that can be used to sort the numbers 1-50. Which criteria have you used and why? Give me an example of a Carroll diagram with four cells that can be used to sort the numbers 1-50 Which criteria have you used and why? What are the important steps when completing a Carroll diagram? What strategies do you use to check your Carroll diagram is complete? What are the important steps when completing a Venn diagram? What strategies do you use to check your Venn diagram is complete? How is a Venn diagram different from a Carroll diagram?Construct and interpret frequency diagrams and simple line graphs Suggest an appropriate frequency diagram to represent particular data, for example decide whether a bar chart, Venn diagram or pictogram would be most appropriate. For pictograms use one symbol to represent several units Decide upon an appropriate scale for a graph e.g. labelled divisions representing 2, 5, 10, 100 Interpret simple pie charts Interpret the scale on bar charts and line graphs, reading between the labelled divisions e.g. reading 17 on a scale labelled in fives Interpret the total amount of data represented, compare data sets and respond to questions e.g. how does our data about favourite television programmes compare to the data from year 8 pupils? For a given graph/table/chart, make up three questions that can be answered using the information represented. What makes the information easy or difficult to represent? How do you decide on the scale to use on the vertical axis? How would a different scale change the graph? Interpret graphs and diagrams, including pie charts, Level 5 and draw conclusions Interpret, by comparing the cells in a 2-way table, Interpret bar charts with grouped data Interpret and compare pie charts where it is not necessary to measure angles Read between labelled divisions on a scale of a graph or chart, for example read 34 on a scale labelled in tens or 3.7 on a scale labelled in ones, and find differences to answer, How much more? Recognise when information is presented in a misleading way, for example compare two pie charts where the sample sizes are different When drawing conclusions, identify further questions to ask. Make up a statement or question for this chart/graph using one or more of the following key words: total, range, mode; fraction, percentage, proportion.Create and interpret line graphs where the intermediate values have meaningDraw and use a conversion graph for Pounds and Euros Answer questions based on a graph showing tide levels; for example, Between which times will the height of the tide be greater than five metres? Use a line graph showing average maximum monthly temperatures for two locations. For example: What was the coldest month in Manchester? During which months would you expect the maximum temperatures in Manchester and Sydney to be about the same?Do the intermediate values have any meaning on these graphs? How do you know? Show graphs where there is no meaning for example, a line graph showing the trend in midday temperatures over a week. Also show examples where interpolation makes sense for example, the temperature in a classroom, measured every 30 minutes for six hours. Convince me that you can use this graph (conversion graph between litres and gallons up as far as 20 gallons) to find out how many litres are roughly equivalent to 75 gallons  Select, construct and modify, on paper & using ICT: Level 6 pie charts for categorical data; bar charts and frequency diagrams for discrete and continuous data; simple time graphs for time series; scatter graphs. and identify which are most useful in the context of the problemUnderstand that pie charts are mainly suitable for categorical data. Draw pie charts using ICT and by hand, usually using a calculator to find angles Draw compound bar charts with subcategories Use frequency diagrams for continuous data and know that the divisions between bars should be labelled Use line graphs to compare two sets of data. Use scatter graphs for continuous data, two variables, showing, for example, weekly hours worked against hours of TV watched.  When drawing a pie chart, what information do you need to calculate the size of the angle for each category? What is discrete/continuous data? Give me some examples. How do you go about choosing class intervals when grouping data for a bar chart/frequency diagram? Whats important when choosing the scale for the frequency axis? Is this graphical representation helpful in addressing the hypothesis? If not, why and what would you change? When considering a range of graphs representing the same data: which is the easiest to interpret? Why? Which is most helpful in addressing the hypothesis? Why?Communicate interpretations and results of a statistical survey using selected tables, graphs and diagrams in supportUsing selected tables, graphs and diagrams for support; describe the current incidence of male and female smoking in the UK, using frequency diagrams to demonstrate the peak age groups. Show how the position has changed over the past 20 years; using line graphs. Conclude that the only group of smokers on the increase is females aged 15 -25. Which of your tables/graphs/diagrams give the strongest evidence to support/reject your hypothesis? How? What conclusions can you draw from your table/graph/diagram? Convince me using the table/graph/diagram. What difference would it make if this piece of data was included? Are any of your graphs/diagrams difficult to interpret? Why? Select, construct and modify, on paper and using ICT, Level 7 suitable graphical representation to progress an enquiry, including frequency polygons and lines of best fit on scatter graphsUse superimposed frequency polygons to compare results. When plotting a line of best fit, find the mean point and make predictions using a line of best fit. Recognise that a prediction based on a line of best fit may be subject to error. Recognise the potential problems of extending the line of best fit beyond the range of known values. How does your graph help with your statistical analysis? What does it show? What are the benefits and any disadvantages of your chosen graph? Did you try any graphs that were not helpful? Why? Convince me that this is the most appropriate graph to use in this case. How do you go about labelling the horizontal axis when constructing a frequency polygon? How can you tell if it is sensible to draw a line of best fit on a scatter graph? Examine critically the results of a statistical enquiry, and justify the choice of statistical representation in written presentationExamine data for cause and effect Analyse data and try to explain anomalies. For example: In a study of engine size and acceleration times, observe that in general a larger engine size leads to greater acceleration. However, particular cars do not fit the overall pattern. Reasons may include because they are heavier than average or are built for rough terrain rather than normal roads. Recognise that establishing a correlation or connection in statistical situations does not necessarily imply that one variable causes change to another (i.e. correlation does not imply causality), and that there may be external factors affecting both. Can you think of an example where there is positive correlation, but it is unlikely that there is causation? Which of the following statements is more precise: 'my hypothesis is true' or 'there is strong evidence to support my hypothesis'. Why? Convince me that there is evidence to support your hypothesis. Why might it not be the case that: Your hypothesis is true if you have found enough evidence to support it You have failed if your hypothesis appears to be flawed. Convince me that you have chosen the most appropriate statistical measure to use in this case. What else did you consider and not use?  Compare two or more distributions and make inferences, Level 8 using the shape of the distributions and measures of average and spread including median and quartilesConstruct, interpret and compare box-plots for two sets of data, for example, the heights of Year 7 and Year 9 pupils or the times that Year 9 boys and Year 9 girls can run 100m. Recognise positive and negative skew from the shape of distributions represented in: frequency diagrams cumulative frequency diagrams box plots What features of the distributions can you compare when using a box plot?a frequency diagram? Make some comparative statements from the shape of each of these distributions. What are the key similarities and differences between these two distributions? Describe two contexts, one in which a variable/attribute has negative skew and the other in which it has positive skew? How can we tell from a box plot that the variable has negative skew? Convince me that the following representations are from different distributions, for example:  What would you expect to be the same/different about the two distributions representing, for example Heights of pupils in Year 1 to 6 and Heights of pupils only in Year 6 Directed Numbers Round decimals to the nearest decimal place Level 5 and order negative numbers in contextPetrol costs 124.9p a litre. How much is this to the nearest penny? Round, e.g. 2.75 to 1 decimal place 176.05 to 1 decimal place 25.03 to 1 decimal place 24.992 to 2 decimal places Order the following places from coldest to warmest: Moscow, Russia: 4(C Oymyakou, Russia: -96(C Vostok, Antarctica: -129(C Rogers Pass, Montana, USA: -70(C Fort Selkirk, Yukon, Canada: -74(C Northice, Greenland: -87(C Reykjavik, Iceland: 5(C Explain whether the following are true or false: 2.399 rounds to 2.310 to 2 decimal places -6 is less than -4 3.5 is closer to 4 than it is to 3 -36 is greater than -34 8.4999 rounds to 8.5 to 1 decimal place How do you go about rounding a number to one decimal place? Why might it not be possible to identify the first three places in a long jump competition if measurements were taken in metres to one decimal place? Show me a length that rounds to 4.3m to one decimal place. Are there other lengths? What is the same / different about these numbers: 72.344 and 72.346 Estimation Check the reasonableness of results with reference Level 4 to the context or size of numbers Use rounding to approximate and judges whether the answer is the right order of magnitude, for example: 2605- 1897 is about 3000-2000 12% of 192 is about 10% of 200 Discuss questions such as: A girl worked out the cost of 8 bags of apples at 47p a bag. Her answer was 4.06. Without working out the answer, say whether you think it is right or wrong. A boy worked out how many 19p stamps you can buy for 5. His answer was 25. Do you think he was right or wrong? Why? A boy worked out 2.38 + 76p on a calculator. The display showed 78.38. Why did the calculator give the wrong answer? Roughly what answer do you expect to get? How did you come to that estimate? Do you expect your answer to be less than or greater than your estimate? Why?Round decimals to the nearest decimal place Level 5 and order negative numbers in contextPetrol costs 124.9p a litre. How much is this to the nearest penny? Round, e.g. 2.75 to 1 decimal place 176.05 to 1 decimal place 25.03 to 1 decimal place 24.992 to 2 decimal places Order the following places from coldest to warmest: Moscow, Russia: 4(C Oymyakou, Russia: -96(C Vostok, Antarctica: -129(C Rogers Pass, Montana, USA: -70(C Fort Selkirk, Yukon, Canada: -74(C Northice, Greenland: -87(C Reykjavik, Iceland: 5(C Explain whether the following are true or false: 2.399 rounds to 2.310 to 2 decimal places -6 is less than -4 3.5 is closer to 4 than it is to 3 -36 is greater than -34 8.4999 rounds to 8.5 to 1 decimal place How do you go about rounding a number to one decimal place? Why might it not be possible to identify the first three places in a long jump competition if measurements were taken in metres to one decimal place? Show me a length that rounds to 4.3m to one decimal place. Are there other lengths? What is the same / different about these numbers: 72.344 and 72.346 Make and justify estimates and approximations of calculations; Level 7 estimate calculations by rounding numbers to one significant figure and multiplying and dividing mentallyExamples of what pupils should know and be able to doProbing questionsEstimate answers to: 5.16 x 3.14 0.0721 x 0.036 (186.3 x 88.6)/(27.2 x 22.8) Talk me through the steps you would take to find an estimate for the answer to this calculation? Would you expect your estimated answer to be greater or less than the exact answer? How can you tell? Can you make up an example for which it would be difficult to decide? Show me examples of multiplication and division calculations using decimals that approximate to 60. Why is 6 2 a better approximation for 6.59 2.47 than 7 2? Why it is useful to be able to estimate the answer to complex calculations? Four Rules Use a range of mental methods of computation with all operations Level 4Calculate mentally a difference such as 8006 - 2993 by 'counting up' or by considering the equivalent calculation of 8006 - 3000 + 7 Work out mentally that: 4005 - 1997 = 2008 because it is 4005 - 2000 + 3 = 2005 + 3 = 2008 Work out mentally by counting up from the smaller to the larger number: 8000 - 2785 is 5 + 10 + 200 + 5000 = 5215 Calculate complements to 1000Which of these calculations can you do without writing anything down? Why is it sensible to work this out mentally? What clues did you look for? How did you find the difference? Talk me through your method. Give me a different calculation that has the same answeran answer that is ten times biggeretc. How did you do it?Recall multiplication facts up to 10 10 and quickly derive corresponding division facts Use their knowledge of tables and place value in calculations with multiples of 10 such as 180 3 Respond rapidly to oral and written questions like: Nine eights How many sevens in 35? 8 times 8 6 multiplied by 7 Multiply 11 by 8 7 multiplied by 0 Respond quickly to questions like Divide 3.6 by 9 What is 88 shared between 8? 0.6 times 7 times2 Know by heart or derive quickly Doubles of two-digit whole numbers or decimals Doubles of multiples of 10 up to 1000 Doubles of multiples of 100 up to 10 000 And all the corresponding halvesIf someone has forgotten the 8 times table, what tips would you give them to work it out? What other links between tables are useful? If you know that 4 x 7 = 28, what else do you know? Start from a two-digit number with at least 6 factors, e.g. 56. How many different multiplication and division facts can you make using what you know about 56? How have you identified the divisions? The product is 40. Make up some questions. How are these different questions linked? The quotient is 5. Make up some questions. How did you go about devising these questions?  Use efficient written methods of addition and subtraction and of short multiplication and division Calculate 1202 + 45 + 367 or 1025 - 336 Work with numbers to two decimal places, including sums and differences with different numbers of digits, and totals of more than two numbers, e.g. 671.7 - 60.2 543.65 + 45.8 1040.6 - 89.09 764.78 - 56.4 76.56 + 312.2 + 5.07 Use, for example, the grid method before moving on to short multiplication. Use efficient methods of repeated subtraction, by subtracting multiples of the divisor, before moving to short divisionGive pupils some examples of work with errors for them to check, for example: 12.3 + 9.8 21.11; 4.07  -1.5 3.57; 36.2 8  288.16 Which are correct/incorrect? How do you know? Explain what has been done wrong and correct the answers. Use a calculator efficiently and appropriately to perform Level 7 complex calculations with numbers of any size, knowing not to round during intermediate steps of a calculation Use a calculator to evaluate more complex calculations such as those with nested brackets or where the memory function could be used. For example: " Use a calculator to work out: a. 45.65 76.8 1.05 (6.4  3.8) b. 4.6 + (5.7  (11.6 9.1)) c. {(4.5)2 + (7.5  0.46)}2 d. 5 "(4.52 + 62) 3How do you decide on the order of operations for a complex calculation when using a calculator? Which calculator keys and functions are important when doing complex calculations? (Explore brackets, memories, +/- key, reciprocal key.) Give pupils some calculations with answers (some correct but some with common mistakes) and ask them to decide which are correct. Ask them to analyse the mistakes in the ones that are incorrect and explain the correct method using a calculator. Why shouldnt you round during the intermediate steps of calculations?  Fractions, Decimals and Percentages Recognise approximate proportions of a whole and Level 4 use simple fractions and percentages to describe these Recognise simple equivalence between fractions, decimals and percentages e.g. 1/2, 1/4, 1/10, 3/4 Convert mixed numbers to improper fractions and vice versa Express a smaller number as a fraction of a larger one. For example: What fraction of: 1 metre is 35 centimetres 1 kilogram is 24 grams 1 hour is 33 minutes?What fractions/percentages can you easily work out in your head? Talk me through a couple of examples. Talk me through how you know that, e.g. 90 is of 120. Approximately what fraction of, e.g. this shape is shaded? Would you say it is more or less than this fraction? How do you know? When calculating percentages of quantities, what percentage do you usually start from? How do you use this percentage to work out others? To calculate 10% of a quantity, you divide it by 10. So to find 20%, you must divide by 20. What is wrong with this statement? Using a 1 - 100 grid, 50% of the numbers are even. How would you check? Give me a question with the answer 20% (or other simple percentages and fractions) Multiply a simple decimal by a single digitCalculate: 2.4 x 7 4.6 x 8 9.3 x 9What would you estimate the answer to be? Is the accurate answer bigger or smaller than your estimate? Why? How would you help someone to understand that, e.g. 0.4 7 = 2.8, 2.4 7 = 16.8? Which of these calculations are easy to work out in your head, and why? 0.5 x 5 0.5 x 8 1.5 x 5 11.5 x 8 Talk me through your method.Order decimals to three decimal places Place these numbers in order of size, starting with the greatest: 0.206, 0.026, 0.602, 0.620, 0.062 Place these numbers on a line from 6.9 to 7.1: 6.93, 6.91, 6.99, 7.01, 7.06 Put these in order, largest/smallest first: 1.5, 1.375, 1.4, 1.3, 1.35, 1.425 Put these in order, largest/smallest first: 7.765, 7.675, 6.765, 7.756, 6.776 What do you look for first when you are ordering numbers with decimals? Which part of each number do you look at to help you? Which numbers are the hardest to put in order? Why? What do you do when numbers have the same digit in the same place? Give me a number between 0.12 and 0.17. Which of the two numbers is it closer to? How do you know? Use equivalence between fractions and order fractions and decimals Level 5Find two fractions equivalent to EMBED Equation.3 Show that EMBED Equation.3 is equivalent to EMBED Equation.3, EMBED Equation.3 andEMBED Equation.3 Find the unknown numerator or denominator in: EMBED Equation.3 = EMBED Equation.3 EMBED Equation.3 = EMBED Equation.3 EMBED Equation.3= EMBED Equation.3 Write the following set of fractions in order from smallest to largest: EMBED Equation.3 Convert fractions to decimals by using a known equivalent fraction and using division. For example: 2/8 = = 0.25 3/5 = 6/10 = 0.6 3/8 = 0.375 using short division Answer questions such as: Which is greater, 0.23 or 3/16? Give me two equivalent fractions. How do you know they are equivalent? Give me some fractions that are equivalent to How did you do it? Can you draw a diagram to convince me that EMBED Equation.3is the same as EMBED Equation.3? Can you show me on a number line? Explain whether the following is true or false: 10 is greater than 9, so 0.10 is greater than 0.9 Explain how you could fill in the missing numbers so that each resulting set of fractions is in ascending order: EMBED Equation.3 EMBED Equation.3 Now show how you could fill in the missing numbers in a different way, so that each set of fractions is in descending order. What are the important steps when putting a set of fractions/decimals in order? Use known facts, place value, knowledge of operations and brackets to calculate including using all four operations with decimals to two placesGiven that 42 ( 386 = 16 212, find the answers to: 4.2 ( 386 42 ( 3.86 420 ( 38.6 16 212 ( 0.42 Use factors to find the answers to:. 3.2 x 30 knowing 3.2 x 10 = 32: 32 x 3 = 96 156 6 knowing 156 3 = 52 52 2 = 26 Use partitioning for multiplication; partition either part of the product: 7.3 x 11 = (7.3 x 10) + 7.3 = 80.3 Use 1/5 = 0.2 to convert fractions to decimals mentally. e.g. 3/5 = 0.2 x 3 = 0.6 Calculate: 4.2 ( (3.6 + 7.4) 4.2 ( 3.6 + 7.4 4.2 + 3.6 ( 7.4 (4.2 + 3.6) ( 7.4 Extend doubling and halving methods to include decimals, for example: 8.12 x 2.5 = 4.06 x 5 = 20.3 Explain how you would do this multiplication by using factors, e.g. 5.8 x 40 What clues do you look for when deciding if you can do a multiplication mentally? e.g. 5.8 x 40 Give an example of how you could use partitioning to multiply a decimal by a two digit whole number, e.g. 5.3 x 23. 37 64 = 2368. Explain how you can use this fact to devise calculations with answers 23.68, 2.368, 0.2368. 73.6 3.2 = 23. Explain how you can use this to devise calculations with the same answer. Explain why the standard compact method for subtraction (decomposition) is not helpful for examples like 10 008 59. Talk me through this calculation. What steps do you need to take to get the answer? How do you know what you have to do first? What are the important conventions for the order of operations when doing a calculation? Use a calculator where appropriate to calculate fractions/percentages of quantities/measurementsUse mental strategies in simple cases, e.g. 1/8 of 20; find one quarter and halve the answer 75% of 24; find 50% then 25% and add the results 15% of 40; find 10% then 5% and add the results 40% of 400kg; find 10% then multiply by 4 Calculate simple fractions or percentages of a number/quantity e.g. \! of 400g or 20% of 300 Use a calculator for harder examples, e.g. EMBED Equation.3of 207; 207 ( 18 = 11.5 43% of 1.36; 0.43 ( 1.36 = 58p 62% of 405 m; 0.62 ( 405 = 251.1mWhat fractions/percentages of given quantities can you easily work out in your head? Talk me through a couple of examples. When calculating percentages of quantities, what percentages do you usually start from? How do you use this percentage to work out others? How do you decide when to use a calculator, rather than a mental or written method, when finding fractions or percentages of quantities? Give me some examples. Talk me through how you use a calculator to find a percentage of a quantity or a fraction of a quantityReduce a fraction to its simplest form by cancelling common factorsCancel these fractions to their simplest form by looking for highest common factors: EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 What clues do you look for when cancelling fractions to their simplest form? How do you know when you have the simplest form of a fraction? Use the equivalence of fractions, decimals and percentages to Level 6 compare proportionsConvert fraction and decimal operators to percentage operators by multiplying by 100. For example: 0.45; 0.45 100% = 45% 7/12; (7 12) 100% = 58.3% (1 d.p.) Continue to use mental methods for finding percentages of quantities Use written methods, e.g. Using an equivalent fraction: 13% of 48; 13/100 48 = 624/100 = 6.24 Using an equivalent decimal: 13% of 48; 0.13 48 = 6.24 Which sets of equivalent fractions, decimals and percentages do you know? From one set that you know (e.g. 1/10 ( 0.1 ( 10%), which others can you deduce? How would you go about finding the decimal and percentage equivalents of any fraction? How would you find out which of these is closest to 1/3: 10/31; 20/61; 30/91; 50/151? What links have you noticed within equivalent sets of fractions, decimals and percentages? Give me a fraction between 1/3 and 1/2. How did you do it? Which is it closer to? How do you know?Calculate percentages and find the outcome of a given percentage increase or decreaseUse written methods, e.g. Using an equivalent fraction: 13% of 48; 13/100 48 = 624/100 = 6.24 Using an equivalent decimal: 13% of 48; 0.13 48 = 6.24 Using a unitary method: 13% of 48; 1% of 48 = 0.48 so 13% of 48 = 0.48 13 = 6.24 Find the outcome of a given percentage increase or decrease. e.g. an increase of 15% on an original cost of 12 gives a new price of 12 1.15 = 13.80, or 15% of 12 = 1.80 12 + 1.80 = 13.80 Talk me through how you would increase/decrease 12 by, for example 15%. Can you do it in a different way? How would you find the multiplier for different percentage increases/decreases? The answer to a percentage increase question is 10. Make up an easy question. Make up a difficult question. Add and subtract fractions by writing them with a common denominator, calculate fractions of quantities (fraction answers); multiply and divide an integer by a fractionAdd and subtract more complex fractions such as 11/18 + 7/24, including mixed fractions. Solve problems involving fractions, e.g. In a survey of 24 pupils 1/3 liked football best, 1/4 liked basketball, 3/8 liked athletics and the rest liked swimming. How many liked swimming? Why are equivalent fractions important when adding or subtracting fractions? What strategies do you use to find a common denominator when adding or subtracting fractions? Is there only one possible common denominator? What happens if you use a different common denominator? Give pupils some examples of +, - of fractions with common mistakes in them. Ask them to talk you through the mistakes and how they would correct them. How would you justify that 4 1/5 = 20? How would you use this to work out 4 2/5? Do you expect the answer to be greater or less than 20? Why? Understand the effects of multiplying and dividing by Level 7 numbers between 0 and 1. Give an approximate answer to: 357 0.3 1099 0.22 1476 x 0.99 57.7 0.65 Know and understand that division by zero has no meaning. For example, explore dividing by successive smaller positive decimals, approaching zero then negative decimals approaching zero. Recognise and use reciprocals. Multiplying makes numbers bigger. When is this statement true and when is it false? Division makes things smaller. When is this statement true and when is it false? How would you justify that dividing by is the same as multiplying by 2? What about dividing by 1/3 and multiplying by 3? What about dividing by 2/3 and multiplying by 3/2? How does this link to division of fractions? 0.8 0.1 16 0.5 1.6 0.5 1.6 0.2 Talk me through the reasoning that took you to the answer to each of these calculations.Add, subtract, multiply and divide fractionsFind the area and perimeter of a rectangle measuring 4 inches by 63/8 inches. Using 22/7 as an approximation for pi, estimate the area of a circle with diameter 28mm. Pupils should be able to understand and use efficient methods to add, subtract, multiply and divide fractions, including mixed numbers and questions that involve more than one operation. How do you go about adding and subtracting more complex fractions e.g. 2 2/5 1 7/8? Give me two fractions which multiply together to give a bigger answer than either of the fractions you are multiplying. How did you do it? Give pupils some examples of +, - and with common mistakes in them (including mixed numbers). Ask them to talk you through the mistakes and how they would correct them. How would you justify that dividing by is the same as multiplying by 2? What about dividing by 1/3 and multiplying by 3? What about dividing by 2/3 and multiplying by 3/2? How does this link to what you know about dividing by a number between 0 and 1?  Understand the equivalence between recurring decimals and fractions Level 8Distinguish between fractions with denominators that have only prime factors 2 and 5 (which are represented as terminating decimals), and other fractions (which are represented by terminating decimals). Decide which of the following fractions are equivalent to terminating decimals: 3/5, 3/11, 7/30, 9/22, 9/20, 7/16 Write 0. EMBED Equation.3  EMBED Equation.3 as a fraction in its simplest termsWrite some fractions which terminate when converted to a decimal. What do you notice about these fractions? What clues do you look for when deciding if a fraction terminates? 1/3 is a recurring decimal. What other fractions related to one-third will also be recurring? Using the knowledge that 1/3 = 0. EMBED Equation.3  How would you go about finding the decimal equivalent of 1/6, 1/30? 1/11 = 0. EMBED Equation.3  EMBED Equation.3 How do you use this fact to express 2/11, 3/11, 12/11 in decimal form? If you were to convert these decimals to fractions; 0.0454545, 0.454545, 4.545454, 45.4545.. Which of these would be easy/difficult to convert? What makes them easy/difficult to convert? Can you use the fraction equivalents of 4. EMBED Equation.3  EMBED Equation.3  and 45. EMBED Equation.3  EMBED Equation.3 to prove the second is ten times greater than the first? Which of the following statements are true/false: all terminating decimals can be written as a fraction all recurring decimals can be written as a fraction all numbers can be written as a fraction Use fractions or percentages to solve problems involving repeated proportional changes or the calculation of the original quantity given the result of a proportional changeSolve problems involving, for example compound interest and population growth using multiplicative methods. Use a spreadsheet to solve problems such as: How long would it take to double your investment with an interest rate of 4% per annum? A ball bounces to of its previous height each bounce. It is dropped from 8m. How many bounces will there be before it bounces to approximately 1m above the ground? Solve problems in other contexts, for example: Each side of a square is increased by 10%. By what percentage is the area increased? The length of a rectangle is increased by 15%. The width is decreased by 5%. By what percentage is the area changed? Talk me through why this calculation will give the solution to this repeated proportional change problem. How would the calculation be different if the proportional change was? What do you look for in a problem to decide the product that will give the correct answer? How is compound interest different from simple interest? Give pupils a set of problems involving repeated proportional changes and a set of calculations. Ask pupils to match the problems to the calculations.  Limits Recognise that measurements given to the nearest Level 7 whole unit may be inaccurate by up to one half of the unit in either directionSuggest a range for measurements such as: 123mm; 1860mm; 3.54kg; 6800m2 Find maximum and minimum values for a measurement that has been rounded to a given degree of accuracy Solve problems such as: The dimensions of a rectangular floor, measured to the nearest metre, are given as 28m by 16m. What range must the area of the floor lie within? Suggest a sensible answer for the area, given the degree of accuracy of the data. .Explain the difference in meaning between 0.6m and 0.600m. When is it necessary to include the zeros in measurements? What range of measured lengths might be represented by the measurement 320cm? What accuracy is needed to be sure a measurement is accurate to the nearest centimetre? Why might you pick a runner whose time for running 100m is recorded as 13.3 seconds rather than 13.30 seconds? Why might you not? Explain how 6 people each weighing 110kg might exceed a weight limit of 660kg for a lift. Place Value Use place value to multiply and divide whole numbers by 10 or 100 Level 4Respond to oral and written questions such as: How many times larger is 2600 than 26? How many 10 notes are in 120, 1200? How many 1 coins, 10p coins, 1p coins? Tins of dog food at 42p each are put in packs of 10. Ten packs are put in a box. How much does one box of dog food cost? 10 boxes? 100 boxes? Work out mentally the answers to questions such as: 329 x 100 = ( 8000 100 = ( 56 x ( = 5600 7200 ( = 72 420 x ( = 4200 3900 ( = 390 Complete statements such as: 4 x 10 = 4 x = 400 10 = 40 x 1000 = 40 000 x 10 = 400Why do 6 x 100 and 60 x 10 give the same answer? What about 30 10 and 300 100? I have 37 on my calculator display. What single multiplication should I key in to change it to 3700? Explain why this works. Can you tell me a quick way of multiplying by 10, by 100? Can you tell me a quick way of dividing by 10, by 100?Use understanding of place value to multiply and divide whole Level 5 numbers and decimals by 10, 100 and 1000 and explain the effectKnow for example: in 5.239 the digit 9 represents nine thousandths, which is written as 0.009 the number 5.239 in words is five point two three nine not five point two hundred and thirty nine the fraction 5 EMBED Equation.3 is read as five and two hundred and thirty-nine thousandths. Complete statements such as: 4 ( 10 = ( 4 ( ( = 0.04 0.4 x 10 = ( 0.4 x ( = 400 0.4 ( 10 = ( 0.4 (( = 0.004 ( (100 = 0.04 How would you explain that 0.35 is greater than 0.035? Why do 25 ( 10 and 250 ( 100 give the same answer? My calculator display shows 0.001. Tell me what will happen when I multiply by 100. What will the display show? I divide a number by 10, and then again by 10. The answer is 0.3. What number did I start with? How do you know? How would you explain how to multiply a decimal by 10 ., how to divide a decimal by 100?  Ratio and Proportion Begin to understand simple ratio Level 4 Pupils use the vocabulary of ratio to describe the relationships between two quantities within a context Given a bag of 4 red and 20 blue cubes, write the ratio of red cubes to blue cubes, and the ratio of blue cubes to red cubes Solve simple problems using informal strategies: For example: A girl spent her savings of 40 on books and clothes in the ratio 1:3. How much did she spend on clothes? Scale numbers up or down, for example, by converting recipes for, say, 6 people to recipes for 2 people: In a recipe for 6 people you need 120 g flour and 270 ml of milk. How much of each ingredient does a recipe for 2 people require? Can you talk me through how you solved this problem? What do you see as the important information in this problem? How do you use it to solve the problem?  Understand simple ratio Level 5Write 16 :12 in its simplest form Solve problems such as: 28 pupils are going on a visit. They are in the ratio of 3 girls to 4 boys. How many boys are there?? How do you know when a ratio is in its simplest form? Is the ratio 1:5 the same as the ratio 5:1? Explain your answer. Convince me that 19:95 is the same ratio as 1:5 The instructions on a packet of cement say, mix sand and cement in the ratio 5:1. A builder mixes 5 kg of cement with 1 bucketful of sand. Could this be correct? Explain your answer. The ratio of boys to girls at a school club is 1:2. Could there be 10 pupils at the club altogether? Explain your answer.Solve simple problems involving ratio and direct proportionThe ratio of yogurt to fruit puree used in a recipe is 5 : 2. If you have 200g of fruit puree, how much yogurt do you need? If you have 250g of yogurt, how much fruit puree do you need? A number of cubes are arranged in a pattern and the ratio of red cubes to green cubes is 2 : 7. If the pattern is continued until there are 28 green cubes, how many red cubes will there be? Three bars of chocolate cost 90p. How much would six bars cost? And twelve bars? Six stuffed peppers cost 9. What will 9 stuffed peppers cost?How do you decide how to link the numbers in the problem with a given ratio? How does this help you to solve the problem? The ratio of boys to girls in a class is 4 : 5. How many pupils could be in the class? How do you know? Give pupils several different simple problems and ask: Which of these problems are linked to, for example the ratio 2: 3. How do you know?  Divide a quantity into two or more parts in a given ratio Level 6 and solve problems involving ratio and direct proportionSolve problem such as: Potting compost is made from loam, peat and sand in the ratio 7:3:2 respectively. A gardener used 1.5 litres of peat to make compost. How much loam did she use? How much sand? The angles in a triangle are in the ratio 6:5:7. Find the sizes of the three angles. If the ratio of boys to girls in a class is 3:1, could there be exactly 30 children in the class? Why? Could there be 25 boys? Why? 5 miles is about the same as 8km. Can you make up some conversion questions that you could answer mentally? Can you make up some conversion questions for which you would have to use a more formal method? How would you work out the answers to these questions?Use proportional reasoning to solve a problem, choosing the correct numbers to take as 100%, or as a wholeUse unitary methods and multiplicative methods, e.g. There was a 25% discount in a sale. A boy paid 30 for a pair of jeans in the sale. What was the original price of the jeans? When heated, a metal bar increases in length from 1.255m to 1.262m. Calculate the percentage increase correct to one decimal place. A recipe for fruit squash for six people is: 300g chopped oranges 1500ml lemonade 750ml orange juice Trina made fruit squash for ten people. How many millilitres of lemonade did she use? Jim used two litres of orange juice for the same recipe. How many people was this enough for? Which are the key words in this problem? How do these words help you to decide what to do? What are the important numbers? What are the important links that might help you solve the problem? How do you decide which number represents 100% or a whole when working on problems? Do you expect the answer to be larger or smaller? Why? What would you estimate the answer to be? Why? Understand and use proportionality Level 7Examples of what pupils know and be able to doProbing questions Sets A, B and C are in direct proportion. Set A Set B Set C 5 11 5 3 2.4 7 1.1 17.6 10.5 Is there sufficient information to find all the missing entries? What is the maximum number of items which could be entered and the task remain impossible? What is the minimum number of entries needed and what is important about their location? For any one empty cell What is the best starting point? How many different starting points are there? Is a 50% increase followed by a 50% increase the same as doubling? Explain your answer. Which is better? A 70% discount or a 50% discount and a further 20% special offer discount? A shop is offering a 10% discount. Is it better to have this before or after VAT is added at 17.5%? Which is the better deal? Buy one get one half price or three for the price of two? How do you go about finding the missing numbers in this table? Miles Kilometres 5 8 ? 20 36 ? Can you do it in a different way? What do you look for in deciding the most efficient way to find the missing numbers? How do you go about checking whether given sets of information are in direct proportion? What hints would you give someone to help them solve word problems involving proportionality? Talk me through the reasoning that took you to this answer. How did you think this through?  Calculate the result of any proportional change using multiplicative methodsExamples of what pupils know and be able to doProbing questionsThe new model of an MP3 player holds1/6 more music than the previous model. The previous model holds 5000 tracks. How many tracks does the new model hold? The previous model cost 119.99 and the new model costs 144.99. Is this less than or greater than the proportional change to the number of tracks? Justify your answer. After one year a scooter has depreciated by 1/7 and is valued at 995. What was its value at the beginning of the year? Weekend restaurant waiting staff get a 4% increase. The new hourly rate is 4.94. What was it before the increase? My friends savings amount to 920 after 7% interest has been added. What was the original amount of her savings before interest was added? Use a multiplicative methods such as: Original Result Multiplier 100% 107% x 100/107  ? 920 x 100/107 How do you go about finding a multiplier to increase by a given fraction (percentage)? What if it was a fractional (percentage) decrease? Why is it important to identify the whole when working with problems involving proportional change? How do you go about finding a multiplier to calculate an original value after a proportional increase/decrease? Given a multiplier how can you tell whether this would result in an increase or a decrease?  Types of Number Recognise and describe number relationships including multiple, factor and square (Level 4)Use the multiples of 4 to work out the multiples of 8. Identify factors of two-digit numbers Know simple tests for divisibility for 2, 3, 4, 5, 6, 8, 9 Find the factors of a number by checking for divisibility by primes. For example, to find the factors of 123 check mentally or otherwise, if the number divides by 2, then 3,5,7,11 Which numbers less than 100 have exactly three factors? What number up to 100 has the most factors? The sum of four even numbers is a multiple of four. When is this statement true? When is it false? Can a prime number be a multiple of 4? Why? Can you give me an example of a number greater than 500 that is divisible by 3? How do you know? How do you know if a number is divisible by 6? etc Can you give me an example of a number greater that 100 that is divisible by 5 and also by 3? How do you know? Is there a quick way to check if a number is divisible by 25? Recognise and use number patterns and relationships (Level 5)Find: A prime number greater than 100 The largest cube smaller than 1000 Two prime numbers that add to 98 Give reasons why none of the following are prime numbers: 4094, 1235, 5121 Use factors, when appropriate, to calculate mentally, e.g. 35 12 = 35 2 6 Continue these sequences: 8, 15, 22, 29, 6, 2, -2, -6, 1, 1D 2, 1D 4, 1D 8, 1, -2, 4, -8 1, 0.5, 0.25 1, 1, 2, 3, 5, 8Talk me through an easy way to do this multiplication/division mentally. Why is knowledge of factors important for this? How do you go about identifying the factors of a number greater than 100? What is the same / different about these sequences: 4.3, 4.6, 4.9, 5.2, 16.8, 17.1, 17.4, 17.7, 9.4, 9.1, 8.8, 8.5, ... Ive got a number sequence in my head. How many questions would you need to ask me to be sure you know my number sequence? What are the questions? 3D Shape Make 3-D models by linking given faces or edges and Level 4 draw common 2-D shapes in different orientations on gridsIdentify and make up the different nets for an open cube. Complete a rectangle which has 2 sides drawn at an oblique angle to the gridWhen presented with a net: Which edge will meet this edge? Which vertices will meet this one? What can you tell me about a 3-D shape from its 2-D net? How would you plot a rectangle that has no horizontal sides on a square grid? How would you convince me that the shape is a rectangle? Visualise and use 2-D representations of 3-D objects Level 6 Visualise solids from an oral description, e.g. Identify the 3-D shape if: The front and side elevations are both triangles and the plan is a square. The front and side elevations are both rectangles and the plan is a circle. The front elevation is a rectangle, the side elevation is a triangle and the plan in a rectangle. Is it possible to slice a cube so that the cross-section is: a rectangle? a triangle? a pentagon? a hexagon? How would the 3-D shape be different if the plan was a rectangle rather than a square? Why? Are there other possible 3-D shapes? Starting from a 2-D representation of a 3-D shape: How many faces will the 3-D shape have? How do you know? What will be opposite this face in the 3-D shape? How do you know? Which side will this side join to make an edge? How do you know? How would you go about drawing the plan and elevation for the 3-D shape you could make from this net?Calculate lengths, areas and volumes in plane shapes and Level 7 right prisms The cross section of a skirting board is the shape of a rectangle, with a quadrant (quarter circle) on the top. The skirting board is 1.5cm thick and 6.5cm high. Lengths totalling 120m are ordered. What volume of wood is contained in the order? Talk me through the steps you took when finding the surface area of this right prism. If you know the height and volume of a right prism, what else do you know? What dont you know? How many different square-based right prisms have a height of 10 cm and a volume of 160 cm3? Why? What do you need to know to be able the find both the volume and surface area of a cylinder? Angles and Triangles Use the properties of 2-D and 3-D shapes Level 4 Recognise and name most quadrilaterals e.g. trapezium, parallelogram, rhombus Recognise right-angled, equilateral, isosceles and scalene triangles and know their properties Use mathematical terms such as horizontal, vertical, parallel, perpendicular Understand properties of shapes, e.g. why a square is a special rectangle Visualise shapes and recognise them in different orientationsCan you describe a rectangle precisely in words so someone else can draw it? What mathematical words are important when describing a rectangle? What properties do you need to be sure a triangle is isosceles; equilateral; scalene? Show a mix of rectangles including squares. Can you tell me which of the shapes are square? Convince me? Can you tell me which shapes are rectangles? Convince me?Use language associated with angle and know and use the Level 5 angle sum of a triangle and that of angles at a pointCalculate missing angles in triangles including isosceles triangles or right angled triangles, when only one other angle is given Calculate angles on a straight line or at a point, such as the angle between the hands of a clock, or intersecting diagonals at the centre of a regular hexagon Understand parallel and perpendicular in relation to edges or faces of 2-D shapes Is it possible to draw a triangle with: i) one acute angle ii) two acute angles iii) one obtuse angle iv) two obtuse angles Give an example of the three angles if it is possible. Explain why if it is impossible. Explain why a triangle cannot have two parallel sides. How can you use the fact that the sum of the angles on a straight line is 180 to explain why the angles at a point are 360? An isosceles triangle has one angle of 30. Is this enough information to know the other two angles? Why?  Identify alternate and corresponding angles; understand a proof Level 6 that the sum of the angles of a triangle is 180 and of a quadrilateral is 360Know the difference between a demonstration and a proof. Understand a proof that the sum of the angles of a triangle is 1800 and of a quadrilateral is 3600. How could you convince me that the sum of the angles of a triangle is 180? Why are parallel lines important when proving the sum of the angles of a triangle? How does knowing the sum of the interior angles of a triangle help you to find the sum of the interior angles of a quadrilateral? Will this work for all quadrilaterals? Why? Area and Perimeter Find perimeters of simple shapes and find areas by counting squares Level 4Find perimeters and areas of shapes other than rectangles. Focus on having a feel for the perimeter and area - not calculating them. Work out the perimeter of some shapes by measuring, in millimetres. Use the terms area and perimeter accurately and consistently Find areas by counting squares and part squares of shapes drawn on squared paper. Begin to find the area of shapes that are made from joining rectangles Use number of squares in a row times number of rows to find the area of a rectangle How do you go about finding the perimeter of a shape? How are the perimeter of a shape and the area of a shape different? How do you remember which is which? Would you expect the area of a paperback book cover to be: 200cm, 600cm, or 6000cm? Explain why. Would you expect the area of a digit card to be: 5 cm, 50cm or 100cm? Explain why. Suggest 2-D shapes/objects where the area could be measured in cm.  Understand and use the formula for the area of a rectangle and Level 5 distinguish area from perimeterFind any one of the area, width and length of a rectangle, given the other two. Find any one of the perimeter, width and length of a rectangle, given the other two. Find the area or perimeter of simple compound shapes made from rectangles The carpet in Walts living room is square, and has an area of 4 m2. The carpet in his hall has the same perimeter as the living room carpet, but only 75% of the area. What are the dimensions of the hall carpet? For a given area (e.g. 24cm2) find as many possible rectangles with whole number dimensions as you can. How did you do it? For compound shapes formed from rectangles: How do you go about finding the dimensions needed to calculate the area of this shape? Are there other ways to do it? How do you go about finding the perimeter? Always, sometimes or never true? If one rectangle has a larger perimeter than another one, then it will also have a larger area. Deduce and use formulae for the area of a triangle and Level 6 parallelogram, and the volume of a cuboid; calculate volumes and surface areas of cuboidsCalculate the areas of triangles and parallelograms Suggest possible dimensions for triangles and parallelograms when the area is known. Calculate the volume and the surface area of a 3cm by 4 cm by 5cm box. Find three cuboids with a volume of 24cm3 Find a cuboid with a surface area of 60cm2 Why do you have to multiply the base by the perpendicular height to find the area of a parallelogram? The area of a triangle is 12cm2. What are the possible lengths of base and height? Right-angled triangles have half the area of the rectangle with the same base and height. What about non-right-angled triangles? What other formulae for the area of 2-D shapes do you know? Is there a formula for the area of every 2-D shape? How do you go about finding the volume of a cuboid? How do you go about finding the surface area of a cuboid? You can build a solid cuboid using any number of interlocking cubes. Is this statement always, sometimes or never true? If it is sometimes true, when is it true and when is it false? For what numbers can you only make one cuboid? For what numbers can you make several different cuboids?Know and use the formulae for the circumference and area of a circleA circle has a circumference of 120cm. What is the radius of the circle? A touring cycle has wheels of diameter 70cm. How many rotations does each wheel make for every 10km travelled? A circle has a radius of 15cm. What is its area? A door is in the shape of a rectangle with a semi-circular arch on top. The rectangular part is 2m high and the door is 90cm wide. What is the area of the door? What is the minimum information you need to be able to find the circumference and area of a circle? Give pupils some work with mistakes. Ask them to identify and correct the mistakes. How would you go about finding the area of a circle if you know the circumference? Calculate lengths, areas and volumes in plane shapes and Level 7 right prismsThe cross section of a skirting board is the shape of a rectangle, with a quadrant (quarter circle) on the top. The skirting board is 1.5cm thick and 6.5cm high. Lengths totalling 120m are ordered. What volume of wood is contained in the order? Talk me through the steps you took when finding the surface area of this right prism. If you know the height and volume of a right prism, what else do you know? What dont you know? How many different square-based right prisms have a height of 10 cm and a volume of 160 cm3? Why? What do you need to know to be able the find both the volume and surface area of a cylinder? Understand the difference between Level 8 formula for perimeter, area and volume in simple contexts by considering dimensionsWork with formulae for a range of 2-D and 3-D shapes and relate the results and dimensions.How do you go about deciding whether a formula is for a perimeter, an area or a volume? Why is it easy to distinguish between the formulae for the circumference and area of a circle? How would you help someone to distinguish between the formula for the surface area of a cube and the volume of a cube? How do you decide whether a number in a calculation represents the length of a dimension?  Compound Measures Understand and use measures of speed Level 7 (and other compound measures such as density or pressure) to solve problemsUse examples of compound measures in science, geography and PE. Understand that: Rate is a way of comparing how one quantity changes with another, e.g. a cars fuel consumption measured in miles per gallon. The two quantities are usually measured in different units, and per, the abbreviation p or an oblique / can be used to mean for every or in every. Solve problems such as: The distance from London to Leeds is 190 miles. An intercity train takes about 2 hours to travel from London to Leeds. What is its average speed? .Make up some easy questions that involve calculating speed, distance or time (density, mass and volume). Make up some difficult questions. What makes them difficult? Talk me through the reasoning of why travelling a distance of 30 miles in 45 minutes is an average of 40mph. How do the units of speed (density, pressure) help you to solve problems? How does the information in a question help you to decide on the units for speed (density, pressure)?  Congruence and Similarity Understand and use congruence and mathematical similarity Level 8Understand and use the preservation of the ratio of side lengths in problems involving similar shapes. Use congruent triangles to prove that the two base angles of an isosceles triangle are equal by drawing the perpendicular bisector of the base. Use congruence to prove that the diagonals of a rhombus bisect each other at right angles. What do you look for when deciding whether two triangles are congruent? What do you look for when deciding whether two triangles are similar? Which of these statements are true? Explain your reasoning. Any two right angled triangles will be similar If you enlarge a shape you get two similar shapes All circles are similar Convince me that: Any two regular polygons with the same number of sides are similar Alternate angles are equal (using congruent triangles) Dimensions Understand the difference between formulae for perimeter, Level 8 area and volume in simple contexts by considering dimensionsWork with formulae for a range of 2-D and 3-D shapes and relate the results and dimensions.How do you go about deciding whether a formula is for a perimeter, an area or a volume? Why is it easy to distinguish between the formulae for the circumference and area of a circle? How would you help someone to distinguish between the formula for the surface area of a cube and the volume of a cube? How do you decide whether a number in a calculation represents the length of a dimension?  Use straight edge and compasses to do standard constructions Level 6Use straight edge and compasses to construct: the mid-point and perpendicular bisector of a line segment the bisector of an angle the perpendicular from a point to a line segment the perpendicular from a point on a line segment. Construct triangles to scale using ruler and protractor (SAS, ASA)and using straight edge and compasses (SSS)Why are compasses important when doing constructions? How do the properties of a rhombus help with simple constructions such as bisecting an angle? For which constructions is it important to keep the same compass arc (distance between the pencil and the point of your compasses)? Why?Find the locus of a point that moves according to a given rule, Level 7 both by reasoning and using ICT.Visualise the result of spinning 2D shapes in 3D around a line that is along a line of symmetry of the shape. Trace the path of a vertex of a square as it is rolled along a straight line. Visualise simple paths such as that generated by walking so that you are equidistant from two trees. Find the locus of: points equidistant from two points, points equidistant from a line, points equidistant from a point the centre of circles which have two given lines as tangents How can you tell for a given locus whether it is the path of points equidistant from another point or a line? What is the same/different about the path traced out by the centre of a circle being rolled along a straight line and the centre of a square being rolled along a straight line? Locus and Constructions Properties of Shapes Use the properties of 2-D and 3-D shapes Level 4 Recognise and name most quadrilaterals e.g. trapezium, parallelogram, rhombus Recognise right-angled, equilateral, isosceles and scalene triangles and know their properties Use mathematical terms such as horizontal, vertical, parallel, perpendicular Understand properties of shapes, e.g. why a square is a special rectangle Visualise shapes and recognise them in different orientationsCan you describe a rectangle precisely in words so someone else can draw it? What mathematical words are important when describing a rectangle? What properties do you need to be sure a triangle is isosceles; equilateral; scalene? Show a mix of rectangles including squares. Can you tell me which of the shapes are square? Convince me? Can you tell me which shapes are rectangles? Convince me?Use a wider range of properties of 2-D and 3-D shapes Level 5 and identify all the symmetries of 2-D shapesUnderstand parallel and perpendicular in relation to edges and faces of 3-D shapes. Find lines of symmetry in 2-D shapes including oblique lines. Recognise the rotational symmetry of familiar shapes, such as parallelograms and regular polygons. Classify quadrilaterals, including trapeziums, using properties such as number of pairs of parallel sides.Sketch me a quadrilateral that has one line of symmetry, two lines, three lines, no lines etc. Can you give me any others? What is the order of rotational symmetry of each of the quadrilaterals you sketched? One of the lines of symmetry of a regular polygon goes through two vertices of the polygon. Convince me that the polygon must have an even number of sides. Sketch a shape to help convince me that: A trapezium might not be a parallelogram A trapezium might not have a line of symmetry Every parallelogram is also a trapezium Classify quadrilaterals by their geometric propertiesKnow the properties (equal and/or parallel sides, equal angles, right angles, diagonals bisected and/or at right angles, reflection and rotation symmetry) of: an isosceles trapezium a parallelogram a rhombus a kite an arrowhead or delta What properties do you need to know about a quadrilateral to be sure it is a kite; a parallelogram; a rhombus; an isosceles trapezium? Can you convince me that a rhombus is a parallelogram but a parallelogram is not necessarily a rhombus? Why cant a trapezium have three acute angles? Which quadrilateral can have three acute angles? Pythagoras Level 7 Understand and apply Pythagoras' theorem when solving problems in 2-D Know that Pythagoras Theorem only holds for triangles that are right-angled. Identify a right angled triangle in a problem Find a missing side in a right-angled triangle. For example: Use Pythagoras theorem to solve simple problems in two dimensions, such as A 5m ladder leans against a wall with its foot 1.5m away from the wall. How far up the wall does the ladder reach You walk due north for 5 miles, then due east for 3 miles. What is the shortest distance you are from your starting point? Identify triangles that must be right-angled from their side-lengths  How do you identify the hypotenuse when solving a problem using Pythagoras theorem? What do you look for in a problem to decide whether it can be solved using Pythagoras theorem? Talk me through how you went about drawing and labelling this triangle for this problem. How can you use Pythagoras theorem to tell whether an angle in a triangle is equal to, greater than or less than 90 degrees? What is the same/different about a triangle with sides 5cm, 12cm and an unknown hypotenuse and a triangle with sides 5cm, 12cm and an unknown shorter side?Level 8 (from trig section) Sketch right angled triangles for problems expressed in words.  How do you decide whether a problem requires use of a trigonometric relationship (sine, cosine or tangent) or Pythagoras theorem to solve it?  Transformations Reflect simple shapes in a mirror line, translate shapes horizontally or vertically and begin to rotate a simple shape or object about its centre or a vertex Level 4Use a grid to plot the reflection in a mirror line presented at 45 for both where the shape touches the mirror line and where it does not Begin to use the distance of vertices from the mirror line to reflect shapes more accurately Give me instructions to reflect this shape into this mirror line. What would you suggest I do first? How do the squares on the grid help when reflecting? Show me. Make up a reflection that is easy to do. Make up a reflection that is hard to do. What makes it hard? How can you tell if a shape has been reflected or translated?Reason about position and movement and transform shapes Level 5Construct the reflections of shapes in mirror lines placed at different angles relative to the shape: Reflect shapes in oblique (45) mirror lines where the shape either does not touch the mirror line, or where the shape crosses the mirror line Reflect shapes not presented on grids, by measuring perpendicular distances to/from the mirror. Reflect shapes in two mirror lines, where the shape is not parallel or perpendicular to either mirror. Rotate shapes, through 90 or 180, when the centre of rotation is a vertex of the shape, and recognise such rotations. Translate shapes along an oblique line. Reason about shapes, positions and movements, e.g. visualise a 3-D shape from its net and match vertices that will be joined visualise where patterns drawn on a 3-D shape will occur on its net Make up a reflection/rotation that is easy to do. Make up a reflection/rotation that is hard to do. What makes it hard? What clues do you look for when deciding whether a shape has been reflected or rotated? What transformations can you find in patterns in flooring, tiling, wallpaper, wrapping paper? What information is important when describing a reflection/rotation? Describe how rotating a rectangle about its centre looks different from rotating it about one of its vertices. How would you describe this translation precisely? Enlarge 2-D shapes, given a centre of enlargement and a positive whole-number scale factor Level 5Construct an enlargement given the object, centre of enlargement and scale factor. Find the centre of enlargement and/or scale factor from the object and image. What changes when you enlarge a shape? What stays the same? What information do you need to complete a given enlargement? If someone has completed an enlargement how would you find the centre and the scale factor? When doing an enlargement, what strategies do you use to make sure your enlarged shape will fit on the paper?Know that translations, rotations and reflections preserve length and angle and map objects onto congruent images Level 6Find missing lengths and angles on diagrams that show an object and its image Match corresponding lengths and angles of object and image shapes following reflection, translation and/or rotation or a combination of these. What changes and what stays the same when you: translate rotate reflect a shape? When is the image congruent? 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