Supporting planning for number in KS4

Transcription

1 Supporting planning for number in KS4 Objectives and key indicators This document provides objectives to support planning for number in Key Stage 4. The objectives in the first column are intended to help pupils progress from level 5 at the end of Key Stage 3 to grade C at the end of Key Stage 4. Those in the second column are intended to help pupils progress from level 6 to grade B, and those in the third column are intended to help pupils progress from level 7 to grade A/A*. Objectives that are highlighted in blue are additional to those that have been taken from the Framework for teaching mathematics: Years 7, 8 and 9. Objectives that are highlighted in bold are key indicators of the target grade. These can be useful in tracking pupils progress.

2 Using and applying number Problem solving Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics across a range of contexts: number, algebra, shape, space and measures, and handling data; generate fuller solutions. Solve substantial problems by breaking them into simpler tasks, using a range of efficient techniques, methods and resources, including ICT; give solutions to an appropriate degree of accuracy. Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics across a range of contexts: number, algebra, shape, space and measures, and handling data; generate fuller solutions. Select and justify appropriate degrees of accuracy for answers to problems. Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics across a range of contexts: number, algebra, shape, space and measures, and handling data; generate fuller solutions. Recognise limitations in the accuracy of measurements and answers to problems. Communicating Communicate methods and techniques clearly through Use mathematical symbols and expressions effective use of expressions, symbols and diagrams. consistently to communicate the method (or steps) towards a solution to a problem or calculation. Use mathematical language and symbols effectively in presenting a convincing reasoned argument. Reasoning Justify solutions, showing some insight into the Explain and justify solutions, showing Justify solutions, showing some insight into the mathematical structure of the problem, e.g. explaining multiplicative methods to find percentage increases and decreases. mathematical structure of the problem, e.g. explaining how multiplicative relationships are used to identify and solve problems involving direct proportion. understanding of the mathematical structure to complex problems. Know when, and explain why, solutions are exact.

3 Numbers and the number system Use rounding to make estimates; round numbers to the nearest whole number or to one or two decimal places. Use rounding to make estimates; round numbers to the nearest whole number or to one, two or three decimal places. Understand upper and lower bounds. Round to one or two significant figures. Round to a given number of significant figures. Estimate calculations by rounding to one significant figure and multiplying and dividing mentally. Use the concepts and vocabulary of common factor, highest common factor, lowest common multiple, prime number and prime factor decomposition. Use significant figures to approximate answers when multiplying or dividing large numbers. Understand how errors can be compounded in calculations. Use index notation for integer powers and simple instances of index laws; know and use the index laws for multiplication and division of positive integer powers. Understand that n 0 = 1 and n -1 = 1/n for positive integers n. Extend understanding of index notation to negative and fractional powers, recognising that the index laws can be applied to these as well. Know that n 1/2 = n and n 1/3 = 3 n for any positive number n. Use inverse operations, understanding that the inverse operation of raising a positive number to power n is raising the result of this operation to power 1 /n. Understand and use rational and irrational numbers. Express numbers in standard index form, both in conventional notation and on a calculator display. Convert between ordinary and standard index form representations. Use standard index form, expressed in conventional notation and on a calculator display; know how to enter numbers in standard index form. Recognise that recurring decimals are exact fractions Distinguish between fractions with denominators that Use an algebraic method to convert a recurring and that some exact fractions are recurring decimals. have only prime factors 2 or 5 (which are represented decimal to a fraction. by terminating decimals), and other fractions (which are represented by recurring decimals).

4 Calculations Understand the effects of multiplying and dividing by numbers between 0 and 1. Recognise and use reciprocals; understand reciprocal as multiplicative inverse; know that any non-zero number multiplied by its reciprocal is 1, and that zero has no reciprocal because division by zero is not defined. Add and subtract fractions by writing them with a common denominator. Multiply and divide a given fraction by an integer, by a unit fraction and by a general fraction. Understand and use efficient methods to add, subtract, multiply and divide fractions, interpreting division as a multiplicative inverse. Use the equivalence of fractions, decimals and percentages to compare proportions. Understand and use proportional changes expressed as fractions, decimals, percentages and ratios. Calculate percentages and find the outcome of a given percentage increase or decrease. Calculate an original amount when given the transformed amount after a percentage change. Understand the multiplicative nature of percentages as operators. Use calculators for reverse percentage calculations by doing an appropriate division. Use proportional reasoning to solve a problem, Calculate an unknown quantity from quantities that Understand and use direct and inverse choosing the correct numbers to take as 100%, or vary in direct proportion. proportion. as a whole.

5 Calculations Use a multiplier raised to a power to represent Divide a quantity into two or more parts in a given ratio; use the unitary method to solve word problems involving ratio and direct proportion. and solve problems involving repeated proportional change, e.g. compound interest. Use calculators to explore exponential growth and decay, using a multiplier and the power key. Use standard index form to make sensible estimates for calculations involving multiplication and/or division. Use calculators efficiently and appropriately to perform complex calculations with numbers of any size, knowing not to round during intermediate steps of a calculation. Calculate with standard index form, using a calculator as appropriate. Use calculators, or written methods, to calculate the upper and lower bounds of calculations, in a range of contexts, particularly when working with measurements. Use an extended range of function keys: the constant, sign change key, function keys for powers, roots, reciprocals, fractions, brackets and memory. Use surds and π in exact calculations, without a calculator; rationalise a denominator such as 1 / 3 = 3 / 3.