Number and Algebra: Equations and expressions
Level 1
AO1: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures. This means students will explain the number strategies they use to others using a combination of words, numbers and pictures. This implies that students will learn to write equations to express their findings, for example 5 + 9 = 14, to express their ideas using their own language in conjunction with mathematical language, e.g. add, subtract, times, fraction, and to develop diagrams to represent their strategies, for example set diagrams or number lines.
Level 2
AO1: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols. This means students will be able to use words, symbols and diagrams to explain their number strategies to others. Recording also allows students to think through solutions to problems and allows them to reduce their working memory load by storing information in written form. This is particularly important for the solving of complex, multi-step problems. Students should be able to write the numerals for whole numbers, to 1000, and simple fractions. They should also be able to write addition, subtraction, multiplication and division equations with understanding of the meaning of these operations and of the equals sign as meaning “equal to”. Similarly they should know which operation to perform on a calculator if the numbers are beyond their mental range. Students should also be familiar with using empty number lines to record addition and subtraction strategies and of drawing arrays to record simple multiplication and division strategies. Formal written algorithms for multi-digit addition and subtraction should not be taught at Level Two until students have the place value knowledge required to understand them.
Level 3
AO1: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality. This means students will use words, symbols and diagrams to explain their number strategies to others. Recording also allows students to think through solutions to problems and allows them to reduce their working memory load by storing information in written form. This is particularly important for the solving of complex, multi-step problems. Students should be able to write the numerals for whole numbers to 1 000 000 at least, simple fractions, percentages and decimals. They should also be able to write addition, subtraction, multiplication and division equations with understanding of the meaning of these operations and of the equals sign as meaning “equal to”. Similarly, they should know which operation to perform on a calculator if the numbers are beyond their mental range. Students should also be familiar with using empty number lines to record addition and subtraction strategies, arrays to record multiplication and division strategies, and strip diagrams or double number lines to solve problems with fractions and percentages. Formal written algorithms for multi-digit addition and subtraction should be taught at Level Three after students have the nested place value knowledge required to understand them.
Level 4
AO1: Form and solve simple linear equations. This means students will solve simple linear equations in the form y = mx + c, where x and y are related variables and where m is a whole number and c is an integer, e.g. q = 3p – c, or a + 5 = 4b. When the value of one variable is given the value of the other can be found by solving the equation, e.g. 3p – 6 = 18. Students should understand the equals sign as a statement of balance and know what operations to both sides of an equation preserve that balance, e.g. take off the same number from both sides. At Level Four students should be able to find the required value using both sensible estimation and improvement, and by formal methods of applying inverse operations, e.g. 3p – 6 = 18 so 3p = 24 (adding six to both sides) so p = 8 (dividing both sides by three).
Level 5
AO1: Form and solve linear and simple quadratic equations.
Students should be able to form the linear equation or simple quadratic (y = ax2 or y = x2 ± c, a and c are integers) to model a given situation (see patterns and relationships). They should understand that solving an equation involves finding the value of a variable when the other variable is defined, and interpret how the solution relates to the original context. Students should be able to solve linear and simple quadratic equations by applying inverse operations with an understanding of the equals sign as a statement of transitive balance, for example (3q + 7)/4 = 16, by multiplying both sides by four, subtracting seven, etc. They should also recognise where it is appropriate to solve an equation through trial and improvement, and find the missing value by systematic calculation.
Level 6
AO1: Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns. This means students will create equations to model everyday situations, e.g. express a taxi charge as a linear equation (flagfall and kilometre rate) or the exponential relationship between the number of repeated folds (in thirds) of a paper strip and the number of sections formed. This includes forming pairs of simultaneous linear equations. Students should be able to form equations from tables of values, using differences between terms, constant first order for linear relations, constant second order differences for quadratic relations and constant ratio for simple exponentials. They should use algebraic manipulation skills to simplify expressions, including rational expressions involving exponents, e.g. 9n4 / 6n3. Students should apply their manipulations skills to solve linear and quadratic equations by applying inverse operations with an appreciation of equality and connect their solutions to corresponding situations of inequality, e.g. If (6x - 8)/4 = 10 has the solution x = 8 then (6x - 8) /4 < 10 has the solution x < 8. They should be able to solve quadratic equations by factorising and have the disposition and capability to check all of their algebraic solutions by substituting values. Solving simple exponential equations should be done by inspection at this level, e.g. 3x = 81 by recognising 34 = 81 so x = 4. Pairs of simultaneous equations may be solved by substitution, elimination and by intercept of graphs.
