The Teaching Of Equations In The Mathematics Curriculum

Mathematics plays an integral role in society today with almost every career using some sort of mathematics (St Paul Public Schools 2011). More importantly mathematics helps the mind to reason and organize complicated problems into clear and concise logical steps; an attribute that is explored in the study of Equations (National Curriculum 1999). Hence the minds of students will develop into logical entities that can resolve difficult situations in their day to day lives. The use of Equations dates back to the 15th century (Wolfram 2011) where Babylonians first solved quadratics in radicals. Through the centuries there has been great advancement into equations, its representations and applications. The knowledge of equations was built upon where the Egyptians delved into linear equations and the Greeks developed the concept of geometric algebra which aided in solving linear equations (Boyer, 1991). Arabs, the Chinese, Indians and Europeans have all had an input in making discoveries relating to equations and all these theories have come under the scope of what is to some extent covered in educational institutions today. This report examines the key attributes of Linear Equations, how it is introduced in the classroom and further built upon as a child progresses into further study and how it is effectively taught in schools from Key stages one to four. The final part of this report highlights any bridges this topic area makes with other topics studied in mathematics, other subjects taught at each key stage and how it is reinforced in a setting outside of the classroom. Why study Linear Equations: Linear equations introduce the concept of representing an equation with an unknown element. Egyptian mathematical methods, extracted from the rhind papyrus showed they could solve problems equivalent to a linear equation in one unknown using rhetorical methods where problems were stated and solved verbally. ()A linear equation is now represented in letters and numbers in the form ax+b=c where x commonly denotes the unknown variable due to its origins from Arabic and Spanish text on algebra. ( )The above is the standard form of a linear equation in one variable, commonly used in the defining a straight line graph. For an equation to be classified as being linear, it must contain no power terms of the unknowns, and is not exacted upon by variable in calculus such as square roots, trigonometric functions or a product of more than one of the unknowns.(kct, 2002) therefore a formal definition of a linear equation is.... () Abiding to these laws however does not prevent a linear equation of having more than one unknown. Hence an equation such as ax+by=c contains two unknown variables that must be found. Such an equation requires more information to solve for both variables thus generating the notion of a system of linear equations, where the number of unknown variables to be found requires an equal number of equations to be able to find each of the unknowns. Because of it`s properties, linear equations are largely used in drawing two line graphs and summarising into words the relationship between two or more elements. Llinear equations can also be transformed into linear inequalities where their solutions are contained within a boundary. This topic area is introduced at key stage 3/4 and paves a way for higher order inequalities. The progression of Linear Inequalities through Key stage one to four At a young age a child's logical understanding is somewhat limited to what they are exposed to by their external environment but after entering the educational system, they structurally develop logical thinking through learning of mathematics. To be able to solve linear equations requires a good understanding in the manipulations of numbers and substantial familiarity with operations; skills that are developed from an early age. Linear equations is represented under the heading of using and applying number in the national curriculum

At key stage 1 (ages 5-7), the national curriculum requires students to be taught problem solving skills, counting and pattern & sequence work. Addition and subtraction operators are also introduced in years 1 and 2 with multiplication and division further into year 2 and 3. At this stage students get the grounding required to be able to solve linear equations. What they experience as being similar to this will be a word problem requiring them to find an unknown number which added to another will give the required total. Any form of algebra is not explicitly introduced at this stage however a good grounding which prepares them for tackling linear equations.

Progressing into key stage 2 (ages 7-11) brings new challenges to the student. Their breadth of knowledge of numbers and its applications increase substantially. Their range of numbers also span to the 100s and 1000s. Real life monetary problems are modelled mathematically to incorporate an aspect of the child`s life outside the classroom. Pupils explore the manipulations of integers, becoming accustomed with symbols such as < (less than), > (greater than) and = (equal), learning to use them correctly. During ks2 pupils use the number system more confidently.( nat cur) hence learning to order numbers both positive and negative Expanding on prior knowledge of the four main operations (+,-, /, x) they are also introduced to brackets and Distributive, associative and commutative laws of addition and multiplication which help them determine the order of operations. Children also consider simple number relationships and formulae, representing in word form then symbols Although these may seem to have no link to linear equations, they equip the pupil with a base knowledge to be able to solve such problems with ease.

Key stage 3 (11-14) introduces an explicit form of linear equations where the national curriculum creates a subsection within its guidelines of what is required to be taught of the subject at this stage. Linear equations are taught as a concise topic at this stage where pupils are introduced to the concept of equations and the use of letter symbols to represent an unknown. Different notations

Key stage 4 ( 14-16) At this stage pupils are academically assessed on all they have learnt so far. Due to differences in a child`s ability the national curriculum has designed a program of study to attend to the needs of each child hence the teaching program is categorised into two tiers: foundation and higher with the higher tier being more challenging. In term of linear equations

Effective teaching of linear equations Linear equations are implicitly introduced into the Maths curriculum in primary teaching. Most primary schools teach pupils three main subjects and use applications such as ICT or investigations to prepare them for secondary education. Because of its implicit nature at the primary stage, it is paramount that pupils have a coherent understanding of the basic concepts of numbers whereas at secondary the subject is explicitly taught and linked to other areas of mathematics. A different teaching style must be adopted at each level of education to ensure effective teaching of the subject area.

Effective teaching of this subject includes the use of games, adopting a cross-curricular approach, use of visuals (graphs),

Linear equations are largely used in graphing due to its linear properties. They are used to represent straight line graphs given a range of unknown values for x. This is further extended into exploring where two lines intercept hence a system of linear equations. Also studied in the curriculum for mathematics are decimals, fractions and patterns and sequences. all these number forms can be used in solving linear equations where the number is a fraction or decimal or a formula is derived to find the next number in a sequence. Apart from links within mathematics, the use of linear equations is also extended to other subject areas taught in schools particulary science balancing chemical equations and also it`s use in the real world