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    Unpacking the big idea of equivalence
    (World Scientific, 2021)
    Equivalence is one of the eight clusters of big ideas proposed for the 2020 secondary school mathematics syllabus in Singapore. In this chapter, I will unpack the meaning of equivalence: the idea of an equivalence relation, the differences between equivalence and equality, and the usefulness of equivalent equations and equivalent statements in solving mathematical problems. In particular, I will examine the solutions of some types of equations which do not seem to produce equivalent equations in subsequent steps, e.g. the introduction of an extraneous solution when solving some equations involving surds or logarithms, and the elimination of a variable when solving a pair of simultaneous equations in two variables, and explain how the solutions of these equations can still produce equivalent equations in subsequent steps. In other words, the transformation or conversion from one equation to another equivalent equation is still the basis of the method for solving any kinds of equations. In addition, this chapter will discuss how to teach secondary school students the big idea of equivalence without teaching the abstract idea of an equivalence relation.
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