Number and Algebra

Furthermore

This video on Deduction is from Revision Village and is aimed at students taking the IB AA Maths Standard level course

Official Guidance, clarification and syllabus links:

Example: Show that \( \frac{1}{4} + \frac{1}{12} = \frac{1}{3} \). Show that the algebraic generalisation of this is \( \frac{1}{m+1} + \frac{1}{m^2+m} \equiv \frac{1}{m} \)

LHS to RHS proofs require students to begin with the left-hand side expression and transform this using known algebraic steps into the expression on the right-hand side (or vice versa).

Example: Show that \( (x-3)^2+5 \equiv x^2-6x+14 \).

Students will be expected to show how they can check a result including a check of their own results.

Example, Show that the algebraic generalisation of the following is true.

$$ \frac{1}{m+1} + \frac{1}{m^2+m} \equiv \frac{1}{m} $$

Let's start by finding a common denominator for the left-hand side of the equation:

$$ \frac{1}{m+1} + \frac{1}{m(m+1)} $$ $$ \frac{m + 1}{m(m+1)} $$

Now, let's simplify the expression:

$$ \frac{1}{m} $$

It is now evident that the original equation:

$$ \frac{1}{m+1} + \frac{1}{m^2+m} \equiv \frac{1}{m} $$

Is generally true, as the left-hand side simplifies to \( \frac{1}{m} \).

Thus, the algebraic generalisation provided does hold true for all values of \( m \).