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Proof in mathematics challenges students to extend their prior learning through precise mathematical reasoning. This is often achieved through the use of algebraic notation to generalise nonspecific cases.

An example of differentiating with proof could be:

- Rather than calculating missing angles in parallel lines students could prove the alternate, corresponding and interior angle theorem using other geometrical properties.
- Instead of calculating the perimeter of rectangles students could derive the formula by generalising the length and width.
- The Quadratic Formula could be derived by completing the square of the general case.
- When teaching recurring decimals students could prove why 0.999 can be considered to be equal to 1.

One of the things that I like most about mathematics is its continuous nature. Linking various aspects together often serves to consolidate the student’s understanding of each topic while adding depth to the learning objective

An example of differentiation by working the problem in reverse could be:

- Rather than working out the area of rectangles students could investigate possible rectangles with a fixed perimeter.
- When solving angle problems the unknown angles could be algebraic expressions so students have to derive and solve equations using geometrical facts.
- Investigate prime and square numbers when learning about factor pairs.

Applying a simple concept to a series of more challenging questions can increase the pace and challenge of a lesson. Students can use the success criteria of a lesson to create a series of questions for their peers to attempt. The person who created the questions would later mark and feedback to their partner who attempted to solve them.

For example:

- When calculating the area of a triangle an easy question could involve a right angled triangle whereas a more difficult problem could involve a compound shape.

- When factorising expressions an easy question could involve a linear and constant term whereas a hard question might consider negative powers.
- If the learning objective is to calculate the mean average an easy question would involve a simple list of positive integers. A difficult question could be to calculate unknown data values using the mean and range.

Whether its angle geometry, ratio or algebra mathematics is full of patterns. Learning to look for them and understanding their origins is key to a deeper level of understanding. They illustrate how all of mathematics is connected and can be generalised.

Some examples could be:

- The constant term in a quadratic is the product of the constant in a pair of polynomials and the coefficient of B is their sum. Could this be generalised for cubics and so on…
- The number of edges of a polygon can be calculated by considering the pattern of interior angles for all regular polygons. The exterior angle of polygons has a relationship with angles on a straight line.
- The Fibonacci sequence links to ratio the number Phi.

How do you differentiate in a maths lesson?

Comments and ideas are very welcome.

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