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MathidiaMathematics Academy
Proof, modelling and Year 10 synthesis
Lesson 75 of 75
Review

Year 10 synthesis

Objective. Solve connected non-routine problems across the Year 10 programme.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Year 10 synthesis means tackling problems that weave together several topics from across the course, such as algebra, geometry, probability and proof, within a single unfamiliar question. The strategy is to read carefully, identify which areas of mathematics are involved, and plan a logical order of steps before calculating, since synthesis questions rarely state which method to use directly. For example, a question might require forming an equation from a geometric diagram, then solving it, then interpreting the answer in context. Checking that the final answer makes sense against the original scenario is essential, since synthesis problems often include extra or indirect information.

Worked example

A rectangle has length (x+3) cm and width x cm. Its area is 40 cm². Form and solve an equation to find x, applying synthesis across algebra and geometry.

Diagram of a rectangle
  1. 1Form the equation: x(x+3) = 40
  2. 2Expand and rearrange: x² + 3x − 40 = 0
  3. 3Factorise: (x+8)(x−5) = 0, giving x = −8 or x = 5
  4. 4Reject the negative length, so x = 5

Answer: x = 5

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

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Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

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Common mistake. Students often apply the first method they recognise without checking it fits every part of the question, e.g. solving an equation correctly but forgetting to interpret the answer back in the real-world context asked for, such as rejecting a negative length.

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