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Mathematics: A Powerful Problem-Solving Tool

Explore the usefulness and interesting aspects of mathematics in solving problems. Gain motivation through real-world examples like calculating percentages and understanding the curious property of numbers. Dive deeper to discover the Difference of Squares and its pictorial proof, making math concepts easier to remember and comprehend.

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Mathematics: A Powerful Problem-Solving Tool

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  1. Mathematics: a Powerful Tool for Solving Problems Doug French The Mathematical Association

  2. MATHEMATICS is USEFUL and INTERESTING. MATHEMATICS is a POWERFUL TOOL for solving PROBLEMS. PROBLEMS provide the MOTIVATION for learning MATHEMATICS.

  3. A Problem with Percentages Two cups of coffee at £1.25 with a 10% discount. How much to pay? 10% of £2.50 = 25p Cost of two cups of coffee: £2.50 – 25p = £2.25 10% of £1.25 = 12.5p = 13p to the nearest penny Cost of one cup of coffee:£1.25 – 13p = £1.12 Cost of two cups of coffee: £2.50 – 26p = £2.24

  4. A Curious Property 7 × 9 = 63 and 8 × 8 = 64 1 × 3 = 3 2 × 4 = 8 3 × 5 = 15 4 × 6 = 24 5 × 7 = 35 6 × 8 = 48 7 × 9 = 63 8 × 10 = 80 9 × 11 = 99 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 5 × 5 = 25 6 × 6 = 36 7 × 7 = 49 8 × 8 = 64 9 × 9 = 81 10 × 10 = 100 20 × 22 = 440 and 21 × 21 = 441 n² – 1 = (n – 1)(n + 1)

  5. The Difference of Two Squares A pictorial proof

  6. The Difference of Squares of Two Consecutive Odd Numbers n 1 3 5 7 9 11 13 n² 1 9 25 49 81 121 169 (2n + 1)² – (2n– 1)² = 4n 2 = 8n Difference of squares of two consecutive odd numbers is a multiple of 8. Difference of squares of ANY two odd numbers is a multiple of 8.

  7. REMEMBERING is NOT ENOUGH. Need to know WHY as well as HOW. It is EASIER to REMEMBER when you UNDERSTAND and there is much LESS to REMEMBER.

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