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Investigative Mathematics

Investigative Mathematics. Objectives. At the end of the lesson, you will be able to :. Know the difference between a linear series pattern, non-linear series pattern and triangular series pattern. Recognise the three different types of pattern question.

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Investigative Mathematics

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  1. Investigative Mathematics Objectives At the end of the lesson, you will be able to : • Know the difference between a linear series pattern, non-linear series pattern and triangular series pattern. • Recognise the three different types of pattern question. • Solve all the three different types of number pattern questions.

  2. Investigative Mathematics The three different types of pattern questions are : • Linear series pattern • Non – Linear series pattern • Triangular series pattern

  3. Investigative Mathematics Linear Series Pattern • Example 1 • Look at the number pattern below • 4, 10, 16, 22, 28 • Find the 10th term. • Find the 55th term. How do I solve this?

  4. Investigative Mathematics Now, let us take a look at the number pattern again. • 4, 10, 16, 22, 28 • Find the 10th term. • Find the 55th term. Step 1 – Find the difference between each of the number 10 – 4 = 6, 16 – 10 = 6, 22 – 16 = 6 Looking at it closely, the difference we get is always 6. Therefore, the first testing formulae we can derive at this stage is +6n where n is just a term.

  5. 4, 10, 16, 22, 28 • Find the 10th term. • Find the 55th term. Step 2 – testing out the formulae +6(n)  if n = 1 Then, +6(n) = +6(1) = 6 Step 3 – adjusting the formulae When n = 1, my answer should be 4, however using the formulae of +6(n) , I got 6 as the answer. Therefore adjustment to the formulae is needed to tally with the answer. To make the answer to be 4, we need to subtract 2 from the first formulae. We get +6(n) – 2

  6. 4, 10, 16, 22, 28 • Find the 10th term. • Find the 55th term. Step 4 – testing out the new formulae +6(n) – 2  if n = 4, my answer would be 22. Let us test +6(n) - 2 = +6(4) - 2 = 24 – 2 = 22. ( correct answer ) Now, if n = 10, then +6(n) - 2 = +6(10) - 2 = 60 – 2 = 58. Now, if n = 55, then +6(n) - 2 = +6(55) - 2 = 330 – 2 = 328.

  7. Investigative Mathematics Now you try Example 2 3, 7, 11, 15, 19…… a) Find the 25th term. b) Find the 100th term.

  8. Investigative Mathematics Non-Linear Series Pattern • Example 3 • Look at the number pattern below • 1, 4, 9, 16, 25 ……. • Find the 15th term. • Find the 50th term. How do I solve this?

  9. Investigative Mathematics • 1, 4, 9, 16, 25 ……. • Find the 15th term. • Find the 50th term. Step 1 – Look at the pattern 4 – 1 = 3, 9 – 4 = 5, 16 – 9 = 7, 25 – 16 = 9 Looking at it closely, is there a pattern? The answer is NO. So…. How? Step 2 – Look at the pattern again. We realised that 1, 4, 9, 16, 25 is actually…. 1² , 2², 3², 4², 5² Now, if n = 50, then 50² = 50 x 50 = 2500 Now, if n = 15, then 15² = 15 x 15 = 225

  10. Investigative Mathematics Non-Linear Series Pattern • Example 4 …..Now you try. • Look at the number pattern below • 2, 5, 10, 17, 26 ……. • Find the 12th term. • Find the 48th term. How do I solve this?

  11. Investigative Mathematics Triangular Series Pattern What is Triangular series? Look at the pattern in the handout given to you. • Example 5 • Look at the number pattern below • 3, 9, 18, 30, • Find the 12th term. • Find the 48th term. How do I solve this?

  12. Example 5 • Look at the number pattern below • 3, 9, 18, 30, • Find the 12th term. • Find the 48th term. Step 1  3 , 9, 18, 30…. Divide all numbers by three Step 2  1, 3, 6, 10 …. Now, look at the difference between each number. 3 - 1 = 2, 6 - 3 = 3, 10 – 6 = 4  now refer to the handout back. This pattern tells you that it is a triangular series. Step 3  In a triangular series, there is a general formulae that is n ( n + 1 ) / 2

  13. Example 5 • Look at the number pattern below • 3, 9, 18, 30, ( ÷ 3 )  1, 3, 6, 10 • Find the 12th term. • Find the 48th term. Step 4  test the formulae. If n = 1, n ( n + 1 ) / 2 = 1 (1 + 1 ) / 2 = 1 If n = 3, n ( n + 1 ) / 2 = 3 (3 + 1 ) / 2 = 6

  14. Example 5 • Look at the number pattern below • 3, 9, 18, 30, ( ÷ 3 )  1, 3, 6, 10 • Find the 12th term. • Find the 48th term. a) If n = 12, n ( n + 1 ) / 2 = 12 (12 + 1 ) / 2 = 78 However, since earlier on we divide it by 3, to get the answer, we have to multiply it by 3 back. Therefore, the answer would be, 78 x 3 = 234 b) If n = 48, n ( n + 1 ) / 2 = 48 (48 + 1 ) / 2 = 1176 1176 x 3 = 3528

  15. Investigative Mathematics Now take a look at the question that you have in the remediation paper

  16. Conclusion • You have learnt • the difference between a linear series pattern, non-linear series pattern and triangular series pattern. • to recognise the three different types of pattern question. • how to solve all the three different types of number pattern questions.

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