Investigative Mathematics

Investigative Mathematics

Objectives

At the end of the lesson, you will be able to :

1. Know the difference between a linear series pattern, non-linear series

pattern and triangular series pattern.

2. Recognise the three different types of pattern question.

3. Solve all the three different types of number pattern questions.


The three different types of pattern questions are :

1. Linear series pattern

2. Non – Linear series pattern

3. Triangular series pattern


Linear Series Pattern

Example 1

Look at the number pattern below

4, 10, 16, 22, 28

a)Find the 10th term.b)Find the 55th term.

How do I solve this?

Now, let us take a look at the number pattern again.


4, 10, 16, 22, 28


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.

Step 2 – testing out the formulae

+6(n) if n = 1 Then, +6(n) = +6(1) = 6

4, 10, 16, 22, 28


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

4, 10, 16, 22, 28


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.


+6(n) - 2 = +6(55) - 2 = 330 – 2 = 328.

Investigative MathematicsNow you try

Example 2

3, 7, 11, 15, 19……

a) Find the 25th term.

b) Find the 100th term.


Non-Linear Series Pattern

Example 3


1, 4, 9, 16, 25 …….



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?

1, 4, 9, 16, 25 …….



Step 2 – Look at the pattern again. We realised that 1, 4, 9, 16, 25 is actually…. 1² , 2², 3², 4², 5²


15² = 15 x 15 = 225


50² = 50 x 50 = 2500


Non-Linear Series Pattern

Example 4 …..Now you try.


2, 5, 10, 17, 26 …….




Triangular Series Pattern

What is Triangular series?

Look at the pattern in the handout given to you.

Example 5


3, 9, 18, 30,



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

Example 5


3, 9, 18, 30,


Example 5


3, 9, 18, 30, ( ÷ 3 ) 1, 3, 6, 10


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

Example 5


3, 9, 18, 30, ( ÷ 3 ) 1, 3, 6, 10


a) If n = 12,

n ( n + 1 ) / 2 = 12 (12 + 1 ) / 2 = 78However, 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

Now take a look at the question that you have in the remediation paper


Conclusion

You have learnt

1. the difference between a linear series pattern, non-linear series pattern

and triangular series pattern.

2. to recognise the three different types of pattern question.

3. how to solve all the three different types of number pattern questions.

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Investigative Mathematics