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Pattern and Function - Number Patterns and Sequences

Grade 3IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A number sequence is an ordered list of numbers that follows a specific pattern or rule. In an increasing sequence, each term is larger than the previous one, often found by adding a constant value.

A sequence showing 5, 10, 15, 20 with an arrow indicating a rule of plus 5.
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Repeating patterns use a core unit of shapes, colors, or numbers that repeats over and over. For example, a 2,4,2,42, 4, 2, 4 pattern repeats the pair (2,4)(2, 4).

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Decreasing sequences are patterns where the numbers get smaller. This usually involves a 'Subtract' rule.

A decreasing sequence 50, 40, 30, 20 showing a subtract 10 rule.
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Function machines (Input/Output) apply a consistent rule to any number put into them. If the rule is ×2\times 2, then an input of 33 becomes an output of 66.

📐Formulae

Term1+Rule=Term2Term_{1} + Rule = Term_{2}

Term1−Rule=Term2Term_{1} - Rule = Term_{2}

Input×Rule=OutputInput \times Rule = Output

Input+Rule=OutputInput + Rule = Output

💡Examples

Problem 1:

Identify the rule and find the next two numbers in the sequence: 12,15,18,21,…12, 15, 18, 21, \dots

Solution:

Step 1: Find the difference between the first two numbers: 15−12=315 - 12 = 3. Step 2: Check if this works for the next pair: 18−15=318 - 15 = 3. The rule is +3+3. Step 3: Add 33 to the last known number: 21+3=2421 + 3 = 24. Step 4: Add 33 to that result: 24+3=2724 + 3 = 27. The next two numbers are 2424 and 2727.

Explanation:

To solve a growing pattern, calculate the difference between consecutive terms to find the constant addition rule.

Problem 2:

An Input/Output machine uses the rule 'Subtract 77'. If the Inputs are 20,25,20, 25, and 3030, what are the corresponding Outputs?

Solution:

Step 1: Apply the rule to the first input: 20−7=1320 - 7 = 13. Step 2: Apply the rule to the second input: 25−7=1825 - 7 = 18. Step 3: Apply the rule to the third input: 30−7=2330 - 7 = 23. The outputs are 13,18,2313, 18, 23.

Explanation:

In a function machine, every input number is transformed by the same mathematical operation to produce an output.

Problem 3:

Observe the growing triangle pattern below. How many dots will be in the 4th4^{th} triangle in the sequence?

Triangular dot patterns showing 1 dot, 3 dots, and 6 dots.

Solution:

The sequence of dots is 1,3,6,…1, 3, 6, \dots The 1st1^{st} triangle has 11 dot. The 2nd2^{nd} triangle has 1+2=31 + 2 = 3 dots. The 3rd3^{rd} triangle has 3+3=63 + 3 = 6 dots. Following the pattern, the 4th4^{th} triangle will have 6+4=106 + 4 = 10 dots.

Explanation:

This is a growing pattern where we add 22, then 33, then 44 to find the next term. These are known as triangular numbers.

Problem 4:

A function machine follows the rule 'Add 100100'. If the machine produces the outputs 450,550,450, 550, and 650650, what were the original inputs?

Function machine diagram with an input arrow, a rule box showing +100, and an output arrow.

Solution:

Output 450−100=350450 - 100 = 350 Output 550−100=450550 - 100 = 450 Output 650−100=550650 - 100 = 550 The inputs were 350,450,350, 450, and 550550.

Explanation:

To find the input when the output and rule are known, we perform the inverse (opposite) operation. The inverse of adding 100100 is subtracting 100100.