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Pattern and Function - Rules for Patterns

Grade 3IB

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

🔑Concepts

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A pattern rule describes how a sequence of numbers or shapes changes. In a growing pattern, the numbers increase by a constant amount each time. We find the rule by subtracting a term from the term that follows it.

A number sequence 5, 8, 11, 14 showing a jump of +3 between each number.
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In a shrinking pattern, the numbers decrease by a constant amount. To find the rule, we subtract the second term from the first term.

A shrinking pattern 40, 30, 20, 10 showing a subtraction rule.
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An Input-Output table uses a function machine rule where every input is changed by the same mathematical operation (addition or subtraction) to produce an output.

A function machine showing an input of 3 being transformed by rule +7 to output 10.
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Repeating patterns follow a core sequence that repeats over and over. Identifying the core helps predict what comes next in the sequence.

📐Formulae

Term+Rule=Next TermTerm + Rule = Next \space Term

Rule=Term2−Term1Rule = Term_{2} - Term_{1} (for growing patterns)

Rule=Term1−Term2Rule = Term_{1} - Term_{2} (for shrinking patterns)

Output=Input+nOutput = Input + n (where nn is a constant)

Output=Input−nOutput = Input - n (where nn is a constant)

💡Examples

Problem 1:

Find the rule and 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 and second term: 15−12=315 - 12 = 3. Step 2: Check the difference between the second and third term: 18−15=318 - 15 = 3. Step 3: Identify the rule as +3+3. Step 4: Add 33 to the last known term (2121) to get the next number: 21+3=2421 + 3 = 24. Step 5: Add 33 to 2424 to get the following number: 24+3=2724 + 3 = 27.

Explanation:

Since the numbers are increasing, we use subtraction between consecutive terms to find the addition rule. We then apply that rule to extend the pattern.

Problem 2:

Complete the Input-Output table for the rule: Subtract 55 (Input−5Input - 5). If the Inputs are 15,20,2515, 20, 25, what are the Outputs?

Solution:

Step 1: Apply the rule to the first input: 15−5=1015 - 5 = 10. Step 2: Apply the rule to the second input: 20−5=1520 - 5 = 15. Step 3: Apply the rule to the third input: 25−5=2025 - 5 = 20. The Outputs are 10,15,2010, 15, 20.

Explanation:

An Input-Output table applies the same mathematical operation to every input value to generate the corresponding output value.

Problem 3:

Observe the pattern of tiles below. How many tiles will be in the 4th shape? Write the rule.

Geometric pattern of tiles increasing from 1 to 3 to 5.

Solution:

7 tiles7 \space tiles

Explanation:

Shape 1 has 11 tile. Shape 2 has 33 tiles. Shape 3 has 55 tiles. The number of tiles is increasing by 22 each time (1+2=3,3+2=51+2=3, 3+2=5). Following the rule +2+2, the 4th shape will have 5+2=75 + 2 = 7 tiles.

Problem 4:

Identify the missing outputs for the rule 'Subtract 88'. Input values are 50,42,3450, 42, 34.

Input-output table with inputs 50, 42, 34 and corresponding outputs.

Solution:

42,34,2642, 34, 26

Explanation:

Apply the rule Output=Input−8Output = Input - 8 to each input: 50−8=4250 - 8 = 42 42−8=3442 - 8 = 34 34−8=2634 - 8 = 26