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Patterns and Function - Identifying and Extending Number Patterns

Grade 4IB

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

🔑Concepts

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A number pattern is a sequence of numbers that follows a specific rule. For example, in an increasing pattern like 5,10,15,205, 10, 15, 20, the rule is to add 55 each time. We can visualize this using steps where each step increases by a fixed amount.

A bar chart showing an increasing pattern of 5, 10, 15, 20 with labels showing a constant increase of 5.
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Input-Output tables show the relationship between two sets of numbers based on a function rule. If the rule is Output=Input−3\text{Output} = \text{Input} - 3, every input number will decrease by 33 to produce the output.

A flowchart showing the process of an input number passing through a rule to become an output number.
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Patterns can also involve multiplication. In a geometric sequence, each term is found by multiplying the previous term by a constant number, such as 2,4,8,162, 4, 8, 16 where the rule is ×2\times 2.

Visual groups of circles representing the numbers 1, 2, and 4 to show a doubling pattern.
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Identifying a missing term requires finding the common difference or common ratio between consecutive terms. For the sequence 50,40,30,…50, 40, 30, \dots, we calculate 40−50=−1040 - 50 = -10, so the rule is to subtract 1010.

📐Formulae

Next Term=Previous Term±Change\text{Next Term} = \text{Previous Term} \pm \text{Change}

Output=Input+n\text{Output} = \text{Input} + n

Output=Input×n\text{Output} = \text{Input} \times n

Difference=Term2−Term1\text{Difference} = \text{Term}_{2} - \text{Term}_{1}

💡Examples

Problem 1:

Identify the rule and find the next two terms in the sequence: 14,21,28,35,…14, 21, 28, 35, \dots

Solution:

Step 1: Find the difference between the first two terms: 21−14=721 - 14 = 7. Step 2: Check if this works for the next pair: 28−21=728 - 21 = 7. The rule is 'Add 77'. Step 3: Add 77 to the last known term: 35+7=4235 + 7 = 42. Step 4: Add 77 to that result: 42+7=4942 + 7 = 49.

Explanation:

By subtracting consecutive terms, we determine the constant additive rule. We then apply this rule repeatedly to extend the sequence.

Problem 2:

Complete the Input-Output table if the rule is Output=Input×4\text{Output} = \text{Input} \times 4. Input values are 2,5,102, 5, 10.

Solution:

Step 1: For Input 22, calculate 2×4=82 \times 4 = 8. Step 2: For Input 55, calculate 5×4=205 \times 4 = 20. Step 3: For Input 1010, calculate 10×4=4010 \times 4 = 40. The outputs are 8,20,408, 20, 40.

Explanation:

This example uses a multiplicative function rule. Each input is multiplied by the same constant (44) to find its corresponding output.

Problem 3:

Observe the pattern of tiles shown below. If the pattern continues, how many tiles will be in the 4th4^{th} figure? Find the rule.

Two figures made of square tiles. Figure 1 has 3 tiles in an L-shape. Figure 2 has 5 tiles.

Solution:

Figure 1=3 tiles\text{Figure 1} = 3 \text{ tiles} Figure 2=5 tiles\text{Figure 2} = 5 \text{ tiles} Figure 3=7 tiles\text{Figure 3} = 7 \text{ tiles} The rule is: +2\text{The rule is: } + 2 Figure 4=7+2=9 tiles\text{Figure 4} = 7 + 2 = 9 \text{ tiles}

Explanation:

By counting the squares in each figure, we see the sequence is 3,5,73, 5, 7. The difference between each term is 22. Adding 22 to the last known term (77) gives us 99.

Problem 4:

A function machine takes an input xx and produces an output yy. If for input 33 the output is 1515, and for input 66 the output is 3030, identify the rule and find the output for input 1010.

A function machine diagram with input 10, the rule 'multiply by 5' inside the box, and a question mark for the output.

Solution:

Input 3→15  ⟹  3×5=15\text{Input } 3 \rightarrow 15 \implies 3 \times 5 = 15 Input 6→30  ⟹  6×5=30\text{Input } 6 \rightarrow 30 \implies 6 \times 5 = 30 Rule: Output=Input×5\text{Rule: } \text{Output} = \text{Input} \times 5 For Input 10:10×5=50\text{For Input } 10: 10 \times 5 = 50

Explanation:

We compare the input to the output. Since 1515 is 55 times 33, and 3030 is 55 times 66, the relationship is a multiplication by 55. Applying this to 1010 gives 5050.