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Patterns and Function - Functional Relationships

Grade 4IB

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

🔑Concepts

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A function is like a machine that takes an input, applies a consistent rule (operation), and produces a specific output. The rule remains the same for every number in a sequence or pattern.

A flow diagram showing an input value going into a rule 'Add 5' to produce an output.
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Pattern rules can be additive (Input+n=OutputInput + n = Output) or multiplicative (Input×n=OutputInput \times n = Output). We identify the rule by looking for a relationship that works for every pair of numbers provided in a set.

A function table showing input 2 and output 6 with the rule 'times 3'.
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Functional relationships can be represented graphically. When the input (xx) increases, the output (yy) often increases in a straight line if the rule involves a constant addition or multiplication factor.

A graph showing points (1,1), (2,2), and (3,3) forming a straight line relationship.
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Inverse relationships exist where the rule might involve subtraction or division. For example, if the input is the total and the output is the remainder after sharing equally.

📐Formulae

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

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

Input−n=Output\text{Input} - n = \text{Output}

y=x±ny = x \pm n

y=x×ny = x \times n

💡Examples

Problem 1:

Look at the following pattern in a function table: Input (2,5,82, 5, 8), Output (10,13,1610, 13, 16). What is the rule, and what would the output be if the input is 1212?

Solution:

Step 1: Find the difference between the first pair: 10−2=810 - 2 = 8. Test the rule +8+8 on the second pair: 5+8=135 + 8 = 13. Test it on the third pair: 8+8=168 + 8 = 16. The rule is +8+8. Step 2: Apply the rule to the new input: 12+8=2012 + 8 = 20.

Explanation:

To solve functional relationship problems, compare the input and output in each row to find a consistent operation. Once the rule is identified, apply it to the target number.

Problem 2:

A pattern of shapes uses 44 toothpicks for 11 square, 88 toothpicks for 22 squares, and 1212 toothpicks for 33 squares. How many toothpicks are needed for 1010 squares?

Solution:

Step 1: Identify the relationship between the number of squares (Input) and toothpicks (Output). 1→41 \rightarrow 4, 2→82 \rightarrow 8, 3→123 \rightarrow 12. Step 2: Determine the rule. Since 1×4=41 \times 4 = 4 and 2×4=82 \times 4 = 8, the rule is Input×4=Output\text{Input} \times 4 = \text{Output}. Step 3: Calculate for 1010 squares: 10×4=4010 \times 4 = 40 toothpicks.

Explanation:

This is a growing pattern where the output is a multiple of the input. Identifying the multiplier allows us to predict much larger terms in the sequence.

Problem 3:

A baker uses 33 cups of flour for every 11 loaf of bread he makes. If he makes 22 loaves, he uses 66 cups, and for 33 loaves, he uses 99 cups. Identify the rule and determine how many cups of flour are needed for 1515 loaves.

Table showing loaves 1 and 2 mapped to flour cups 3 and 6.

Solution:

15×3=4515 \times 3 = 45

Explanation:

Looking at the relationship: 1→31 \rightarrow 3, 2→62 \rightarrow 6, 3→93 \rightarrow 9. The output is always 33 times the input. The rule is Input×3=OutputInput \times 3 = Output. Therefore, for 1515 loaves, we calculate 15×3=4515 \times 3 = 45.

Problem 4:

Observe the sequence of triangles. The first figure has 33 sides, the second (two triangles side-by-side sharing no edges) has 66 sides, and the third has 99 sides. If this pattern continues, how many sides will the 77th figure have?

Diagram showing figure 1 with one triangle and figure 2 with two separate triangles.

Solution:

7×3=217 \times 3 = 21

Explanation:

Each figure adds one more triangle. Since each triangle has 33 sides, we multiply the figure number (input) by 33 to find the total sides (output). For the 77th figure, 7×3=217 \times 3 = 21.