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Pattern and Function - Expressing Patterns as Rules

Grade 5IB

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

🔑Concepts

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A pattern is a recurring sequence that follows a specific logic. In mathematics, we identify the 'Position' (Input) and the 'Value' (Output) to find the relationship between them.

A visual growing pattern where each step adds one square block vertically.
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The rule for a linear pattern often takes the form Output=(Input×m)+cOutput = (Input \times m) + c. The value mm is the 'common difference' (how much the output increases each time), and cc is the adjustment needed to match the first term.

Diagram
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Function machines help visualize how an input is transformed into an output through a specific operation or set of operations.

A function machine diagram showing an input entering a box with a rule and an output exiting.
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Using a table of values is the most effective way to identify the common difference. If the difference between consecutive outputs is constant, the pattern is linear.

A table showing inputs 1, 2, 3 and outputs 5, 8, 11.

📐Formulae

Output=(Input×Common Difference)±AdjustmentOutput = (Input \times \text{Common Difference}) \pm \text{Adjustment}

y=mx+by = mx + b

Common Difference=Termn−Termn−1\text{Common Difference} = \text{Term}_n - \text{Term}_{n-1}

Term=(Position×Multiplier)+ConstantTerm = (Position \times \text{Multiplier}) + \text{Constant}

💡Examples

Problem 1:

Find the rule and the 20th20^{th} term for the sequence: 4,7,10,13,…4, 7, 10, 13, \dots

Solution:

  1. Find the common difference: 7−4=37 - 4 = 3. So, the multiplier is 33.
  2. Test the multiplier on the first position: 1×3=31 \times 3 = 3.
  3. Determine the adjustment: To get from 33 to the actual first term (44), we need to add 11.
  4. Write the rule: y=3x+1y = 3x + 1.
  5. Calculate the 20th20^{th} term: y=(3×20)+1=60+1=61y = (3 \times 20) + 1 = 60 + 1 = 61.

Explanation:

We first identify how much the pattern grows each time to find the multiplier, then adjust the formula to match the starting value of the sequence.

Problem 2:

A pattern of tiles grows according to an Input-Output table where Input 11 gives Output 55, Input 22 gives 99, and Input 33 gives 1313. What is the rule?

Solution:

  1. Identify the change in Output: 9−5=49 - 5 = 4 and 13−9=413 - 9 = 4. The common difference is 44.
  2. Create a trial rule: Output=Input×4Output = Input \times 4.
  3. Check the first entry: 1×4=41 \times 4 = 4. We need the output to be 55, so we add 11.
  4. Final Rule: y=4x+1y = 4x + 1.

Explanation:

By comparing the expected output (from the multiplier) to the actual output in the table, we find the constant that needs to be added or subtracted.

Problem 3:

Observe the pattern of matchsticks used to create triangles. How many matchsticks are needed for the 10th10^{th} figure in the sequence?

A pattern of triangles made of sticks. Figure 1 is 1 triangle, Figure 2 is 2 triangles sharing a side, etc.

Solution:

  1. Identify the values: Figure 1 uses 33 sticks, Figure 2 uses 55 sticks, Figure 3 uses 77 sticks.
  2. Find the difference: The difference between terms is 22 (5−3=25-3=2, 7−5=27-5=2).
  3. Determine the adjustment: Input×2+c=OutputInput \times 2 + c = Output. For Figure 1: 1×2+1=31 \times 2 + 1 = 3. The rule is y=2n+1y = 2n + 1.
  4. Calculate for n=10n = 10: 2×10+1=212 \times 10 + 1 = 21.

Explanation:

We first find the common difference (2) to use as our multiplier. Then we adjust it to fit the first term. Finally, we substitute the position number into our rule.

Problem 4:

A sequence is represented by a function graph. Determine the rule that relates the xx (Input) to the yy (Output) and find the value of yy when x=5x = 5.

A coordinate graph showing points (1,4), (2,6), and (3,8) connected by a straight line.

Solution:

  1. From the graph points: (1,4),(2,6),(3,8)(1, 4), (2, 6), (3, 8).
  2. The difference in yy values is 22.
  3. Rule: y=2x+2y = 2x + 2.
  4. For x=5x = 5: y=2(5)+2=12y = 2(5) + 2 = 12.

Explanation:

By plotting the inputs and outputs on a coordinate plane, we can see they form a straight line. The 'slope' of the line is our multiplier, and where it would cross the y-axis (at x=0x=0) is our constant.