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Patterns and Function - Input and Output Tables

Grade 4IB

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

🔑Concepts

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An input-output table shows a relationship between two sets of numbers. Each number that enters the 'function machine' (the input) is changed into a different number (the output) based on a specific, consistent rule.

A flow diagram showing an input x passing through a function rule to become an output y.
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The rule is the mathematical operation (addition, subtraction, multiplication, or division) that is applied to every input to get its corresponding output. For example, if the rule is +5+ 5, every input value must have 55 added to it.

A box showing a rule of plus 7 turning an input of 3 into an output of 10.
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To find a missing value in a table, first identify the pattern by comparing the input and output pairs. Check if the output is getting larger (suggests addition or multiplication) or smaller (suggests subtraction or division) relative to the input.

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The same rule must work for every single row in the table. If a rule works for the first row but not the second, it is not the correct rule for that function.

📐Formulae

Output=Input+kOutput = Input + k

Output=Input−kOutput = Input - k

Output=Input×kOutput = Input \times k

Output=Input÷kOutput = Input \div k

y=x±ky = x \pm k

y=x×ky = x \times k

💡Examples

Problem 1:

Look at the following table and find the rule and the missing output: Input (xx): 2,4,6,82, 4, 6, 8 | Output (yy): 10,20,30,?10, 20, 30, ?

Solution:

Step 1: Compare the first pair. 2→102 \rightarrow 10. Possible rules are 2+8=102 + 8 = 10 or 2×5=102 \times 5 = 10. Step 2: Test both rules on the second pair. 4+8=124 + 8 = 12 (Incorrect, the output is 2020). 4×5=204 \times 5 = 20 (Correct). Step 3: Verify with the third pair. 6×5=306 \times 5 = 30 (Correct). Step 4: Apply the rule to find the missing value. 8×5=408 \times 5 = 40.

Explanation:

By checking multiple pairs, we confirmed that the rule is 'Multiply by 55'. Therefore, the missing output for the input 88 is 4040.

Problem 2:

Identify the rule for this function table: Input (xx): 15,20,25,3015, 20, 25, 30 | Output (yy): 8,13,18,238, 13, 18, 23.

Solution:

Step 1: Observe the change. The output is smaller than the input, so the rule involves subtraction or division. Step 2: Test subtraction for the first pair. 15−7=815 - 7 = 8. Step 3: Test the same rule for the next pairs. 20−7=1320 - 7 = 13 (Correct), 25−7=1825 - 7 = 18 (Correct), and 30−7=2330 - 7 = 23 (Correct).

Explanation:

Because subtracting 77 from every input value consistently results in the correct output value, the rule is y=x−7y = x - 7.

Problem 3:

Determine the missing output value for the following function table:

Input (x)Output (y)12316420524?\begin{array}{|c|c|} \hline \text{Input (}x\text{)} & \text{Output (}y\text{)} \\ \hline 12 & 3 \\ \hline 16 & 4 \\ \hline 20 & 5 \\ \hline 24 & ? \\ \hline \end{array}

A function table with input x and output y showing 20 becoming 5 and 24 needing an output.

Solution:

24÷4=624 \div 4 = 6

Explanation:

Compare the inputs and outputs: 12→312 \rightarrow 3, 16→416 \rightarrow 4, and 20→520 \rightarrow 5. In each case, the output is smaller than the input. Dividing the input by 44 gives the output (12÷4=312 \div 4 = 3, 16÷4=416 \div 4 = 4). Therefore, the rule is y=x÷4y = x \div 4. Applying this to the last row, 24÷4=624 \div 4 = 6.

Problem 4:

Identify the rule and find the missing input value for this table:

Input (x)Output (y)318530?42954\begin{array}{|c|c|} \hline \text{Input (}x\text{)} & \text{Output (}y\text{)} \\ \hline 3 & 18 \\ \hline 5 & 30 \\ \hline ? & 42 \\ \hline 9 & 54 \\ \hline \end{array}

Diagram showing an unknown input passing through a multiply by 6 rule to result in 42.

Solution:

42÷6=742 \div 6 = 7

Explanation:

Look at the relationship: 3×6=183 \times 6 = 18, 5×6=305 \times 6 = 30, and 9×6=549 \times 6 = 54. The rule is y=x×6y = x \times 6. To find the missing input when the output is 4242, we perform the inverse operation: 42÷6=742 \div 6 = 7.