To find the pattern rule in a table, examine the relationship between corresponding values and identify a consistent operation.
Examining Relationships
The key to identifying the rule in a table is to understand the mathematical connection between the input values (usually in the top row or column) and their corresponding output values. This involves:
- Initial Observation: Begin by looking at a couple of entries in the table.
- Identifying Potential Operations: Determine if there’s a potential addition, subtraction, multiplication, or division, or even a combination of these, that could be the rule. As mentioned in the video reference (0:53), "So I can look at these two and I can say I'm either from the top one to the bottom one I'm either adding. Two. So I'm adding. Two or I'm multiplying by 3 cuz 1 what gives me 3 1 . 3".
- Testing the Rule: Once you have a candidate rule, test it against other rows or columns in the table. If the rule works consistently, you’ve found the pattern.
- Adjusting the Rule: If the initial rule doesn’t work, try a different approach, or consider more complex relationships like combinations of operations or patterns that grow based on exponents.
Common Patterns
Here are some typical patterns you might encounter:
- Addition or Subtraction: The output increases or decreases by a constant amount.
- Multiplication or Division: The output is multiplied or divided by a constant amount.
- Combination: A combination of operations, for example, multiplying by a number then adding another number.
- Exponents: The output is derived by raising the input to a power, such as squaring or cubing.
Examples
Consider this example from the reference (0:53) of a table where the input values are in the top row and the output values are in the bottom row. Let's say we have:
| Input | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Output | 3 | 6 | 9 | 12 |
Here's how to approach finding the pattern:
- Compare the first pair: To get from 1 to 3, you could add 2 or multiply by 3.
- Check with other pairs: For the second pair, to get from 2 to 6 you could add 4 or multiply by 3. Since multiplying by 3 works for both, it’s likely the correct operation.
- Test the rule: Confirm that multiplying each input number by 3 yields its corresponding output number.
Here's another table example:
| Input | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Output | 4 | 6 | 8 | 10 |
- Compare the first pair: To get from 1 to 4, you could add 3.
- Check with other pairs: For the second pair, to get from 2 to 6 you could add 4. Since there is no consistent add, check for multiplication: to get from 1 to 4 you could multiply by 4, and to get from 2 to 6 you could multiply by 3. The pattern of either adding or multiplying by a consistent number does not work.
- Test a more complex rule: Try multiplying by 2 then adding 2. Does 1*2+2=4? Yes. Does 2*2+2=6? Yes. Does 3*2+2=8? Yes. Does 4*2+2=10? Yes. The rule is to multiply the input by 2 and then add 2.
Practical Insights
- Start Simple: Begin by testing simple operations like addition or multiplication.
- Look for Consistency: The rule must apply to all corresponding pairs in the table.
- Test Your Rule: Once you identify a rule, test it with multiple sets of values.
- Be Flexible: Sometimes, the rule is a combination of operations.
By systematically approaching the data in the table, you can find the rule that defines its pattern.
