Patterns in Mathematics
Patterns are repeated arrangements or regularities or relationships among numbers, shapes, or operations.
Patterns help us to predict, generalize, and understand how numbers and shapes behave.
Patterns can occur in the following:
- Numbers
- Shapes (such as a growing square or triangle design)
- Algebraic expressions
- Everyday situations, such as the arrangement of tiles, bricks, or stair steps,etc.,
By studying patterns, we begin to see connections that allow problem-solving to be both faster and deeper.
Investigating a pattern:
To investigate a pattern means looking for how something changes or grows and trying to find a rule or formula that describes it.
This process involves the following:
This process involves the following:
1: Observing the arrangement or sequence carefully.
2: Recording your observations, the first few terms, differences, or shapes that you see.
2: Recording your observations, the first few terms, differences, or shapes that you see.
3: Predicting what might come next.
4: Formulating a general rule using variables to describe the pattern.
5: Testing the general rule with new terms to see if it always works.
4: Formulating a general rule using variables to describe the pattern.
5: Testing the general rule with new terms to see if it always works.
There are often several ways to observe and interpret a pattern in mathematics. Approaching a problem with different perspective encourages creative and imaginative thinking.
Example:
Consider the following pattern:
\( \begin{matrix}
{\color{Yellow}\Delta } & {\color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta} & {\color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta} & {\color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta} & ... \\ & & & & \\
\text{Step 1} & \text{Step 2 } & \text{Step 3} & \text{Step 4} & ... \\
\end{matrix}\)
{\color{Yellow}\Delta } & {\color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta} & {\color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta} & {\color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta \color{Yellow} \Delta} & ... \\ & & & & \\
\text{Step 1} & \text{Step 2 } & \text{Step 3} & \text{Step 4} & ... \\
\end{matrix}\)
Observe the pattern and describe a general rule for it.
Explanation:
Step - 1: The number of triangles is \(1\).
\(\Rightarrow\) \(\frac{1 \times 2}{2}\) \(=\) \(\frac{2}{2} = 1\)
Step - 2: The number of triangles is \(3\).
That is, \(1 + 2 =3\).
\(\Rightarrow\) \(\frac{2 \times 3}{2}\) \(=\) \(\frac{6}{2} = 3\)
Step - 3: The number of triangles is \(6\).
That is, \(3 + 3 = 6\).
\(\Rightarrow\) \(\frac{3 \times 4}{2}\) \(=\) \(\frac{12}{2} = 6\)
Step - 4: The number of triangles is \(10\).
That is, \(6 + 4 = 10\).
\(\Rightarrow\) \(\frac{4 \times 5}{2}\) \(=\) \(\frac{20}{2} = 10\)
Observe that, the numbers \(2, 3, 4,...\) are consecutively added in order, to the previous step.
This above pattern can be algebarically expresses as \(\frac{n(n+1)}{2}\).
Therefore, the general rule for the given pattern is \(\frac{n(n+1)}{2}\).
Important!
The example pattern given here can be solved using different algebraic expression.