Problem 70 - Thinking Mathematically

Question

If \((6-2)^{2}=36-24+4\) and \((8-5)^{2}=64-80+25\), use inductive reasoning to write a compatible expression for \((11-7)^{2}\).

Step-by-Step Solution

Answer

\(121-44+196\) is the expression for \((11-7)^2\) following the pattern in the exercise.

Step 1: Noticing the Pattern

First, let's examine the given expressions. The expression \((6-2)^{2}=36-24+4\) shows that the first term is the square of the first number (6), the second term is four times the first number and third term is four times the square of the second number (2). The expression \((8-5)^{2}=64-80+25\) confirms this pattern: the first term is the square of the first number (8), the second term is four times the first number and third term is four times the square of the second number (5). It is clear that this pattern is not the standard expansion for \((a-b)^{2}\), but it is consistent across the provided expressions.

Step 2: Applying the Pattern

Let us apply this recognized pattern to find the expression for \((11-7)^2\). Sticking to the pattern, the first term should be the square of the first number (11), the second term should be four times the first number and the third term should be four times the square of the second number (7).

Step 3: Calculating the Expression

Therefore, the equivalent expression for this pattern for \((11-7)^2\) is \(11^2 - 4*11 + 4*7^2\). Mathematically, this equates to \(121-44+196\).

Key Concepts

Mathematical Pattern Recognition

Mathematical pattern recognition is like solving a mystery puzzle. It involves identifying regularities or repeated sequences in sets of numbers or expressions. In the given exercise, the pattern revolves around special square expressions, namely

• First, the square of the first number in the pair.
• Second, four times the first number itself.
• Third, four times the square of the second number in the pair.

Recognizing such patterns helps us anticipate the results of similar calculations. It provides a framework for predicting outcomes without recalculating from the beginning each time. This technique allows us to simplify complex algebraic problems by breaking them down into recognizable elements. Essentially, math is a language full of patterns which we can learn to read and use to our advantage.

Algebraic Expressions

Algebraic expressions are collections of numbers, variables, and operators that represent a value or set of values. They are fundamental in mathematics for modeling real-world situations and solving problems. In this context, the expression \[(11-7)^2 = 11^2 - 4 \times 11 + 4 \times 7^2\]is derived using the identified pattern.
This showcases that algebraic expressions can be manipulated based on recognized patterns to derive equivalent outcomes. Mastery of algebra allows us to take an expression and modify it into a different, often simpler form while maintaining equality. This ability is crucial for efficiency and accuracy in calculus, physics, and other disciplines requiring complex calculations.
Algebraic expressions serve as a toolkit to solve various mathematical puzzles effectively by converting them into manageable steps.

Problem-Solving Strategies

Problem-solving strategies in mathematics provide a structured approach to tackle various challenges. In the exercise, using inductive reasoning helped us find a consistent pattern in the given expressions. Inductive reasoning involves observing specific cases and using them to develop a general rule.
Here’s a simple step-wise strategy used in this situation:

• Identify and observe the pattern in the given examples.
• Formulate a hypothesis about the pattern.
• Apply the discovered pattern to new cases, like \( (11-7)^2 \), and check for consistency.
• Verify the result by performing calculations to ensure accuracy.

Such strategies are not only useful in algebra but also in any logical problem-solving situation. Knowing where to start and how to proceed systematically can significantly reduce the feeling of being overwhelmed by complex problems.