Welcome back to MAT 135! Today marked the conclusion of Chapter 2, where we moved beyond simple observation and started creating mathematical models to describe the world around us. We focused on three distinct areas: constructing equations for patterns, utilizing deductive reasoning for word puzzles, and the problem-solving strategy of working backwards.

1. Finding Equations for Patterns

Recognizing a pattern is only the first step; the real power of mathematics comes from describing that pattern with an equation. This allows us to predict the future (or the $n$-th term) without having to write out every single step.

When analyzing a sequence of numbers, we look for the relationship between the position number ($n$) and the value. For example, consider the sequence:

$$ 5, 8, 11, 14, ... $$

We can see that the pattern is adding $3$ each time. However, to find the 100th term, we don't want to add $3$ ninety-nine times. Instead, we use the formula for an arithmetic sequence:

$$ a_n = a_1 + (n-1)d $$

Where:

  • $a_n$ is the $n$-th term.
  • $a_1$ is the first term (5).
  • $d$ is the common difference (3).

This simplifies to $a_n = 3n + 2$. Now, finding the 100th term is easy: $3(100) + 2 = 302$.

2. Solving Wordgram Puzzles

We also had some fun with Wordgrams (also known as alphametics or cryptarithms). These are puzzles where letters stand for digits ($0-9$). The goal is to determine the numerical value of each letter to make the mathematical equation true.

Key Rules for Wordgrams:

  • Each letter represents a unique digit (e.g., if $A=2$, $B$ cannot be $2$).
  • The leading letter of a word cannot be zero.

Solving these requires deductive reasoning. You often have to look at the constraints of addition. For example, if you add two single-digit numbers, the maximum sum is $9+9=18$ (or $19$ with a carry). This helps us determine if a carried number is a $1$ or a $0$.

3. The Strategy of Thinking Backwards

Finally, we discussed the strategy of Working Backwards. This is incredibly useful when you know the final outcome of a series of operations but need to find the starting number.

To solve these, we start at the end and reverse every operation as we move to the beginning:

  • If the problem says "add 5", we subtract 5.
  • If the problem says "multiply by 2", we divide by 2.

Example:
"I thought of a number, multiplied it by 3, and subtracted 4. The result was 11. What was my number?"

Working backwards from 11:

  1. Undo "subtract 4": $11 + 4 = 15$
  2. Undo "multiply by 3": $15 \div 3 = 5$

The starting number was $5$. You can verify this by working forward: $3(5) - 4 = 11$.

Class Resources

Don't forget to review the lecture slides and check the details for our upcoming project. These concepts are foundational for the algebra we will cover in the coming weeks!