3.5 Mathematical Problem-Solving Strategies for the LET

George Polya’s Four-Step Problem-Solving Model

Educational psychologist George Polya formulated a universal four-step framework for mathematical problem solving. Understanding this methodology empowers LET examinees to approach non-routine mathematical items systematically.

The Four-Step Framework

1. Understand the Problem:
   • Read the problem carefully.
   • Identify the target variable (what is asked?).
   • List all given conditions, explicit values, and implicit constraints.
   • Determine if there is sufficient information to solve.
2. Devise a Plan:
   • Select an appropriate strategy (e.g., set up an equation, draw a diagram, look for a pattern, work backward, or plug in options).
3. Carry Out the Plan:
   • Execute the mathematical operations accurately.
   • Keep track of units of measurement and algebraic signs.
4. Look Back (Reflect and Check):
   • Check if the answer satisfies all initial conditions.
   • Confirm that the magnitude and units of the answer are reasonable.

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High-Yield Test-Taking Heuristics for the LET

1. Working Backward (Back-Solving)

When a word problem involves complex algebraic setups, testing multiple-choice options from middle values (Option B or C) can save critical minutes.

Worked Example 1: Back-Solving Strategy

Problem: If 7 is added to three times a certain number, and this sum is multiplied by 2, the result is 50. Find the number.

Testing Options: [A] 4, [B] 6, [C] 8, [D] 10

• Test Option B (6):
  1. Three times 6 = 18.
  2. Add 7: $18 + 7 = 25$.
  3. Multiply by 2: $25 \times 2 = 50$.
     Matches 50 exactly! Option B is correct.

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2. Venn Diagrams and Inclusion-Exclusion Principle

Survey and multi-group categorization items on the LET are best solved using set theory and Venn diagrams.

$\text{Two-Set Formula: } |A \cup B| = |A| + |B| - |A \cap B|$ $\text{Total Sample Space: } N = |A \cup B| + \text{Neither}$

Worked Example 2: Venn Diagram Application

Problem: In a class of 50 LET examinees, 30 students passed the English mock test, 25 passed the Math mock test, and 10 passed both subjects. How many students failed both subjects?

Step 1: Use the Inclusion-Exclusion formula to find students passing AT LEAST ONE subject ($|A \cup B|$).
$|A \cup B| = 30 + 25 - 10 = 45$

Step 2: Subtract passing students from total class size.
$\text{Failed Both} = 50 - 45 = 5\text{ students}$

Final Answer: 5 students

Advanced Polya Strategy: Working Backward & Model Method

In addition to standard algebraic equations, two powerful heuristics frequently tested on the LET are working backward and the bar model method (visual representation).

• Working Backward: Best applied when a problem describes a sequence of operations starting with an unknown initial quantity and ending with a known final amount. Instead of setting up a forward equation, start from the final result and invert each mathematical operation in reverse order (addition becomes subtraction, multiplication becomes division).
• Bar Modeling (Concrete-Pictorial-Abstract): Particularly effective for ratio, fraction, and comparative word problems. By drawing rectangular bars to represent proportional quantities, examinees can visually identify unit values without getting tangled in complex multi-variable algebra.

Sequences, Series, and Pattern Recognition

Pattern recognition items evaluate mathematical reasoning through arithmetic and geometric progressions.

Arithmetic Progressions (AP)

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant ($d = a_{k+1} - a_k$).

$\text{n-th Term Formula: } a_n = a_1 + (n - 1)d$ $\text{Sum of First } n \text{ Terms: } S_n = \frac{n}{2} \left[ 2a_1 + (n - 1)d \right] = \frac{n}{2} (a_1 + a_n)$

Worked Example 3: Arithmetic Series Sum

Problem: Find the sum of all even integers from 2 to 100 inclusive.

Step 1: Identify values.
$a_1 = 2, d = 2, a_n = 100$.

Step 2: Find number of terms $n$.
$100 = 2 + (n - 1)2 \implies 98 = 2(n - 1) \implies 49 = n - 1 \implies n = 50$

Step 3: Calculate sum $S_{50}$.
$S_{50} = \frac{50}{2} (2 + 100) = 25 \times 102 = 2,550$

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Geometric Progressions (GP)

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero ratio ($r = \frac{a_{k+1}}{a_k}$).

$\text{n-th Term Formula: } a_n = a_1 \cdot r^{n-1}$ $\text{Finite Sum Formula: } S_n = \frac{a_1 (1 - r^n)}{1 - r} \quad (r \neq 1)$ $\text{Infinite Geometric Sum Formula: } S_\infty = \frac{a_1}{1 - r} \quad (\text{where } |r| < 1)$

Worked Example 4: Infinite Geometric Series

Problem: Evaluate the sum of the infinite geometric series: $12 + 6 + 3 + 1.5 + \dots$

Step 1: Identify first term $a_1$ and common ratio $r$.
$a_1 = 12$, $r = \frac{6}{12} = 0.5$.

Step 2: Verify $|r| < 1$.
$|0.5| < 1$, so infinite sum converges.

Step 3: Calculate $S_\infty$.
$S_\infty = \frac{12}{1 - 0.5} = \frac{12}{0.5} = 24$

Final Answer: $24$