PROBLEM SOLVING AND REASONING

🧠 PROBLEM SOLVING AND REASONING

(Foundation topic alert 🚨 — this part is important kasi ito ‘yung backbone ng logical thinking.)

I. Inductive Reasoning

🔎 What is Inductive Reasoning?

Inductive reasoning is when you:

• Look at specific examples

• Notice a pattern

• Then form a general conclusion

BUT ⚠ — the conclusion is called a conjecture, meaning:

Think of it as:
“Hmm, based on what I’ve seen… mukhang ganito ‘yan.”

🧩 Example 1: Predicting the Next Number

Sequence:
3, 6, 9, 12, 15, ?

What’s happening?

• Each number increases by 3.

So:
15 + 3 = 18

Easy pattern spotting lang.

Another one:
1, 3, 6, 10, 15, ?

Differences:

• 3 − 1 = 2

• 6 − 3 = 3

• 10 − 6 = 4

• 15 − 10 = 5

Notice something? 👀
The differences increase by 1.

So next difference:
5 + 1 = 6
15 + 6 = 21

Boom. Inductive reasoning.

🧪 Example 2: The “Magic” Procedure

Procedure:

1. Pick a number

2. Multiply by 8

3. Add 6

4. Divide by 2

5. Subtract 3

Try with 5:

• 5 × 8 = 40

• +6 = 46

• ÷2 = 23

• −3 = 20

20 is 4 × 5

Try 10:
Final answer = 40
Also 4 × 10.

So we form a conjecture:

That’s inductive reasoning — we tested examples and noticed a pattern.

🪀 Galileo and the Pendulum

From the table in the handout:

Observation:
The period is the square root of the length.

So:
If length = 49
Period = √49 = 7

Another observation:
If length becomes 4× bigger,
period becomes 2× bigger.

Real-life connection:
Scientists observe patterns first before forming formulas.
That’s inductive reasoning in action.

📝 Quick Summary – Inductive Reasoning

• Specific → General

• Based on patterns

• Conclusion = conjecture

• Not guaranteed true

🧠 Memory trick:
I = I observe patterns

II. Deductive Reasoning

🧠 What is Deductive Reasoning?

Deductive reasoning is the opposite direction:

• Start with a general rule

• Apply it to a specific case

• Get a logically certain conclusion

This one is more solid.
If logic is correct → conclusion must be true.

Think:
“Since this rule is true, therefore…”

🔢 Example: Proving the Procedure

Let original number = n

Multiply by 8 → 8n
Add 6 → 8n + 6
Divide by 2 → 4n + 3
Subtract 3 → 4n

Final answer = 4n

So now it’s proven.
No more guessing. This is deductive reasoning.

🧩 Logic Puzzle Example

Four neighbors:

• Sean

• Maria

• Sarah

• Brian

Jobs:

• Editor

• Banker

• Chef

• Dentist

Using clues and elimination (like detective mode 🕵‍♂️):

Final answer:

• Sean → Banker

• Maria → Editor

• Sarah → Chef

• Brian → Dentist

This works because we applied logical rules step-by-step.

🆚 Inductive vs Deductive

🧠 Memory trick:
DEductive = DEfinite answer

III. Polya’s Four-Step Problem Solving Strategy

George Polya basically said:

The 4 Steps:

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Review the solution

1⃣ Understand the Problem

Ask:

• What is given?

• What is asked?

• Is there extra info?

• What’s the goal?

This part is important kasi dito madalas nagkakamali students — hindi binabasa nang maayos.

2⃣ Devise a Plan

You can:

• Draw a diagram

• Make a table

• Work backwards

• Look for a pattern

• Write an equation

• Guess and check

Different problems = different strategy.

3⃣ Carry Out the Plan

Execute carefully.

• Work neatly

• Double-check

• If it fails, try another method

Math is trial-and-adjust sometimes.

4⃣ Review the Solution

Ask:

• Does the answer make sense?

• Is it realistic?

• Can it apply to other problems?

This is where you avoid careless mistakes.

🎓 Example: Gauss and 1–100

1 + 2 + 3 + … + 100

Instead of adding one by one:

Pair them:
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

There are 50 pairs.

50 × 101 = 5050

Formula:
n(n + 1) / 2

Super efficient thinking.

🏆 Example: Wins and Losses

A team won 2 out of 4 games.

Possible orders:
WWLL
WLWL
WLLW
LWWL
LWLW
LLWW

Total = 6 ways

Organized listing = no duplicates.

📝 Quick Summary – Polya

U-D-C-R

Understand
Devise
Carry out
Review

Memory trick:
“U Don’t Cheat Results”

IV. Sequences and Patterns

📌 What is a Sequence?

An ordered list of numbers.

Example:
5, 14, 27, 44, 65…

Each number is called a term.

Notation:
a₁ = first term
a₂ = second term
aₙ = nth term

🔢 nth-Term Formula (Arithmetic Sequence)

Example:
1, 4, 7, 10…

Difference = 3

Formula:
nth term = dn + (a − d)

So:
3n + (1 − 3)
= 3n − 2

If n = 10:
3(10) − 2 = 28

No need to list all terms.

📊 Difference Table

Used when pattern isn’t obvious.

Example:
2, 5, 8, 11, 14…

First differences:
3, 3, 3, 3

Since constant → arithmetic sequence.

Next term:
14 + 3 = 17

Harder example:
5, 14, 27, 44, 65…

First differences:
9, 13, 17, 21

Second differences:
4, 4, 4

Since second differences are constant → quadratic pattern.

Next:
21 + 4 = 25
65 + 25 = 90

Boom.

📝 Quick Summary – Sequences

• Constant first difference → linear

• Constant second difference → quadratic

• nth term helps jump directly

Memory trick:
If first diff is fixed → line
If second diff is fixed → curve

🧠 FINAL RECAP (Big Picture)

Inductive Reasoning
→ Observe patterns
→ Form conjecture

Deductive Reasoning
→ Apply general rule
→ Prove conclusion

Polya’s Strategy
→ U-D-C-R

Sequences
→ Look at differences
→ Use nth-term formula

🎯 Super Memory Trick

Think of the whole lesson as:

Observe → Prove → Strategize → Predict

Inductive = Observe
Deductive = Prove
Polya = Strategize
Sequences = Predict