Unit 8: Problem Solving · Lesson 7
Problem Solving - Lesson 7
What this lesson is for
- Students will understand and practice Problem Solving - Lesson 7
- Students will solve problems related to this topic
Train up a child in the way he should go: and when he is old, he will not depart from it.
Proverbs 22:6
Using Logical Reasoning
Logical reasoning uses facts, rules, and relationships to reach conclusions step by step. It eliminates impossible answers until only the correct one remains.
Types of Logical Reasoning
Deductive Reasoning
Moving from general rules to specific conclusions.
- All multiples of 10 end in 0. 350 ends in 0. Therefore, 350 is a multiple of 10.
Inductive Reasoning
Moving from specific observations to general conclusions (pattern recognition).
- 2+2=4, 3+3=6, 4+4=8... All even numbers + even numbers = even.
Process of Elimination
Systematically ruling out impossible options.
- Often used in logic puzzles: "Amy is not in row 2. Beth is in row 3. Amy is not Beth. Therefore Amy is in row 1."
Logic Puzzle Example
Problem: Alex, Ben, and Cara each have a different pet (cat, dog, fish). Alex doesn't have a dog. Ben doesn't have a cat or fish. Who has each pet?
Use a grid:
| | Cat | Dog | Fish | |---|---|---|---| | Alex | ? | No | ? | | Ben | No | ? | No | | Cara | ? | ? | ? |
- Ben: no cat, no fish → Ben has dog
- Alex: no dog → Alex has cat or fish
- Cara: whatever Alex doesn't have
Since Ben has dog: Alex has cat or fish. Cara has the other. No additional clue... but all must be different: If Alex has cat → Cara has fish ✓ (consistent) If Alex has fish → Cara has cat ✓ (also consistent — need more info)
Without more clues, both are valid. Shows importance of all clues!
Using Clues Systematically
In logic problems:
- List all possibilities.
- Use each clue to eliminate impossibilities.
- Fill in certainties.
- Repeat until solved.
Everyday Logic
- "If all quadrilaterals with 4 right angles are rectangles, and a square has 4 right angles, then a square is a rectangle." ✓
- "The sale is 25% off. The item costs $40. Discount = 25% × $40 = $10. Sale price = $30."
Christian Worldview
God gave humans the unique ability to reason. Logic is part of the image of God (imago Dei) in us. When we use logical reasoning to solve problems, we exercise one of the most distinctly human gifts God gave.
"Come now, and let us reason together, saith the LORD." — Isaiah 1:18 (KJV)
God invites us to reason with Him! Logical reasoning is not opposed to faith — God Himself appeals to our capacity to reason.
Practice
- All prime numbers greater than 2 are odd. 37 is prime and greater than 2. Is 37 odd?
- Three students (May, Pat, Sam) took three prizes (1st, 2nd, 3rd). May didn't win 1st. Pat won 2nd. Who won what?
- Using deduction: if all rectangles have 2 pairs of parallel sides, and a square is a rectangle, what can you say about a square?
Guided practice
Four friends—Amy, Ben, Cara, and Dan—each have a different favorite subject: math, science, reading, or history. Amy doesn't like math or science. Ben likes either math or science. Cara likes reading. Dan doesn't like math. Who likes math?
Show the steps and the answer
Cara likes reading (given). Amy doesn't like math or science, so Amy likes history. Dan doesn't like math, so Dan likes science. Ben likes either math or science; since Dan has science, Ben likes math.
Answer: Ben
Three families—Smiths, Johnsons, Williamses—each planted a different crop: corn, beans, or squash. The Smiths didn't plant beans. The Johnsons planted either corn or beans. The Williamses didn't plant corn or beans. Who planted beans?
Show the steps and the answer
The Williamses didn't plant corn or beans, so the Williamses planted squash. That leaves corn and beans for the Smiths and Johnsons. The Smiths didn't plant beans, so the Smiths planted corn, and the Johnsons planted beans. Check: Smiths (corn) ✓, Johnsons (beans) ✓, Williamses (squash) ✓.
Answer: Johnsons
Three siblings—Lily, Marcus, Nora—have ages 8, 10, and 12. Lily is younger than Marcus. Nora is older than Marcus. Who is the oldest?
Show the steps and the answer
Nora > Marcus > Lily (from the clues). So order is: Lily (8), Marcus (10), Nora (12). Nora is oldest.
Answer: Nora
Homework
Complete the practice problems for Problem Solving - Lesson 7.
Note for the parent
Today's lesson covers Problem Solving - Lesson 7. Estimated time: 50 minutes. Encourage independent work but be available for questions.
The first three of this lesson's 10 questions
Deductive reasoning moves from:
Show the answer
Answer: General rules to specific conclusions
Inductive reasoning moves from:
Show the answer
Answer: Specific examples to general conclusions
Logic puzzle: Ben has no cat, no fish. Ben must have:
Show the answer
Answer: Dog
Another 7 questions follow in the lesson itself, each one graded the moment it is answered.