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
Logic and deductive reasoning allow us to draw certain conclusions from given facts. Deductive reasoning begins with general statements (premises) and draws specific conclusions. If the premises are true and the logic is valid, the conclusion must be true. Inductive reasoning moves from specific observations to general conclusions, which are probable but not certain.
A syllogism is a classic form of deductive argument. Major premise: All squares are rectangles. Minor premise: ABCD is a square. Conclusion: ABCD is a rectangle. This conclusion must be true. Contrast with inductive reasoning: "The first 100 swans I saw were white; therefore all swans are white." This conclusion is probable but was famously shown to be false when black swans were discovered in Australia. Conditional statements have the form "If p, then q." The converse switches p and q (if q then p), the inverse negates both (if not p then not q), and the contrapositive negates and switches (if not q then not p). A conditional and its contrapositive are logically equivalent. Proverbs 8:14 declares wisdom saying "Counsel is mine, and sound wisdom: I am understanding; I have strength." Sound reasoning is a form of wisdom.
Practice: (1) Write the converse, inverse, and contrapositive of: "If it rains, then the ground is wet." (2) Write a valid syllogism using the following: all mammals are warm-blooded, and all dogs are mammals.
Review and Apply: Ask your student: "What problem-solving strategy did we use today? Now apply it to a problem I give you without any hints." Present one new multi-step problem and observe their approach — do they plan before calculating? Check their work? Stay organized? "The thoughts of the diligent tend only to plenteousness; but of every one that is hasty only to want." — Proverbs 21:5 (KJV). Problem solving is a discipline of patience and method — virtues God values. Remind your student that persisting through a hard problem builds both mathematical skill and character. Close with prayer.
Guided practice
Premise 1: All vegetables grown in the church garden are organic. Premise 2: Carrots are grown in the church garden. Conclusion: Are the carrots organic? Explain your reasoning.
Show the steps and the answer
This is deductive reasoning. If all vegetables are organic (Premise 1) and carrots are vegetables in that garden (Premise 2), then by logic, those carrots must be organic.
Answer: Yes, the carrots are organic.
Observe: Day 1 was sunny, Day 2 was sunny, Day 3 was sunny. Based on inductive reasoning, what might you predict for Day 4?
Show the steps and the answer
Inductive reasoning looks at specific cases (Days 1-3) and predicts a general pattern (Day 4). However, this is not certain—weather can change.
Answer: Day 4 will likely be sunny.
Use deductive reasoning: If a number is divisible by 2 and divisible by 3, then it is divisible by 6. Is 18 divisible by 6?
Show the steps and the answer
18 is divisible by 2 (18 ÷ 2 = 9). 18 is divisible by 3 (18 ÷ 3 = 6). By the rule, 18 must be divisible by 6. (18 ÷ 6 = 3.)
Answer: Yes
Homework
Complete the practice problems for Problem Solving - Lesson 7.
Note for the parent
Today's lesson covers Problem Solving - Lesson 7. Estimated time: 55 minutes. Encourage independent work but be available for questions.
The first three of this lesson's 10 questions
Deductive reasoning proceeds from:
Show the answer
Answer: General premises to specific conclusions
Inductive reasoning proceeds from:
Show the answer
Answer: Specific observations to general patterns
A valid deductive argument: All mammals breathe air; a whale is a mammal. Therefore:
Show the answer
Answer: A whale breathes air
Another 7 questions follow in the lesson itself, each one graded the moment it is answered.