Inductive Reasoning and Pattern Conjectures - Topic 1, Lesson 4 (Comprehensive Notes)

I CAN… use inductive reasoning to make conjectures about mathematical relationships.
VOCABULARY:- Conjecture
- Counterexample

When points on a circle are connected, the line segments divide the circle into a number of regions, as shown.
A. How does the number of regions change when another point is added?
B. Look for Relationships: Using the pattern you observed, make a prediction about the number of regions formed by connecting 5 points on a circle. Make a drawing to test your prediction. Is your prediction correct?
EXPLORE & REASON:- Points: I 2 3
- Regions: 1 2 8 16 4
Note: The content on this page includes a visual pattern exercise about regions formed by chords in a circle; exact region counts for each point count are shown as part of the activity in the original resource.

A. 88, 82, 76, 70, 64, …
ESSENTIAL QUESTION: How is inductive reasoning used to recognize mathematical relationships?
EXAMPLE 1: 5852,44 -2 - lo 15th Term - 15(6) = 90
Note: Some content on this page is garbled in the transcript; the key idea is the extension of a pattern using inductive reasoning.

B. 3, 5, 9, 15, 23, …
EXAMPLE 1: IF4FFs334
Note: The example line appears garbled in the transcript; the intended idea is another pattern extension exercise.

Try It! 1. What appear to be the next two terms in each sequence?- a. 800, 400, 200, 100, …
- b. 18, 24, 32, 250, 25
E = 18.3
Note: Some parts of the content are garbled, but the prompt is to predict next terms for given sequences.

A conjecture is an unproven statement or rule that is based on inductive reasoning.
Question: What conjecture can you make about the number of dots in the nth term of this geometric pattern?
EXAMPLE 2: (content garbled in transcript)
Observed aim: formulate a conjecture about a geometric pattern's nth term.

Conjecture: The nth term of the sequence will contain n(n + 1), or n^2 + n dots.
EXAMPLE 2: n(n+1) = n^2 + n (the intended relation is shown, though the transcript contains garbled characters: "n(n⑫ +1) n2+ n").

EXAMPLE 2 (continued):- a. How many dots are in the 5th and 6th terms of the pattern?
- b. What conjecture can you make about the number of dots in the nth term of the pattern?
Content in the transcript is garbled around the exact numbers; the task remains to determine nth-term behavior from a dot-pattern.

Partners:- ① 2, 6, 7, 21, 22, g 66 H 03 + 103
- ② A, D, G, J, M, P 1 .1 . 1
- ③ 1, pu non, 8, 27, 64, 000
Notes: Several entries are garbled; the intended activity is pattern discovery in short sequences.

Topic 1 Lesson 4: Use a Conjecture to Make a Prediction
Prompt: Based on the data in the table, how many residents would you expect to vote in the 7th town council election?
EXAMPLE 3: APPLICATION & (data in transcript garbled) – includes various numbers and symbols that appear corrupted.

Try It! 5. For each conjecture, test the conjecture with several more examples or find a counterexample to disprove it.
EXAMPLE 5:- a. For every integer n, the value of n^2 is positive.
- b. A number is divisible by 4 if the last two digits are divisible by 4.
Transcript shows garbled examples like -84, 016, 98.-7 8, but the intended ideas are:- Test whether n^2 > 0 for all integers n.
- Test divisibility rule: last two digits test for divisibility by 4.