Question

name: _____________________________date:_____
directions: work together in your group to find patterns, make conjectures, and explain your reasoning. be ready to share your answers with the class.

part a: number patterns

1. 3, 6, 12, 24, 48, ...

1. 1, 4, 9, 16, 25, ...

1. 2, 5, 10, 17, 26, ...

part b: shape patterns

1. △, ◇, △, ◇, △, ... what comes next?

1. ●, ●●, ●●●, ●●●●, ... predict the next 2 terms.

part c: conjectures

1. the sum of the first 3 even numbers is 12.

the sum of the first 4 even numbers is 20.
make a conjecture: the sum of the first n even numbers is ______.

1. a sequence goes: 2, 6, 12, 20, 30, ...

find the next two terms and describe the pattern.

part d: real - life application

1. a plant grows 5 cm every week.

week 1 = 5 cm, week 2 = 10 cm, week 3 = 15 cm.
predict: how tall will the plant be on week 12?

part 2
observe the pattern in each question and make a conjecture (a general rule).

1. number pattern

2, 4, 8, 16, _, _
what is the pattern? state a conjecture. _______________________________

1. odd numbers

1 + 3 = 4
3 + 5 = 8
5 + 7 = 12
what do you notice about the sum of two odd numbers? ____________________

1. square numbers

$1^2 = 1$
$2^2 = 4$
$3^2 = 9$
$4^2 = 16$
what pattern do you observe? make a conjecture. _________________________

Explanation:

Part A: Number Patterns

1)

Step1: Identify the pattern (multiply by 2)

Each term is \( \text{previous term} \times 2 \).

Step2: Find the next term

\( 48 \times 2 = 96 \)

Step1: Recognize square numbers

Terms: \( 1^2, 2^2, 3^2, 4^2, 5^2 \), so next is \( 6^2 \).

Step2: Calculate \( 6^2 \)

\( 6^2 = 36 \)

Step1: Pattern: \( n^2 + 1 \) (for term \( n \))

Term 1: \( 1^2 + 1 = 2 \), Term 2: \( 2^2 + 1 = 5 \), Term 3: \( 3^2 + 1 = 10 \), Term 4: \( 4^2 + 1 = 17 \), Term 5: \( 5^2 + 1 = 26 \). Next term (n=6): \( 6^2 + 1 = 37 \).

Answer:

96

2)