Lesson 1: Comparing and Classifying Numerical Patterns

Property

Given two starting numbers and two rules, we can generate two numerical patterns. By comparing the corresponding terms in each pattern, we can identify the relationship between them. For a pattern starting at $a_1$ with rule "add $c$", the terms are $a_1, a_1+c, a_1+2c, \dots$.

Examples

• Pattern A: Start with 0, add 4: $0, 4, 8, 12, \dots$
• Pattern B: Start with 0, add 8: $0, 8, 16, 24, \dots$

Each term in Pattern B is twice the corresponding term in Pattern A.

• Pattern X: Start with 0, add 3: $0, 3, 6, 9, \dots$
• Pattern Y: Start with 5, add 3: $5, 8, 11, 14, \dots$

Each term in Pattern Y is 5 more than the corresponding term in Pattern X.

Explanation

This skill involves creating sequences of numbers by following a specific rule, such as "add 4". When you generate two different patterns, you can look for a relationship between them. To do this, you compare the first term of the first pattern to the first term of the second, the second term to the second term, and so on. The relationship might be that the terms in one pattern are always a certain amount more than, or a multiple of, the terms in the other.

Property

If two numerical patterns have the same additive rule, the difference between their corresponding terms is constant. This constant difference is equal to the difference between their starting values. For a term at any position $n$, the relationship is:

Examples

Property

An additive pattern has a constant difference between corresponding terms.

A multiplicative pattern has a constant ratio between corresponding terms.

Examples

• Multiplicative: A car travels at 50 miles per hour ($distance=50 \times time$). Doubling the time doubles the distance. The ratio $d/t$ is always 50.
• Additive: A child is 4 years younger than their sibling ($Child = Sibling - 4$). In 10 years, they will both be 10 years older, and the age difference remains 4 years.
• Multiplicative: Earning 2 dollars for every box sold ($Earning=2 \times box$). Additive: Starting with 10 dollars and earning 2 dollars per box ($Earning=10+2 \times box$).