Generating and Comparing Numerical Patterns
Students learn to generate two numerical patterns from given starting values and addition rules and compare corresponding terms to identify the relationship between the two patterns, as taught in Illustrative Mathematics Grade 5, Chapter 7: Shapes on the Coordinate Plane. For example, Pattern A starting at 0 with rule add 4 (0, 4, 8, 12) and Pattern B starting at 0 with rule add 8 (0, 8, 16, 24) shows each term in B is always twice the corresponding term in A.
Key Concepts
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.
Common Questions
How do you generate a numerical pattern?
Start with the given number and repeatedly apply the rule (like add 4 or add 3) to generate each successive term in the sequence.
How do you compare two numerical patterns?
Generate both patterns, then look at corresponding terms (first term with first term, second with second) and identify what operation connects them, such as one pattern always being twice the other.
What relationship can you find between Pattern A (add 4: 0,4,8,12) and Pattern B (add 8: 0,8,16,24)?
Each term in Pattern B is twice the corresponding term in Pattern A; this happens because the add-8 rule is double the add-4 rule, so the terms grow at double the rate.
How can patterns be represented on a coordinate plane?
You can plot corresponding terms as ordered pairs (term from Pattern A, term from Pattern B) on a coordinate plane; if the relationship is constant multiplication, the points will form a straight line.
Why is comparing numerical patterns an important skill?
Comparing patterns develops understanding of proportional relationships, prepares students for algebra and functions, and builds the ability to identify and describe mathematical structure.