Patterns and Relations: Rules, Tables, and Simple Equations
What you must understand
Patterns in Grade 5 come in two families. Repeating patterns cycle through a core, and you predict any term by finding the cycle length and the remainder. Growing (increasing or decreasing) patterns change by a constant amount or follow a rule, and you describe the rule in words and connect it to a table of values, a graph, and an equation such as y = 3n + 2. You must extend a pattern, find a missing term, and explain why a rule works. Basic algebraic thinking includes using a letter for an unknown, solving one-step equations like n + 7 = 15 or 4n = 32, and checking by substitution.
Traps to watch
- Counting the starting term as step 1 when the rule uses a different starting value.
- Confusing the pattern's change amount with its starting value: in 5, 8, 11, 14 the change is +3 but the first term is 5, so the rule is 3n + 2, not 3n + 5.
- Off-by-one errors in repeating patterns: with a cycle of 4, term 8 ends the second cycle, so term 9 begins a new one.
- Treating an equation as a left-to-right instruction rather than a balance.
Worked example
A pattern starts at 5 and adds 3 each time: 5, 8, 11, 14, ... Find the 20th term. The rule is 3n + 2, because term 1 is 3(1) + 2 = 5. For n = 20: 3(20) + 2 = 62. Check with the table: term 4 is 14, and 3(4) + 2 = 14.
Rule of thumb
To turn a table into a rule, find the constant change first, multiply it by the term number, then adjust with a constant so the first row matches.