Patterns and Relations: Patterns, Linear Relations, and Equations
Grade 7 pattern work has two goals: describe a pattern with a rule, and solve equations that model a situation.
A recursive rule says how to get the next term ("start at 5, add 3"). An explicit rule, or general term, lets you jump straight to any term using its position number n. For a linear pattern, the constant difference becomes the multiplier of n, and the value at n = 0 becomes the constant. The pattern 5, 8, 11, 14 has difference 3, and since term 1 is 5, the rule is 3n + 2. Test n = 1: 3(1) + 2 = 5.
Linear relations graph as straight lines. The rate of change (slope) is the change in the dependent variable for each one-unit increase in the independent variable; the starting value is where the graph crosses the vertical axis. In a table, equal steps in x must produce equal steps in y for the relation to be linear.
An expression has no equals sign; an equation does. Solve equations by applying the inverse operation to both sides so the balance is preserved, then check by substituting your answer back into the original equation.
Traps: substituting a negative number without brackets (3x with x = −2 is −6, not 1); dividing only one term on a side instead of the whole side; confusing the term number n with the term value; and assuming a relation is linear when the differences in the table are not constant.
Worked example: solve 3x − 7 = 14. Add 7 to both sides: 3x = 21. Divide both sides by 3: x = 7. Check: 3(7) − 7 = 21 − 7 = 14.