Squares in a row, built from matchsticks. Figure 1 uses 4 sticks, Figure 2 uses 7, Figure 3 uses 10. Step 1: Look at the GROWTH: 4 -> 7 -> 10. Each new square adds exactly 3 sticks, because the new square shares one side with the square before it. Step 2: Growth of 3 means the term number is multiplied by 3: 3n. Step 3: Test it: 3(1) = 3, but Figure 1 has 4 sticks. There is 1 extra stick - the very first side that starts the row. So the relation is t = 3n + 1. Step 4: Check all three figures: 3(1) + 1 = 4, 3(2) + 1 = 7, 3(3) + 1 = 10. Table: term numbers 1, 2, 3 across the top; values 4, 7, 10 below them - the values climb by 3 every column, and the graph of these points (later in this unit) will be a straight line. That is what "linear" means. Say it plainly: a pattern rule tells you how to get the NEXT term. A linear relation tells you how to get ANY term directly from its term number.

Dots in rows: Figure 1 has 5 dots, Figure 2 has 9, Figure 3 has 13. Write the relation. -> d = 4n + 1 (growth is 4, and 4(1) = 4 needs +1 more; check: 4(3) + 1 = 13)