Here are two tables representing two different situations.

• A student runs errands for a neighbor every week. The table shows the pay he may receive, in dollars, in any given week.

  number of errands	pay in dollars	difference from previous week	factor from previous week
  0	10	-	-
  1	15	5	1.5
  2	20	5	1.33
  3	25	5	1.25
  4	30	5	1.2

• A student at a high school heard a rumor that a celebrity will be speaking at graduation. The table shows how the rumor is spreading over time, in days.

  day	people who have heard the rumor	difference from previous day	factor from previous day
  0	1	-	-
  1	5	4	5
  2	25	20	5
  3	125	100	5
  4	625	500	5

Once we recognize how these patterns change, we can describe them mathematically. This allows us to understand their behavior, extend the patterns, and make predictions.

Notice that in the situation with the student running errands, the difference is constant from week to week, while the factor changes. In the situation about a rumor spreading, the difference changes from day to day, but the factor is constant. This can give us clues to how we might write out the pattern in each situation.