SolutionAn equation is provided, when possible, to provide the teacher with succinct information about the relationship. Students may frame their arguments by creating tables or graphs and using them to support their reasoning, and some samples are provided. The important characteristic of a linear relation is for the quantity to exhibit a constant difference over consistent intervals.
a. Linear. Let c = cost in dollars and p = weight in pounds.
b. Linear. Let m = number of months and h = height in inches. Note: since the tree will not keep growing at the same rate forever, this equation only works for reasonable values of m. Students may discuss how they might limit what values were "allowed" for m.
c. Not linear. Let n = number of folds and L = length in cm.
where n can only be natural numbers (you can't fold a piece of paper 4/5 of a time, for example).
d. Linear. Let P = perimeter in any units of length and s = side length in the same units.
e. Not linear. At certain time intervals, his distance traveled is linear, but he does not have a constant rate of change for his whole trip.
f. Not linear. Let r = radius in units and A = area in square units.
g. Linear. Let m = time in minutes and T = number of tests that have been graded.
h. Linear. Let s = step and N = number of squares. N = 3s + 4, where s is a natural number.
i. Not linear. Let s = step and P = number of pebbles. where s is a natural number.