Surfboard Sums

In this activity, students interpret the graph showing how the value of material goods such as a surfboard usually decreases over time.

There are three distinct periods that need to be considered. During the first 3 month period, the surfboard loses value more rapidly than at any other time. In fact, it loses $200 in value, which is almost 30% of its price when new. Its rate of loss of value (rate of depreciation) is about $67 per month (found from $200 ÷ 3 = 66.666). For the next 9 months, the surfboard loses value steadily from $500 to $350. The line for this part of the graph is much less steep than the line for the first 3 months, and the rate of depreciation over the 9 months is almost $17 per month (found from $150 ÷ 9 = 16.666). For the third period, that is, for the second 12 months that Vinny owns the surfboard, there is a loss of $100 as a result of a steady rate of depreciation that is a little more than $8 per month (found from $100 ÷ 12 = 8.333).

In questions 3 and 4, the students use the graph to make sensible predictions, or extrapolations, for periods beyond the data shown on the graph. They are also asked to justify their predictions. While it is safe to simply extend the graph by assuming that the rate of depreciation remains the same for the period beyond 24 months, a more in-depth analysis is possible by looking at the shape of the graph over the first 24 month period.

The gradual flattening of the graph as a consequence of the rate of depreciation reducing as time goes on suggests that the value of the surfboard will not decrease by an amount greater than for the previous 3 months, that is, $25. So, if its value at 24 months is $250, then by 27 months, its value is not likely to be less than $225. For the same reason, if the value of the surfboard at 27 months is $225, it is likely to fall in value by not more than $25 in the next 3 months. So Vinny is likely to get at least $200 for the surfboard by selling it at 30 months.

It is unlikely that many students will make the argument above without some initial prompting. One way to help might be to first extend the straight line representing the part of the graph between 12 and 24 months to the point representing the value of the surfboard at 27 months and the point when the value of the surfboard is $200. The students might then decide whether it is more likely that these points will actually lie on the line, above the line, or even below the line. It is argued above that it is more likely that these points will be on or above this line. The students should be encouraged to explain their reasoning for their decisions.

Answers to Activity

1. a. $700
b. The value of the surfboard is continuously decreasing.
2. a.

Time from new (months)	Value of surfboard ($)
0	700
3	500
6	450
9	400
12	350
15	325
18	300
21	275
24	250

b. i. During the first 3 months
ii. In the 3 month periods between 12 and 24 months
c. During the first 3 months, the value decreases by $200 in 3 months or approximately $67 per month. Between 12 and 24 months, the value decreases by $25 in each 3 month period or approximately $8 per month.
3. Probably about $225. The gradual flattening of the graph suggests the value of the surfboard will not decrease by an amount greater than that for the previous 3 months, that is, $25. So, if its value at 24 months is $250, then by 27 months, its value is likely to be at least $225.
4. 30 months is the latest time that Vinny could sell his surfboard for $200 or more. If the surfboard’s value at 27 months is $225, it is unlikely to fall in value by more than $25 in the next 3 months.