Is every increasing relationship proportional?
Compare a constant-rate rule with a fixed-fee rule using tables and points.
Begin with the activity ↓Is every increasing relationship proportional?
If each notebook costs 2,000 won with no extra fee, 1, 2 and 3 notebooks cost 2,000, 4,000 and 6,000 won. Doubling count doubles cost. In direct proportion, scaling one quantity by a factor scales the other by that same factor.
If each notebook costs 2,000 won with no extra fee, 1, 2 and 3 notebooks cost 2,000, 4,000 and 6,000 won. Doubling count doubles cost. In direct proportion, scaling one quantity by a factor scales the other by that same factor.
Zero notebooks cost zero won. At nonzero counts, price divided by count is always 2,000 won per notebook. Check that this rule, with no extra fee, applies to all permitted counts. A few matching rows alone do not prove every possible case.
Now consider a rule charging a fixed 1,000 won plus 2,000 won per notebook. One, two and three notebooks cost 3,000, 5,000 and 7,000 won. Cost increases by the same 2,000 won each step, but doubling count from one to two does not double price.
Under this fixed-fee rule, zero gives 1,000 won. The activity explicitly applies the fee even at zero notebooks. If a real store charges nothing without an order, specify that separate zero-count condition. The rule determines which points belong on a graph.
Use count horizontally and price in thousands of won vertically. The no-fee rule includes (0, 0), (1, 2), (2, 4); the fee rule includes (0, 1), (1, 3), (2, 5). Both sets rise regularly. Check the origin and constant ratio to distinguish them.
Notebooks are counted in whole units, so this activity plots integer-input points without joining them. Measurable lengths can allow intermediate values. A future function course can extend this to rules assigning one output per input; the introductory course is available through the continuation link below.
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Hold on to this distinction
An increase of 2,000 won per step alone does not ensure direct proportion. Compare costs for one and two notebooks, then inspect zero and rates at nonzero counts.
Change it and check
Switch between fixed fees of zero and 1,000 won. Select a point, find its table row, and state the coordinates and unit price.
| Count | Price (thousand won) | Average rate (thousand won/notebook) |
|---|---|---|
| 0 | 0 | No division by zero |
| → 1 | 2 | 2 |
| 2 | 4 | 2 |
| 3 | 6 | 2 |
| 4 | 8 | 2 |
(1, 2) · Direct proportion: the nonzero-input rate is always 2 thousand won per notebook.
Each notebook costs 2,000 won; the fixed fee applies at every input. Horizontal units are notebooks and each vertical interval is 1,000 won. Do not divide by a zero count. Passing through the origin alone does not establish direct proportion for an arbitrary relationship.
Try with paper and objects
Make two tables for zero through four notebooks, using thousands of won for prices. Plot the points without joining them and test whether doubling from one to two notebooks doubles cost.
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8 · thousand won
4 × 2 = 8: the vertical coordinate is eight, representing 8,000 won.
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