Why can the same formula give separate dots in one situation and a line in another?
My prediction and reason
The projection cost model is y = 2x + 4 for inputs from 0 to 4. Predict whether x = 2.5 is allowed when x counts whole items, and when x measures continuous rental hours.
Evidence and observations
In the projection model y = 2x + 4, complete a table for x = 0, 1, 2, 3, 4. Record the fixed cost and cost added per input unit.
Switch between whole-number inputs 0, 1, 2, 3, 4 and all real inputs in [0, 4]. Test x = 2.5 and record whether (2.5, 9) belongs to each graph.
Return to the paper’s relation (1, 2), (1, 4), (2, 3). Record all outputs at input 1 and explain the failure of uniqueness.
Explain or give a counterexample
Transfer
A straight-line model passes through (0, 5) and (2, 9). Find its formula. If the input counts tickets from 0 to 4, state its domain and decide whether to connect the plotted points.
Exit task
Let C(n) = 2n + 5 for n = 0, 1, 2, 3. Find the range and explain whether the graph contains (1.5, 8).
5 min predict dots or lines; 12 min link the table and graph; 8 min derive the line; 12 min paper conversions; 3 min exit check.
Why can the same formula give separate dots in one situation and a line in another?
The projection cost model is y = 2x + 4 for inputs from 0 to 4. Predict whether x = 2.5 is allowed when x counts whole items, and when x measures continuous rental hours.
Evidence tasks
- In the projection model y = 2x + 4, complete a table for x = 0, 1, 2, 3, 4. Record the fixed cost and cost added per input unit.
- Switch between whole-number inputs 0, 1, 2, 3, 4 and all real inputs in [0, 4]. Test x = 2.5 and record whether (2.5, 9) belongs to each graph.
- Return to the paper’s relation (1, 2), (1, 4), (2, 3). Record all outputs at input 1 and explain the failure of uniqueness.
Reference reasoning
Tables, graphs and formulas describe the same function only when they preserve the allowed inputs and outputs. In the projection model y = 2x + 4, the real interval [0, 4] produces a line segment, while the whole-number domain produces five separate points. The point (2.5, 9) belongs only to the real-input version. The paper’s worked example is a different line: given that it is straight, (0, 2) fixes its intercept and (2, 8) gives slope 3, hence y = 3x + 2. Two points alone do not prove an unknown relation is linear. Two distinct heights at the same x violate output uniqueness.
Paper example: find the straight line through these two points, then find f(3). This is a different line from the projection model.
(0,2), (2,8);f(x)=mx+b - The point (0, 2) gives b = 2. The slope is m = (8 − 2)/(2 − 0) = 3.
- Thus f(x) = 3x + 2 and f(3) = 11.
Do these ordered pairs define y as a function of x?
{(1,2),(1,4),(2,3)} - Input 1 has outputs 2 and 4.
- No: the same allowed input has two different outputs.
Transfer and exit task
A straight-line model passes through (0, 5) and (2, 9). Find its formula. If the input counts tickets from 0 to 4, state its domain and decide whether to connect the plotted points.
Let C(n) = 2n + 5 for n = 0, 1, 2, 3. Find the range and explain whether the graph contains (1.5, 8).
The range is {5, 7, 9, 11}. The graph does not contain (1.5, 8), because 1.5 is not an allowed input.
- Lists four attained outputs rather than the interval [5, 11].
- Uses the domain to explain the missing point, even though it satisfies the formula.
Questions
- Does joining two plotted points add new allowed inputs?
- What information tells us that using a straight line is justified?
Expected responses
- Students may automatically join a purchase-count graph. Ask what half an item means in the given model.
- Students may infer a unique general formula from a short table. State the linear assumption explicitly.
Confusions
- A few table entries do not establish the rule at unlisted inputs without an additional assumption.
- Vertical-line testing concerns one fixed input, not whether different inputs share an output.
Board plan
- Representations agree on both the rule and the domain.
- For this linear model: an interval of real inputs gives a line segment; whole-number inputs give separate points.
- Given a line: slope = output change ÷ input change.
