Choose a method
Write an equation with named quantities before substituting numbers. To recover an input, subtract the initial amount and then divide by the rate.
Avoid this trap
An initial amount is not a rate. A question about an increase asks for a difference, while a question about a total includes the starting value.
Check and transfer
Test your rule at input zero and at one known observation. Use the units of the slope to decide whether to multiply or divide.
Worked example with a numerical check
A school newspaper has 10 subscribers before a campaign. Each classroom visit adds 4 subscribers. After 6 visits, how many subscribers does the model predict? The campaign adds the same count at each visit. Multiply the per-visit gain by six, then add the initial count. The requested value is 34.
Contrast and diagnose
A pottery workshop charges $14 to join plus $4 per bowl. The total charge is $42. How many bowls were made? Remove the joining charge from the total. Divide the remaining amount by the price per bowl. The requested value is 7. An incorrect approach: Add rather than subtract the fixed fee.
Transfer practice
The rule p = 4h + 6 models points earned after h hours. How many additional points are earned when h increases from 2 to 5? Try first. Check: 12. Evaluate the same rule at both hour values. Subtract the earlier total from the later total; the initial points cancel.
A school newspaper has 10 subscribers before a campaign. Each classroom visit adds 4 subscribers. After 6 visits, how many subscribers does the model predict?
- The campaign adds the same count at each visit.
- Multiply the per-visit gain by six, then add the initial count.
- The requested value is 34.
Compare the other answer choices
Omit the initial subscribers.
Add the starting count at every visit.
Multiply initial count by the campaign gain.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications