Lesson plan Grade 8 Math
Prairie Junction Booster Station: Linear Functions from a Table
A 50-minute grade 8 lesson where students build a linear function from a booster pump station's volume data table, finding rate of change and initial value.
Objectives
- Students will determine the rate of change and initial value of a linear relationship from a data table.
- Students will construct a linear function modeling a pump station's tank volume over time.
- Students will use a constructed function to predict a volume at a given time and a time at a given volume.
Vocabulary
Key words: rate of change, initial value, linear function, data table
Teacher overview
Prairie Junction Booster Station #2 (a fictional pump station used for this lesson) fills a standpipe on a predictable schedule, which makes it a clean setting for building a linear function directly from a data table rather than from a stated rate. A common misconception in this kind of task is calculating the rate of change using only the first and last rows of a table instead of checking that the rate is the same between every consecutive pair of rows, which matters because a table with an inconsistent rate is not actually linear and cannot be modeled with one function.
Procedure
Hook (5 min): Ask students how a plant operator could predict the tank's volume at any future time if they only had a few readings from earlier in the day, without waiting to check the tank again. Take two or three guesses without commenting on their accuracy.
Direct instruction (15 min): Present this data table from Prairie Junction Booster Station #2:
| Time, t (min) | Volume, V (gal) |
|---|---|
| 0 | 50,000 |
| 10 | 65,000 |
| 20 | 80,000 |
| 30 | 95,000 |
Model finding the rate of change by comparing consecutive rows: (65,000 − 50,000) ÷ (10 − 0) = 1,500 gallons per minute. Check that the rate is the same between rows 2 and 3, and rows 3 and 4, confirming the table is linear. Identify the initial value as the volume at t = 0, which is 50,000 gallons. Write the function V(t) = 50,000 + 1,500t, and confirm it against every row of the table.
Guided practice (15 min): In pairs, students build a function from a second station's data table:
| Time, t (min) | Volume, V (gal) |
|---|---|
| 0 | 20,000 |
| 10 | 33,000 |
| 20 | 46,000 |
Students find the rate of change (1,300 gallons per minute), identify the initial value (20,000 gallons), write the function V(t) = 20,000 + 1,300t, and check it against the third row.
Independent practice (10 min): Using the function from guided practice, each student finds the tank's volume at t = 45 minutes and solves for the time when the tank reaches 85,000 gallons.
Closure / exit ticket (5 min): Students answer one question on an index card: "A third station's table shows 40,000 gallons at t = 0 and 44,000 gallons at t = 5. Write its linear function." Collect and scan for a correctly identified rate of change (800 gallons per minute) before dismissal.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
- Answer: 78,500 gal.
Using V(t) = 20,000 + 1,300t at t = 45: V = 20,000 + 1,300(45) = 20,000 + 58,500 = 78,500 gallons. - Answer: 50 minutes.
Solving 20,000 + 1,300t = 85,000 gives 1,300t = 65,000, so t = 50 minutes. - Exit ticket answer: V(t) = 40,000 + 800t.
The rate of change is (44,000 − 40,000) ÷ (5 − 0) = 800 gallons per minute, and the initial value at t = 0 is 40,000 gallons.
Differentiation
Support
Provide the rate-of-change calculation already worked out for the guided-practice table, so students only need to identify the initial value and assemble the function. Allow a calculator for all division and multiplication.
Extension
Give students a fourth station's table with only three rows and one missing entry, and ask them to use the linear function they build from the other two rows to fill in the missing value before checking it makes sense with the pattern.
ELL supports
Pre-teach "rate of change" and "initial value" with a simple labeled diagram: an arrow rising over time labeled "rate of change" and a starting point on the vertical axis labeled "initial value." Provide a sentence frame for the exit ticket: "The rate of change is ___ gallons per minute, and the initial value is ___ gallons, so V(t) = ___."