iPhone Battery Graphs: Increasing, Decreasing, Positive, and Negative Functions
Materials Needed
- Paper or digital graphing tool
- Pencil and colored pencils or digital drawing tools
- Calculator, optional
- Timer or phone stopwatch
- Exit ticket below
Learning Target
I can identify when graphs of functions are increasing and decreasing and when functions are positive and negative.
Objectives
By the end of this 15-minute lesson, the learner will be able to:
- Determine whether a linear graph is increasing or decreasing by reading it from left to right.
- Identify where a function is positive, negative, or zero using the x-axis.
- Interpret key features of a linear function in an iPhone-related situation.
- Graph a simple linear function from an equation or written description.
Success Criteria
- I read a graph from left to right.
- I identify an upward-sloping line as increasing and a downward-sloping line as decreasing.
- I identify points above the x-axis as positive and points below the x-axis as negative.
- I correctly identify the x-intercept, where the function equals zero.
- I can explain what the graph means in the iPhone situation.
Introduction: Bell Work and Hook — 2 Minutes
Bell Work: Reinforce Prior Knowledge
Imagine your iPhone battery is at 80% and loses 10% every hour. Answer these questions:
- What is the starting value?
- What is the rate of change?
- Write an equation for the battery percentage, B, after h hours.
Expected answers:
- Starting value: 80%
- Rate of change: −10 percentage points per hour
- Equation: B(h) = 80 − 10h
Hook: Ask, “How could a graph help you predict when your iPhone battery will reach 0%?”
Body: Gradual Release of Responsibility
I Do: Teacher or Parent Modeling — 4 Minutes
Use the function:
B(h) = 80 − 10h
Explain each part:
- 80 is the starting battery percentage, or the y-intercept.
- −10 is the rate of change. The battery decreases by 10 percentage points each hour.
- h represents hours.
- B(h) represents the battery percentage.
Create a table:
| Hours, h | Battery, B(h) |
|---|---|
| 0 | 80 |
| 1 | 70 |
| 2 | 60 |
| 3 | 50 |
| 4 | 40 |
| 8 | 0 |
Plot the ordered pairs and connect them with a line or line segment.
Model the Key Features
- Increasing or decreasing: The graph moves downward from left to right, so the function is decreasing.
- Positive: The function is positive when the graph is above the x-axis. This occurs for 0 ≤ h < 8.
- Zero: The battery reaches 0% at h = 8. This is the x-intercept.
- Negative: The equation gives negative values after 8 hours, but a real battery percentage cannot be negative. In this context, the useful domain ends at 8 hours.
Think-aloud: “I look from left to right. Since the line goes down, it is decreasing. I look at the x-axis to determine positive and negative values: above is positive, on the axis is zero, and below is negative.”
Quick Check: Ask the learner, “What does the point (3, 50) mean?”
Expected response: After 3 hours, the iPhone battery is at 50%.
We Do: Guided Practice — 4 Minutes
Work together with the function:
P(t) = 20t − 100
This represents the battery percentage change during charging, where t is the number of hours after charging begins. For this practice, focus on the mathematics, even though a real phone may charge at a changing rate.
- Identify the slope.
- Identify the y-intercept.
- Complete the table.
| Time, t | P(t) = 20t − 100 |
|---|---|
| 0 | _____ |
| 2 | _____ |
| 5 | _____ |
| 6 | _____ |
Guide the learner to identify:
- Slope: 20
- y-intercept: −100
- Table values: −100, −60, 0, and 20
- The function is increasing because the slope is positive.
- The function is positive when t > 5.
- The function is negative when t < 5.
- The function is zero at t = 5, the x-intercept.
Think-Pair-Share or Talk-Aloud: Have the learner explain why the function changes from negative to positive at t = 5. If working independently, the learner can record a 20-second explanation or write it in one sentence.
You Do: Independent iPhone Challenge — 3 Minutes
A phone plan charges a one-time activation fee of $30 and then costs $8 each month. The total cost is represented by:
C(m) = 8m + 30
Complete the following:
- Make a table for m = 0, 1, 2,\text{ and }5.
- Plot the points and draw the linear graph.
- Is the function increasing or decreasing? Explain.
- When is the function positive, negative, or zero?
- What does the y-intercept mean in this situation?
Expected responses:
- Table values: 30, 38, 46, and 70
- The function is increasing because the slope is positive.
- The function is positive for all realistic values of m ≥ 0.
- The function is never zero or negative in the realistic domain.
- The y-intercept, 30, represents the $30 activation fee before any months have passed.
Choice option: Instead of the phone plan, the learner may create a linear function about phone storage, screen time, battery use, or monthly app spending. The learner must include an equation, a table with at least four points, a graph, and an explanation of whether the function is increasing or decreasing.
Conclusion: Closure and Assessment — 2 Minutes
Exit Ticket
Given the function F(x) = −5x + 25, answer:
- Is the function increasing or decreasing?
- When is the function positive?
- When is the function negative?
- At what x-value is the function zero?
- What does the slope tell you?
Answer key:
- Decreasing
- Positive when x < 5
- Negative when x > 5
- Zero at x = 5
- The function decreases by 5 units for every 1-unit increase in x.
Recap
Ask the learner to complete these statements:
- “A graph is increasing when it moves _____ from left to right.” up
- “A graph is decreasing when it moves _____ from left to right.” down
- “A function is positive when its graph is _____ the x-axis.” above
- “A function is negative when its graph is _____ the x-axis.” below
- “A function equals zero at its _____.” x-intercept
Assessment Plan
- Formative assessment: Bell work responses, table completion, verbal explanations, and the guided-practice quick check.
- Summative assessment: Independent iPhone challenge and exit ticket.
- Feedback: Give specific feedback such as, “You correctly used the slope to identify the function as increasing. Now label where the graph crosses the x-axis.”
Differentiation and Support
- For additional support: Provide a partially completed table, label the x- and y-axes, and use the sentence frame: “The graph is _____ because it moves _____ from left to right.”
- For visual learners: Color the graph above the x-axis green, the graph below the x-axis red, and the x-intercept blue.
- For verbal learners: Have the learner explain the graph aloud using the words increasing, decreasing, positive, negative, and zero.
- For hands-on learners: Use a ruler, sticky notes, or floor grid to physically model a line moving upward or downward.
- For advanced learners: Ask the learner to determine the domain and range that make sense in the real-world context and explain why mathematical values may not always be realistic.
- For digital learning: Enter each equation into a graphing calculator or online graphing tool and compare the table, equation, and graph.