A comprehensive 8th-grade mathematics lesson introducing function concepts, representations, and real-world modeling. Students progress from distinguishing functions to evaluating equations, discovering patterns in tables, and analyzing real-world systems.
Hiking Elevation vs. Time Time (min) Elevation (ft) 1 2 3 4
13. Describe the behavior in Segment 1:
Is the hiker ascending, resting, or descending? Explain the rate:
14. What occurs during Segment 2?
Why does the elevation line remain completely flat?
15. Compare Segment 1 and Segment 3:
How do their rates of change (slopes) differ in sign and steepness?
5 Part 5: Mathematical Error Analysis
Misconception
Leo's Claim:"The relation \(\{ (3, 4), (5, 4), (8, 4) \}\) cannot be a function because the number \(4\) repeats three times!"
Is Leo correct?
Yes No
Explain Leo's error:
Page 2 of 2 • Function Power Practice Great work building your mathematical stamina!
$20 base monthly access + $5 per GB used.
Rule: \(y = 5x + 20\)
Plan B: Summit Wireless
$40 base monthly access + $2.50 per GB used.
Rule: \(y = 2.5x + 40\)
Complete the Comparison Table
Data Used (\(x\) GB)
Plan A: \(y = 5x + 20\)
Plan B: \(y = 2.5x + 40\)
Cheaper Option?
0 GB
$20
$40
Plan A ($20 cheaper)
4 GB
$40
8 GB
$60
$60
TIE ($60 each)
12 GB
Break-Even Point:
At what point on a coordinate graph would these two lines intersect? Write as an ordered pair \((x, y)\):
Intersection:
Consumer Recommendation:
If a student uses 15 GB each month, which plan should they purchase? Justify:
4
Quest 4: The Runner's Journey Analysis
Distance vs. Time Time (min) Distance (mi) I II III IV
Match each segment to the runner's action:
The runner jogged at a steady warm-up pace of 6 mph.
The runner stopped at a red crosswalk light for 2 minutes.
The runner sprinted uphill at their fastest forward speed.
The runner turned around and sprinted directly back home.
Page 2 of 2 • Function Quests Application Congratulations! Mathematical Model Complete.
4 Part 5: Mathematical Error Analysis (Leo's Misconception)
Leo is INCORRECT. The definition of a function specifies that each input can only have one output. It is completely valid for different inputs (\(3, 5, 8\)) to yield the exact same output (\(4\)). A repeated output represents a horizontal line, which easily passes the Vertical Line Test!
Page 1 of 2 • Teacher Answer Key Page 2: Real World Quests Solutions & Scoring Rubric
Each additional combo plate costs exactly $3 to make.
\(y\)-Intercept Context:
The initial fixed fee of $60 to enter the festival, paid even if 0 plates are prepared.
2 Quest 2: Search Drone Altitude Descent
A. Function Rule:
\(A(t) = 320 - 40t\)
or \(A(t) = -40t + 320\)
B. At \(t = 5\) Seconds:
\(320 - 40(5) = 320 - 200\)
\(\mathbf{120}\) feet altitude
C. Landing Time (\(A = 0\)):
\(0 = 320 - 40t \implies 40t = 320\)
\(t = \mathbf{8}\) seconds
3 Quest 3: Mobile Data Plan Showdown
Table Completions: At 4 GB: Plan A = $40, Plan B = $50 (Plan A cheaper by $10). At 8 GB: Both plans = $60 (Tie). At 12 GB: Plan A = $80, Plan B = $70 (Plan B cheaper by $10).
Intersection Point:
\((8, 60)\)
At 8 GB of data, both providers charge exactly $60.
Recommendation for 15 GB:
Choose Plan B! Plan A costs \(5(15) + 20 = \mathbf{\$95}\). Plan B costs \(2.5(15) + 40 = \mathbf{\$77.50}\). Plan B saves $17.50!
4 Quest 4: Mystery Runner Segment Match
Segment I: Steady warm-up pace
Segment II: Stopped at light (flat line)
Segment III: Sprinting uphill (steepest)
Segment IV: Returning home (negative slope)
Teacher Diagnostic Guide & Key Common Errors:
Notation Confusion: Watch for students reading \(f(5)\) as \(f \times 5\). Remind them: \(f\) is the machine name, and \(5\) is the fuel going into it.
Rate vs. Intercept: On Table B (Problem 12), students often mistakenly take the first \(y\)-value (\(15\)) as the initial value \(b\). Highlight that \(x\) must be \(0\).
Decreasing vs. Negative Distance: Clarify that in Quest 4 Segment IV, a downward slope on a distance-from-home graph means traveling back home, not negative speed.
Page 2 of 2 • Teacher Answer Key & Guide Function Foundry Resource Suite
