Rate Change Slides RATE CHANGE RESCUE
Identifying calculation errors and mastering the Average Rate of Change.
Spot the Error
Analysis Lab
Anchor Tips
WARM-UP: Spot the Error
5 MINUTES
Student Work:
"Find the slope between (2, 5) and (4, 10)"
\[ m = \frac{4 - 2}{10 - 5} \] \[ m = \frac{2}{5} \]
What happened here?
Analyze the calculation above. Did this student find the rate of change correctly?
Think-Pair-Share:
Check the numerator (top). What values are being subtracted?
Check the denominator (bottom).
How would you fix it?
Video Lab: Tables & Formula
Connecting Slope to Average Rate of Change
3:19 - 4:08
Embedded media
Watch & Verify
Identify the points for the interval [-1, 3] on the table.
Watch the subtraction of (-5). How does Justin handle the double negative?
Why would the answer be different for the interval [0, 1]?
THE ERROR ANALYSIS LAB
You are now the Instructor. You have been handed 5 problems graded by a "student" who struggled with their calculations.
Identify
Find exactly where the calculation went wrong.
Explain
Write down why the mistake happened (the misconception).
Solve
Provide the correct average rate of change.
Open your Worksheets! Grab your Red Pens!
REFLECT & DEBRIEF
Most Common Mistake?
Which error did you see the most? Why do you think students keep making that specific mistake?
The "Watch Out" Tip
If you had to give one piece of advice to a friend taking this quiz tomorrow, what would it be?
Closure Activity
We are going to build our class **Anchor Chart**.
The ARoC Formula
Graph vs. Table Examples
3 "Watch Out!" Tips
Error Analysis Worksheet The Error Analysis Lab
Topic: Average Rate of Change (ARoC)
INSTRUCTOR:
DATE:
Lab Objective: Below are 5 "Completed Problems" submitted by a student. Each submission contains exactly one error . As the grader, you must: (1) Locate the mistake, (2) Explain the misconception using math vocabulary, and (3) Provide the correct solution. Use your red pen!
01
Interval: [2, 4]
(2, 6)(4, 12)
Find ARoC on [2, 4]
STUDENT WORK:
\[ \text{Rate} = \frac{4 - 2}{12 - 6} = \frac{2}{6} = \frac{1}{3} \]
GRADER ANALYSIS:
Explain the mistake here...
CORRECTED CALCULATION:
GRADE
X
02
Interval: [0, 2]
STUDENT WORK:
\[ \text{Rate} = \frac{6 - 4}{2 - 0} = \frac{2}{2} = 1 \]
GRADER ANALYSIS:
Explain the mistake here...
CORRECTED CALCULATION:
GRADE
X
03
Interval: [-2, 1]
Points identified:
(-2, 10) and (1, 4)
STUDENT WORK:
\[ \text{Rate} = \frac{10 - 4}{1 - (-2)} = \frac{6}{3} = 2 \]
GRADER ANALYSIS:
Explain the mistake here...
CORRECTED CALCULATION:
GRADE
X
04
Interval: [1, 5]
STUDENT WORK:
\[ \text{Rate} = \frac{5 - 1}{5 - 1} = \frac{4}{4} = 1 \]
GRADER ANALYSIS:
Explain the mistake here...
CORRECTED CALCULATION:
GRADE
X
05
Interval: [0, 8]
Function values:
f(0) = 20, f(8) = 14
STUDENT WORK:
\[ \text{Rate} = \frac{14 - 20}{8 - 0} = \frac{-6}{8} = 0.75 \]
GRADER ANALYSIS:
Explain the mistake here...
CORRECTED CALCULATION:
GRADE
X
Lab Grading Key Answer Key: Lab Report
Teacher Facilitation Guide
OFFICIAL KEY
1
The "Inverted Slope" Error
The Mistake
The student swapped the x and y values in the formula. They calculated run / rise instead of rise / run.
Correct Solution
\[ \frac{12 - 6}{4 - 2} = \frac{6}{2} = 3 \]
2
The "Subtraction vs. Sign" Error
The Mistake
The student ignored the negative sign of the y-value (-4). They treated 6 - (-4) as 6 - 4.
Correct Solution
\[ \frac{6 - (-4)}{2 - 0} = \frac{10}{2} = 5 \]
3
The "Inconsistent Order" Error
The Mistake
The student subtracted in different directions: y1 - y2 in the numerator, but x2 - x1 in the denominator.
Correct Solution
(Using x2 first consistently)
\[ \frac{4 - 10}{1 - (-2)} = \frac{-6}{3} = -2 \]
4
The "Domain Only" Error
The Mistake
The student used the x-values (1 and 5) in both the numerator and denominator instead of finding the corresponding function values f(1) and f(5).
Correct Solution
\[ \frac{90 - 10}{5 - 1} = \frac{80}{4} = 20 \]
5
The "Sign/Magnitude" Error
The Mistake
The student lost the negative sign in the final answer. -6/8 should be -0.75, not 0.75.
Correct Solution
\[ \text{Rate} = -0.75 \text{ or } -\frac{3}{4} \]
Teacher Note
Encourage students to use parentheses when substituting negative numbers: y2 - (y1). This is the #1 way to prevent the Error #2 seen above.
Watch Out Anchor Chart WATCH OUT!
Average Rate of Change Master Guide
The Formula
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
"Slope formula applied to an interval"
The Flip Trap
Don't put X on top. Remember: Rise over Run.
Y's go on top!
The Negative Hole
Subtracting a negative? Use parentheses to avoid signs disappearing!
y - (-4) = y + 4
Pick Your Path
Stay consistent. If you start with Point 2 on top, you must start with it on bottom.
Point 2 → Point 1
Point Precision
Only use the endpoints of the interval [a, b]. Ignore everything in between!
Endpoints only.
Master Pro Tip:
Average Rate of Change is the slope of the SECANT LINE connecting two points on a curve. If the curve goes down, the rate is negative. If it goes up, it's positive.
