Soccer Stats: Average Rate of Change and Linear Functions

Materials Needed

• Pencil and paper or a digital document
• Calculator, optional
• Practice worksheet below
• Graph paper or a graphing tool, optional
• Timer

Learning Objectives

By the end of this 15-minute lesson, I can:

• Calculate average rate of change using a table, graph, or equation.
• Explain what the rate of change means in a real-world soccer situation.
• Identify the rate of change, or slope, and the initial value of a linear function.

Success Criteria

I am successful when I can:

• Use the formula average rate of change = change in output ÷ change in input.
• Correctly subtract the output values and input values in the same order.
• Include units in my answer.
• Interpret the answer as what happens for every one-unit increase in the input.
• Identify the initial value as the output when the input is 0.

Key Vocabulary and Formula

• Input: The independent variable, such as time or number of practices.
• Output: The dependent variable, such as goals scored or distance traveled.
• Average rate of change: How much the output changes compared with the input over an interval.
• Initial value: The output value when the input is 0.
• Slope: The rate of change of a linear function.

Formula:

Average rate of change = (output₂ − output₁) ÷ (input₂ − input₁)

Lesson Sequence: 15 Minutes

1. Introduction and Hook — 2 Minutes

Soccer scenario: A player tracks the distance they run during practice. After 10 minutes, the player has run 1.2 miles. After 30 minutes, the player has run 3.6 miles.

Ask:

• How quickly is the player running on average?
• What would the rate mean in this situation?

Explain: “Today we will calculate how quickly a quantity changes. In soccer, this could describe distance per minute, goals per game, or shots per practice.”

Tell learners what they will learn: “First, I will model how to calculate and interpret rate of change. Then we will solve one together. Next, you will try examples independently. Finally, we will check your understanding with an exit ticket.”

2. I Do: Teacher or Parent Models — 4 Minutes

Example A: Rate of Change from a Table

The table shows the number of goals a soccer team scored during different numbers of games.

Step 1: Identify the two points.

(2, 5) and (6, 13)

Step 2: Find the change in output.

13 − 5 = 8 goals

Step 3: Find the change in input.

6 − 2 = 4 games

Step 4: Divide.

Average rate of change = 8 ÷ 4 = 2 goals per game

Interpretation: The team scored an average of 2 additional goals for every additional game during this interval.

Example B: Initial Value and Slope from an Equation

A soccer training app models a player's points with the equation:

p = 4t + 10

• Slope/rate of change: 4 points per training session
• Initial value: 10 points, because p = 10 when t = 0

Interpretation: The player starts with 10 points and earns 4 more points per training session.

Quick Check: If the equation were p = −2t + 18, what would the slope mean?

Expected response: The player’s points decrease by 2 points per session, starting at 18 points.

3. You Do: Independent Try — 3 Minutes

Complete the following problem independently. Show each step.

A soccer player records the number of successful passes during practice.

1. Calculate the average rate of change.
2. Include the units.
3. Explain what the answer means in this situation.

Expected work:

(43 − 18) ÷ (15 − 5) = 25 ÷ 10 = 2.5 successful passes per minute

Interpretation: The player completed an average of 2.5 additional successful passes for every additional minute of practice during this interval.

4. We Do: Guided Practice — 3 Minutes

Work through this problem aloud with a parent, tutor, or teacher. If working independently, pause after each question and explain your reasoning out loud.

A graph of a soccer ball’s height contains the points (2, 8) and (5, 20), where x is time in seconds and y is height in feet.

1. What is the change in height?
2. What is the change in time?
3. What is the average rate of change?
4. What does the rate mean?

Solution:

• Change in height: 20 − 8 = 12 feet
• Change in time: 5 − 2 = 3 seconds
• Average rate of change: 12 ÷ 3 = 4 feet per second
• Interpretation: The ball’s height increased by an average of 4 feet for every additional second between 2 and 5 seconds.

Discussion: Why is it important to include the interval? Encourage the learner to explain that the rate describes what happened between the two selected input values, not necessarily during the entire motion.

5. Practice Worksheet — 2 Minutes in Lesson or Complete Immediately Afterward

Directions: Show your work. For each rate of change, include units and write a sentence interpreting the answer when requested.

1. A soccer team scores 7 goals in 3 games and 15 goals in 7 games. Find the average rate of change in goals per game.
2. A player runs 2 miles after 20 minutes and 5 miles after 50 minutes. Find the average rate of change in miles per minute.
3. Use the table to find the average rate of change.

   Minutes Practiced, x	Shots on Goal, y
   10	24
   25	54

4. A goalkeeper makes 12 saves after 2 games and 30 saves after 5 games. Find the average rate of change and interpret it.
5. A soccer camp charges according to the equation C = 15h + 25, where h is the number of training hours. Identify the rate of change and the initial value. Explain what each means.
6. A player’s confidence score is modeled by s = −3g + 24, where g is the number of goals missed. Identify the slope and initial value. Interpret the slope.
7. A graph contains the points (1, 6) and (4, 18), where x is the number of games and y is the number of successful tackles. Find the average rate of change.
8. A soccer ball travels 18 feet at 2 seconds and 42 feet at 6 seconds. Find the average rate of change in feet per second and interpret the answer.
9. The table represents a linear relationship.

   Games, x	Team Points, y
   0	4
   1	7
   2	10
   3	13

   
   Identify the initial value and rate of change.
10. Real-world problem: A soccer club has $120 in its equipment fund. It adds $35 each week from snack sales. Write an equation for the fund, identify the initial value and rate of change, and find the amount after 6 weeks.

Practice Worksheet Answer Key

1. 2 goals per game.
2. 0.1 mile per minute.
3. 2 shots on goal per minute.
4. 6 saves per game; the goalkeeper makes an average of 6 additional saves per game.
5. Rate of change: $15 per hour. Initial value: $25. The camp charges $25 before training and $15 for each training hour.
6. Slope: −3 points per missed goal. Initial value: 24. The score starts at 24 and decreases by 3 points for each missed goal.
7. 4 successful tackles per game.
8. 6 feet per second; the ball travels an average of 6 additional feet each second over the interval.
9. Initial value: 4 points. Rate of change: 3 points per game.
10. Equation: F = 35w + 120. Initial value: $120. Rate of change: $35 per week. After 6 weeks: 35(6) + 120 = $330.

Exit Ticket — Final 1 Minute

Answer without looking at the examples.

1. A soccer player has 10 assists after 4 games and 22 assists after 10 games. What is the average rate of change?
2. What does the rate of change mean in this situation?
3. For the equation y = 5x + 8, identify the slope and initial value.

Exit Ticket Answer Key

1. (22 − 10) ÷ (10 − 4) = 12 ÷ 6 = 2 assists per game.
2. The player earns an average of 2 additional assists for every additional game during the interval.
3. Slope: 5. Initial value: 8.

Assessment and Feedback

• Formative assessment: Listen for correct identification of input, output, interval, units, and interpretation during the examples and guided practice.
• Independent assessment: Review the “You Do” problem and practice worksheet calculations.
• Summative check: Use the exit ticket. A strong performance includes at least 2 out of 3 correct responses and correct units on the rate question.
• Feedback prompt: If an answer is incorrect, ask: “What are the two output values? What are the two input values? Did you subtract in the same order?”

Differentiation and Choice

• Scaffold: Use the sentence frame: “The output changes by ___ for every ___ increase in the input.”
• Scaffold: Highlight input values in one color and output values in another color.
• Scaffold: Provide the structure:
  Average rate of change = (new output − old output) ÷ (new input − old input).
• Choice: Solve practice problems with a calculator, graph paper, or a digital spreadsheet.
• Extension: Create a soccer scenario with a rate of change of 3 goals per game and an initial value of 2 goals. Write its equation and make a table with four input-output pairs.
• Challenge: Explain whether a negative rate of change could make sense in a soccer context, such as remaining energy, distance to the goal, or unanswered goals needed.

Closure

Ask the learner to complete this recap aloud:

“To find average rate of change, I subtract the __________ values and divide by the difference in the __________ values. In a linear function, the slope represents __________, and the initial value is the output when the input is __________.”

Expected recap: output; input; rate of change; 0.