Harry Potter: Scatter Plots and Lines of Fit

Materials Needed

• Graph paper or a digital graphing tool
• Pencil and colored pens or highlighters
• Calculator
• Practice worksheet and exit ticket
• Optional: Harry Potter-themed objects, such as spell cards or house points

Learning Objectives

By the end of the lesson, I can:

• Identify the relationship between two variables in a scatter plot.
• Use a linear equation to model real-world data.
• Interpret the slope and y-intercept in context.
• Use a line of fit to make predictions.
• Identify reasonable constraints, such as possible input and output values.

Success Criteria

I am successful when I can:

• Write or use a linear model in the form y = mx + b.
• Explain what m and b mean in the situation.
• Make a prediction using the model and show my substitution.
• State whether my answer makes sense in the real-world context.

Introduction: Hook and Purpose — 2 Minutes

Hook: Harry is preparing for his O.W.L. exams. Does practicing more spells generally lead to a higher spell-casting score?

Explain that a scatter plot shows pairs of data, such as hours of practice and spell-casting score. A line of fit shows the overall trend in the data and can be used to make predictions.

Today’s plan: First, watch a model example. Next, solve a problem independently. Finally, work through a similar problem together and complete an exit ticket.

Body

I Do: Teacher or Parent Modeling — 4 Minutes

Use the following model:

Situation: A group of young witches and wizards records the number of hours they practice a spell and their spell-casting score.

A reasonable line of fit for this data is:

y = 8x + 50

1. Interpret the slope:

   The slope is 8. For each additional hour of practice, the predicted spell-casting score increases by about 8 points.

2. Interpret the y-intercept:

   The y-intercept is 50. With 0 hours of recorded practice, the predicted score is 50 points.

3. Make a prediction:

   Predict the score after 6 hours of practice.

   y = 8(6) + 50

   y = 48 + 50 = 98

   The predicted score after 6 hours is 98 points.

4. State a constraint:

   The data were collected for 1–5 hours of practice, so predicting at 6 hours is an extrapolation. The model is most reliable for a reasonable range such as 0 ≤ x ≤ 6. Practice hours cannot be negative, and a score should not exceed the maximum possible score.

Quick Check: Ask, “What does the 8 mean in this situation?” The expected response is: “The score increases by about 8 points for every additional hour of practice.”

You Do: Independent Practice — 3 Minutes

Suppose a line of fit for the number of potion-making attempts and potion quality score is:

y = 6x + 40

Complete the following independently:

1. Identify the slope and explain what it means.
2. Identify the y-intercept and explain what it means.
3. Predict the potion quality score after 7 attempts.
4. Decide whether a prediction for 100 attempts would be reasonable. Explain why or why not.

Expected answers:

• The slope is 6. The score increases by about 6 points per attempt.
• The y-intercept is 40. The predicted score is 40 before any attempts are recorded.
• y = 6(7) + 40 = 82, so the predicted score is 82.
• A prediction for 100 attempts is probably unreasonable because it is far outside the data range and may predict an impossible score.

We Do: Guided Practice — 3 Minutes

Work through this problem together. The student may choose to graph the data by hand or use a digital graphing tool.

Situation: Hermione tracks how many library books she reads and the number of research points she earns for a project.

Use the line of fit:

y = 5x + 8

1. What does the slope mean?
2. What does the y-intercept mean?
3. How many research points are predicted after reading 6 books?
4. Would the model be useful for predicting the points earned after reading 50 books? Why?

Discuss and solve:

• The slope is 5, so each additional book is associated with about 5 more research points.
• The y-intercept is 8, so the model predicts 8 points when 0 books are read.
• y = 5(6) + 8 = 38, so the prediction is 38 points.
• A prediction for 50 books is probably not reliable because it is far beyond the observed data and may not fit the project’s scoring limits.

Feedback prompt: Ask the student to explain not only the calculation but also whether the answer makes sense in context.

Closure and Recap — 1 Minute

Ask the student to complete these statements aloud or in writing:

• A scatter plot is used to show __________________________.
• The slope of a line of fit tells us _______________________.
• The y-intercept tells us _________________________________.
• A prediction is more reliable when _______________________.

Key takeaway: A line of fit describes a trend in real-world data. The slope and y-intercept must be interpreted using the context, and predictions should stay within reasonable constraints.

Exit Ticket

A Quidditch player tracks practice time and the number of successful throws.

The line of fit is:

y = 4x + 12

1. What does the slope of 4 mean in this context?
2. What does the y-intercept of 12 mean?
3. Predict the number of successful throws after 8 hours of practice.
4. Why might a prediction for 200 hours of practice be unreasonable?

Exit ticket answer key:

1. The number of successful throws increases by about 4 for every additional hour of practice.
2. The model predicts 12 successful throws at 0 hours of recorded practice.
3. y = 4(8) + 12 = 44, so the prediction is 44 successful throws.
4. Two hundred hours is likely outside the data range and may produce an impossible or unrealistic prediction.

10-Question Practice Worksheet

Name: ____________________________    Date: __________________

Directions: Show your work. Explain your answers using the Harry Potter context whenever possible.

Part A: Understanding Lines of Fit

1. A line of fit for hours spent practicing a spell, x, and spell score, y, is y = 7x + 45. Identify the slope and y-intercept.
2. In the equation y = 7x + 45, explain what the slope means in this situation.
3. In the same equation, explain what the y-intercept means in this situation.
4. Use y = 7x + 45 to predict the score after 5 hours of practice.
5. Use y = 7x + 45 to determine how many hours of practice are needed for a predicted score of 80.

Part B: Scatter Plots, Predictions, and Constraints

Use this data set for Questions 6–8.

6. Plot the data points on a coordinate plane. Label the x-axis and y-axis.
7. Describe the relationship shown by the scatter plot: positive, negative, or no association? Explain.
8. A reasonable line of fit is y = 7x + 47. Use it to predict the accuracy score after 6 spells are practiced.
9. State a reasonable constraint for the number of spells practiced. Explain why a prediction far outside the data range may be unreliable.
10. Real-world problem: A student earns house points by completing magical community service. The line of fit is p = 10h + 25, where h is the number of service hours and p is the number of house points.
    
    a. Interpret the slope.
    b. Interpret the y-intercept.
    c. Predict the number of house points after 12 service hours.
    d. If the maximum possible score is 150 points, what is a reasonable constraint on the output?

Practice Worksheet Answer Key

1. Slope: 7. Y-intercept: 45.
2. The predicted score increases by 7 points for every additional hour of practice.
3. The predicted score is 45 when 0 hours of practice are recorded.
4. y = 7(5) + 45 = 80. The predicted score is 80.
5. 80 = 7x + 45; 35 = 7x; x = 5. Five hours are needed.
6. The points are (1, 54), (2, 61), (3, 67), (4, 76), and (5, 82).
7. Positive association; accuracy generally increases as the number of spells practiced increases.
8. y = 7(6) + 47 = 89. The predicted score is 89.
9. A reasonable constraint is approximately 0 ≤ x ≤ 6, or the observed range of 1–5 if no extrapolation is desired. Predictions far outside the data range may not represent the real situation.
10. a. House points increase by 10 for each additional service hour.
    b. The model predicts 25 points at 0 service hours.
    c. p = 10(12) + 25 = 145 points.
    d. A reasonable output constraint is 0 ≤ p ≤ 150.

Differentiation and Adaptations

Support for Learners Who Need More Scaffolding

• Provide the sentence frame: “The slope means that for every additional ___, the predicted ___ changes by ___.”
• Highlight mx as the rate of change and b as the starting value.
• Allow the student to use a calculator or graphing tool.
• Complete a table showing substitution before asking the student to solve independently.

Extension for Advanced Learners

• Have the student create a new Harry Potter-themed data set with at least six points.
• Ask the student to calculate or estimate a line of fit using a graphing tool.
• Compare two possible lines of fit and explain which model is more useful and why.
• Ask the student to identify whether a prediction is interpolation or extrapolation.

Flexible Learning Options

• Hands-on: Use spell cards as data points and arrange them on a floor coordinate grid.
• Digital: Enter the data into a spreadsheet or graphing calculator.
• Verbal: Explain the slope and intercept aloud to a parent or tutor.
• Creative: Design a “Hogwarts Data Report” that includes a scatter plot, equation, prediction, and constraints.