Mathematics · 10th Grade · Quadratic Functions and Modeling · Weeks 28-36

Comparing Quadratic and Linear Models

Students will compare and contrast quadratic and linear functions in real-world contexts, identifying when each model is appropriate.

• Common Core State Standards
• CCSS.Math.Content.HSF.LE.A.1
• CCSS.Math.Content.HSF.BF.A.1

About This Topic

Comparing linear and quadratic models is a modeling and reasoning task that sits at the intersection of the CCSS Functions standards. Students must move beyond recognizing the shape of each function type and develop judgment about which model fits a given situation based on rates of change, context, and data patterns. A linear model captures constant change; a quadratic model captures situations where the rate of change is itself changing at a constant rate.

In the US 10th-grade curriculum, students typically arrive with solid fluency in linear functions from 8th and 9th grade. This unit asks them to apply that fluency comparatively, examining first and second differences, regression outputs, and contextual clues to determine model appropriateness. Common real-world contexts include analyzing projectile versus constant-velocity motion, comparing linear salary growth to quadratic revenue models, or examining area relationships as dimensions change.

Active learning is especially valuable here because model selection requires students to articulate reasoning, not just compute. Discussion-based tasks and data analysis projects push students to use mathematical language precisely and defend their choices.

Key Questions

1. Differentiate between situations best modeled by linear functions versus quadratic functions.
2. Analyze data sets to determine whether a linear or quadratic model is a better fit.
3. Construct a real-world scenario that can be modeled by a quadratic function and explain why it's not linear.

Learning Objectives

• Analyze real-world data sets to determine whether a linear or quadratic model is the most appropriate fit.
• Compare and contrast the graphical and algebraic characteristics of linear and quadratic functions.
• Explain the concept of constant versus changing rates of change in the context of linear and quadratic models.
• Create a real-world scenario that is best modeled by a quadratic function and justify why a linear model is insufficient.
• Evaluate the appropriateness of linear and quadratic models for predicting outcomes in given scenarios.

Before You Start

Introduction to Functions

Why: Students need a foundational understanding of what a function is, including input/output relationships and function notation.

Linear Functions and Their Graphs

Why: Students must be proficient in identifying, graphing, and analyzing linear functions, including their constant rate of change (slope).

Graphing Quadratic Functions

Why: Students should have prior experience graphing basic quadratic functions (parabolas) and understanding their general shape.

Key Vocabulary

Linear Function	A function whose graph is a straight line, characterized by a constant rate of change (slope).
Quadratic Function	A function whose graph is a parabola, characterized by a changing rate of change (second differences are constant).
Rate of Change	The measure of how much one quantity changes with respect to another; for linear functions, this is constant, while for quadratic functions, it changes.
First Differences	The differences between consecutive y-values in a data set; constant first differences indicate a linear relationship.
Second Differences	The differences between consecutive first differences; constant second differences indicate a quadratic relationship.

Watch Out for These Misconceptions

Common Misconception

Any curved data should be modeled with a quadratic function.

What to Teach Instead

Students see a scatter plot that is not perfectly linear and immediately assume quadratic is the right fit. Many non-linear patterns, including exponential growth and logarithmic curves, are not well-modeled by quadratics. Examining residual plots and considering the underlying context is essential before committing to a model. Sorting tasks that include exponential scenarios help students see that "not linear" does not mean "quadratic."

Common Misconception

A quadratic model is always more accurate than a linear model.

What to Teach Instead

Students may assume that a more complex model is always better. A quadratic model may fit a small sample well by chance without capturing the true relationship, which is a form of overfitting. Context and theory should guide model choice alongside statistical measures of fit. Comparing both models' residuals on the same dataset makes the tradeoff concrete.

Common Misconception

If the data has a turning point, it must be quadratic.

What to Teach Instead

Students over-apply the vertex concept: any dataset that goes up and then comes down gets labeled quadratic. Other functions, including absolute value, piecewise, and periodic functions, also have turning points. The key quadratic signature is constant second differences in the data table, not simply the presence of a turning point in the graph.

Active Learning Ideas

Think-Pair-Share: Data Set Decision

Provide three sets of data points printed on cards: one clearly linear, one clearly quadratic, and one ambiguous. Each student makes an initial model choice individually, marking their reasoning. Pairs share and try to reach agreement on the ambiguous case. Class debrief focuses specifically on the ambiguous set, surfacing multiple defensible positions.

20 min·Pairs

Small Group: Real-World Scenario Sort

Groups receive 8-10 scenario cards describing real contexts (e.g., steady salary raise, ball drop, fixed-perimeter area problem). Groups sort them into linear, quadratic, and neither/unsure, then prepare a one-sentence justification for each card. Final share-out reveals disagreements and prompts class discussion about edge cases.

25 min·Small Groups

Gallery Walk: Residual Plot Analysis

Post four stations, each showing a scatter plot with both a linear and quadratic regression line fit to the same data, along with residual plots for each. Students rotate and record which model fits better at each station and why, using residual pattern language. Debrief connects visual residual patterns to the formal model comparison process.

30 min·Small Groups

Whole Class: Desmos Differences Investigation

Display a table of values with no graph. Students predict whether the relationship is linear or quadratic based on first and second differences. The teacher reveals the graph after students commit to a prediction. Repeat with three to four datasets, each designed to reinforce the distinction between constant first differences (linear) and constant second differences (quadratic).

20 min·Whole Class

Real-World Connections

• Athletic coaches and sports analysts use quadratic models to predict the trajectory of projectiles like baseballs or basketballs, understanding that gravity causes their speed to change over time, unlike a linear model which assumes constant speed.
• Engineers designing bridges or roller coasters often use quadratic equations to model parabolic curves, ensuring structural integrity and passenger experience, as these curves represent changing acceleration and forces.
• Farmers might use linear models to predict crop yield based on fertilizer amounts if the relationship is directly proportional, but would use quadratic models if yield increases at a decreasing rate after a certain point, due to saturation or other limiting factors.

Assessment Ideas

Quick Check

Present students with two data tables, one showing constant first differences and the other showing constant second differences. Ask them to identify which table represents a linear relationship and which represents a quadratic relationship, and to explain their reasoning using the terms 'first differences' and 'second differences'.

Discussion Prompt

Pose the scenario: 'Imagine you are advising a city planner. One proposal is for a new bus route with a constant speed, and another is for a new park's fountain spray pattern. Which scenario would likely be modeled by a linear function, and which by a quadratic function? Justify your answers by describing the expected rates of change in each situation.'

Exit Ticket

Provide students with a brief description of a real-world situation (e.g., 'The height of a ball thrown upwards over time' or 'The cost of producing widgets based on the number produced'). Ask them to write one sentence explaining whether a linear or quadratic model is more appropriate and one sentence explaining why, referencing the expected pattern of change.

Frequently Asked Questions

How do you decide whether to use a linear or quadratic model for a data set?

Check the first and second differences in the data. Constant first differences indicate a linear relationship; constant second differences indicate a quadratic relationship. Also examine a scatter plot and consider the context: does the situation involve a constant rate of change (linear) or a rate of change that is itself changing at a constant rate (quadratic)?

What is a real-world example that requires a quadratic model instead of a linear one?

Projectile motion is the clearest example: the height of a thrown object changes at a non-constant rate due to gravity, producing a parabolic path. Another common example is area optimization, where doubling one dimension of a rectangle with a fixed perimeter does not simply double the area. In both cases, the rate of change is not constant.

How can residual plots help determine which model fits data better?

A well-fitting model produces residuals scattered randomly around zero with no visible pattern. When a linear model is applied to quadratic data, the residuals will show a curved pattern, revealing that the linear fit is systematically missing the true trend. A quadratic fit on the same data will produce residuals that appear more randomly distributed.

How does active learning support students in comparing linear and quadratic models?

Model selection is fundamentally a reasoning task, not a computation task. Students who debate model choices in small groups, justify decisions in writing, and analyze real data sets develop the judgment that multiple-choice practice cannot build. Structured discussion tasks give students practice with evidence-based mathematical argument and the language to express it precisely.