Mathematics · 8th Grade · Functions and Modeling · Weeks 10-18

Comparing Functions

Comparing properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

• Common Core State Standards
• CCSS.Math.Content.8.F.A.2

About This Topic

Comparing functions is a higher-order skill that requires students to extract key properties such as rate of change and initial value from whichever representation is given, and then make meaningful comparisons. In 8th grade, students might compare a function shown as a graph to one shown as a table, or an equation to a verbal description. This cross-format comparison directly reflects how data appears in real contexts, where different sources rarely use the same format.

The standard CCSS.Math.Content.8.F.A.2 asks students to compare properties of two functions, not just identify properties in isolation. This distinction matters instructionally: a student must be able to extract slope from a graph, extract slope from a table, and then decide which function has a greater rate of change. The comparison step is where reasoning becomes explicit.

Active learning is particularly effective here because comparing functions across representations is a social task. When students must explain to a partner why one function grows faster using evidence from two different formats, they practice the kind of mathematical argumentation that deepens understanding well beyond computation.

Key Questions

1. Compare the rates of change of two functions presented in different formats.
2. Analyze which function would be more suitable for a particular real-world application.
3. Justify conclusions about two functions based on their various representations.

Learning Objectives

• Compare the rates of change for two functions presented in different formats (graphical, tabular, algebraic, verbal).
• Analyze which of two functions, represented differently, is more suitable for a given real-world scenario.
• Justify conclusions about the relative properties (e.g., rate of change, initial value) of two functions using evidence from their distinct representations.
• Calculate the rate of change and initial value for functions presented in tabular or graphical formats.
• Translate between different representations of a function (e.g., from a table to an equation, from a graph to a verbal description).

Before You Start

Representing Linear Functions

Why: Students need to be able to interpret and generate equations, graphs, and tables for linear functions before they can compare them.

Calculating Rate of Change and Initial Value

Why: Understanding how to find these key properties within a single representation is fundamental to comparing them across different representations.

Key Vocabulary

Rate of Change	The constant rate at which the dependent variable changes with respect to the independent variable. It is often represented as 'rise over run' or 'change in y over change in x'.
Initial Value	The value of the dependent variable when the independent variable is zero. For linear functions, this is the y-intercept.
Linear Function	A function whose graph is a straight line. Its rate of change is constant.
Representation	A way of showing a mathematical relationship, such as an equation, a graph, a table of values, or a verbal description.

Watch Out for These Misconceptions

Common Misconception

Students often assume the function with the steeper visual slope in a graph is always the 'better' or 'faster' function, without reading the axis scales.

What to Teach Instead

Always have students check axis labels and scale before comparing graphs. Pair activities where one partner intentionally uses a rescaled graph help students notice that visual steepness alone is not sufficient for comparison.

Common Misconception

When comparing a graph to a table, students sometimes compute unit rate from the table but interpret it differently from visual slope, not realizing they represent the same property.

What to Teach Instead

Have students explicitly convert both representations to slope form (rise over run or change in y over change in x) before comparing. A side-by-side annotation activity makes the equivalence visible.

Active Learning Ideas

Think-Pair-Share: Which Is Faster?

Give each pair two functions in different formats (one as a graph, one as a table). Each student independently determines the rate of change for their assigned function, then the pair compares and identifies which grows faster. Pairs share their reasoning process with the class.

20 min·Pairs

Matching Activity: Same Function, Different Look

Provide a set of cards showing the same linear function in four formats (equation, graph, table, verbal description). Students match complete sets and then compare two different functions to identify which has the steeper rate of change and the higher initial value. Groups must cite specific evidence from each format.

30 min·Small Groups

Gallery Walk: Real-World Function Comparison

Post six stations, each showing two functions representing real scenarios (e.g., two phone data plans, two savings accounts). Students determine which function represents the better deal under given conditions, writing their reasoning on a response sheet. Debrief focuses on how representation format affected difficulty.

35 min·Small Groups

Real-World Connections

• Financial analysts compare different investment plans, one described by an equation for projected earnings and another by a table of historical performance, to recommend the best option for a client.
• City planners evaluate two proposed public transportation routes. One route's travel time is represented by a graph showing distance over time, while the other's is given by a table of average speeds between stops, to determine which is more efficient for commuters.
• Engineers designing a new product might compare the cost of materials versus production volume. One supplier offers a cost function as an equation, while another provides a table of costs for different quantities, to find the most economical choice.

Assessment Ideas

Exit Ticket

Provide students with two functions: Function A is a graph, and Function B is a table of values. Ask students to write one sentence comparing their rates of change and one sentence explaining which function represents a faster growth rate.

Quick Check

Present students with a scenario, for example, two different cell phone plans. Plan 1 is described by an equation, and Plan 2 is presented as a table. Ask students to calculate the cost for a specific number of minutes for each plan and determine which plan is cheaper for that usage.

Discussion Prompt

Pose the question: 'Imagine you are comparing two ways to save money. One method is described by the equation y = 5x + 50, where y is the total savings and x is the number of weeks. The second method is shown in a table where after 3 weeks you have $65, and after 7 weeks you have $85. Which method will help you save more money after 10 weeks, and how do you know?'

Frequently Asked Questions

How does active learning help students compare functions across different representations?

Comparing functions is fundamentally a reasoning task. Active approaches like matching activities and gallery walks require students to verbalize how they extracted a rate of change from each format and why their comparison holds. That narration makes implicit processes explicit, which is precisely where students' gaps in cross-representation fluency tend to appear.

What properties should students compare when analyzing two functions?

Students should compare rate of change (slope), initial value (y-intercept), and the overall behavior of the function. For real-world contexts, they should also consider which function produces a greater output for a specific input value, since that often drives the applied decision.

What formats might functions be presented in for CCSS 8.F.A.2?

Functions can appear algebraically (as an equation), graphically (as a plotted line), numerically (as a table of values), or as a verbal description of a real-world relationship. Students must be able to extract comparable properties regardless of which format is used.

How do you compare two functions when they are in different formats?

Start by identifying the same property (like slope) in both functions using the method appropriate for each format: calculate rise-over-run from a graph, find the constant rate of change in a table, or read the coefficient of x in an equation. Once both are in comparable form, the comparison is straightforward.

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