Patterning and First Differences

Students learn best when they can see, touch, and talk about growing patterns rather than just read about them. Active learning invites students to build, sort, and compare representations, which strengthens their ability to recognize linear relationships and connect first differences to real-world contexts like motion and costs.

• Grade 8
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Calculate the first differences for a given table of values and identify if the pattern is linear.
2. Compare the algebraic rules and tables of values for two different linear relationships to determine which has a greater rate of change.
3. Explain how a constant first difference in a table of values signifies a linear relationship.
4. Identify the rate of change (slope) from a table of values by analyzing its first differences.
5. Represent a linear pattern described verbally as a table of values and an algebraic rule.

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Manipulative Build: Growing Patterns

Provide color tiles or linking cubes. Students construct first five terms of patterns like staircases or borders, sketch each, record values in tables, and calculate first differences. Groups share findings to classify as linear or not.

Prepare & details

Explain how first differences in a table of values indicate whether a relationship is linear.

Facilitation Tip: During Manipulative Build, circulate and ask students to predict the next term before adding tiles, reinforcing the connection between physical growth and numeric tables.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 25 min·Pairs

Card Sort: Representation Matches

Prepare cards showing tables with first differences, graphs, equations, and descriptions of linear functions. In pairs, match pairs of equivalent representations, then compare rates of change between two sets. Discuss mismatches.

Prepare & details

Apply the concept of first differences to identify the rate of change in a growing pattern.

Facilitation Tip: For Card Sort, have students justify each match aloud, especially when verbal descriptions and graphs don’t immediately align, to deepen transfer of concepts.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 40 min·Pairs

Table Comparison: Function Duel

Give pairs of functions, one as a table and one algebraic. Students complete missing table values, find first differences for both, and compare rates and starting points. Extend to graphing quick sketches.

Prepare & details

Compare different linear patterns by analyzing their tables of values and algebraic rules.

Facilitation Tip: In Table Comparison, challenge pairs to defend their choice of which pattern grows faster using both first differences and rate of change language.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 30 min·Whole Class

Real Data: Motion Tables

Students walk at constant speeds, time and record distances in tables as a class. Calculate first differences to find rates. Compare two walkers' data side-by-side.

Prepare & details

Explain how first differences in a table of values indicate whether a relationship is linear.

Facilitation Tip: In Real Data Motion Tables, prompt groups to discuss why the first differences might not be perfectly constant, introducing the idea of measurement error and real-world variation.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Start with concrete tools like tiles or counters to ground students in the concept of growing patterns before moving to abstract representations. Emphasize the process of calculating first differences by hand to build number sense and avoid over-reliance on technology. Avoid rushing to symbolic rules; instead, let students discover the relationship between constant differences and linearity through structured exploration and discussion.

What to Expect

Successful learning looks like students confidently building patterns with manipulatives, accurately computing and comparing first differences, and clearly explaining why constant differences indicate linearity. They should move between tables, graphs, rules, and verbal descriptions without hesitation, showing deep conceptual understanding.

Watch Out for These Misconceptions

Common Misconception

During Manipulative Build, watch for students assuming that a pattern starting with zero tiles means the first difference begins at zero.

What to Teach Instead

Prompt students to record their starting point and first difference in a shared table on the board, emphasizing that the y-intercept and first difference are independent values to compare across different patterns.

Common Misconception

During Manipulative Build, watch for students believing all growing patterns must be linear.

What to Teach Instead

Have students build a second pattern using tiles that accelerates growth (e.g., adding an extra tile each step), then calculate first differences to see they change, prompting a class discussion on non-linear patterns.

Common Misconception

During Card Sort, watch for students assuming first differences only apply to numeric tables.

What to Teach Instead

Ask groups to calculate the rate of change from the algebraic rule and compare it to the first differences in the table, using sticky notes to annotate connections between representations.

Assessment Ideas

Quick Check

After Manipulative Build, provide three different tables of values. Ask students to calculate the first differences for each table and circle the tables that represent linear relationships, justifying their choice with one sentence in their notebooks.

Exit Ticket

After Card Sort, give each student a card with a verbal description of a linear pattern (e.g., 'A taxi charges $4.00 plus $2.00 per kilometer'). Ask them to create a table of values for the first 5 kilometers and identify the rate of change from their table before leaving class.

Discussion Prompt

During Table Comparison, present two different linear relationships, one as a table of values and one as an algebraic rule (e.g., y = 3x + 1). Ask students to determine which relationship grows faster and explain their reasoning using the concepts of first differences and rate of change in a short written response.

Extensions & Scaffolding

• Challenge students to create a non-linear growing pattern using tiles and write a verbal description that hides its quadratic nature, then have peers identify the flaw in assuming linearity.
• For students who struggle, provide partially completed tables with scaffolding questions like, 'What do you notice about the gaps between terms?'
• Deeper exploration: Ask students to research a real-world scenario (e.g., population growth, savings interest) and analyze whether it is linear or not, using first differences as evidence.

Key Vocabulary

First Differences	The result of subtracting consecutive terms in a sequence or consecutive y-values in a table of values. Constant first differences indicate a linear relationship.
Linear Relationship	A relationship between two variables where the graph is a straight line. In a table of values, this is indicated by constant first differences.
Rate of Change	The constant rate at which the dependent variable (y) changes with respect to the independent variable (x). For linear relationships, this is the same as the slope and is found by calculating the first differences.
Algebraic Rule	A mathematical equation, typically in the form y = mx + b, that describes the relationship between two variables in a linear pattern.