Recognizing Linear Patterns

Overview

In this first lesson of the unit, students distinguish linear from nonlinear relationships by examining tables, graphs, equations, and real-world situations. They connect linearity to a constant rate of change and prepare to model relationships using equations in later lessons.

Learning intentions

Students will be able to:

• Identify whether a relationship is linear or nonlinear.
• Determine whether a table has a constant rate of change.
• Interpret what a constant rate of change means in a real-world situation.
• Explain how an equation of the form (y = mx + b) represents a linear relationship.

Success criteria

• I can use differences or slope to determine whether a relationship is linear.
• I can explain why a graph showing a straight line represents a linear function.
• I can identify a nonlinear relationship from a table, graph, equation, or situation.
• I can describe what the rate of change means in context.

Curriculum links

• Interpret equations of the form (y = mx + b) as linear functions and recognize that their graphs are straight lines.
• Construct and analyze linear relationships using constant rate of change from tables, graphs, and descriptions.
• Describe whether a functional relationship is linear or nonlinear by analyzing a graph.
• Compare functions represented in different ways.

Lesson structure (45 minutes)

1. 0–5 min · Hook and notice. Teacher opens with the hook and comparison slides showing two situations: a taxi fare that increases by the same amount per mile and the area of a square as its side length increases. Ask, “Which relationship would produce a straight-line graph, and how do you know?” Students make an individual prediction, then share evidence with a partner.

2. 5–12 min · Direct instruction. Teacher uses the linear relationships teaching slides to define a linear relationship as one with a constant rate of change. Model checking a table such as (x: 0,1,2,3) and (y: 4,7,10,13), noting that (y) increases by 3 each time. Connect this to (y=3x+4), identifying 3 as the rate of change and 4 as the initial value. Students record the definition and annotate the table.

3. 12–17 min · Nonlinear comparison. Teacher displays a table for (y=x^2) and a graph of a curved relationship using the linear-versus-nonlinear comparison slides. Ask, “What changes from interval to interval?” and “What visual evidence do you see?” Students calculate successive differences, identify that they are not constant, and explain why the relationship is nonlinear.

4. 17–30 min · Partner analysis. Teacher distributes the linear pattern analysis worksheet and assigns pairs a set of tables, graphs, equations, and real-world descriptions. Students label each relationship linear or nonlinear, justify each decision, and calculate the rate of change for linear examples. Circulate and ask, “What is changing?” “Is that change consistent?” and “Where do you see the initial value?” Students must write evidence rather than relying only on a visual guess.

5. 30–38 min · Representation match and discussion. Teacher displays the discussion prompts in the partner analysis and discussion slides. Pairs compare one table with one equation and one graph, then explain how the representations show the same constant rate of change. Invite two pairs to share different strategies, addressing the misconception that every increasing relationship is linear.

6. 38–45 min · Check for understanding and exit ticket. Teacher presents the final prompt in the review and exit-ticket slides. Students complete the final worksheet questions independently: determine whether (y=2x+5) is linear and explain the meaning of 2 and 5; then classify a table with changing differences as linear or nonlinear and justify the answer. Collect responses to plan the next lesson.

Resources

• the complete Recognizing Linear Patterns slide deck
• the linear pattern analysis worksheet
• Projector or interactive display
• Whiteboard and markers
• Student notebooks
• Pencils and graph paper
• Calculators, if needed for access support

Assessment

• During instruction, use cold-call questions and mini whiteboard responses to check whether students connect constant differences with linearity.
• During partner work, listen for accurate use of “rate of change,” “initial value,” “constant,” and “nonlinear,” and review students’ written justifications.
• Use the independent exit questions to identify students who can classify relationships but need further support interpreting rate of change or initial value.

Differentiation

• Support students with a table-difference guide: calculate each change in (y), compare the results, and use the sentence frame, “This relationship is ___ because the rate of change is ___.”
• Provide enlarged graphs, clearly spaced tables, and a reference showing (y=mx+b), with (m) as rate of change and (b) as initial value. Read directions aloud and allow students to explain reasoning verbally before writing.
• For multilingual learners, preview “linear,” “nonlinear,” “constant,” “rate,” and “initial value” with simple examples and sentence frames. Pair students strategically.
• Extend early finishers by asking them to create one linear and one nonlinear real-world situation, represent each with a table, and explain how a classmate could tell them apart.