Modeling Sequences

Overview

Today students will model arithmetic sequences using a linear function rule, then use that rule to find missing terms and interpret the sequence’s initial value and rate of change in context. The lesson builds toward constructing and interpreting linear functions from tables and graphs, but focused on arithmetic sequence patterns.

Learning intentions

• Students will be able to recognize an arithmetic sequence from a pattern of equal differences.
• Students will be able to represent an arithmetic sequence with a linear function rule.
• Students will be able to determine the rate of change and initial value from a description, table, or graph.
• Students will be able to predict terms and explain what the parameters mean in the situation.

Success criteria

• I can find the common difference by comparing consecutive terms.
• I can write a function rule for an arithmetic sequence of the form y = mx + b.
• I can use my rule to find a requested term and justify the prediction.
• I can explain what the rate of change and initial value mean for the situation.

Curriculum links

• Functions: Construct a function to model a linear relationship and interpret its rate of change and initial value from a table or description.
• Expressions and Equations: Use standard number sense for computations with small integer values (as needed in term calculations).
• The Number System: Estimate/compare irrational numbers is not the focus today; class work emphasizes rational arithmetic with sequence terms.

Lesson structure (75 minutes)

1. 0–7 min · Hook (sequence warm-up). Teacher displays 5 terms of an unknown pattern: 4, 9, 14, 19, 24 and asks students what stays the same and what changes. Students think-pair-share, then volunteer the “difference” they notice.

2. 7–20 min · Mini-lesson: arithmetic sequences as linear functions. Teacher models the idea that an arithmetic sequence has a constant difference and that constant difference corresponds to a constant rate of change in a linear model; for example, for 4, 9, 14, 19, 24 the difference is 5. Students complete a short guided chart: index n (starting at 1) and term a(n), then identify m and b in y = mx + b language.

3. 20–33 min · Direct instruction: find initial value and rate of change. Teacher connects “initial term” to the starting point of the function and “common difference” to the slope: if a(1) is the first term, then b = a(1) when the input x = 1. Students work in pairs on two sequences:

• Sequence A: 3, 7, 11, 15, __
• Sequence B: 18, 13, 8, 3, __ They compute the common difference, write a rule, and fill the missing term.

4. 33–48 min · Table-to-rule modeling practice. Teacher gives a table (no graph yet):

• n: 1, 2, 3, 4, 5
• s(n): 12, 17, 22, 27,? Teacher asks: “What is the rate of change? What is the starting value at n=1? What rule could generate the table?” Students independently write y = mx + b and compute the missing value, then check with a partner.

5. 48–60 min · Graph/table interpretation check. Teacher sketches or projects a simple scatter of plotted points for the same sequence (n vs s(n)) and labels that points lie on a line. Teacher asks how to read slope and starting value from the points (rise over run; y-value when x=1). Students respond in writing: identify m and b, and state what they mean for the scenario.

6. 60–70 min · Application: word problem (context interpretation). Teacher presents: “A club charges a $12 membership plus $5 per event. After n events, the total cost is T(n).” Students determine T(1), the rate of change, write T(n) = 5n + 7 or the correct equivalent form based on how inputs are defined (n=1 corresponds to first event included), and compute the total after 6 events. Teacher circulates to check parameter interpretation.

7. 70–75 min · Exit ticket (quick assessment). Teacher collects:

• Arithmetic sequence: 7, 11, 15, 19, __
• Write a rule y = mx + b where x=1 corresponds to the first term.
• What does m represent in one sentence?

Resources

• Printed practice sheet with 3 tasks (table, missing term, word problem)
• Projector/board for displaying sequences and tables
• Small calculators allowed for checking only (computation should be simple)
• Graph paper or blank coordinate grids for points
• Student notebook or response sheet
• Timer for transitions

Assessment

• Formative checks during pair work: listen for correct common difference and correct mapping to m and b
• Targeted teacher prompts (“How do you know your difference is constant?” “What does b represent in this situation?”)
• Exit ticket scored for: common difference, correct rule, correct missing term, and correct interpretation of m

Differentiation

• Support: Provide a partially filled table template (n row, term row, and a difference column) and sentence starters: “The rate of change is… because…” and “The initial value is…”
• Support: Offer an optional reminder that if x=1 is the first term, then b equals the first term’s value.
• Extension: For early finishers, ask for an alternate rule form (equivalent expressions) and to verify by generating at least two terms.
• EAL/SEN: Allow verbal explanation plus labeled calculations; reduce cognitive load by keeping numbers small and giving one worked example of “difference → slope.”