Finding the Term Rule

Overview

In Lesson 8 of 26, students move from describing a linear pattern by repeated addition to finding any term using a position-based rule. Using cubes, a term-position chart, guided notes, Mathology practice, and Mathletics, students connect the constant increase to a rule such as (4n+1).

Learning intentions

Students will:

• identify the constant difference in a linear pattern;
• describe a pattern using a table of values and words;
• develop and use a rule to find any term, including a distant term;
• verify a rule by substituting the position of a known term.

Success criteria

• I can find and explain the constant difference.
• I can use a term-position chart to connect position and term.
• I can write a rule in the form “term = ___ × position + ___.”
• I can use my rule to find a distant term and check it with a known term.

Curriculum links

• Patterns and Relations — oral and written patterns and their corresponding relations.
• Patterns and Relations — construct and analyse tables of values from a relation.
• Communication, Connections, Reasoning, Problem Solving, and Visualization.
• Report-card evidence: Knowledge and Understanding, Mental Math and Estimation, and Problem Solving.

Lesson structure (50 minutes)

1. 0–5 min · Hook and retrieval. Display a growing cube pattern in the hook and retrieval slides: 4, 7, 10, 13. Ask, “Without building it, how could we find Term 20?” Students silently predict, then explain the pattern to a partner. Briefly revisit the meaning of term, position, and constant difference.

2. 5–13 min · Concrete modelling. Build the first four terms with linking cubes: 1 cube plus 3 more each time, 1 + 3, 1 + 3 + 3, and so on. Teacher records a term-position chart and models repeated addition. Students build or sketch the terms, identify the part that repeats, and state the difference mentally. Use the sequence rule cards briefly as a visual prompt for identifying “Add a constant” and “Linear n-th term.”

3. 13–23 min · Explicit teaching and guided notes. Open the rule-building slides and show a short, pre-selected age-appropriate video segment about finding the nth term; pause before the worked answer so students predict the next step. Students complete the guided notes and term-rule worksheet while the teacher models:

• find the constant difference;
• multiply the difference by the position;
• adjust by the starting value;
• write the rule: term = difference × position + adjustment. For the pattern 4, 7, 10, 13, model (3n+1). Check (n=1): (3(1)+1=4).

4. 23–33 min · Guided practice. Teacher and students solve two patterns together using cubes and the chart in the guided-practice slides. Example A: 5, 9, 13, 17; Example B: 8, 14, 20, 26. Students first state the difference, then complete the rule template: “The difference is __. The rule begins __n. I add/subtract __ because __.” Prompt students to verify the rule with Term 1 and Term 3. Ask each student to explain one step aloud.

5. 33–41 min · Independent application. Distribute the remaining questions in the independent practice section. Students find rules and use them to calculate specified distant terms, including Term 25 or Term 40. Students who finish early create a linear pattern with a negative adjustment, write its rule, and swap with a partner to solve. Teacher conferences individually, recording evidence for Knowledge and Understanding and Problem Solving.

6. 41–47 min · Digital practice and feedback. Students complete a focused linear-pattern activity in Mathletics, working independently with headphones where available. Teacher supports students on modified programs and checks that answers are supported by a rule rather than guessing. Students correct one error by identifying whether the problem was the difference, adjustment, multiplication, or substitution.

7. 47–50 min · Evidence checkpoint and exit. Display the checkpoint and reflection slides. Students complete: “The pattern is 6, 10, 14, 18… Find Term 30. Write your rule and verify it using Term 2.” Students add a confidence rating and hand in the response. Collect the worksheet and note who can independently identify a rule, use it for a distant term, and verify it.

Resources

• the complete 50-minute teaching deck
• the guided notes and term-rule worksheet
• Linking cubes or small connecting blocks
• Whiteboard, markers, and individual mini-whiteboards
• Term-position chart displayed or projected
• Mathology linear-pattern lesson and practice materials
• Mathletics devices and headphones
• Pre-selected video segment on finding the nth term
• Calculators, if required by an individual learning plan

Assessment

• Knowledge and Understanding: Observe whether students identify the constant difference, complete the term-position chart, and state a correct rule. Record progress using “independent,” “developing,” or “requires prompting.”
• Mental Math and Estimation: Listen for accurate mental identification of repeated increases and reasonable estimates before calculating distant terms.
• Problem Solving: Use the checkpoint to assess whether students can select a rule, find Term 30, and verify it with a known term. Record whether each step is complete, partially complete, or not yet demonstrated.
• Use student explanations, worksheet responses, and Mathletics results as evidence for PowerSchool descriptors.

Differentiation

• Provide cubes, a colour-coded chart, enlarged guided notes, and the rule template: “Term = ___ × position + ___.” Keep the constant difference in one colour and the adjustment in another.
• For students with dyslexia, provide uncluttered, dyslexia-friendly copies with a clear sans-serif font, generous spacing, reduced copying, read-aloud instructions, and permission to answer orally or by pointing before recording.
• For students with a math disability or modified program, use patterns with small positive differences, pre-complete the first row, limit the number of questions, and allow a calculator after the structure of the rule is explained.
• Seat distractible students close to the teacher, assign a concrete role such as cube builder or chart checker, use short timed tasks, and provide private praise for on-task mathematical talk. Extend capable students with a rule beginning at a non-unit position or a pattern with a negative adjustment.