Extending Number Patterns

Overview

This is lesson 7 of 26 in Linear Patterns and Algebra. Students build on prior work identifying pattern rules by extending arithmetic sequences forward and backward, finding missing terms, and interpreting negative common differences using concrete number-line movement and symbolic notation.

Learning intentions

Students will:

• Extend arithmetic sequences forwards and backwards.
• Identify and use the common difference, including negative changes.
• Find missing terms by reasoning about equal jumps.
• Record a sequence rule and explain their thinking using mathematical language.

Success criteria

• I can identify the common difference in an arithmetic sequence.
• I can extend a sequence in either direction, including when the difference is negative.
• I can find a missing term and explain how I know it is correct.
• I can check my answer by applying the same change between every pair of terms.

Curriculum links

• Patterns and Relations — oral and written patterns and their corresponding relations.
• Patterns and Relations — constructing tables of values from a relation and analysing patterns.
• Mathematical processes — communication, connections, mental mathematics and estimation, problem solving, reasoning, and visualization.
• Report card evidence — Knowledge and Understanding, Mental Math, and Problem Solving.

Lesson structure (50 minutes)

1. 0–5 min · Engage and retrieve. Teacher displays the opening question, “What number comes next, and how could you prove it?” using the opening pattern challenge, then presents the sequence 6, 10, 14, 18 and asks students to identify the change mentally. Students discuss the rule with a partner and show the next two terms using fingers or mini-whiteboards.

2. 5–13 min · Explicit teaching. Teacher uses the common-difference examples to model that an arithmetic sequence changes by the same amount each time; demonstrate 3, 7, 11, 15 as “add 4,” then 20, 15, 10, 5 as “subtract 5.” Connect positive and negative changes to right and left movement on a number line. Students record a brief guided note and explain why the common difference is +4 or −5.

3. 13–23 min · Number-line floor activity. Teacher places a floor number line from −10 to 30 and models a starting number, a jump size, and the direction of travel. Students take turns physically moving forward or backward to represent sequences such as −2, 1, 4, 7 and 12, 8, 4, 0; the observing students predict the next position and justify it. Teacher uses colour-coded arrows for the jump and addresses misconceptions immediately.

4. 23–31 min · Sequence card investigation. Teacher distributes the Sequence Rule Cards and selected Mathology sequence cards, modelling one card before pairs begin. Students sort or discuss examples by the type of change, extend each sequence in both directions where possible, and explain the common difference to the group. Teacher prompts with, “What stays the same?” and “How can you check?”

5. 31–42 min · Guided and independent practice. Teacher completes the first two problems on the extending number patterns worksheet with the class: one forward sequence and one sequence containing a missing term. Students then complete the remaining problems independently, including negative changes and backward extension. Early finishers use Mathletics for arithmetic-sequence practice; teacher works beside students requiring additional support.

6. 42–50 min · Evidence checkpoint and reflection. Teacher asks each student to complete one forward, one backward, and one missing-term sequence on the final section of the extending number patterns worksheet, recording whether the common difference was identified independently. Students explain one answer orally or in writing and rate their confidence from 1 to 3. Teacher collects the work for outcome-based documentation.

Resources

• the complete pattern lesson deck
• the extending number patterns worksheet
• the Sequence Rule Cards
• Mathology sequence cards
• Floor number line and coloured arrows
• Mini-whiteboards, markers, and erasers
• Student devices and Mathletics
• Short, teacher-previewed arithmetic-sequences video from the planned YouTube search
• Pencils, highlighters, and visual guided notes

Assessment

• Knowledge and Understanding: During modelling and card work, check whether each student identifies the repeated change, distinguishes positive from negative common difference, and extends terms accurately. Record independently, with prompting, or not yet.
• Mental Math: Listen for efficient mental calculation of equal jumps, such as adding or subtracting 3, 4, 5, or 10. Note whether students use the number line, count-on/count-back, or a secure mental strategy.
• Problem Solving: Use the checkpoint to assess whether students can solve forward, backward, and missing-term sequences and explain their reasoning. Record independence, accuracy, and whether the student checks the common difference.
• Collect the checkpoint as evidence for Knowledge and Understanding, Mental Math, and Problem Solving in PowerSchool descriptors.

Differentiation

• Provide an individual number line, physical arrows, colour-coded jumps, and a completed example with the first jump highlighted.
• Reduce the number of terms and use smaller numbers for students on modified programmes or with a math disability; read each direction aloud and check understanding before independent work.
• Use the dyslexia-friendly worksheet format: clear sans-serif font, generous spacing, one problem per line, minimal visual clutter, bold operation signs, and optional highlighting of each jump.
• Offer sentence starters such as “The common difference is ___ because…” and allow oral explanation, manipulatives, or pointing before written recording.
• Challenge confident students to create a sequence with a negative common difference, hide two terms, and prove that their sequence has only one possible solution.