Foundations of Mathematical Thinking · Junior Infants

Active learning ideas

Sequences and Series: Arithmetic Sequences

Active learning builds number sense in young learners by letting them touch, move, and see arithmetic sequences in real time. When children stack blocks or hop in patterns, they connect abstract numbers to concrete actions, making the idea of a common difference visible and memorable.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Strand 3: Algebra - A.1.10

• 15–30 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Stations Rotation
• 25 min · Small Groups

Hands-On: Block Stacking Patterns

Give each small group 20 linking cubes in two colors. Start a sequence like red, red-blue, red-red-blue (adding one blue each time). Groups copy it with cubes, find the common difference, and extend it five steps further. Discuss as a class what they notice.

Differentiate between various types of numerical sequences.

Facilitation Tip

During Block Stacking Patterns, encourage students to describe their stack aloud as they work to link the visual pattern with the numerical rule.

What to look for

Present students with three number patterns. Ask them to circle the patterns that are arithmetic sequences and underline the common difference for each. For example: 2, 4, 6, 8 (common difference: 2); 1, 3, 6, 10; 10, 7, 4, 1 (common difference: -3).

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 02

• Stations Rotation
• 20 min · Pairs

Movement: Hop and Count

Model hopping forward two steps while chanting '2, 4, 6'. Pairs take turns leading: one calls a starting number and difference (e.g., start 5, add 3), the other hops and counts aloud. Switch roles twice, then predict the tenth hop together.

Explain how to find the common difference in an arithmetic sequence.

Facilitation Tip

In Hop and Count, stand beside students to model the first two hops yourself, then let them lead the next steps to build ownership.

What to look for

Give each student a card with the first three terms of an arithmetic sequence, like 5, 10, 15. Ask them to write down the common difference and then write the next two terms in the sequence.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 03

• Stations Rotation
• 30 min · Small Groups

Card Sort: Sequence Match-Up

Prepare cards with partial sequences like 1 _ 5 _ and rule cards 'add 2'. In small groups, students fill blanks with number cards and match to rules. Extend by creating their own sequence for peers to solve.

Predict the next terms in an arithmetic sequence given the first few terms.

Facilitation Tip

For Sequence Match-Up, arrange the cards on a large table so students can physically group them before recording their matches.

What to look for

Hold up two fingers, then three, then four. Ask students: 'What is the pattern here? How many fingers am I adding each time?' Repeat with taking fingers away, like five, four, three. Guide them to identify the common difference and predict the next step.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 04

• Stations Rotation
• 15 min · Whole Class

Whole Class: Pattern Chant

Teach a clapping chant: clap once, twice, three times (common difference of one clap). Students echo and predict the next line. Vary differences (e.g., add two claps) and have volunteers lead the class.

Differentiate between various types of numerical sequences.

Facilitation Tip

During the Pattern Chant, pause after each line to let children whisper the next step to a partner before the whole class says it aloud.

What to look for

Present students with three number patterns. Ask them to circle the patterns that are arithmetic sequences and underline the common difference for each. For example: 2, 4, 6, 8 (common difference: 2); 1, 3, 6, 10; 10, 7, 4, 1 (common difference: -3).

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Teach arithmetic sequences by starting with objects children can manipulate, then move to visuals, and finally to symbols. Avoid rushing to abstract recording; give students time to articulate the rule in their own words before introducing terms like ‘common difference.’ Research shows that young learners grasp inverse operations better when they experience both addition and subtraction in the same lesson, so alternate between increasing and decreasing patterns to build flexible thinking.

By the end of these activities, students will identify arithmetic sequences, state the common difference, and predict the next term with confidence. They will demonstrate this through actions, discussions, and simple recordings, showing they recognize both increasing and decreasing patterns.

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Watch Out for These Misconceptions

• During Block Stacking Patterns, watch for students who assume every stack must grow taller. Redirect by asking, 'What if we take one block away each time? Let’s try it with these red blocks.'

  Hold up a completed stack of 5 blocks. Remove one block and ask, 'How many are left? What did I subtract? Will the next stack have more or fewer blocks?' Repeat the process together to show the rule works in reverse.

• During Block Stacking Patterns, watch for students who assume the common difference is always 1. Redirect by asking, 'How many blocks did you add this time? Show me with your fingers.'

  Use bead strings with marked skips of 2 or 5 beads. Ask students to count the gap between beads and say the rule aloud before stacking counters to match.

• During Card Sort: Sequence Match-Up, watch for students who think sequences only use numbers. Redirect by asking, 'Does this pattern have to be all numbers? What if we use shapes?'

  Place colored block cards face up. Ask students to sort them into a pattern and describe the rule using words like ‘add a circle each time.’ Then swap a number sequence card and repeat to show the pattern idea travels across materials.

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Methods used in this brief

• Stations Rotation