Patterns and Functions: Input-Output Tables

Active learning helps young students see the connection between concrete actions and abstract rules in input-output tables. When they physically manipulate objects or move through sequences, the rule becomes visible and memorable. This hands-on approach builds confidence and clarity before moving to symbolic representation.

• Junior Infants
• Foundations of Mathematical Thinking
• 4 activities
• 15 min–30 min

Learning Objectives

1. Identify the pattern or rule connecting input and output values in a given table.
2. Predict the output value for a new input based on an identified pattern in a table.
3. Construct a simple algebraic rule to represent a linear pattern from a table.
4. Explain the relationship between input, rule, and output in a functional context.

Inquiry Circle

⏱ 25 min·Pairs

Manipulative Tables: Counter Patterns

Provide trays with counters and simple input-output cards (e.g., input 2, output 4: double). Pairs build their table by placing inputs, applying the rule, and recording outputs on a large chart. They swap rules and predict the next three entries. End with sharing one prediction.

Prepare & details

Analyze how to identify the rule that connects input and output values.

Facilitation Tip: During Manipulative Tables, have students record each step in a notebook so they can see the progression of inputs and outputs side by side.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 30 min·Small Groups

Function Machine Game: Small Group Relay

Create a 'machine' from a cardboard box with a rule inside (e.g., add 1). One student inputs a number verbally or with fingers, the next processes it secretly, and a third checks the output. Groups rotate roles and adjust rules midway. Record results on a class table.

Prepare & details

Predict the output for a given input based on an identified pattern.

Facilitation Tip: For the Function Machine Game, place the relay stations far enough apart to encourage physical movement but close enough for peer discussion.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 20 min·Whole Class

Whole Class Chain: Human Patterns

Students line up as a chain. Teacher gives first input (clap 1), next adds rule (clap 2 more), passing output down. Predict what the last student claps. Repeat with new rules like 'double', drawing the table on the board as a class.

Prepare & details

Construct an algebraic rule to represent a linear pattern from a table.

Facilitation Tip: In Whole Class Chain, pause after each child’s turn to ask the class to predict the next output before moving on.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 15 min·Individual

Individual Drawing Boards: Picture Tables

Give laminated sheets with input pictures (e.g., 1 car). Students draw outputs using crayons (e.g., 3 cars: add 2). Circle the rule from options. Share drawings in a gallery walk.

Prepare & details

Analyze how to identify the rule that connects input and output values.

Facilitation Tip: With Individual Drawing Boards, provide colored pencils so students can visually separate inputs, operations, and outputs.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Teaching This Topic

Teach this topic by letting students experience the rule first and name it second. Avoid giving rules upfront; instead, ask them to describe what they did with the counters or blocks. Research shows that when children discover patterns themselves, they retain the concept longer. Limit whole-group explanations until after exploration, so misconceptions surface naturally during activities.

What to Expect

Successful learning looks like students articulating the rule aloud, predicting outputs accurately, and explaining their reasoning with examples. They should use terms like 'input', 'output', and 'rule' with confidence. Tables should be completed consistently, showing they trust the pattern rather than guessing.

Watch Out for These Misconceptions

Common Misconception

During Manipulative Tables, watch for students who guess outputs without touching the counters or writing a rule.

What to Teach Instead

Prompt them to show you how they arrived at the output using the counters. Ask, 'What did you do to the 3 blocks to get 6?' and have them demonstrate the action.

Common Misconception

During Function Machine Game, notice when students change the rule midway through the relay, especially after a pause.

What to Teach Instead

Call a brief huddle after two rotations to ask, 'Was the machine the same for every turn? How do you know?' Have them check previous entries to confirm consistency.

Common Misconception

During Whole Class Chain, observe students reversing the direction of the rule, saying 'You put in 3 and get 6, so 6 must go to 3.'

What to Teach Instead

Point to the arrow on the board and say, 'The arrow shows the direction. What did we do to 3 to get 6?' Use gestures to reinforce the one-way flow.

Assessment Ideas

Quick Check

After Manipulative Tables, present a new table with pictures of apples and juice boxes (1→2, 2→3, 3→4). Ask students to whisper the rule to a partner, then hold up fingers to show how many juice boxes 5 apples would make.

Exit Ticket

During Function Machine Game, collect each team’s completed relay sheet. Check that the rule is written clearly and that the output for an input of 4 is correct (e.g., 8 if doubling).

Discussion Prompt

After Whole Class Chain, show a table with a missing output (1→3, 2→4, 3→?). Ask three students to explain their answers, focusing on the language they use to describe the rule.

Extensions & Scaffolding

• Challenge students who finish early to create their own input-output table with a rule using drawing boards, then swap with a partner to solve.
• For students who struggle, provide a partially completed table with visuals (e.g., pictures of blocks doubling) and ask them to fill in one missing pair at a time.
• Deeper exploration: Introduce a second rule in the same table (e.g., one column for adding 2, another for doubling) to compare how different rules behave with the same inputs.

Key Vocabulary

Input	The number or item that goes into the function machine or table.
Output	The number or item that comes out of the function machine or table after the rule is applied.
Rule	The instruction or operation that changes the input into the output. For example, 'add 2' or 'double it'.
Pattern	A repeating or predictable sequence of numbers or objects.