Engaging in Open-Ended Investigations

Active learning works for this topic because open-ended investigations require students to think flexibly and communicate ideas, skills that are best developed through doing rather than listening. When students physically manipulate materials or discuss strategies, they build confidence in their problem-solving abilities and see mathematics as a creative process.

• Year 6
• Mathematics
• 4 activities
• 30 min–50 min

Learning Objectives

1. Formulate at least two distinct, testable mathematical questions arising from a given open-ended scenario.
2. Analyze and compare at least two different strategies for approaching a mathematical investigation.
3. Evaluate the validity of potential solutions based on evidence gathered during an investigation.
4. Create a structured report that clearly communicates the methods, findings, and conclusions of a mathematical investigation.

Ready-to-Use Activities

Project-Based Learning

⏱ 35 min·Pairs

Pair Hypothesis Challenge: Number Patterns

Pairs select a starting number and rule, like 'multiply by 3 then add 2,' to generate sequences. They hypothesize what happens after 20 terms and test with calculators. Groups share predictions and verify against actual results.

Prepare & details

Hypothesize potential solutions to an open-ended mathematical problem.

Facilitation Tip: For the Pair Hypothesis Challenge, provide a timer so students practice moving from quick ideas to deliberate testing before forming conclusions.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Project-Based Learning

⏱ 45 min·Small Groups

Small Group Exploration: Shape Puzzles

Provide tangram sets; groups pose questions like 'What shapes cover the most area without gaps?' They sketch hypotheses, build models, measure areas, and compare pathways in a shared chart.

Prepare & details

Analyze the different pathways one could take to investigate a mathematical concept.

Facilitation Tip: During Small Group Exploration, circulate to ask, 'What have you tried so far?' to guide students toward documenting their process, not just the outcome.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Project-Based Learning

⏱ 50 min·Whole Class

Whole Class Inquiry: Data Pathways

Collect class data on hand spans; students hypothesize sorting methods by mean or range. Explore tally charts versus graphs, discuss efficiency, then vote on best report format.

Prepare & details

Construct a clear and concise report detailing the findings of a mathematical investigation.

Facilitation Tip: In the Whole Class Inquiry, display a large grid where groups can mark their routes so all students see how different strategies lead to varied solutions.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Project-Based Learning

⏱ 30 min·Individual

Individual Report Builder: Game Strategies

Students play a dot-to-dot game, hypothesize winning paths, track trials, and write solo reports. Share one key finding in a class gallery walk for feedback.

Prepare & details

Hypothesize potential solutions to an open-ended mathematical problem.

Facilitation Tip: For the Individual Report Builder, provide a template with clear sections for question, method, findings, and conclusion to scaffold structure.

Setup: Flexible workspace with access to materials and technology

Materials: Project brief with driving question, Planning template and timeline, Rubric with milestones, Presentation materials

• Apply
• Analyze
• Evaluate
• Create
• Self-Management
• Relationship Skills
• Decision-Making

Teaching This Topic

Experienced teachers approach this topic by modeling curiosity themselves—posing questions without rushing to answers and normalizing uncertainty. They avoid stepping in too soon, instead asking, 'What have you noticed so far?' to keep students engaged in sense-making. Research suggests that students thrive when the teacher acts as a co-learner, celebrating partial progress and redirecting attention to process over product.

What to Expect

Successful learning looks like students confidently asking their own questions, testing multiple approaches, and clearly explaining their reasoning. You’ll see persistence when solutions aren’t immediate, respectful debate during group work, and pride in sharing unique findings with peers.

Watch Out for These Misconceptions

Common Misconception

During Pair Hypothesis Challenge, watch for students assuming there is only one correct pattern or rule.

What to Teach Instead

Prompt pairs to compare their starting points by asking, 'How did you decide what to look for first?' Encourage them to list all patterns they found before choosing one to explore further.

Common Misconception

During Small Group Exploration, watch for students following steps in a fixed order without testing alternatives.

What to Teach Instead

Place a 'dead end' shape or constraint in their puzzle set and ask, 'What happens if you change the order of your steps? Can you prove why this works better?' This forces them to reconsider their approach.

Common Misconception

During Individual Report Builder, watch for students treating the report as a series of answers without explaining their reasoning.

What to Teach Instead

Provide a sticky note template labeled 'What I learned from this' and require students to attach it to their report, summarizing one insight about their problem-solving process.

Assessment Ideas

Quick Check

After Pair Hypothesis Challenge, ask students to write down one question they investigated, one strategy they used, and one thing they would try differently next time.

Discussion Prompt

After Small Group Exploration, facilitate a debrief where groups share the most surprising result they found and explain why it challenged their initial thinking.

Peer Assessment

During Whole Class Inquiry, have students swap their draft reports and use a checklist to provide one specific suggestion for improving clarity in the 'methods' section.

Extensions & Scaffolding

• Challenge: Ask students who finish early to create a similar investigation for peers to solve, including a 'hidden pattern' only some will discover.
• Scaffolding: Provide a partially completed report template with sentence starters like, 'One thing we tested was...' to support students who struggle with structuring their thinking.
• Deeper exploration: Invite students to design a new open-ended problem based on their findings, then swap with another student to solve it.

Key Vocabulary

Open-ended problem	A mathematical problem that allows for multiple correct answers or multiple valid approaches to finding a solution.
Hypothesis	A proposed explanation or prediction for a mathematical outcome, based on initial observations or understanding.
Investigation pathway	A specific method or sequence of steps chosen to explore a mathematical concept or problem.
Mathematical reasoning	The process of using logical thinking and evidence to understand mathematical ideas and solve problems.
Data representation	Ways of organizing and displaying mathematical information, such as tables, charts, or graphs, to make it understandable.