Mathematics · Year 6 · Problem Solving and Investigations · Term 4

Engaging in Open-Ended Investigations

Engaging in open-ended mathematical investigations, formulating questions, and exploring solutions.

About This Topic

Open-ended mathematical investigations invite Year 6 students to formulate their own questions, hypothesize potential solutions, and explore diverse pathways to uncover mathematical ideas. This topic aligns with Australian Curriculum Mathematics standards in Problem Solving and Reasoning, where students tackle ill-defined problems like finding patterns in number tiles or optimizing routes on a grid. They learn to adapt strategies, evaluate progress, and communicate findings clearly in reports.

These activities build essential skills such as persistence, critical analysis of methods, and precise mathematical language. Students connect investigations across strands like number, geometry, and measurement, seeing mathematics as a flexible toolkit for real-world challenges. Reporting findings reinforces structure: introduction, methods, results, and conclusions.

Active learning benefits this topic most because students drive their own inquiries through collaboration and experimentation. They test hypotheses in pairs or groups, discuss dead ends, and refine reports based on peer feedback. This process turns passive learners into confident mathematicians who embrace uncertainty and multiple solutions.

Key Questions

1. Hypothesize potential solutions to an open-ended mathematical problem.
2. Analyze the different pathways one could take to investigate a mathematical concept.
3. Construct a clear and concise report detailing the findings of a mathematical investigation.

Learning Objectives

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

Before You Start

Identifying Patterns

Why: Students need to be able to recognize patterns to formulate initial hypotheses and identify trends within their investigations.

Basic Data Collection and Recording

Why: Students must have experience gathering and organizing simple data sets to use as evidence in their investigations.

Communicating Mathematical Ideas

Why: Students should have prior experience explaining mathematical thinking, which is foundational for constructing investigation reports.

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.

Watch Out for These Misconceptions

Common Misconception

All problems have one correct answer.

What to Teach Instead

Many open-ended tasks yield multiple valid solutions; active group discussions reveal diverse pathways and build tolerance for ambiguity. Students compare strategies, seeing value in varied approaches.

Common Misconception

Investigations follow a fixed sequence of steps.

What to Teach Instead

Real inquiries involve branching paths and revisions; hands-on trials show students how to pivot from failed hypotheses. Collaborative debriefs highlight flexibility over rigidity.

Common Misconception

Reports are just lists of answers.

What to Teach Instead

Effective reports explain thinking and justify choices; peer review stations help students structure narratives with evidence. This active feedback refines clarity and depth.

Active Learning Ideas

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.

35 min·Pairs

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.

45 min·Small Groups

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.

50 min·Whole Class

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.

30 min·Individual

Real-World Connections

• Urban planners use open-ended problem-solving to design efficient public transport routes, considering factors like population density, traffic flow, and environmental impact.
• Game designers often engage in open-ended investigations to create engaging game mechanics and balance gameplay, testing numerous variations to find optimal player experiences.
• Scientists exploring new phenomena, such as the behavior of exotic particles or the formation of distant galaxies, must formulate questions and devise novel methods to gather and interpret data.

Assessment Ideas

Quick Check

Present students with a scenario, for example, 'How can we arrange 24 chairs in a rectangular formation for a school play to maximize audience visibility?' Ask students to write down two different questions they could investigate about this scenario and one potential strategy for each question.

Discussion Prompt

After students have completed a short investigation, ask: 'What was the most challenging part of your investigation? How did you decide which pathway to follow? What would you do differently if you started again?' Facilitate a class discussion comparing approaches.

Peer Assessment

Students swap their draft investigation reports. Provide a checklist: 'Does the report clearly state the question? Are the methods described step-by-step? Are the findings supported by data or examples? Is the conclusion logical?' Students provide one specific suggestion for improvement.

Frequently Asked Questions

How do you introduce open-ended investigations in Year 6 maths?

Start with familiar problems like 'How many ways can you make 24 with four numbers?' Model hypothesizing aloud, then release to pairs for exploration. Provide prompt cards for question formulation and pathway analysis. Circulate to probe reasoning, ensuring reports include methods and reflections. This scaffolds independence while aligning to ACARA standards.

What Year 6 examples work for mathematical investigations?

Try Fibonacci patterns in nature, grid path efficiencies, or fraction pizzas with variable toppings. Students hypothesize outcomes, test with manipulatives, and report comparisons. These span number, geometry, and probability, encouraging multiple solutions and clear communication of findings.

How can active learning help with open-ended maths investigations?

Active approaches like pair testing and group pathway mapping let students experience trial and error firsthand, building resilience. They discuss hypotheses in real time, refining ideas through talk, and construct reports from shared evidence. This engagement deepens understanding of ambiguity and boosts problem-solving confidence over worksheets.

How to assess open-ended mathematical investigations?

Use rubrics focusing on question quality, hypothesis testing, pathway analysis, and report structure. Collect artefacts like journals and group charts alongside oral explanations. Peer assessments on clarity add accountability, aligning with proficiency strands in reasoning and communicating.