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Problem-solving maths investigations for year 6.

Hamilton provide an extensive suite of problem-solving maths investigations for Year 6 to facilitate mathematical confidence, investigative inquiry and the development of maths meta skills in 'low floor – high ceiling' activities for all.

Explore all our in-depth problem solving investigations for Year 6 .

Use problem-solving investigations within every unit to encourage children to develop and exercise their ability to reason mathematically and think creatively.

Investigations provide challenges that offer opportunities for the development of the key mathematical skills while deepening conceptual understanding. They are designed to be accessible in different ways to all children. An added bonus is the substantial amount of extra calculation practice they often incorporate! The problems are designed to help children identify patterns, to explore lines of thinking and to reason and communicate about properties of numbers, shapes and measures.

Hamilton provide a mix of our own specially commissioned investigations, that include guidance for teachers together with a child-friendly sheet to guide your pupils through the investigation, as well as links to investigations on other highly regarded websites.

I am very grateful for Hamilton Trust resources, particularly the maths investigations. Julia, teacher in Wiltshire

You can find Hamilton's investigations for Year 6:

• Individually, they are incorporated into every unit in our Year 6 flexible maths blocks .
• Collectively, they appear on our resources page where you can explore all our in-depth problem solving investigations for Year 6 .

Do read our extensive range of advice for more information about the investigations and for tips on how to use them effectively.

Hamilton’s problem-solving investigations are 'low floor, high ceiling' activities that give all children opportunities to develop mastery and mathematical meta-skills. Explore a set for a whole year group.

Hamilton’s Problem-solving Investigations provide school-wide solutions to the challenges of building investigative skills from Early Years to Year 6.

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A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

We offer curriculum-linked resources for students aged 3-18, designed to nurture curious, resourceful and confident learners of school mathematics.

All of our resources are free to use by everyone.

We are based in the Faculty of Mathematics at the University of Cambridge.

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For teachers.

Browse our collection of rich activities and resources to help you develop your students' mathematical thinking

For Students

Take a look at our collections of tasks that will encourage you to think like a mathematician

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Read our guidance on ways to support children working mathematically at home

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Learn about our exciting new intiative to embed non-routine problem-solving opportunities in your maths curriculum

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A selection of rich tasks ideal for developing subject content knowledge, mathematical thinking, and problem-solving skills.

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Year 6 Maths Problems

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

22 February 2018

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Year 6 Maths Scheme of Work (based on White Rose Maths)

A scheme of learning with planning and worksheets included for Year 6 Maths. Included are 90 lessons that cover: Place Value Rounding Negative Numbers Roman Numerals Indices Inverse operations Written addition and subtraction Mental addition and subtraction Written multiplication and division Order of operations Finding fractions of amounts Comparing fractions Converting between improper and mixed number fractions Add and subtract fractions Multiply fractions together Multiply fractions by whole numbers Divide fractions by whole numbers Percentages Fraction, decimal and percentages Ratio Algebra including missing information, expressing problems, satisfying equations, satisfying two variables and sequences Multiplying and dividing by 10, 100 and 1000 Multiplying decimal numbers Finding percentages of amounts x 2 Fraction, decimal and percentage equivalents Converting measurements Miles and kms Time Shape with same area but different perimeter and vice versa Volume Area Area of triangles Addition and subtraction (Same as 1st planning scheme as it is revision) Multiplication (Same as 1st planning scheme as it is revision) Division (Same as 1st planning scheme as it is revision) Worksheets are differentiated three ways with a mastery aim running throughout. Having used a similar format in my own lessons, these sheets often provide enough challenge for two days lessons, with children taking on the easier work then moving on to harder challenges within the other sheets. These have all been made with mastery in mind and so follow this pattern. Worksheet One - Works on practise and fluency (LA) Worksheet Two - Practise, fluency and reasoning (MA) Worksheet Three - Practise, fluency, reasoning and problem solving (HA) Worksheet Four - Same as worksheet three as a spare for anybody who finishes problem solving Answers are included for each sheet.

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• Authentic tasks
• F - 10 Resources

Authentic tasks are designed to help students see mathematics as worthwhile and important. When students understand the purpose of a given problem in mathematics, they are more likely to persist when challenged. Authentic tasks generally have an ‘open middle’ which means that students can use different representations and solutions to communicate their knowledge and reasoning.

These curated links provide MAV members with access to nine authentic tasks from some of our primary consultants’ favourite resources. The 11 criteria provide MAV members with a research-informed context to consider each task’s potential impact on student thinking, ways of working, attitudes towards mathematics, their knowledge and understanding.

The following criteria was used to select the tasks based on their potential:

Used with permission © Martin Holt Educational Consultant 2017

If you would like to learn more about this approach to assessing or using tasks contact [email protected]

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Unit 1: Ratios

Unit 2: arithmetic with rational numbers, unit 3: rates and percentages, unit 4: exponents and order of operations, unit 5: negative numbers, unit 6: variables & expressions, unit 7: equations & inequalities, unit 8: plane figures, unit 9: coordinate plane, unit 10: 3d figures, unit 11: data and statistics.

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Year 6 tasks

The Ministry is migrating nzmaths content to Tāhurangi.             Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz).  When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024.  e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available. 

For more information visit https://tahurangi.education.govt.nz/updates-to-nzmaths

To solve maths problems at year 6 it is very important for your child to be able to instantly recall addition and subtraction facts up to 20, and multiplication and division facts up to 10 x 10.

e-ako maths provides free online basic fact learning tools which will help your child learn their basic facts, as well as a collection of games to improve their speed and accuracy. If you think this would benefit your child, click to read how to access this resource  (show instructions).

• To access e-ako maths, your child will need an account. You can make a new account for free, but check first whether they already have one that they use at school. If they do, they can use the same username and password to use e-ako maths at home.
• You can log in to e-ako maths or register a new account from the e-ako maths home page . If you are registering a new account to use at home you will need to click the "No join code? Click here." button and then "I am learning at home".

• The two large green boxes at the top are 'Learning tools' which will test the basic facts your child knows, and then help them learn the ones they don't already know.
• The blue buttons below are a collection of games designed to practise basic facts to improve speed and accuracy. 
• In Year 3 your child should mostly be concentrating on learning and practising their addition and subtraction facts, the buttons on the left.

Click to read about the purpose of the tasks below  (show purpose).

The tasks in this section are intended:

• for students in year 6 
• to complement similar class tasks
• to be a selection only of enjoyable learning opportunities for your child (and for you) 
• to generate useful explanations and discussions between you and your child.

The tasks in this section are NOT: 

• intended to be comprehensive in their coverage of all maths at year 6
• drill and practice exercises to learn basic facts.

These Year 6 tasks recognise that students:

• are mostly working at level 3 of the NZ Curriculum, growing in their understanding of number, algebra, geometry, measurement and statistics
• are all different, and some find parts of maths easier or harder than others
• are learning a range of approaches to solving problems that often involve several steps
• learn from and appreciate tasks presented in a variety of ways
• respond well to supportive questions from adults, such as, ‘What is the question asking you to do?’, or ‘What do you already know that might help you (solve the problem)?’
• enjoy exploring open tasks for which there can be a range of appropriate approaches and solutions (accordingly, set answers are not included for a number of these tasks).

We suggest that you ask your child what they are learning in maths at school, and select a task from this menu.

Click for  versions with Māori content .

Measurement

Financial literacy.

Town of Campbell receives help for new drilling project

CAMPBELL, Wis. (WEAU) - The town of Campbell may be close to solving a problem that has affected the town for years. The town received $500,000 in funding from La Crosse County for a drilling project that looks to remove PFAS from the town’s water system. Town supervisor Lee Donahue says the PFAS is anywhere from ground level to 85 feet below.

“That will be very helpful for us and for neighboring communities. It’s also important to determine what the status is of the rock layer between the upper contaminated aquifer and the lower non contaminated aquifer,” says Donahue.

The town is also receiving help from UW-Madison geoscience professor, Michael Cardiff, and his research team.

“We’re going to collect core from the well. We’re going to get samples of rock and look at those samples to try to analyze them and see what their permeability is, which is basically how fast water can flow through them,” says Cardiff.

Donahue wants to use a municipal system that goes to the lower aquifer. She says this project is about making sure that the lower aquifer is sustainable for those future plans.

“What we’re really trying to figure out is how is the PFAS moving and is this protective layer between the aquifers? Is it really going to provide a safety element to prevent that contamination from leaking to the lower aquifer? Answering questions like that,” says Donahue.

Donahue adds that while the water in the lower aquifer isn’t fit for consumption without filtration, it will be much easier to filter water from there.

“There are still some compounds in this water that would need to be filtered out, but it’s much less expensive to filter out the iron, the manganese, and the radium than it is to filter PFAS,” says Donahue.

The research and drill teams will be around for at least the next week working on the project.

Donahue hopes that if this project successfully finds no PFAS in the lower aquifer, the town will be able to begin its municipal water system in the next three to six months.

Copyright 2024 WEAU. All rights reserved.

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Multimodal Design for Interactive Collaborative Problem-Solving Support

• Conference paper
• First Online: 01 June 2024
• Cite this conference paper

• Hannah VanderHoeven   ORCID: orcid.org/0000-0003-3234-6797 26 ,
• Mariah Bradford   ORCID: orcid.org/0009-0009-2162-3307 26 ,
• Changsoo Jung   ORCID: orcid.org/0000-0002-2232-4300 26 ,
• Ibrahim Khebour   ORCID: orcid.org/0009-0009-4374-7263 26 ,
• Kenneth Lai   ORCID: orcid.org/0000-0003-2870-7019 27 ,
• James Pustejovsky   ORCID: orcid.org/0000-0003-2233-9761 27 ,
• Nikhil Krishnaswamy   ORCID: orcid.org/0000-0001-7878-7227 26 &
• Nathaniel Blanchard   ORCID: orcid.org/0000-0002-2653-0873 26  

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14689))

Included in the following conference series:

• International Conference on Human-Computer Interaction

When analyzing interactions during collaborative problem solving (CPS) tasks, many different communication modalities are likely to be present and interpretable. These modalities may include speech, gesture, action, affect, pose and object position in physical space, amongst others. As AI becomes more prominent in day-to-day use and various learning environments, such as classrooms, there is potential for it to support additional understanding into how small groups work together to complete CPS tasks. Designing interactive AI to support CPS requires creating a system that supports multiple different modalities. In this paper we discuss the importance of multimodal features to modeling CPS, how different modal channels must interact in a multimodal AI agent that supports a wide range of tasks, and design considerations that require forethought when building such a system that most effectively interacts with and aids small groups in successfully completing CPS tasks. We also outline various tool sets that can be leveraged to support each of the individual features and their integration, and various applications for such a system.

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Acknowledgments

This work was partially supported by the National Science Foundation under subcontracts to Colorado State University and Brandeis University on award DRL 2019805. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government. All errors and mistakes are, of course, the responsibilities of the authors.

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VanderHoeven, H. et al. (2024). Multimodal Design for Interactive Collaborative Problem-Solving Support. In: Mori, H., Asahi, Y. (eds) Human Interface and the Management of Information. HCII 2024. Lecture Notes in Computer Science, vol 14689. Springer, Cham. https://doi.org/10.1007/978-3-031-60107-1_6

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What is decision making?

Decisions, decisions. When was the last time you struggled with a choice? Maybe it was this morning, when you decided to hit the snooze button—again. Perhaps it was at a restaurant, with a miles-long menu and the server standing over you. Or maybe it was when you left your closet in a shambles after trying on seven different outfits before a big presentation. Often, making a decision—even a seemingly simple one—can be difficult. And people will go to great lengths—and pay serious sums of money—to avoid having to make a choice. The expensive tasting menu at the restaurant, for example. Or limiting your closet choices to black turtlenecks, à la Steve Jobs.

Get to know and directly engage with senior McKinsey experts on decision making

Aaron De Smet is a senior partner in McKinsey’s New Jersey office, Eileen Kelly Rinaudo  is McKinsey’s global director of advancing women executives and is based in the New York office, Frithjof Lund is a senior partner in the Oslo office, and Leigh Weiss is a senior adviser in the Boston office.

If you’ve ever wrestled with a decision at work, you’re definitely not alone. According to McKinsey research, executives spend a significant portion of their time— nearly 40 percent , on average—making decisions. Worse, they believe most of that time is poorly used. People struggle with decisions so much so that we actually get exhausted from having to decide too much, a phenomenon called decision fatigue.

But decision fatigue isn’t the only cost of ineffective decision making. According to a McKinsey survey of more than 1,200 global business leaders, inefficient decision making costs a typical Fortune 500 company 530,000 days  of managers’ time each year, equivalent to about $250 million in annual wages. That’s a lot of turtlenecks.

How can business leaders ease the burden of decision making and put this time and money to better use? Read on to learn the ins and outs of smart decision making—and how to put it to work.

Learn more about our People & Organizational Performance Practice .

How can organizations untangle ineffective decision-making processes?

McKinsey research has shown that agile is the ultimate solution for many organizations looking to streamline their decision making . Agile organizations are more likely to put decision making in the right hands, are faster at reacting to (or anticipating) shifts in the business environment, and often attract top talent who prefer working at companies with greater empowerment and fewer layers of management.

For organizations looking to become more agile, it’s possible to quickly boost decision-making efficiency by categorizing the type of decision to be made and adjusting the approach accordingly. In the next section, we review three types of decision making and how to optimize the process for each.

What are three keys to faster, better decisions?

Business leaders today have access to more sophisticated data than ever before. But it hasn’t necessarily made decision making any easier. For one thing, organizational dynamics—such as unclear roles, overreliance on consensus, and death by committee—can get in the way of straightforward decision making. And more data often means more decisions to be taken, which can become too much for one person, team, or department. This can make it more difficult for leaders to cleanly delegate, which in turn can lead to a decline in productivity.

Leaders are growing increasingly frustrated with broken decision-making processes, slow deliberations, and uneven decision-making outcomes. Fewer than half  of the 1,200 respondents of a McKinsey survey report that decisions are timely, and 61 percent say that at least half the time they spend making decisions is ineffective.

What’s the solution? According to McKinsey research, effective solutions center around categorizing decision types and organizing different processes to support each type. Further, each decision category should be assigned its own practice—stimulating debate, for example, or empowering employees—to yield improvements in effectiveness.

Here are the three decision categories  that matter most to senior leaders, and the standout practice that makes the biggest difference for each type of decision.

• Big-bet decisions are infrequent but high risk, such as acquisitions. These decisions carry the potential to shape the future of the company, and as a result are generally made by top leaders and the board. Spurring productive debate by assigning someone to argue the case for and against a potential decision can improve big-bet decision making.
• Cross-cutting decisions, such as pricing, can be frequent and high risk. These are usually made by business unit heads, in cross-functional forums as part of a collaborative process. These types of decisions can be improved by doubling down on process refinement. The ideal process should be one that helps clarify objectives, measures, and targets.
• Delegated decisions are frequent but low risk and are handled by an individual or working team with some input from others. Delegated decision making can be improved by ensuring that the responsibility for the decision is firmly in the hands of those closest to the work. This approach also enhances engagement and accountability.

In addition, business leaders can take the following four actions to help sustain rapid decision making :

• Focus on the game-changing decisions, ones that will help an organization create value and serve its purpose.
• Convene only necessary meetings, and eliminate lengthy reports. Turn unnecessary meetings into emails, and watch productivity bloom. For necessary meetings, provide short, well-prepared prereads to aid in decision making.
• Clarify the roles of decision makers and other voices. Who has a vote, and who has a voice?
• Push decision-making authority to the front line—and tolerate mistakes.

Introducing McKinsey Explainers : Direct answers to complex questions

How can business leaders effectively delegate decision making.

Business is more complex and dynamic than ever, meaning business leaders are faced with needing to make more decisions in less time. Decision making takes up an inordinate amount of management’s time—up to 70 percent for some executives—which leads to inefficiencies and opportunity costs.

As discussed above, organizations should treat different types of decisions differently . Decisions should be classified  according to their frequency, risk, and importance. Delegated decisions are the most mysterious for many organizations: they are the most frequent, and yet the least understood. Only about a quarter of survey respondents  report that their organizations make high-quality and speedy delegated decisions. And yet delegated decisions, because they happen so often, can have a big impact on organizational culture.

The key to better delegated decisions is to empower employees by giving them the authority and confidence to act. That means not simply telling employees which decisions they can or can’t make; it means giving employees the tools they need to make high-quality decisions and the right level of guidance as they do so.

Here’s how to support delegation and employee empowerment:

• Ensure that your organization has a well-defined, universally understood strategy. When the strategic intent of an organization is clear, empowerment is much easier because it allows teams to pull in the same direction.
• Clearly define roles and responsibilities. At the foundation of all empowerment efforts is a clear understanding of who is responsible for what, including who has input and who doesn’t.
• Invest in capability building (and coaching) up front. To help managers spend meaningful coaching time, organizations should also invest in managers’ leadership skills.
• Build an empowerment-oriented culture. Leaders should role model mindsets that promote empowerment, and managers should build the coaching skills they want to see. Managers and employees, in particular, will need to get comfortable with failure as a necessary step to success.
• Decide when to get involved. Managers should spend effort up front to decide what is worth their focused attention. They should know when it’s appropriate to provide close guidance and when not to.

How can you guard against bias in decision making?

Cognitive bias is real. We all fall prey, no matter how we try to guard ourselves against it. And cognitive and organizational bias undermines good decision making, whether you’re choosing what to have for lunch or whether to put in a bid to acquire another company.

Here are some of the most common cognitive biases and strategies for how to avoid them:

• Confirmation bias. Often, when we already believe something, our minds seek out information to support that belief—whether or not it is actually true. Confirmation bias  involves overweighting evidence that supports our belief, underweighting evidence against our belief, or even failing to search impartially for evidence in the first place. Confirmation bias is one of the most common traps organizational decision makers fall into. One famous—and painful—example of confirmation bias is when Blockbuster passed up the opportunity  to buy a fledgling Netflix for $50 million in 2000. (Actually, that’s putting it politely. Netflix executives remember being “laughed out” of Blockbuster’s offices.) Fresh off the dot-com bubble burst of 2000, Blockbuster executives likely concluded that Netflix had approached them out of desperation—not that Netflix actually had a baby unicorn on its hands.
• Herd mentality. First observed by Charles Mackay in his 1841 study of crowd psychology, herd mentality happens when information that’s available to the group is determined to be more useful than privately held knowledge. Individuals buy into this bias because there’s safety in the herd. But ignoring competing viewpoints might ultimately be costly. To counter this, try a teardown exercise , wherein two teams use scenarios, advanced analytics, and role-playing to identify how a herd might react to a decision, and to ensure they can refute public perceptions.
• Sunk-cost fallacy. Executives frequently hold onto underperforming business units or projects because of emotional or legacy attachment . Equally, business leaders hate shutting projects down . This, researchers say, is due to the ingrained belief that if everyone works hard enough, anything can be turned into gold. McKinsey research indicates two techniques for understanding when to hold on and when to let go. First, change the burden of proof from why an asset should be cut to why it should be retained. Next, categorize business investments according to whether they should be grown, maintained, or disposed of—and follow clearly differentiated investment rules  for each group.
• Ignoring unpleasant information. Researchers call this the “ostrich effect”—when people figuratively bury their heads in the sand , ignoring information that will make their lives more difficult. One study, for example, found that investors were more likely to check the value of their portfolios when the markets overall were rising, and less likely to do so when the markets were flat or falling. One way to help get around this is to engage in a readout process, where individuals or teams summarize discussions as they happen. This increases the likelihood that everyone leaves a meeting with the same understanding of what was said.
• Halo effect. Important personal and professional choices are frequently affected by people’s tendency to make specific judgments based on general impressions . Humans are tempted to use simple mental frames to understand complicated ideas, which means we frequently draw conclusions faster than we should. The halo effect is particularly common in hiring decisions. To avoid this bias, structured interviews can help mitigate the essentializing tendency. When candidates are measured against indicators, intuition is less likely to play a role.

For more common biases and how to beat them, check out McKinsey’s Bias Busters Collection .

Learn more about Strategy & Corporate Finance consulting  at McKinsey—and check out job opportunities related to decision making if you’re interested in working at McKinsey.

Articles referenced include:

• “ Bias busters: When the crowd isn’t necessarily wise ,” McKinsey Quarterly , May 23, 2022, Eileen Kelly Rinaudo , Tim Koller , and Derek Schatz
• “ Boards and decision making ,” April 8, 2021, Aaron De Smet , Frithjof Lund , Suzanne Nimocks, and Leigh Weiss
• “ To unlock better decision making, plan better meetings ,” November 9, 2020, Aaron De Smet , Simon London, and Leigh Weiss
• “ Reimagine decision making to improve speed and quality ,” September 14, 2020, Julie Hughes , J. R. Maxwell , and Leigh Weiss
• “ For smarter decisions, empower your employees ,” September 9, 2020, Aaron De Smet , Caitlin Hewes, and Leigh Weiss
• “ Bias busters: Lifting your head from the sand ,” McKinsey Quarterly , August 18, 2020, Eileen Kelly Rinaudo
• “ Decision making in uncertain times ,” March 24, 2020, Andrea Alexander, Aaron De Smet , and Leigh Weiss
• “ Bias busters: Avoiding snap judgments ,” McKinsey Quarterly , November 6, 2019, Tim Koller , Dan Lovallo, and Phil Rosenzweig
• “ Three keys to faster, better decisions ,” McKinsey Quarterly , May 1, 2019, Aaron De Smet , Gregor Jost , and Leigh Weiss
• “ Decision making in the age of urgency ,” April 30, 2019, Iskandar Aminov, Aaron De Smet , Gregor Jost , and David Mendelsohn
• “ Bias busters: Pruning projects proactively ,” McKinsey Quarterly , February 6, 2019, Tim Koller , Dan Lovallo, and Zane Williams
• “ Decision making in your organization: Cutting through the clutter ,” McKinsey Quarterly , January 16, 2018, Aaron De Smet , Simon London, and Leigh Weiss
• “ Untangling your organization’s decision making ,” McKinsey Quarterly , June 21, 2017, Aaron De Smet , Gerald Lackey, and Leigh Weiss
• “ Are you ready to decide? ,” McKinsey Quarterly , April 1, 2015, Philip Meissner, Olivier Sibony, and Torsten Wulf.

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