maths problem solving cycle

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• The Three Stages of the Problem-Solving Cycle

Essentially every problem-solving heuristic in mathematics goes back to George Polya’s How to Solve It ; my approach is no exception. However, this cyclic description might help to keep the process cognitively present.

A few months ago, I produced a video describing this the three stages of the problem-solving cycle: Understand, Strategize, and Implement. That is, we must first understand the problem, then we think of strategies that might help solve the problem, and finally we implement those strategies and see where they lead us. During two decades of observing myself and others in the teaching and learning process, I’ve noticed that the most neglected phase is often the first one—understanding the problem.

The Three Stages Explained

• What am I looking for?
• What is the unknown?
• Do I understand every word and concept in the problem?
• Am I familiar with the units in which measurements are given?
• Is there information that seems missing?
• Is there information that seems superfluous?
• Is the source of information bona fide? (Think about those instances when a friend gives you a puzzle to solve and you suspect there’s something wrong with the way the puzzle is posed.)
• Logical reasoning
• Pattern recognition
• Working backwards
• Adopting a different point of view
• Considering extreme cases
• Solving a simpler analogous problem
• Organizing data
• Making a visual representation
• Accounting for all possibilities
• Intelligent guessing and testing

I have produced videos explaining each one of these strategies individually using problems we have solved at the Chapel Hill Math Circle.

• Implementing : We now implement our strategy or set of strategies. As we progress, we check our reasoning and computations (if any). Many novice problem-solvers make the mistake of “doing something” before understanding (or at least thinking they understand) the problem. For instance, if you ask them “What are you looking for?”, they might not be able to answer. Certainly, it is possible to have an incorrect understanding of the problem, but that is different from not even realizing that we have to understand the problem before we attempt to solve it!

As we implement our strategies, we might not be able to solve the problem, but we might refine our understanding of the problem. As we refine our understanding of the problem, we can refine our strategy. As we refine our strategy and implement a new approach, we get closer to solving the problem, and so on. Of course, even after several iterations of this cycle spanning across hours, days, or even years, one may still not be able to solve a particular problem. That’s part of the enchanting beauty of mathematics.

I invite you to observe your own thinking—and that of your students—as you move along the problem-solving cycle!

[1] Problem-Solving Strategies in Mathematics , Posamentier and Krulik, 2015.

About the author: You may contact Hector Rosario at [email protected].

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Teaching Problem Solving in Math

• Freebies , Math , Planning

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

There’s also this visuals strategy wheel practice.

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

• read the problem carefully
• restated the problem in our own words
• crossed out unimportant information
• circled any important information
• stated the goal or question to be solved

We did this over and over with example problems.

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

• taking our time
• working the problem out
• showing all our work
• estimating the answer
• using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

• switch strategies or try a different one
• rethink the problem
• think of related content
• decide if you need to make changes
• check your work
• but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

• compare your answer to your estimate
• check for reasonableness
• check your calculations
• add the units
• restate the question in the answer
• explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

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Problem-solving cycle.

The Problem-Solving Cycle (PSC) is a National Science Foundation funded project that has developed a research-based professional development (PD) model.  This model is highly adaptable and can be specifically focused on problems of practice that are of interest to the participating teachers and administrators.  Additionally, it can be tailored to highlight federal, state, district, and school-based initiatives that are ever-changing and ongoing in the life of a teacher.

The PSC project is a research-practice partnership with the San Francisco Unified School District.  The current focus is on creating teacher leaders in middle school math classrooms and studying the effect on student learning.  

PSC Project Products:

Borko, H., Carlson, J., Jarry-Shore, M., Barnes, E., & Ellsworth, A. (2017, May). All students & teachers as math learners: A partnership to refine and implement two interconnected models. Presented at Stanford University’s CSET’s Pondering Excellence in Teaching Talk Series, Stanford, CA.

Borko, H., Carlson, J., Deutscher, R., & Ryan, J. (2018, May). A research-practice partnership to build district capacity. Video presented at 2018 STEM For All Video Showcase. http://stemforall2018.videohall.com/presentations/1299  

Borko, H. (2021 August). The Problem-Solving Cycle and Teacher Leadership Preparation Program: Developing and Researching a Model for Bringing Mathematics Professional Development to Scale . Research Seminar [Zoom] presented at IPN Leibniz Institute for Science and Mathematics Education, University of Kiel. 

Borko, H., Carlson, J., Mangram, C., Anderson, R., Fong, A., Million, S., Mozenter, S., & Villa, A. M. (2017). The role of video-based discussion in model for preparing professional development leaders. International Journal of STEM Education, 4 (1), 1-15.

Borko, H., Carlson, J., Deutscher, R., Boles, K. L., Delaney, V., Fong, A., Jarry-Shore, M., Malamut, J., Million, S., Mozenter, S., & Villa, A. M. (2021). Learning to Lead: an Approach to Mathematics Teacher Leader Development. International Journal of Science and Mathematics Education , 1-23.

Conference Presentations

Borko, H. (2015, February). Design-based implementation research in schools: Benefits & challenges . Paper presented at AACTE, Washington, D.C.

Borko, H., & Carlson, J. (2016, April) Design-based implementation research: adapting a professional development leadership model with a school district” Paper presented at AERA in a symposium entitled A Behind-the-Scenes Look at Effective Video-Based Professional Development , Washington, D.C.

Borko, H. (2016, June). Preparing mathematics teachers to facilitate the problem-solving cycle professional development . Paper presented at the Symposium and Workshop on Video Resources for Mathematics Teacher Development at the Weizmann Institute, Rehovot, Israel.

Mozenter, S. (2017, February). Video-based discussions: Meeting the multiple demands of PD for content teachers serving English language learners. Presented at National Association for Bilingual Education, Dallas, TX.

Borko, H., & Villa III, A. M. (2017, March). Facilitating Video-Based Mathematics Professional Development. Presented at Teacher Development Group Leadership Seminar, Portland, OR.

Villa III, A. M., & Jarry-Shore, M. (2017, March). Facilitating video-based mathematics professional development. Research symposium at National Council of Teachers of Mathematics Research Conference, San Antonio, TX.

Carlson, J., Jarry-Shore, M., Barnes, E., & Ellsworth, A. (2017, March). All students & teachers as math learners: A partnership to refine and implement two interconnected models.   Presented at Stanford-SFUSD Partnership Annual Meeting, Stanford, CA.

Jarry-Shore, M., Fong, A., Dyer, E., Gomez Zaccarelli, F., & Borko, H. (2018, February).  Video for equity: Designing video-based discussions of student authority.  Presentation at Association of Mathematics Teacher Education, Houston, TX.

Fong, A., Dyer, E., & Gomez Zaccarelli, F. (2018, February).  A shared vision for teacher improvement: Adapting professional development for local context by leveraging district-developed tools.  Presentation at Association of Mathematics Teacher Education, Houston, TX.

Mozenter, S., Gomez Zaccarelli, F., & Ellsworth, A. (2018, February ).  Video-based discussions in service of student agency, authority, and identity. Presentation at the Association of Teacher Education, Las Vegas, NV.

Mozenter, S., Ellsworth, A., & Gomez Zaccarelli, F. (2018, March). Video-based discussions in service of student agency, authority, & identity. Presentation at the American Association of Colleges for Teacher Education, Baltimore. MD.

Borko, H., & Villa III, A. M. (2018, March ). Building district capacity to address student access & equity: A research-practice partnership to develop teacher leaders. Presentation at the Teacher Development Group Leadership Seminar, Portland, OR.

Borko, H., Carlson, J., & Treviño, E. (2018, April).  A research-practice partnership to develop district capacity: Learning with & from each other.  Paper presented at the American Educational Research Association, New York, NY. 

Mozenter, S., Borko, H., & Jarry-Shore, M. (2018, June). Complicating the connection: Immigrant-background teachers . Paper presented at Teaching & Teacher Education Special Interest Group of the European Association for Research on Learning and Instruction, Kristiansaand, Norway.

Treviño, E. Brown, A., Villa III, A.M., & Borko, H. (2018, November).  Deconstructing student math content knowledge and groupwork through video-based discussion. Presentation at California Mathematics Council - Northern Section Conference Asilomar, Pacific Grove, CA.

Jarry-Shore, M. (2018, November ). The in-the-moment noticing of the novice mathematics teacher. Paper and presentation at the North American chapter of the International Group for the Psychology of Mathematics Education, Greenville, SC.

Villa III, A.M., & Boles, K. (2019, February).  Actualizing agency, authority, identity, and access to content in two contrasting cases of mathematical groupwork . Presented at Association of Mathematics Teacher Education, Orlando, FL.

Borko, H., & Villa III, A.M. (2019, February/March). Building teachers’ capacity to promote students’ access to rigorous and meaningful mathematics through video-based discussions. Presentation at the Teacher Development Group Leadership Seminar, Portland, OR.

Gomez Zaccarelli, F., Villa III, A.M., Mozenter, S., Boles, K., Deutscher, R., Borko, H., & Carlson, J. (2019, April).  How students are oriented toward a mathematical task and their peers: Access to content, agency, authority, and identity. Paper presented at the American Educational Research Association, Toronto, Canada. 

Mozenter, S., & Borko, H. (2019, April ). “ Not many people ask me this kind of question.” Three contrasting cases of immigrant-background teachers . Paper presented at the American Educational Research Association, Toronto, Canada.

Borko, H., Carlson, J., & Deutscher, R. (2019,  April ). Learning environments to support teacher leaders’ learning to lead video-based discussions. Poster presented in the structured poster session at the American Educational Research Association, Toronto, Canada .

Villa III, A.M., Boles, K.L., & Borko, H. (2019, November ).  Teacher leader learning through participation in and facilitation of professional development addressing problems of practice . Paper and presentation at the North American chapter of the International Group for the Psychology of Mathematics Education, St. Louis, MO.

Boles, K. L., Jarry-Shore, M., Muro Villa III, A., Malamut, J., & Borko, H. (2020, June). Building capacity via facilitator agency: Tensions in implementing an adaptive model of professional development. In M. Gresalfi, & I. S. Horn (Eds.),  The Interdisciplinarity of the Learning Sciences, 14th International Conference of the Learning Sciences (ICLS)  (pp. 2585-2588). Nashville, TN: International Society of the Learning Sciences. 

Jarry-Shore, M., & Allen, T. (2020, December). Noticing Struggle to Support Student Understanding [Conference Presentation]. California Mathematics Council - North Conference, Pacific Grove, CA, United States.

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Building fluency through problem solving

Editor’s Note:

This is an updated version of a blog post published on January 13, 2020.

Problem solving builds fluency and fluency builds problem solving. How can you help learners make the most of this virtuous cycle and achieve mastery?

Fluency. It’s so important that I have written not one , not two , but three blog posts on the subject. It’s also one of the three key aims for the national curriculum.

It’s a common dilemma. Learners need opportunities to apply their knowledge in solving problems and reasoning (the other two NC aims), but can’t reason or solve problems until they’ve achieved a certain level of fluency.

Instead of seeing this as a catch-22, think of fluency and problem solving as a virtuous cycle — working together to help learners achieve true mastery.

Supporting fluency when solving problems

Fluency helps children spot patterns, make conjectures, test them out, create generalisations, and make connections between different areas of their learning — the true skills of working mathematically. When learners can work mathematically, they’re better equipped to solve problems.

But what if learners are not totally fluent? Can they still solve problems? With the right support, problem solving helps learners develop their fluency, which makes them better at problem solving, which develops fluency…

Here are ways you can support your learners’ fluency journey.

Don’t worry about rapid recall

What does it mean to be fluent? Fluency means that learners are able to recall and use facts in a way that is accurate, efficient, reliable, flexible and fluid. But that doesn’t mean that good mathematicians need to have super-speedy recall of facts either.

Putting pressure on learners to recall facts in timed tests can negatively affect their ability to solve problems. Research shows that for about one-third of students, the onset of timed testing is the beginning of maths anxiety . Not only is maths anxiety upsetting for learners, it robs them of working memory and makes maths even harder.

Just because it takes a learner a little longer to recall or work out a fact, doesn’t mean the way they’re working isn’t becoming accurate, efficient, reliable, flexible and fluid. Fluent doesn’t always mean fast, and every time a learner gets to the answer (even if it takes a while), they embed the learning a little more.

Give learners time to think and reason

Psychologist Daniel Willingham describes memory as “the residue of thought”. If you want your learners to become fluent, you need to give them opportunities to think and reason. You can do this by looking for ways to extend problems so that learners have more to think about.

Here’s an example: what is 6 × 7 ? You could ask your learners for the answer and move on, but why stop there? If learners know that 6 × 7 = 42 , how many other related facts can they work out from this? Or if they don’t know 6 × 7 , ask them to work it out using facts they do know, like (5 × 7) + (1 × 7) , or (6 × 6) + (1 × 6) ?

Spending time exploring problems helps learners to build fluency in number sense, recognise patterns and see connections, and visualise — the three key components of problem solving.

Developing problem solving when building fluency

Learners with strong problem-solving skills can move flexibly between different representations, recognising and showing the links between them. They identify the merits of different strategies, and choose from a range of different approaches to find the one most appropriate for the maths problem at hand.

So, what type of problems should you give learners when they are still building their fluency? The best problem-solving questions exist in a Goldilocks Zone; the problems are hard enough to make learners think, but not so hard that they fail to learn anything.

Here’s how to give them opportunities to develop problem solving.

Centre problems around familiar topics

Learners can develop their problem-solving skills if they’re actively taught them and are given opportunities to put them into practice. When our aim is to develop problem-solving skills, it’s important that the mathematical content isn’t too challenging.

Asking learners to activate their problem-solving skills while applying new learning makes the level of difficulty too high. Keep problems centred around familiar topics (this can even be content taught as long ago as two years previously).

Not only does choosing familiar topics help learners practice their problem-solving skills, revisiting topics will also improve their fluency.

Keep the focus on problem solving, not calculation

What do you want learners to notice when solving a problem? If the focus is developing problem-solving skills, then the takeaway should be the method used to answer the question.

If the numbers involved in a problem are ‘nasty’, learners might spend their limited working memory on calculating and lose sight of the problem. Chances are they’ll have issues recalling the way they solved the problem. On top of that, they’ll learn nothing about problem-solving strategies.

It’s important to make sure that learners have a fluent recall of the facts needed to solve the problem. This way, they can focus on actually solving it rather than struggling to recall facts. To understand the underlying problem-solving strategies, learners need to have the processing capacity to spot patterns and make connections.

The ultimate goal of teaching mathematics is to create thinkers. Making the most of the fluency virtuous cycle helps learners to do so much more than just recall facts and memorise procedures. In time, your learners will be able to work fluently, make connections, solve problems, and become true mathematical thinkers.

Jo Boaler (2014). Research Suggests that Timed Tests Cause Math Anxiety. Teaching Children Mathematics , 20(8), p.469.

Willingham, D. (2009). Why don’t students like school?: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for Your Classroom. San Francisco: Jossey-Bass.

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Overview of the Problem-Solving Mental Process

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

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You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Supporting critical numeracy and maths skills in teaching and learning

In today’s article, Dave Tout, Justine Sakurai and Carly Sawatzki discuss numeracy and its relationship with mathematics, and the importance of real-world contexts. They’ll also share a problem-solving cycle to help students develop their skills and a classroom example of health numeracy, using trampolining as a focus for mathematical investigation .

Over the last couple of years, we have worked collaboratively to help shape and write Victorian senior secondary curricula that better supports the development of critical numeracy and maths skills in our school students. This is the first of two articles where we will share with you some of the research and theory that guided us, our thoughts about the purpose of education, and how these ideas influenced how we wrote those curriculum frameworks.

This first article will describe the background to our approach, and the second article will describe how we have attempted to espouse this in a curriculum document.

But first, what is, or are, our starting points? There are a few key elements behind our approach, including that:

• numeracy is the use and application of mathematical knowledge in context;
• numeracy is a social practice, which means there is always a clear purpose for learning that is connected to the real world;
• there is an implicit numeracy demand within most real-world problems; and,
• underpinning all this is a problem solving approach.

What is numeracy (and its relationships with mathematics)?

From our perspective, numeracy surrounds us in our everyday lives. Numeracy is about using mathematics to make sense of the world and applying mathematics in a context for a social purpose. For most young people and adults, numeracy gives meaning to mathematics, and mathematical knowledge and skills contribute to efficient and critical numeracy.

Students need a range of mathematical knowledge, skills, understandings and dispositions to solve problems in real contexts across personal, further learning, work, and community settings. To become numerate you need to know some mathematics. As Lynn Steen eloquently said:

...numeracy is not the same as mathematics, nor is it an alternative to mathematics. Today's students need both mathematics and numeracy. Whereas mathematics asks students to rise above context, quantitative literacy is anchored in real data that reflect engagement with life's diverse contexts and situations. (Steen, 2001, p.10)

Numeracy is not just about numbers and arithmetic. The mathematical knowledge and skills needed by all students includes number and quantity, measurement, shape, dimensions and directions, data and chance, and mathematical relationships and thinking. It also includes the ability to dip into your toolkit and choose and use the most appropriate analogue tools and digital technologies.

An Australian model of numeracy that we believe illustrates this perspective well is shown in Figure 1 below. This model incorporates four dimensions of contexts, mathematical knowledge, tools, and dispositions that are embedded in a critical orientation to using mathematics.

Why are real-world contexts so important?

Students often report that the mathematics they encounter at school feels disconnected from the real world. They express their frustration via the question, ‘When am I going to use this?’

Curriculum writers certainly intend that teachers bring their curriculum to life through contextualised lessons that connect with students’ real-world experiences. The complex challenges of modern life and work necessitate that schools deliver contextualised learning opportunities – students need higher levels of mathematics and numeracy than ever before and they need practise applying mathematics and numeracy to a range of familiar and unfamiliar authentic problems and issues (AAMT & AiGroup 2014; Binkley et al., 2012; FYA 2017; Gravemeijer et al., 2017).

As students become numerate, they develop the ability to make considered, mathematically-informed decisions, whether they be related to personal financial matters, planning travel arrangements, understanding and interpreting big data such as with the current COVID-19 epidemic, following instructions about a health or medical matter, or understanding the personal and social implications of problematic gambling.

Another of our key underpinning beliefs is that the exploration of real-world issues and problems is more valuable, satisfying and useful for students than the too frequent, often meaningless and repetitive practise of standard mathematical facts, procedures and processes. If students have little experience grappling with the messiness of real-world situations and problems, and if they can only apply mathematical procedures when problems are packaged in very familiar, structured ways (like in traditional maths classrooms and textbooks), then how can we expect them to value, see, use and apply maths in the world outside the classroom?

The AAMT and AiGroup research project referenced above documented this, with one of the teachers involved commenting on this disconnect:

This is one of the most interesting aspects/concepts of this project. The relationship between workplace mathematical skills and school mathematics could be described as ‘distant’ at best. Teacher observation (AAMT & AiGroup, 2014)

Being illiterate is considered an appalling state in modern Australia, yet evidence of significant numbers of students, especially young women, exiting the education system innumerate does not receive the same attention (ABS, 2013; OECD, 2017).

We believe therefore that we, as maths educators, need to support our students to be able to engage with and problem solve when maths is embedded in real-world situations and contexts, and this includes within our maths classrooms, as well as across the curriculum.

An underpinning problem-solving cycle

We believe that students need to develop the skills to problem solve, to investigate and solve a problem where the mathematics is embedded within a real-world context. The contexts should be the starting point, and students need well-planned and guided experiences with a structured problem-solving cycle , so that they know how to move from the real-world context to the mathematical world and apply their mathematical knowledge to find answers and solutions to the problem at hand.

An important aspect of numeracy is the ability to critically reflect on, evaluate and review your outcomes, and finally to be able to communicate and report on what you did and found.

In our curriculum endeavours, our suggested problem-solving cycle is modelled and adapted from the one used in the OECD's Programme for International Student Assessment (PISA) mathematical literacy assessment framework (OECD, 2019). This is consistent with the Figure 1 model. Our model has four distinct components, as represented in Figure 2 below.

The four stages in the problem-solving cycle are:

• Formulate : where you need to identify, select, and interpret the mathematical information embedded in a real-world context and decide and plan what mathematics you need to use and what questions you might ask.
• Act on and use mathematics : in this stage you need to do the maths ­­– perform the mathematical actions and processes so you can complete the task; this includes the use of a range of tools and technologies.
• Evaluate and reflect : here you are expected to check and reflect on both the mathematical processes you used and the reasonableness of your results and outcomes, especially in relation to the real-world context.
• Communicate and repor t: finally, there’s no point to the activity and investigations if you don’t document and report on your outcomes and any results. Here you need to use a combination of informal and formal mathematical representations.

Below is an example of an investigation taken from the context of health and fitness that illustrates the directions that teaching and learning can take when a numeracy problem-solving cycle is used in practice.

A classroom example – health numeracy

Health and safety related contexts provide interesting and useful opportunities to develop numeracy. Mathematical data and evidence can inform an understanding of risks, costs and benefits associated with such things as:

• Health and exercise
• Vaccination
• Prescription medications
• Alcohol and other drugs of addiction
• Medicare and health insurance

When an individual is informed, they are able to make better personal choices. This is not only good for the individual, their family and community, but can limit costs to the economy.

An example of a health and exercise context that might interest young people is trampolining. The backyard trampoline is a great tool in promoting health and fitness, and commercial trampolining centres are popular amongst young people.

The mock media report below includes statistical and numerical representations and language that underpin risk assessment and behaviour choices, and provides a context from which to initially introduce and study the problem. The ability to engage critically with and determine the trustworthiness of health reports presented by politicians, medical experts, journalists and social media influencers is essential.

Students might discuss:

• What is the probability (overall risk) of a spinal or head injury from trampolining?
• Do the benefits of trampolining outweigh the risks?
• What can be done to prevent accidents and injuries?
• What advice should be given to parents and children?

Once they identify the issues, they can mathematise the questions by unpacking the key concepts of probability and statistics using knowledge of fractions and percentages.

Student thinking may be extended by comparing safety figures between commercial centres, or health data from 20 years ago to today. Once students have considered the mathematics, reflecting on their findings should help them situate the problem and decide if the mathematics makes sense in the context.

A lesson may consider both aspects of the trampoline debate; cause of injuries, and building health.

A lesson sequence on trampoline health for younger students might consider asking students to keep a journal over a week or a month – how many times did they jump on their own or a friend’s trampoline? Students could consider how long each jumping session lasted for. They could break it down into ‘guesstimates’ of how much time was doing tricks , and how much time was straight jumps. They could conduct a survey and ask other students too. Their data could be displayed as a visual proportion on a line to aid in student development of estimation and proportional reasoning.

For an investigation, ask your students to estimate how many trampoline jumps would equal a two kilometre run ? How would they work this out? Trampolining provides a real-life context that is relevant and applicable to students’ daily lives, through which they can learn using the problem-solving cycle.

There are many paths that may be taken as you and your students mathematise this problem. They may want to approach this problem using distance as a perspective by converting jumps into lengths? Another method may be for them to time how long it takes to run as opposed to jump? There are no limits to the approaches that may be employed in applying a mathematical lens to the problem. Be as creative and as physically active as you like when carrying out your mathematical investigations.

Once you have completed the mathematical tasking, the results must be looked at and interpreted within the context of the trampolining context. Ask your students to evaluate and reflect on their thinking: Has the question been answered and do the answers make sense in relation to the question? Finish by asking them to write up the results – get them to make a poster or a video.

Whatever angle you choose to take when designing problem-based tasks for your students, remember to keep it real, relatable, and relevant!

Ashby, K., Pointer, S., Eager, D., & Day, L. (2015). Australian trampoline injury patterns and trends. Australian and New Zealand journal of public health , 39 (5), 491-494. https://doi.org/10.1111/1753-6...

Australian Association of Mathematics Teachers (AAMT) & Australian Industry Group (AiGroup). (2014). Tackling the School–Industry Mathematics Divide . Commonwealth of Australia. https://www.chiefscientist.gov... (PDF, 355KB)

Australian Bureau of Statistics. (2013). Programme for the International Assessment of Adult Competencies (Catalogue No. 4228.0). https://www.abs.gov.au/

Binkley, M., Erstad, O., Herman, J., Raizen, S., Ripley, M., Miller-Ricci, M., & Rumble, M. (2012). Defining twenty-first century skills. In P. Griffin, B. McGaw, & E. Care (Eds.), Assessment and Teaching of 21st Century Skills (pp. 17-66). Springer.

Carbonell, R. (2018, September 15). Double bounced: Why jumpy insurers are hopping out of the trampoline business. ABC News . https://www.abc.net.au/news/2018-09-15/trampoline-boom-loses-its-bounce-with-insurers/10249630

Foundation for Young Australians. (2017). The New Basics: Big data reveals the skills young people need for the New Work Order . FYA. https://www.fya.org.au/wp-content/uploads/2016/04/The-New-Basics_Update_Web.pdf (PDF, 1.5MB)

Geiger, V., Goos, M., & Forgasz, H. (2015). A rich interpretation of numeracy for the 21st century: A survey of the state of the field. ZDM Mathematics Education, 47 (4), 531-548. https://doi.org/10.1007/s11858-015-0708-1

Goos, M., Geiger, V., & Dole, S. (2014). Transforming professional practice in numeracy teaching. In Y. Li, E. Silver, & S. Li (Eds.), Transforming Mathematics Instruction (pp. 81-102). Springer. https://doi.org/10.1007/978-3-319-04993-9_6

Gravemeijer, K., Stephan, M., Julie, C., Lin, F. L., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education , 15 (1), 105-123. https://doi.org/10.1007/s10763-017-9814-6

Michie, F. (2015, September 11). Rise in serious trampoline injuries in children worries trauma specialists, prompts new research. ABC News . https://www.abc.net.au/news/2015-09-11/rise-in-trampoline-injuries-worries-trauma-specialists/6766536

OECD. (2017). Building Skills for All in Australia: Policy Insights from the Survey of Adult Skills. OECD Skills Studies, OECD Publishing. https://doi.org/10.1787/9789264281110-en .

OECD. (2019). PISA 2018 Assessment and Analytical Framework. PISA, OECD Publishing. https://doi.org/10.1787/b25efab8-en .

Sharwood, L. N., Adams, S., Blaszkow, T., & Eager, D. (2018). Increasing injuries as trampoline parks expand within Australia: a call for mandatory standards. Australian and New Zealand Journal of Public Health , 42 (2), 153-156. https://doi.org/10.1111/1753-6405.12783

Steen, L. (2001). Mathematics and Numeracy: Two Literacies, One Language. The Mathematics Educator , (6)1, 10-16.

With a colleague, consider a problem-based task you’ve used in one of your own lessons. Was this problem real, relatable and relevant to your students, their context and experiences? Now, work together to design a new task for an upcoming lesson or topic area that meets these requirements.

Related articles

Teaching and learning

A range of suggested learning activities have been provided for each module in Units 1–4. It should be noted that the activities included cover a range of the Learning goals and Applications for each Module, but not all of them. Some activities could be completed within one class and others could be completed over an extended period. They include learning activities that involve group work, class discussion, and practical application of skills. Many of the learning activities could be adapted for use in other Modules or Units, or developed into assessment tasks. All are intended to be examples that teachers will use and/or adapt to suit the needs of their own students. It should be noted that teachers are encouraged to develop teaching and learning activities specifically suited to the needs to their students and context.

Included external links are for teacher reference purposes. They do not constitute VCAA endorsement of the views or materials contained on these sites and teachers need to ensure that any information or activities are appropriately adapted to meet the requirements of the VPC Numeracy Curriculum Design (1 January 2023 – 31 December 2027).

Unit 1 and 2

Exemplar 1 – module 1: personal numeracy - movies, focus areas: location and systematics, focus area: location.

Learning Goal

• find location and direction in relation to everyday familiar places within the vicinity
• find location and direction with everyday, simple and familiar maps and technologies
• use everyday oral directions using informal language such as left/right, up/down, front/back, under/beside/over

Application

• orally describe location of familiar, local places
• use interactive and paper maps to locate highly familiar places or objects
• give and follow simple oral directions to highly familiar locations

Focus Area: Systematics

• find common and familiar information and data inputs
• read data outputs
• summarise information
• input simple data into familiar apps
• read simple output data
• interpret simple output data

The Problem-solving cycle

Personal Numeracy : The focus of the context for this unit is using technology for planning and scheduling.

This plan demonstrates the Problem-solving cycle as a five week learning program.

Unit plan descriptor

This unit has students exploring Personal Numeracy with the Focus areas: Location and Systematics. There are many hands-on activities where students gather their own understanding of planning a day out.

This unit explores all three Learning Requirements concurrently as mandated by the curriculum and supports the learning of all activities.

The four stages of the Problem-solving cycle are supported by the inclusion of the multiple learning activities.

All the activities are contextualised with the issue of using technology for planning and scheduling.

Students are taken through the stages of the Problem-solving cycle over a five week period.

When students are completing the technology components, they are working towards successfully building their Mathematical toolkit – Learning Requirement Three. Opportunities presented in these tasks include: using online applications for planning and paper and interactive maps to locate places of interest. This is not an extensive list and teachers are encouraged to use as many technologies as are available within the confines of the classroom.

Integrated unit suggestion

VPC Literacy Unit 1 : This unit could form part of an integrated unit with VPC Literacy Unit 1 Module 1 – Literacy for personal use and Module 2 – Understanding and creating digital texts.

Suggested resources/required equipment

General classroom stationery supplies which support student learning and teaching in mathematics. These may include, but not be limited to:

• Student workbooks or paper
• Pens and pencils

Access to the internet and computers or tablets is essential.

Technologies may include:

• Google Maps
• PTV website
• Cinema booking sites
• Online interactive maps
• Phones for apps and calculations where permissible by the Principal

This list is not exhaustive, and teachers are encouraged to use extra materials and resources that support the learning for their students in their classrooms.

Unit 1: Module 1: Personal Numeracy with Focus Areas: Location and Systematics

This section details the activities.

Please note: These activities must not be taught in isolation from the Problem-solving cycle, or the Mathematical toolkit.

These activities are detailed in this table to help with implementation, but must be read in conjunction with the planning table.

What technology can we use?

Students explore the different websites and applications that are used for planning and scheduling, identifying what information is required to be entered into them (input data) and what information we then receive (output data).

• The teacher leads a brainstorm to generate a list of the technology we use for the purpose of planning and scheduling. For the purpose of this unit the list should include websites and apps such as; Maps, PTV, Village/Hoyts cinema etc.
• The teacher demonstrates the use of Google Maps, the PTV app and a cinema booking app. Whilst doing this ask student to look out for and make note of the information you are entering into the websites/applications and what information is given back to us.
• The class define the terms; inputs – information goes into the technology and outputs – information comes out of the technology.
• Students share their ideas and create a combined class list of inputs and outputs.

Use the Technology

Students practice inputting data into the websites and applications and interpreting the output data using given scenarios.

• Determine the fastest route to drive from school to Melbourne Aquarium. Take a screen shot of the directions given. Now determine if the suggested route changes if you want to arrive at the aquarium at 9am tomorrow. Take a screenshot of the fastest route.
• Determine the suggested route to walk from your house to school. Take a screenshot of the suggested route. Is it the same as the route you take to school?
• Determine how you could catch public transport from school to Melbourne Aquarium. Compare the travel time to the driving directions previously determined.
• Find out what movies are showing at your local cinema on Friday night.

Directional Language – Part 1

Students explore the words used to give direction or indicate a location. Students then practice using directional language by describe the location of objects around the room.

• Left, Right
• Front, Back
• Under, Over
• Besides, Next to
• Above, Below
• Inside, Outside
• The teacher places objects around the room. In pairs, students describe the location of an object to their partner who has to guess which object they are describing.
• The teacher leads a discussion to reflect on the art of orally giving directions and what makes directions good or bad.

Directional Language – Part 2

Students continue to practice their use of directional language. First students describe a chosen location within the school. Then they give oral directions to a different location within the school that a partner has to follow.

• In pairs, students think of a specific place within the school and orally describes to their partner the location of the chosen place in relation to other familiar places within the school. Their partner has to try and correctly guess the chosen place.
• In pairs, students individually think of a common location in the school and writes directions on how to get there from the classroom. They give their directions to their partner, keeping the final location a secret. Then, using only the written directions, students move around the school to reach their partners secret location.
• The teacher leads a discussion to reflect on the task – did the directions lead to the correct location? What worked well or didn’t work well?

Interactive vs Paper Maps

Students explore a variety of paper maps for familiar locations and their interactive online equivalents. Students use the maps to find specific locations and to give directions to another student.

• The teacher shows students a variety of paper maps for locations such as the local shopping centre, Melbourne zoo, Melbourne aquarium etc. as well as the interactive online equivalents.
• The teacher leads a discussion with students comparing the features that are shown on both types of maps. Discuss the positives and negatives of each type of map (paper and interactive).
• Students find specific locations on each of the maps such as the toilets, information desk, food outlet, bus stop, entrances and exits, first aid etc.
• The teacher leads a discussion on which type of map was easier to use, guiding students to think about what made the maps easier or harder to use.
• Students select a location of interest on one of the maps and use the map and other locations on the map to give directions to another student, who needs to follow the directions to find the location.

Determining Input and Output Information

Using the scenario of planning a trip to the movies, travelling by public transport. With support student determine appropriate input information required to plan the trip and the expected output information they could receive.

• The teacher gives students the scenario of planning a trip to the movies, where they have to catch public transport to the cinemas.
• Where they are going (input)
• Day, date, time (input)
• What movie they want to see (input)
• Movie time (output)
• Location of seats in cinema (output)
• Time to meet at the cinema (input)
• Public transport route (output)
• Travel time (output)
• Time to leave (output)
• The teacher supports the students to make decisions about the input information they need to plan their trip to the movies, ensuring the decisions they have made are reasonable.

Should We Blindly Follow?

Class discussion on things that can go wrong if directions or information from planning apps is not first evaluated for appropriateness. Students are supported to evaluate review the solutions for the scenarios from Activity 2.

• The teacher leads a discussion with students on the perils of blindly following directions or outputs from planning applications. Students share a time or story when they were led astray by navigation app or something similar.
• The teacher introduces Step 3 of the Problem-Solving Cycle and discuss the importance of review and reflecting on results to make sure they are reasonable and appropriate.
• Do the solutions make sense? How do you know?
• Do you need to make any adjustments to the information you have input?
• What errors could others have made?

Sharing Our Plans

Students explore how we communicate plans and practice using a variety forms of communications, such as written notes, text messages, phone calls etc.

• Who needs to know what we are doing?
• What information do they need to know?
• How can we explain our plans?
• Students communicate their solutions to the scenarios from Activity 2 using a variety forms of communication such as; a written note, a text message, phone call etc.

Assessment task

Plan a trip to the movies.

This assessment task combines the Learning Requirements 1, 2, and 3 cohesively as mandated in the curriculum guidelines.

The assessment adheres to the curriculum requirement to include all three Learning Requirements. Learning requirement 2 allows students to use the Problem-solving cycle within the context and skills outlined in Learning Requirement 1, and Learning Requirement 3 involves students using their Mathematical toolkit to support Learning Requirement 1 and 2. 

Assessment Task : Students must plan a trip to the local cinemas to see a movie with a friend and they must travel by public transport.

To complete the tasks students are required to complete the following:

• Select a movie session to go to, based on the input information decided on in Activity 6 and follow the process of booking tickets, including selecting where to sit.
• Write a SMS message to your friend informing them of the movie details and describing where you’ll be sitting.
• Determine how to get to the cinemas via public transport, making note of the route, departure location and time, arrival location and time, and travel time.
• Role play a phone conversation with your friend, who is lost (e.g. somewhere in the centre or at the bus stop) and can’t find their way to the cinemas. You can use a map of the cinema complex/shopping centre to help you give the directions. Your friend should follow your directions, by drawing them on paper map of the cinema complex/shopping centre to make sure they can find you at the cinemas.

Problem-solving cycle:

Support students through the Problem Solving cycle as they complete the requirements of the task.

Step1 – Identify the mathematics

Students write the purpose of the task in their own words, including the specific parameters (input information) they determined in Activity.

Students list the specific mathematical skills or knowledge they will need to complete the task including any associated costs, times, public transport routes, and directions. Prompt students to think about the skills or knowledge they have developed by completing the unit activities.

Step 2 – Act on and use mathematics

Students use the appropriate technology and applications to complete each of the task requirements needed to plan their trip to the movies. Ask students to keep evidence of their planning – such as notes or screen shots. 

Step 3 – Evaluate and reflect

Students review their plans to make sure they are reasonable and appropriate and make any changes to their plans if it is required. Students could share these with a peer to check for reasonableness.

Step 4 – Communicate and report

Students present their plan to their teacher, explaining the details of their trip to the movies and be ready to justify all choices made.

For assessment, students should submit:

• The movie details including location, session time and a screenshot of the chosen seat location.
• A copy of the text message to their friend.
• The details of the public transport route they will take including the form of public transport, departure location and time, arrival location and time, total travel time and a screenshot of the PTV app
• A copy of the map that shows the oral directions given during the role play

Exemplar 2 – Module 1: Personal Numeracy - Sharing our plans

Focus area: location & systematics.

On completion of this module the student should be able to:

• find location and direction in relation to everyday, familiar places within the vicinity
• use everyday oral directions using informal language such as left/right, up/down, front/back, under/beside/over.

Demonstration of the learning goals requires students to apply a variety of skills. The following applications assist students to demonstrate they have met the learning goal.

• apply a variety of skills. The following applications assist students to demonstrate they have met the learning goal.
• give and follow simple oral directions to highly familiar locations.

On completion of this module the student should to be able to:

• summarise information.
• interpret simple output data.

This unit has been designed for students 16-18 years old diagnosed with Autism Spectrum Disorder and Intellectual Disability in an independent specialist school. They are attending school full time and will undertake a Certificate I VET Certificate one day per week.

Personal Numeracy : Personal numeracyrelates to the mathematical requirements for personal organisational matters involving money, time and travel for participation in community-based activities and events.

Where am I going?

In this unit, students will learn about the location of familiar landmarks and how to get to them. In the first half of the unit, students will learn how to create maps with real images and objects, describe the appropriateness of maps vs technology, follow and give verbal and visual instructions whilst walking around their local community.

Once students have mastered these skills in the local community they will investigate technology such as Google Maps and the PTV app to find their way to locations that are further afield e.g. local cafes, shopping centres and their work experience.

Over the course of the unit, students will move from participating in excursions lead by staff to having more independence to choose locations, establishing supports that help them regulate and lead others through activities.

Numeracy Unit 1 – Module 2 ; This unit may be integrated with elements of Module 2: Financial Numeracy to support students to learnt about how to pay for their travel (in the case of this cohort, paying for a service such as travel is quite an abstract concept and so will not be taught in this way).

Numeracy Unit 2 – Module 3 ; the unit may be integrated with Module 3: Health and recreational numeracy, with the Focus area Quantity and measures as students explore time and length concepts.

• iPad or iPhone with 4G data enabled
• Myki cards for each student
• PTV app loaded onto digital device
• Companion cards for all students with additional needs
• Paper timetables of local public transport
• Local area maps

These activities are detailed in this table to help with implementation but must be read in conjunction with the planning table.

Design a map of the school

Introduction

• Students are sitting at their desks in the classroom looking at the IWB
• Students check their visual or written schedule and mark off the previous activity
• Students identify the session
• Students read through visual lesson components for the session displayed beside the board (some students may have individual lesson components they can cross off along the way)
• Students watch an online video about prepositions (search YouTube)
• Students are introduced to the Problem-solving cycle - phrased as a series of questions

Introduce the context and identify the mathematics

What mathematics language do we use when accessing the world around us?

• Students refresh each of the main prepositions they are familiar with by watching a series of video models
• Students come to the board and use the board pens to demonstrate their understanding of positions by moving visuals around the screen
• Students demonstrate their understanding of prepositions by using real objects to arrange items in different positions e.g. put the water bottle on the table, put your pencil in the cup
• Students watch a motivating video from YouTube about mapping skills

Act on and use the mathematics

• Students watch a visual forewarning of the activity and understand the technology (iPad) that they will need and how to use it
• Students walk around the school and take photos of key points in the school e.g. classroom doors, playground equipment, toilets
• Students return to class, print the photos they have taken and use them to fill in the gaps on a large map of the school
• Students are asked a short series of summary questions about their learning and are asked to engage with the map to answer questions using prepositions

Design a map of the neighbourhood

• Students watch an online video about prepositions

Act on and Use mathematics

• The teacher leads a discussion about size and appropriateness of map size and features
• Students walk around the neighbourhood and take photos of key points in the school e.g. houses, gardens, stop signs, roundabouts, shops, post-boxes etc.
• Students return to class, print the photos they have taken and use them to fill in the gaps on a large map of the neighbourhood

Evaluate and reflect

• Students are asked to discuss the practicality of the A0 size map and suggest alternatives that would be easier to carry

Using your map of the neighbourhood

• Students demonstrate their understanding of prepositions by using real objects to arrange items in different positions e.g. put the water-bottle on the table, put your pencil in the cup
• Students watch a motivating video from YouTube about using GoogleMaps
• Students use the neighbourhood map they made in the last activity to plan a route to particular destinations (changes each lesson) in the local environment e.g. shops, post-box
• Students explore Google Maps app and other technologies which can be used to orient to particular places. Use a compass and map to create a list of instructions on how to get to particular destinations and check answer in Google Maps, practising directional language learnt including up, down, left, right, etc.

Communicate and report

• Students then use Google Maps live view and their written answers to navigate to the place on the map
• Students are asked to discuss the practicality of their map vs Google Maps on an iPad

QR Code Scavenger Hunt

• Students watch a video about prepositions
• Students watch a motivating video from YouTube about using QR codes

Act on and use mathematics

• Students follow a QR code scavenger hunt (with imbedded motivating videos) which has been designed by the teacher in the local environment
• Once students have grasped the concept – they will create their own QR code scavenger hunt (motivating videos selected by the teacher but clues about the next destination from students)
• Students invite their peers in other classes to complete the scavenger hunt

Communicate and reflect

• Students are asked a short series of summary questions about their learning and are asked to reflect on the success of the event and what could improve it next time

Planning a route

• Students watch an online video about directions
• Students demonstrate their understanding of prepositions by using real objects to follow different directions to move about the classroom
• Students watch an online motivating video from YouTube about using QR codes
• Students work together to select a preferred destination such as a café to visit within walking distance of the school.
• Students plan directions to get there and test them out in person using GoogleMaps and by walking themselves
• Students write an invitation to another class to join them at the café
• (Final lesson) students walk to café together following directions at each point along the way

Planning short excursions

Identify the mathematics

• Students watch a motivating video from YouTube about using the PTV app
• Students read through a forewarning about checking the PTV app to see when trams are arriving
• Initially the scheduling component of the activity (how long is it going to take to walk to the tram stop) is mitigated from the activity by the teacher
• Students plan and participate in a series of short excursions to destinations selected by the teacher that are 1-2 tram/bus stops away from school
• Two lessons are spent planning the activity and one is spent engaging in the activity
• Students plan next stages of their outing
• Students are asked a short series of summary questions about their learning and are asked to engage with the PTV app to get the tram appropriately
• Students reflect on what they learnt during the process about timetables, catching public transport and time

Planning long excursions

• Students watch a video about directions
• Students discuss the Problem-solving cycle - phrased as a series of questions
• Students take more responsibility to plan a longer excursion to larger destinations with specific purposes e.g. Bunning, Kmart
• Students will access the PTV app at the tram stop to set a visual timer for themselves so they are able to better comprehend how long they have to wait
• Students take pictures at the different steps along the way and then return to class to match the photo they have taken with each step of the social story.
• Students are asked a short series of summary questions about their learning and are asked to engage with the PTV app to get the tram appropriately and the factors that need to be considered when planning a longer excursion e.g. checking the weather, knowing how long the walk may take, public transport running late / cancelled etc.

Core Conclusion

• Students pack away materials relevant to the lesson and move to get their schedules

Plan and share a final excursion

• Students will plan and invite another class to join them on an excursion to their chosen destination, and execute the excursion with support from the teaching staff
• Using the skills gained from this unit, students will agree and plan an excursion to an agreed destination, such as a café.
• Students will go into the other class, invite them with a small speech and written invitation with information, and execute the excursion on the day.

Staff should support the students with the four steps of the Problem-Solving Cycle.

Step 1 – Identify the mathematics

With support from the teachers, students will

• Find the location and find it on the map
• Agree on a meeting time at the café
• Use their maps and/or PTV app to plan the travel route to the café – and include the needs of everyone on the trip (does someone in this class or the other class have mobility issues that need to be considered?, is there a second bus scheduled in case we miss the first one? etc)
• Check how much it will cost to travel via public transport
• Look up the menu to share with the other class to allow students to have enough money on the day and to check for any dietary purposes
• Find the phone number of the café to make the booking
• Create the invitations for the other class – complete with all details of the school day beforehand, travelling as a group together using public transport/walking, approximate times out of school, and returning to school time, and the appropriate permission form will accompany this (created by the teacher).

Step 2 – Act on the mathematics

• Set tasks in small groups, and guide students with their goals. Check in regularly to provide positive support and role-modelling for problem solving purposes.
• Have regular check-ins as whole class meetings to see where people are up to – how they are travelling – to give it an inclusive vibe but also to allow students to help problem solve with each other.

Step 3 – Evaluate and Reflect

• Bring the class together – and have their work on display for everyone to review together. Allow only positive feedback and supported feedback – bring in the concept of positive team work and constructive criticism that leads to improvements.
• Ask students to justify the choices they have made – not to put them on the spot – but to allow them to talk through what they have done – so they can see their work in its entirety (this is a useful tool when you are highlighting something that needs tinkering or something that has been forgotten).

Make a list of excursion checks – such as but not limited to:

• Has proof-reading of the invitation been done?
• Has the date been checked on the calendar?
• Is the time the same as the booking with the café?
• Have the travel plans been checked? Reminders sent out about any travel costs?
• Have the permission forms been made to be attached to the invite?
• Has a first aid kit been booked to take out?
• Has someone checked the weather to make sure we know what to wear and/or pack?
• Excursion day information – where are we meeting before starting our excursion? What time are we leaving?
• Is the café and menu link available and on the invite?
• Is the invite created?

Step 4 – Communicate and Report

• Time to bring it all together – create an invitation that is appealing but with all the important information!
• Plan a group of students to go into the other class and invite them publicly – and pass out the invitations with permission forms attached. This group of students can explain the excursion, the travel plans, explain the café and what style of food is on offer, etc

Prior to the excursion, the class can appoint teams of students different jobs – to help working as a team to get there and back. There could be two shifts – one getting the group to the café, and one returning the group back to school – if there are enough students wanting to do it – or students can work in bigger groups and have two roles for the day?

Possible ideas for jobs – which are all supported by the teaching staff:

• Event Planner / Logistics Specialist – (overall leader who is a good problem solver, calming, has a good understanding of all the tasks)
• Personal Assistant (checking the roll with the teacher)
• Travel guides (executing the travel plans and getting everyone to the destination)
• Concierges (students who approach the café upon arrival and let them know they are there)

Exemplar 3 – Module 2: Financial Numeracy - Winner, winner, chicken dinner

Focus areas: number and change, focus area: number.

• Place value and numbers up to 1000
• Whole numbers and monetary amounts up to $1000
• Addition and subtraction (with no borrowing or decomposition) of whole numbers and familiar monetary amounts into the 100s
• Common, simple unit fractions such as 1/2, 1/4 and 1/10
• Common decimals and percentages such as 0.5, 0.25, 50%, 25%
• Identify place value and read whole numbers up to 1000
• Perform calculations of addition and subtraction with simple whole number amounts and familiar monetary amounts (into the 100s)
• Recognise and understand very common simple unit fractions, decimals and percentages.

Focus Area: Change

• Pattern prediction with shapes
• Repeating patterns with one element such as with shapes or $2, $4, $6, $8, …
• Changes and number matching with simple numbers for examples, prices increasing or decreasing, matching corresponding numbers.
• Recognise changes in numerical values such as prices increasing or decreasing with a common fixed price discount
• Number matching and comparison of simple numbers in context such as matching prices from receipts to on-the-shelf items
• Predict pattern continuation with shapes. For example, triangle, square repeating pattern
• Demonstrate patterns with one element. For example, $2, $4, $6, $8, …

Financial Numeracy : The focus of the context for this unit is money management.

This plan demonstrates the problem-solving cycle as a five week learning program.

This unit has students exploring Financial Numeracy with the Focus areas: Number and Change. There are many hands-on activities where students gather their own understanding of managing money.

The four stages of the Problem-Solving Cycle are supported by the inclusion of the multiple learning activities.

All the activities are contextualised with the issue of managing money.

Students are taken through the stages of the Problem-Solving Cycle over a five week period.

When students are completing the technology components, they are working towards successfully building their Mathematical toolkit – Learning Requirement Three. Opportunities presented in these tasks include: using the FairWork pay calculator, online shopping sites and restaurant menus. This is not an extensive list and teachers are encouraged to use as many technologies as are available within the confines of the classroom.

VPC WRS Unit 1 : This unit could also form part of an integrated unit with VPC Work Related Skills Unit 1 relating to workplace conditions (pay) and applying for an employment opportunity.

• Calculators
• Post-It notes
• Copies of a range of financial documents, including payslips
• Images of supermarket shelf labels
• FairWork Pay calculator
• Online shopping sites
• Online restaurant menus
• Phones for apps and calculations where permissible explicitly granted by the Principal

Numbers Numbers Everywhere?

In this activity students explore a variety of financial documents such as household bills, supermarket receipts, bank account statements etc. in order to recognise and understand the financial information displayed on them.

Step 1 - Identify the mathematics

In this stage students identify the task and purpose, and to then identify the mathematics involved

The teacher displays a variety of financial documents around the classroom and lead a discussion with the following questions:

• What information would you expect to see on financial documents?
• What skills or knowledge would we need to be able to understand them?
• What sorts of numbers might be found on the documents?

Step 2 - Act on and use the mathematics

In this stage students choosing the mathematics and the mathematical tools to use, and performing the required calculations and processes.

• The teacher instructs students to move around the room to examine each of the financial documents and identify the numerical information that is displayed. Students write down numbers they find, separating them into two categories; amounts or other numerical information (e.g. account numbers, dates, phone numbers etc.).
• The teacher leads a discussion asking students to share the numerical information they found on the documents. Discuss how to read each of the numbers correctly and what each of the numbers mean.
• the total amount due is two hundred and forty dollars. This is how much you have to pay ninety-one days, this is how long the billing period is 3% this is how much the total is reduced by if paid on time

Step 3 - Evaluate and reflect

In this stage students consider the appropriateness and reasonableness of their results and adjust if necessary, including redoing any calculations

• Have you correctly read all the numerical information?
• Have you made any errors with place value?
• Check your annotations with someone else who has the same document – are they similar?

Step 4 - Communicate and Report

In this stage students consider the best method/s to produce their findings, and to ensure they have communicated it sufficiently so that the audience is clear on the numbers and message being presented.

• Students partner up with someone with a different financial document. They use their annotations to share and explain how to read and understand the financial document to their partner.

Earning Money

In this activity has students explore wages, pay rates and penalty rates.

• Students examine payslips to identify the key information such as hours worked, pay rate, gross pay, tax withheld, any other deductions or allowances, and net pay.
• Students also perform calculations and use the FairWork website to ensure they have been paid correctly.
• The teacher asks students if anyone has a part-time or casual job. Lead a discussion about what it is like to earn money and if there are other ways that we can earn money other than having a job.
• What information would you expect to see on a payslip?
• What words or terminology do we need to know or understand?
• What mathematical calculations would be needed to check our payslips?
• What tools can we use to make sure we are getting the correct pay rates?

In this stage students choose the mathematics and the mathematical tools to use, and performing the required calculations and processes.

• The teacher provides students with a sample payslip for a relevant job such as fast food restaurant, supermarket or retail shop – and if students have a job they can use one of their own payslips. Ask students to identify the key information that is on their payslip including: hours worked, pay rate, gross pay, tax withheld (if any), any other deductions or allowances, net pay, and YTD pay.
• The teacher asks students to perform the relevant calculations to check that the payslip is correct and no errors have been made.
• The teacher leads a discussion with students about penalty rates and what double pay, time and half etc might mean. Demonstrate how to use the FairWork pay calculator to find the minimum pay rate and penalty rates. Note: if you click on ‘why’, then ‘details’ it shows the % loading added to the base rate. Lead a discussion on what the %loading figures mean in terms of fractions and decimals.
• Students use the FairWork pay calculator to check their own pay rates to ensure they are being paid correctly.
• The teacher supports students to review their work and decide if it is reasonable and to check over the calculations performed.
• The teacher asks students to share if their pay slips are correct. They will need to be able to explain and justify their calculations and information they have found.

Spending Our Money

In this activity students explore what things they spend money on and how to make decision about what to spend.

• The teacher leads a brainstorm with students asking them to list all the things they spend money on or want to spend money one. The class categorise these as committed expenses - must pay, not negotiable e.g. phone bill, rent or board or discretionary expenses – can be changed or adjusted e.g. take-out food, going out with friends.
• Select items to purchase: Students compare supermarket shelf items to determine which items are better value for money. Students also compare the cost of selected items from different shops.
• Calculate staff discount: Students calculate the cost of selected items if they were to use a staff discount of 10%, 25% and 50%.
• Calculate change: Students practice a variety of strategies for calculating change to determine what is the easiest method for themselves to use
• Knowing much they have to spend
• Knowing how much things cost and working out best value for money.
• Different strategies for calculating staff discounts, such as 10%, 25%, 50%
• Different strategies for calculating change

Task 1: Comparing items to purchase

• The teacher shows students a selection of shelf labels of supermarket items.
• Students identify the information that is shown including the item price, the size or weight of the item and the unit price.
• Students select a series of similar items e.g. same item but different brands, or same brand but different package size and determines which is ‘better value for money’. They should do this for at least 3 different series of items. Note: this could be done through an excursion to a local supermarket, asking students to take photos of shelf labels and bring them to class, or the teacher providing a selection of appropriate shelf label photos.
• Students choose three different items they would like to purchase such as clothing, shoes, concert ticket, mobile phone, gaming console etc.
• Students research how much it would cost to purchase each item from at least two different shops or suppliers, making note of cost of the items, any additional costs such as shipping charges, and any discounts.

Task 2: Calculating staff discount

• The teacher asks students who have jobs and what their staff discount is.
• The teacher explicitly teaches different strategies for calculating 10%, 25% and 50% discounts, including identifying the fraction and decimal equivalents.
• Students select items from their place of work, or a suitable place of work and calculate how much it would cost if they had a 10%, 25% and 50% staff discount.
• If students have a different staff discount amount (e.g. 5% or 20%), the teacher supports them to calculate this discount as well.

Task 3: Calculating change

• The teacher explains to students that knowing and understanding number patterns helps us to calculating change.
• The teacher provides students with a series of repeating shape patterns and asks them to continue the pattern.
• The teacher provides students with a series of repeating number patterns, counting by 5s, 10s, 20s, 50s, and 100s. Ask students to identify the number pattern, continue the pattern and find missing numbers.
• The teacher leads a discussion about the idea that doing ‘vertical subtraction’ isn’t always practical when trying to work out change. Explicitly teach different strategies for calculating change, such as counting on, mental subtraction left to right etc.
• The teacher provides students with a range of scenarios for which they have to calculate change.
• Do your answers seem reasonable?
• Have you performed the calculations correctly?
• Do you want someone to check over your work, or are you happy with it?
• Task 1 – the items they selected as being the best value for money in both parts A and B, explaining and justifying their choices.
• Task 2 – how much they save using their staff discount
• Task 3 – their preferred method for calculating change and how they use it.

Winner Winner Chicken Dinner

The assessment adheres to the curriculum requirement to include all three Learning Requirements. Learning Requirement 2 allows students to use the Problem-Solving Cycle within the context and skills outlined in Learning Requirement 1, and Learning Requirement 3 involves students using their Mathematical toolkit to support Learning Requirements 1 and 2.

Assessment Task: This task requires students to plan a dinner out with friends at a chosen restaurant and they have been ‘given’ a $100 voucher. In part 2 of the task students need to calculate how much a public holiday surcharge of 10% of the bill would be. Then decided if they would need to change their menu items in order to not overspend.

• The teacher informs students that they have been ‘gifted’ a $100 voucher to their favourite restaurant and they are going to take 3 friends out to dinner. They need to decide what they are going to order, making sure there is enough food for everyone. Remind them not to forget drinks!
• The goal is to spend as close to $100 as possible.
• Inform students that on public holidays the restaurant has a 10% surcharge on the total bill. Instruct students to calculate how much the surcharge would be if they went on a public holiday and to adjust their choices, if required, to ensure they do not overspend.

Ensure students understand the requirements of the task and ask them:

• What information do you need to complete the task?
• What mathematical calculation will you perform?
• What happens if you spend under or over the voucher amount?
• Students select a restaurant to spend their voucher at, and look up the menu. They use the menu to choose what they will order, showing all calculations as they go.
• Students then calculate the public holiday surcharge amount, and make the necessary changes to their order.

In this stage students consider the appropriateness and reasonableness of their results and adjust if necessary, including redoing any calculations.

• Do your menu choices seem reasonable?
• Did you under or over spend? Do you need to make any adjustments?
• The teacher instructs students to write a menu plan to give to their friends explain the menu choices they have made, including the change to the menu if they go on a public holiday.

Exemplar 1 – Module 3: Health and Recreational Numeracy - Making change

Focus area: shape & quantity and measures, focus area - shape.

On completion of this module the student should to be able to understand:

• common and familiar one- and two-dimensional shapes such as lines, triangles, circles and squares
• common properties of different one- and two-dimensional shapes such as size, colour, number and type of sides (straight/curved).
• recognise common and familiar one- and two-dimensional shapes
• name common and familiar one- and two-dimensional shapes
• construct common and familiar two-dimensional shapes
• categorise similar shapes according to common classifications.

Focus Area – Quantity & Measures

• use common and familiar basic metric measurements and quantities such as length, mass, capacity/volume, time and temperature in everyday ways such as personal height and weight, door height, liquid measurement, temperatures
• recognise common and familiar units such as m, cm, Kg, L, degrees C
• recognise 12-hour digital time, including minutes and hours on digital clocks, and hours, quarter-, and half-hours on analogue clocks
• recognise day and month dates.
• estimate lengths of highly familiar objects or items
• order and compare simple everyday measures and quantities
• recognise familiar and commonly used units of metric measurement
• read common and familiar dates and times using digital and analogue clocks.

This unit has been designed for students 16-18 years old diagnosed with autism spectrum disorder and intellectual disability in an independent specialist school. They are attending school full time and will undertake a Cert 1 VET Certificate one day per week.

Health & Recreational Numeracy

Health and recreational numeracy relates to accessing, understanding and using foundational mathematical information to be aware of issues related to health and well-being, or when engaging in different recreational activities.

In Unit 2, students explore Module 3 Health and Recreational Numeracy with the Focus Areas of Shape and Quantity and Measures. There are many hands-on activities where students gather their own data to explore measurement in different contexts

In order to support the student cohort of learners with autism and intellectual disability this module has been broken down further into 3 separate integrated units to support student learning by teaching skills in functional contexts. These units are:

1. Morning Meeting : Students start each day by addressing core areas of their program including day, date, season and time. Concepts are supported through different colours for each day and month to support students with low reading comprehension skills. Students are presented with a schedule of activities for the day in their preferred receptive style – this may be a list or series of pictures which represent each activity. These activities are paired with digital or analogue times which are varied over the course of the year as student’s master the concepts. Students will use the internet to research the daily temperature and pair this knowledge visually with what it means e.g. hot = wear shorts, cold = wear a jumper. Teacher questioning will test all of these concepts which may be answered verbally, visually or with support of Augmentative and Alternative Communication.

2. Melt and Pour : This unit is integrated with a range of modules as listed below. Students will create a social enterprise called Glow for Giant Steps. As part of this unit, students will make soap/candles/bath products which require them to use their measurement skills to weigh, assess liquid quantities, temperatures and other measurements. Students will first refresh their knowledge of 2D and 3D shapes before exploring what weight of solid soap they need to fill various 3D shapes. Initially, the students will use 3D shape models to produce shapes for their business, which over time can be changed to more decorative styles.

3. Where am I going? : This unit is integrated with Unit 1 – Module 1 Personal Numeracy. Once students are familiar with exploring their local environments they will start to explore how they can travel further afield by using public transport. Students will use the PTV and Google Maps apps to work out how to get to a destination testing their time telling skills to catch transport on time to arrive a work experience or other activities.

This unit explores all three Learning Requirements in accordance with the the VCAA’s publication Victorian Pathways Certificate: Numeracy – Curriculum Design.

The four stages of the Problem-solving cycle (Learning Requirement 2) are supported with the inclusion of the multiple learning activities. All the activities are contextualised with the issue of a social Enterprise – Glow for Giant Steps .

When students are completing the technology components, they are working towards successfully building their Mathematical toolkit – Learning Requirement 3. Opportunities presented in these tasks include: using a calculator to perform calculations, online applications to create and conduct surveys and using spread-sheet software to perform these calculations and using spread-sheet software to present graphs and tables. This is not an extensive list and teachers are encouraged to use as many technologies as are available within the confines of the classroom.

Through the Melt and Pour activity students will create a social enterprise called Glow for Giant Steps making candles and moulded soaps. The measurement skills taught in this unit may be integrated across:

PDS Unit 2 ; to find opportunities to connect with the community and practice socially appropriate interactions via operating a Pop-Up stall in the school carpark to sell soaps.

WRS Unit 1 & 2 ;

• Unit 1 to explore soap and candle making as an interest/capability
• Unit 2 to host a launch party for their business

Literacy Unit 1 & 2 ;

• Unit 1 – Module 2: to create and make a presentation at the launch party to explain to others the purpose of their business
• Unit 2 – Module 1: to share these made products with other organisations such as Orange Sky

Beneficial resources: Smart phone – using smart assistants to support understanding of time Computer – for researching weather, transport times

Required equipment: Supplies for melt and pour soap making

Daily Numeracy – Morning Meeting

• Students are sitting at their desks in the classroom looking at the Interactive Whiteboard (IWB)
• Students watch an online hello song (search greeting songs on YouTube)

Introduce the Context and Issues

We need to know the date so that we can work effectively and plan our lives and work activities

Act on and Use the Mathematics

• Look at the calendar to establish the day/date/month/ year. This calendar should be presented visually on the wall with visual images to indicate key activities for each day of the week e.g. work experience on Friday.
• Students respond to teacher to label day, date, month, year.
• Students use a smart assistant or the internet to research the weather and temperature that day and complete a short sequence of activities to explain what that means in terms of dressing e.g. wear a jumper when it is 10 degrees.
• Students are lead through their schedules for the day looking at each of the activities they are completing. Each of these activities should be paired with a time to enable questioning e.g what is happening at 11:00? When is Lunch? When is home time?
• Students place their schedules in their folders

Daily Numeracy – Afternoon Meeting

Core introduction

• Students watch an online goodbye song (search greeting songs on YouTube)

Re-introduce the Context and Issues

• It is important to reflect on what we have done each day so that we can prepare and modify our strategies for a more efficient tomorrow
• Students respond to teacher to label day, date, month, year
• Did the calendar match our experiences today?
• Students refer to the visual calendar to identify the key activity of their day and point to it on their individual schedules
• Students create a sentence reflective of their learnings that day e.g. “Today on Tuesday I went to work experience and learnt how to make coffees”
• The teacher pulls up a series of pictures that were taken during the session and students are supported to copy these images into a PowerPoint presentation and make a comment about the images

Activity 3A

Melt and pour - part a.

• Students watch a familiar video about 2-D shapes

Identify the Mathematics

• Students discuss how they can apply their knowledge of shapes in practical ways as they build their social enterprise Glow for Giant Steps
• Students are reintroduced to and revise their knowledge of shape and measurement
• Students revise their knowledge of certain shapes with a bingo activity
• Students revise their knowledge of shape in the real world to ‘find’ shapes in the environment and describe them through pictures and playing online games https://pbskids.org/peg/games/highlight-zone
• Students look at objects that are motivating to them and describe them in terms of their shape using language e.g. ‘a tram is rectangular’, ‘the ball is a sphere’.
• Students reflect on the different shapes that they can see and what language is used to describe them

Activity 3B

Melt and pour - part b.

• Students watch a familiar video about 2D shapes
• Students play various games  https://pbskids.org/games/shapes which explore shape and measurement and size
• Students look at laminated shapes and organise them by their different attributes e.g. all shapes with 4 sides, all shapes with curved edges
• Students look at objects that are motivating to them and describe them in terms of their shape e.g. the tram is rectangular

Activity 3C

Melt and pour – part c.

• Students watch videos about making melt and pour soap

Act on and use the Mathematics

• Students use their knowledge of shapes to label and discuss 3D shape products that have been prepared by the teacher
• Students explore different soaps that have been prepared in advance that are different shapes e.g. cube, rectangular prism, pyramid
• Students describe these shapes in terms of their nets and 2D qualities
• Students follow visual forewarnings to make soap including use of the microwave to measure digital time, digital and analogue thermometers to measure the temperature of the soap, measuring cylinders and syringes. Students measure liquid quantities and weights of soaps before and after melting to calculate volume of different sized objects.
• Students reflect on the unit of measurement that was the focus of that session and explore other ways it could be applied to their world

* integrated with WRS Unit 2 – students will make and sell these soaps to the local community. A log of sales will be kept to establish which is the most popular soap shape

Activity 3D

Melt and pour - part d.

• Students continue making soap and exploring the relationship between length, volume and weight
• The teacher will address temperature in the melting process
• Students will consult the data collection to soap sales to graph which has been the most popular shaped soap over the sales period
• The class will discuss the making of different soap shapes and discuss which shapes should be focussed on for sale

Communicate and report Students will respond to questions describing each of the soap shapes available and suggest reasons why some soaps had more successful sales than others and modify the selection of soaps that are made the following weeks to reflect the popularity of each shape.

Integrated with Unit 1 **This is integrated with Unit 1 – Module 1 – Personal Numeracy

• Students discuss the Problem-solving cycle – how can we use mathematics to help us access public transport?
• https://www.youtube.com/watch?v=dAZYsT-n-2s
• Students use PTV app/google maps to navigate to public transport – calculating times it will take them to get to various places as well as using schedules to work out what time tram/bus/train they are going to get.
• Students plan a longer excursion to larger destinations with specific purposes e.g Bunnings, Kmart

Evaluate and Reflect

• Students are asked a short series of summary questions about their learning and are asked to engage with the PTV app to get the tram appropriately and the factors that need to be considered when planning a longer excursion e.g. checking the weather, knowing how long the walk may take

Communicate and Report

• Students report how they will modify their excursion next time based on discussion of the above factors
• Students discuss liquid quantities of water based on their volume knowledge from Activity 3
• Students watch a video about saving water
• Students explore the different ways that they use water each day during their lives e.g. showers, drinking water, watering plants at the community garden.
• Students estimate how much water they use within a day and track this over a week by filling in a spread sheet
• Students use the internet to calculate how much water a shower uses (e.g 9L/minute) to calculate the quantity of water they use each day.
• Students reflect on their number and think about ways in which they can reduce their water use

Making Change: ‘Providing Access to Soap and Water’

Task - Students will explore the idea that ‘not everyone has access to soap and water’, and write a newsletter article that combines:

• The important of soap and water for hygiene
• The important information from the Guest Speaker – Orange Sky
• Research statistics showing how Orange Sky is making a big difference for its communities
• Ways that people can help in the community
• Highlighting their social enterprise – Glow for Giant Steps – soap making integrated WRS project – summary of soap making process, photos, sales figures, timeline it takes, etc

Step 1 – Identify the Mathematics

• Students explore their access to water and describe how easy it is to access for them
• Students recall their knowledge of hygiene practices and reflect on how they use soap and water across the day (especially during the COVID pandemic)
• Students are told that not everyone has access to soap and water e.g. homeless people and how that can cause adverse health effects
• Students are told of different ways in which they can help others
• Guest speaker from Orange Sky – students are prompted with different questions to ask the employee from Orange Sky about what issues they see people face and how they help others

Step 2 – Act on the Mathematics

When the Guest Speaker arrives, make sure students have their prepared questions ready – and ensure they have a pen with them to write down their answers.

Research statistics to support the information – such as rates of people using Orange Sky services, how long they have been providing the services, health and hygiene affected back lack of water and soap, how we can provide help to Orange Sky and other organisations to help be a part of this solution.

Step 3 – Evaluate and Report

Students decide what they want to report on – Melt and Pour Soap Making project (sales figures, soap making process, photos, up-coming sales etc), research found about Orange Sky and questions answered by Guest Speaker, ways that we can support Orange Sky and other organisations.

Teacher to provide template and work with groups to support the writing process, drafting and final edits. 

• Students reflect on how they can use their business to support organisations such as Orange Sky.

Exemplar 2 – Module 3: Health and Recreational Numeracy - Paper planes

Focus areas: shape and quantity & measures.

• Common and familiar one- and two-dimensional shapes such as lines, triangles, circles, squares, etc.
• Common properties of different one- and two-dimensional shapes such as size, colour, number and type of sides (straight/curved).
• Recognise common and familiar one- and tw0-dimensional shapes
• Name common and familiar on- and two-dimensional shapes
• Construct common and familiar two-dimensional shapes
• Categorise shapes according to common classifications
• Use common and familiar basic metric measurements and quantities such as length, mass, capacity/volume, time and temperature such as personal height and weight, door height, liquid measurements, temperatures.
• Recognise common and familiar units such as m, cm, Kg, L, degrees C
• Recognise 12-hour time, including minutes and hours on digital clocks and hours, quarter- and half-hours on analogue clocks
• Recognise day and month dates
• Estimate lengths of highly familiar objects or items
• Order and compare simple everyday measures and quantities
• Recognise familiar and commonly used units of metric measurements
• Read common and familiar dates and times using digital and analogue clocks.

Health and Recreational Numeracy : The focus of the context for this unit is a paper plane competition

This unit has students exploring Health and Recreational Numeracy with the Focus areas: Shape and Quantity and Measures. There are many hands-on activities where students gather their own understanding of designing paper planes for a competition.

This unit could form part of an integrated unit with VPC Literacy through the reading and writing of instructions or through a film analysis of the movie Paper Planes .

This unit could form part of an integrated unit with VPC Personal Development Skills focusing on team-work, communication, time management and problem solving skills.

• Coloured paper
• Origami paper
• Measuring tools (e.g. tape measures, kitchen scales, body weight scales, measuring jug, stop watch, thermometer, digital thermometer)
• Digital and analogue clocks
• Post-it notes
• Glue or tape
• Digital calendars or diaries
• Digital measuring tools
• Phones for apps and to take photos where permissible by the Principal

Too cool to rule

Students use a variety of analogue and digital measuring tools to estimate then measure a selection of items from the classroom. The items selected must include length, mass, capacity/volume, time, and temperature.

• The teacher shows students a range of measuring equipment including rulers, tape measures, kitchen scales, body weight scales, measuring jug, stop watch, thermometer, digital thermometer etc.
• The teacher leads a brainstorm with students asking them to identify different objects or things that could be measured using the different measuring tools and what units of measurement would commonly be used.
• Students classify the measuring tools as either analogue or digital, and discuss the benefits or disadvantages of using each type.
• The teacher demonstrates, or ask students to come up and demonstrate, how to use each of the measuring tools and read them accurately.
• The teacher instructs students to select an item in the room to measure using each of the different measuring tools. They must first estimate, then use the measuring tool accurately.
• The items measuring must also include at least one instance of each of the following;
• length (e.g. of the table, or personal height)
• mass (e.g. weight of their pencil case or personal weight)
• capacity/volume (e.g. amount of water in their drink bottle)
• time (e.g. to walk across the room or to complete the activity as a whole)
• temperature (e.g. personal temperature or temperature of a hot drink).

It’s all a matter of time

Students explore time as a unit of measurement including the different units of time, how they relate to each other and the different tools used to measure, record or display time. Students then compete a series of scenarios that require them to recognise time on both digital and analogue clocks and recognise day and month dates.

• The teacher asks students to identify what the time is right now. Ask them to explain how they know this, and what does that ‘time’ actually mean.
• The teacher leads a discussion with students about time as a unit of measure, including the different units of time such as seconds, minutes, hours, days, weeks, months, years etc.
• The class discuss the relationship between the different units of time and the different tools or equipment we use to measure, record or display time.
• The teacher explicitly teaches students how to read hours, quarter- and half-hours on an analogue clock, and provides real-life scenarios where work-places use these clocks, such as hospitals and hospitality, to give relevance and importance.
• The teacher provides a series of scenarios that requires them to recognise time on clocks, and recognise day and month dates, such as:
• Looking up the public holiday dates for the year and writing them in on a calendar or diary
• Looking at a bus or movie timetable and matching the digital time to the time shown on analogue clocks
• Creating a calendar of all the birthdays in the class, and seeing how many people have birthdays on Mondays, Tuesdays etc. throughout the year
• Looking at the time shown on the classroom clock, and the world clock app and writing the different times down in words

Shape Scavenger Hunt

Students go on a scavenger hunt around the school to find examples of common shapes. Students take photos of the shapes, then identify the properties of each of the shapes. As a class, students group similar shapes together.

Students go on a scavenger hunt around the school to find examples of common one-and two-dimensional shapes, including lines, triangles, circles, squares etc. They take a photo of each shape they have found and return to the classroom.

Students draw each of the shapes found. Support students to name each shape and write down its properties including, size, colours, number and type of sides (straight/curved) how many corners etc.

As a class, group each similar shape together according to common and agreed upon classifications.

Paper planes – What do they look like?

Students investigate the shape and design of different types of paper planes, writing down the shapes they can see in the design and the approximate dimensions of the plane.

• The teacher leads a discussion with students on the wide variety of shapes that paper planes come in.
• Students find 4 different paper plane designs or provide a variety of premade paper plans to look at.
• Students either draw or cut and past pictures of the designs.
• Next to each picture students write:
• The shapes they can see in the design
• The approximate dimensions of the plane (length, width and height)

What are the rules?

Students determine the agreed upon rules, such as what can be used to make the plane, how each category will be conducted, what will be used to measure the distance or flight time etc.

• Students create a flyer or poster that clearly shows the time, day and date of the competition.
• Let students know that before they begin designing and testing their planes they need to come up with the rules of the competition. There are going to be three categories in which they will be competing. They are:
• Furthest distance travelled
• Longest flight time
• Best plane tricks (if time permits)
• The teacher leads a brainstorm with the class to determine the agreed upon rules, such as what can be used to make the plane, how each category will be conducted, what will be used to measure the distance or flight time etc.
• For each category students must copy down the:
• Rules of the competition
• How they will test/judge the competition
• List of items that are needed
• Discuss with students when the paper plane competition will take place and ask students to create a flyer or poster that clearly shows the time, day and date of the competition.

Design Time

Students research different paper plane designs and select a design for each category (3 planes in total). While making their paper planes, students draw a top down and side view of each, write the shapes that can be seen during the making of each, estimate the length and width of each plan then measure the actual length and width.

• Students are required to design and make a paper plane for each category, making sure they adhere to the agreed upon rules and restrictions of the competition. They can find and use designs from any resources.
• Step 1: Students research different designs of paper plans and selects one for each category (3 planes in total). They should practice making the planes before deciding on a final design.
• Step 2: After deciding upon the paper plane designs students are required to:
• Write down or print a copy of the step-by-step instructions on how to make the chosen designs.
• Draw a top down and side view of each paper plane.
• Write the shapes that can be seen during the making of each paper plane.
• Estimate the length and width of the paper plan.
• Make your final version of each paper plane and write the actual measurements of length and width.

Competition Time

• Before taking part in the competition students estimate how far their plane will travel (longest distance category) and how long their plan will fly before it touches a surface (longest flight time category)
• Students take turns flying their planes and are given three chances for each of the competition categories.
• Students need to make sure to record the results from each of their ‘runs’ e.g.

• Ask students to look at each of their run times and select the best one. Students add their results to the class result sheet.  Lead a discussion with students looking at the class results and make a decision on who was the winner for each category.
• Before starting the competition remind students of the rules and restrictions of each category.

Design a prize

This assessment task combines the Learning Requirements 1, 2, and 3 cohesively as per the curriculum guidelines.

Assessment Task:

• Student research and create an origami award or prize that can be given to the paper plane competition winners.
• Research different origami designs such as medals, stars or trophies and select the design you will make.
• Estimate the measurements of your origami award. Write the actual measurements once you make the final product.
• Draw a top down and side view of your origami award.
• As you make the origami award, write down the shapes that can be seen during the making of it and in the final product.

Discuss with the students:

• Ask students to create an origami award or prize to can be given to the paper plane competition winners.

Support students through the Problem-Solving Cycle as they complete the requirements of the task.

Students write the purpose of the task in their own words.

Students list the specific mathematical skills or knowledge they will need to complete the task. Prompt students to think about the skills or knowledge they have developed by completing the unit activities

Students use make their origami awards, completing each of the task requirements.

Students review their final product and make any adjustments or remake it if necessary.

Students share their origami award with the class, explaining why it should be chosen as the prize for the competition winners. The class votes on best award, and the top three awards are selected to be given to the winners in each of the categories.

• A copy of the origami instructions
• Their measurement estimations and final measurements
• Top down and side view drawings
• List of the shapes found in the final product.

Exemplar 3 – Module 4: Civic Numeracy - Saving water

Focus areas: data & likelihood, focus area: data.

• Understand simple data collection by hand or with tables
• Understand simple cases of data, graphs and infographics
• Collect and display simple data
• Read simple graphs such as bar or pie graphs
• Read simple tables
• Identify and locate key facts from simple data

Focus Area: Likelihood

• Use everyday language to talk about the likelihood of an event occurring such as possible, impossible, unlikely, likely, certain, ‘Buckley’s chance’, “pigs might fly”, “dead-set”
• Understand language and relative magnitude of simple and high familiar chance events.
• Recognise and use the everyday language of chance and likelihood
• Use everyday language to compare and order different and simple magnitude of chance

Civic Numeracy : The focus of the context for this unit is conservation of water.

This plan demonstrates the Problem-solving cycle as a six week learning program.

This unit has students exploring Civic Numeracy with the Focus areas: Data and Likelihood (Learning Requirement One). There are many hands-on activities where students gather their own understanding of reading information from tables, graphs and infographics and using everyday language to describe the likelihood of weather events.

This unit explores all three learning requirements concurrently as mandated by the curriculum and supports the learning of all activities.

All the activities are contextualised with the issue of water consumption and saving water.

Students are taken through the stages of the Problem-Solving Cycle over a six week period, as outlined in Learning Requirement Two.

When students are completing the technology components, they are working towards successfully building their Mathematical toolkit – Learning Requirement Three. Opportunities presented in these tasks include: using online applications for reading data, tables and graphs and collecting and displaying simple data. This is not an extensive list and teachers are encouraged to use as many technologies as are available within the confines of the classroom.

VPC Literacy : This unit could form part of an integrated unit with VPC Literacy through the reading and writing of articles and persuasive pieces related to climate change.

VPC PDS : This unit could form part of an integrated unit with VPC Personal Development Skills focusing on climate change, sustainability and other issues related to drought.

• Pens, pencils and markers
• Digital data, tables and graphs via websites such as https://www.makeeverydropcount.com.au/ , www.melbournewater.com.au , http://www.bom.gov.au/ ,
• Household water calculator ( https://smartwatermark.org/watercalculator )
• Phones for apps where permissible by the Principal

Data Overload

Students explore a variety of graphs, charts, tables and infographics related to weather, climate change, water and drought exploring the different ways data can be presented and identifying the key features of graphs and tables.

• The teacher presents a variety of graphs, tables and infographics related to weather, climate change, water and drought.
• Students complete a ‘think-pair-share’, first looking quietly at the graphs, tables and infographic and making some notes about what they can see or notice, then sharing with a table partner, and finally participating in a whole class discussion.
• During the class discussion, highlight the different ways data can be presented and the key features of graphs and tables including; titles, scale, key etc.

Interpreting Data – Can you trust what you see?

Students work in small groups to read and interpret information from weather graphs and tables. Students then explore how graphs can be misleading and review their interpretations to see if they need to make any adjustments on the basis of some of the graphs being misleading.

• The teacher presents a selection of graphs, tables and infographics related to weather, climate change, water and drought These could be the from Activity 1, but also include some graphs with deliberately misleading information such as not starting at zero, expanding or compressing scale to make changes seem more or less significant, not including a full data range e.g. only a few months of the year not the whole year, incorrect graph for data type etc.
• The teacher supports students to group similar data sets together e.g. temperature, rainfall etc and in small groups write down what information they can gather or interpret from the graphs.
• The teacher leads a discussion, drawing students’ attention to the deliberately misleading graphs. Point out why the graphs are misleading and discuss why misleading graphs might be used.
• Students look back at their interpretations and see if they need to make any adjustments due to some of the graphs being misleading.

What’s the weather doing?

Students explore the weather forecast and describe the likelihood of upcoming weather events. Students then further examine how the likelihood of rain is forecasted and interpret the likelihood of rain in the coming week.

• Use the BOM website or other weather forecast resources to look at the weather forecast for the upcoming week.
• The teacher leads a discussion with students to identify the information that is provided on the forecast.
• Students use everyday language to describe the likelihood of upcoming weather events e.g. it is unlikely to be windy, you will certainly need an umbrella, the chance of snow is impossible etc.
• Draw students’ attention to the how the chance of rain is shown on the forecast. Explain that the words; isolated, scattered, patchy and widespread used to be used. Discuss what they think each of the old forecast terms mean in relation to the likelihood of rain occurring.
• E.g. if the chance of rain is 30% then there is a 1 in 3 chance of getting wet or a slight chance, but there is also a 70% chance or high chance of staying dry. Also explain that the possible rainfall shows how much rain is forecasted and is represented as a range between two values. The first value means the location has a 50% chance of receiving at least that amount of rain and the second represents a 25% chance of receiving at least that amount. E.g. for a forecast of ‘4 to 15mm’ the area has a 50% chance of receiving 4mm or more and a 25% chance of receiving 15mm or more.
• Students review the rain forecast for the upcoming week and interpret the information to describe of the likelihood of rain each day.

Do we even need to be saving water?

Students explore the information that is provided about Melbourne’s water storage levels and temperature and rainfall annual data and trends to determine how Melbourne’s water levels have changed over time and form connections between changing weather events and water supply levels.

• The teacher leads a discussion with students on what the weather has to do with water storage levels and the conservation of water.
• The teacher supports students to explore the information that is given about Melbourne’s water storage levels and how they have changed over time. ( https://www.melbournewater.com.au/water-data-and-education/water-storage-levels#/ ) and temperature and rainfall annual data and trends ( http://www.bom.gov.au/climate/ )
• What do you notice about Melbourne’s’ water storage levels?
• Do we have enough water?
• Do you notice any connection between weather/climate and water storage levels?
• Does our water supply meet demand? Is this likely to change over time?
• Can one person make a difference?

Determining our water consumption

Students bring in a copy of a water bill from home and examine the information that is shown on it to determine the water consumption of the household. Students then complete the ‘Home water calculator’ (https://smartwatermark.org/watercalculator/) to further explore the water consumption of their household in more detail. Students share their results and information with the class to create a class data set on how much water is used in households. Students display this data using appropriate graphs and tables.

• The water use, a representation of the daily average water use over time, comparing with usage from the last 4 bills.
• Average use per person, indicates the average daily water use per person and includes a comparison to the same time period from the previous year and the average consumption of other properties in the same postcode.
• Students write down what they notice about the water usage in their home, including reference to the Target 155 goal.
• Bathroom: how many showers per day and average length of showers, number of shower heads and if they are water efficient or not, how many baths taken per week, number of toilets.
• Laundry: type of washing machine and how many loads per week, how many times washing is done by hand per week
• Kitchen: Do you have a dishwasher and how many loads per week, how many times wash dishes using the sink per week, any dripping taps or leaking toilets.
• Garden: how many minutes hand water with a hose in summer per week, any sprinklers, drip irrigation or spray irrigation, the number/length used and hours used per week, how many hours hose is used for outside washing per week, number of water tanks.
• What areas use the most water?
• How does your household compare with the average household? Goal water usage?
• Where could your household make changes to reduce your water consumption?
• What is the likelihood your household would adopt changes?

Water saving tips

Students brainstorm and explore water saving tips and strategies that can be applied to their households to reduce the amount of water they use. Students develop specific advice and recommendations, relating these water saving strategies to the likelihood of certain weather events.

• The teacher leads a brainstorm with students about ways people can save water. Ask student to think about the impact has on saving water e.g. water fights/play in summer, re-washing clothes if they get rained on, hot showers/baths if cold, watering the garden etc.
• Students can explore water saving tips on the ‘make every drop count’ count website and their local water supply company’s website.
• Students are required to come up with specific advice and recommendations for their household to reduce their water consumption. They should also relate these water saving strategies to the likelihood of certain weather events e.g. check the weather before watering the garden, if there is a medium or high chance of rain then don’t water the garden.

Make your recommendations

Students are required to present their water saving recommendations in a suitable format, such as a poster, flyer, report etc.

• Students must include:
• The current water consumption of the household.
• The target water consumption and why the household should try and reduce their water consumption.
• Specific water saving strategies (at least 5), why they have chosen them and how they will help reduce water consumption.
• A link between the water saving strategies and the likelihood of certain weather events.

Support students through the Problem-Solving Cycle as they complete the requirements of the task. 

Students list the specific mathematical skills or knowledge they will need to complete the task. Prompt students to think about the skills or knowledge they have developed by completing the unit activities prior this task.

Students use class-time to compile their information and address each component in tasks.

For assessment, students should submit their final submission (poster, flyer, report etc.) with:

• A display of the current water consumption of the household.

Unit 3 and 4

Exemplar 1 – module 1: personal numeracy.

On completion of this module students should have the knowledge to be able to:

• find locations and give directions in relation to everyday, familiar places within their extended vicinity
• find locations and give directions using simple navigation with everyday, familiar maps and technologies
• use informal, and some formal, language of location and direction, including simple angle measures and representations such as: quarter and half turns, left and right, N, S, W, E.

Application of the learning goal requires students to demonstrate the following skills:

• provide oral and written instructions to describe the location of familiar, local places and landmarks
• use interactive, digital technologies and paper maps to locate familiar places or landmarks and places of significance, and describe suitable routes
• give and follow simple oral and written directions to familiar locations
• use everyday language of angles and compass directions (N, S, W, E) to describe familiar locations and directions such as half turn, U-turn.
• use common and familiar information including data 
• read and interpret data inputs and outputs
• plan and schedule.
• input data into familiar apps
• read input and output data
• plan and schedule with common and familiar data.

Personal Numeracy : The contexts explored in this unit include technology for planning, scheduling and mapping locations, and planning and undertaking cooking. Students will combine both for their assessment task.

This is a six-week learning program.

Students explore personal numeracy with the focus areas: location and systematics. For the focus area location, students work with maps and explore directional language in written and oral forms. For the focus area systematics, students work with inputting data and exploring the outputs.

The contexts used in these activities include, but are not limited to, using local parks nearby, making use of multi-sport courts, completing and creating scavenger hunts, re-purposing fairy tales into creative maps, exploring recipe and cooking apps and websites, preparing a two-course meal plan for ten people and ordering takeaway food via apps and websites.

The assessment task combines these tasks and students work in pairs to present a day out at the local park where they plan activities and a BBQ lunch. The class votes on the best proposal that fits within the budget and parameters set by the teacher.

This unit explores and demonstrates all three learning requirements as mandated by the curriculum. The four stages of the Problem-solving cycle are supported and demonstrated in multiple activities. The Mathematical toolkit is highlighted throughout the activities, as indicated by the third learning requirement.

General classroom stationery supplies which support student learning and teaching in mathematics. These may include, but are not limited to:

Access to the internet and computers/tablets is essential.

The Mathematical toolkit technologies explored include, but are not limited to:

• Multiple cooking apps and websites
• Multiple on-line food delivery apps and websites
• Spreadsheet software
• Phones for apps and calculations where permissible by the principal

Teachers are encouraged to use extra materials and resources that support the learning that caters directly for their students needs. All timelines are given as a guideline only.

Unit 3: Module 1: Personal Numeracy Focus areas: Location and Systematics

Read it, see it.

Students practise using directional language within their school grounds by working in pairs and using the school map. They write instructions on how to get from a common location to their new location, using common direction language such as left/right, up/down etc. They swap the instructions with their partner and complete the task. Each pair meet up and provide feedback on the written description and check if the final location was correct. Students repeat task using feedback to improve written instructions.

Students brainstorm local venues nearby school, such as the local shops, parks, primary schools, other businesses etc. Using that list, students each select one location and write the directions from the school to the location using an online mapping tool, such as Google Maps.

Students take it in turn to share their directions to the class or in small groups and their peers guess the locations. This could also be done via different stations around the room with the directions at different tables, and students could submit their answers via electronic form or have an answer card ready.

Shout it out loud

Students visit sporting courts that have multiple courts, such as basketball, netball, tennis, volleyball etc. Students study the lines and develop warm-up routines using the different lines. Students pair up and shout out their instructions to their partners (e.g. jog along the white netball line in a southerly direction, turn right, sprint the top line, turn right, complete lunges along the first netball third and cross the court doing star jumps). Students might also incorporate the different equipment of the relevant sports.

Provide students with a detailed paper map of a well-known place, such as The Royal Melbourne Zoo or the Royal Melbourne Show. Prepare a scavenger hunt where students shout out answers or answer via Kahoot  and use their map skills to find:

• How many toilets there are.
• What compass direction the train station is.
• How many information services stands there are etc.

Students show their own skills by producing their own scavenger hunts on maps. Provide three to four different maps in the classroom and have students generate six to ten questions. Students with the same maps swap and complete the hunts. A timer could be visible on the front whiteboard if the class wants to make it a competition. Students then swap feedback about their maps with a Plus ,Minus Interesting (PMI) graphic organiser.

Open a digital map as a class and explore the features. Compare the view on a laptop to a smaller screen, such as a tablet. Create a story for the students to follow with directional language and see if they finish at the intended stop. Ask a few students to create a story, share it with the class using their directional language and see if the class can follow the instructions and finish at the correct stop. 

Made up maps

Students need to recreate a known story, or write their own, that has multiple locations and is an adventurous tale.

Students draw a map on grid paper that correlates with the story and provide co-ordinates on each side to be able to locate the locations (e.g. A5, B8 etc.).

Students read and reconstruct the story on cards. They use directional language on the last line of each card to give directions to the next location, which lands them in a certain co-ordinate  and this is on the next card. The students repeat this process until the end of the story and have their cards in order to check with the writer. 

Chefs in the kitchen

Explore different food apps by exploring the different inputs, including:

• different levels of recipes by skill to prepare/cook
• different levels of recipes by time to prepare/cook
• cooking requirements, such as vegan, dairy intolerant etc
• different ingredients which then output different recipes.

Students collect a variety of different recipes based on different situations to practice using the input functions.

A busy kitchen

Students form a team that will cook for 10, as if for cooking in a pop up restaurant or a big family meal. Each student sets their own two-course menu that will be prepared.

Problem-Solving Cycle

Step 1: Identify the mathematics Each student writes up the kitchen preparations list for the 1pm sit-down and identifies each step that needs to be completed. Consider how long will it take and how best to record it?

Step 2: Act on and use the mathematics Students start creating the plan of how the kitchen will operate to complete and plate the lunch to start at 1pm. Students list all jobs in order that need to be completed: approximate job times (such as grating carrots 5 minutes), pre-heating oven 10 minutes etc. Students list how long cooking will take and if batch cooking will be needed.

Step 3: Evaluate and reflect How will students be able to double-check their times and expect them to be reasonable? Is there an expert they could consult? Can they have a small interview with the teacher? Have they double checked the recipe?

Step 4: Communicate and report Students consider how their plan will be presented to the kitchen team. Will there be different teams working on different aspects? Will it need colour coding? How big will the font be? Will it be in columns to separate person, job and time? The class should brainstorm ideas to help work out these aspects.

Food orders

Discuss students’ experiences of using food ordering apps. Students explore and compare two food services online, cost their family favourite takeaway and see if there are discounts ordering via the restaurant directly.

Step 1: Identify the mathematics Discuss with students their experiences using food delivery apps and websites. Students consider their successful and not so successful experiences. Each student recalls their family’s favourite takeaway meal and the shop/restaurant from which it is usually ordered. They then open one of the food delivery apps/websites to see how much the order would cost and if there are any specials/discounts. Students compare this order with another competitor in the area or with the same shop on a different food ordering app/website, and their specials/discounts. Compare whether ordering directly from the shop would bring any extra savings.

Step 2: Act on and use the mathematics Students complete their research and compile it on a spreadsheet. Remind students to have the food in one column and the cost in another so the spreadsheet can add the costs together.

Step 3: Evaluate and reflect Students look over their costs and check their figures. Using a spreadsheet, students check their formulas so they are adding the correct columns together.

Step 4: Communicate and report Students consider the best way to produce this information. Would it be best to communicate it in the spreadsheet? Enhance it with graphs? Talk to students about conventions used in formatting spreadsheets, including making the totals stand out, bolding the headings, placing the investigation question up the top etc.  

Planning a day at the park

This task demonstrates assessment of the three components: Numeracy in context, Problem-solving cycle and Mathematical toolkit cohesively as per the curriculum guidelines. Students use the Problem-solving cycle within the context and skills outlined in Personal numeracy, and students use their Mathematical toolkit to support Personal numeracy.

Assessment Task : Students plan and cater for a BBQ lunch at a local park that has appropriate facilities for lunch and activities. Students work in pairs to present their plans to the class. These are voted on and then executed on the day.

Students are required to complete the following:

• Participate in the brainstorm
• Present a menu and costing for the BBQ lunch
• Present the chosen park and the public transport route to and from school
• Present a draft email script
• Present a list of jobs for each team member when at the park
• Present the scavenger hunt game to play at the park using directional language and compass points
• Present a map of the park with the set up

Support students through the problem-solving cycle as they complete the requirements of the task.

Present the idea to the students and complete a brainstorm together for ideas for the day. Help students organise their thoughts into different categories: food/drinks, activities/equipment, school logistics/communication, public transport, etc. This will form as a checklist for work to be completed. Students consider and list the mathematics required for each job.

Students work in their pairs to work through and put their plans together. The teacher might need to help conduct a class survey (such as BBQ preferences or food allergies, or provide budget limits for supplies) to support planning. Students should keep evidence of all planning, such as a screenshot of the supermarket estimate, the public transport routes and ticket costs, the draft email to PE departments to borrow equipment etc. The teacher will support students with the school logistics (permission forms, organising the class to be out on the day etc.).

This step supports students to stop and review their work and consider if the work is authentic, if they need to review or repeat some component of the work. Students should:

• Check if their shopping list adds up correctly. Does it appear reasonable and within budget?
• Check their public transport time going to the park, the route, stop and journey time. Students have a backup ready in case they miss it.
• Check their public transport time returning to school, the route, stop and journey time. Students have a backup ready in case they miss it.
• Print and review their draft email to the PE staff that makes a request to borrow the equipment.
• Check that everyone is included on the setup plan for being at the park and jobs list. See if any additional equipment is needed? (sunscreen, garbage bags, first aid-kit etc.).
• Create a scavenger hunt to play before lunch. Has it been tested?

Students present their plans to the class and explain how their day out at the park would look. Students consider how to arrange their presentations so they present information in an easy to read format that includes all details required.

Students provide feedback and vote on which team’s excursion they attend.

For assessment: Students submit their final presentations

Exemplar 2 – Module 2: Financial Numeracy

Focus area: number and change.

• place value and reading numbers up to 10 000
• whole numbers and monetary amounts up to $10 000
• common decimals and fractions and percentages such as ¼, ⅒, 50%, 0.25, 0.75 and other common decimals up to two decimal places, such as money and time
• addition and subtraction with borrowing and decomposition
• multiplication and division related to small whole-value numbers
• the order of the four arithmetical operations.
• identify place value and read whole numbers up to 10 000
• perform calculations of addition and subtraction with numbers up to 10 000
• recognise and use common decimals, fractions, and percentages such as ¾, 10%, 75% or 0.75, and other common decimals up to two decimal places
• find and use multiplication and division facts related to small whole-value number values only
• calculate simple problems using the order of the four arithmetical operations with whole-value numbers only.

On completion of this module students should have the knowledge to be able to understand:

• familiar and simple patterns or sequences in patterns and in a series of numbers
• familiar mathematical language and terms used in numerical pattern prediction
• changes and reconciliation in sets of numbers into the 1000s
• repeating patterns with two or more elements such as simple pricing structures.
• identify and describe relationships and patterns of change in sets of simple numerical values
• demonstrate repeating patterns with one element, e.g. $2, $4, $8, $16…
• reconcile and compare simple numbers in context, e.g. prices, warehouse stock levels
• predict simple pattern continuation using familiar mathematical terms
• demonstrate repeating arithmetical and spatial patterns of familiar and simple numbers with more than one element.

Financial Numeracy : The contexts explored in this unit include looking at our income statements and how we spend our money, and looking for patterns using coins and notes.

This is an eight-week learning program.

Students explore financial numeracy with the focus areas: number and change. For the focus area number, students explore place value using Australian coins and notes, work with money values through shopping activities, applying the four arithmetic operations and discounts using fractions and percentages. For the focus area change, students work with coins and notes to identify patterns with money, looking for simple patterns in real-life examples and costing the designs, and looking at inventory templates for predictions of stock control.

The contexts used in these activities include, but are not limited to, exploring the information found on income statements, checking the calculations, looking into the benefits of family’s meal planning over the week and creating their own planning to suit their family’s needs and preferences.

For the assessment task, students use these skills to explore the finances needed before moving out of home and into a rental property. Students would investigate the on-going costs needed to create a budget based on their apprentice wages found on the Fair Work Australia website.

This unit explores and demonstrates all three learning requirements as mandated by the curriculum. The four stages of the Problem-solving cycle are supported and demonstrated in multiple activities and the assessment task. The Mathematical toolkit is highlighted throughout the activities, as indicated by the third learning requirement.

• Catalogues from supermarkets
• Australian money coins and notes resource kit
• Graph paper

Access to the internet and computers / tablets is essential.

• Better Health website
• Meal Planning apps or websites to create shopping lists
• Meal planning apps or websites to help with ideas and recipes
• Supermarket apps or websites
• Splitting bill apps
• Garden centre websites
• Fair Work Australia website to find wages
• Real estate websites or apps to find rentals
• Utility websites or apps as calculators for estimates
• VixRoads website or app for yearly expenses
• Public transport websites or apps for yearly expenses
• Laptop calculator to replicate phone calculator

Teachers are encouraged to use extra materials and resources that support the learning that caters directly for their students’ needs. All timelines are given as a guideline only.

Unit 3: Module 2: Financial Numeracy Focus areas: Number and Change

Please note: These activities must not be taught in isolation from the Problem-solving cycle or the Mathematical toolkit.

Income statements

Students look at a variety of pay slips and identify the key features: employee’s name, company name and address, hourly rate, hours worked, tax paid, gross wage, net page. Each feature is discussed to understand how they differ. Encourage students to examine their own pay slip while using the one on the whiteboard as a guide.

Place pay slips in different stations around the room. Students work in pairs to examine these different pay slips and information. Remove total pays from the pay slips so students can calculate hours worked X hourly rate, and calculate the net amount when they subtract the tax from that pay (leave the amount of tax there). Students present in pairs their answers to the class with their calculations for a class discussion.

Shopping discounts

Students discuss how dinners are planned and made in their family. They discuss family meal planning and how to organise who cooks, who shops (e.g. shopping once a week vs every few days vs online and delivery). Use a PMI graphic organiser throughout the discussion and how family schedules can affect dinners. Explore the stereotypes of who cooks dinners and ask if any students like to cook and help with dinner. Create a concept map to put all ideas together at the end.

Explore the need to eat healthy meals and how we do this. Students go to BetterHealth and explore the four areas of interest and write a small summary about the ideas listed on the website and their views opinion about their advice. Students collate their information on a detailed mind map as they read through the information.

Topics listed on Better Health include:

• Australian Dietary guidelines
•  5 major food groups, what makes an ‘occasional food’
• how to order a healthier meal at restaurants and takeaway venues
• high sugar/salt foods and looking at labels
•  recommended daily intakes for women and men
•  alcohol in moderation, eating healthy on a budget
•  pantry staples, changing your mindset about eating.

Read through the Better Health information about Meal Planning (BetterHealth, 2022). Discuss the benefits and any negatives that the students can think of.

Students write a weekly dinner meal plan for their family, listing the meal and recipe. From this list, they form an ingredients list and consolidate any repeated items. Students explore meal planning apps that help store meal plans and make shopping lists.

Students open a supermarket shopping app or website and put their items into a shopping cart for pricing. Students print their list as evidence, then delete the items in the cart.

With the list, students calculate their total prices including any discounts, 5%, 10% or 15%. Students organise their shopping list into sections: meat, fruit/veg, dairy, bakery, shelf/pantry items. They estimate their items into fractions and percentages of each section.

Look at a supermarket layout or take students on an excursion. See how the fresh produce is on the outside, the shelf/pantry items make up the shelves and the specials/promotions are on the ends. Talk about why this might be. Look at and discuss the different promotions in catalogues or take photos on-site to discuss. Calculate the percentage saving or look at different sales techniques. How are items stocked? Do businesses buy different shelves at eye level? What is effective about a brand’s marketing techniques on packaging etc.? How do they grab attention for sales? How can you save money shopping on a budget?

Look at current catalogues and see if students can change any of their products on their lists for those on sale/promotion to make their final costs cheaper. See who is a savvy shopper and how much they can alter their budget. Use catalogues from all supermarkets. They are easy to find online or look up the sale items on the supermarket websites.

Splitting bills

What restaurants have students been to? What did they like about them? What didn’t they like? What restaurants would they like to try? The class list several different restaurants on the whiteboard and print the menus/display on laptops.

Place these menus around the room in different stations. Create an ordering slip for each menu for each table and place it with the menus. Students spread themselves out around the stations, look at the menus and write down their order for each restaurant (note: when students have finished writing their order place a line under it to show it is finished). If they really do not want to go to that restaurant, it’s OK not to write an order. Encourage the restaurant selection at the start of the activity to be inclusive of all students. Have enough restaurants for either one per student or one per pair.

Students work independently or in pairs to take one order and calculate the total bill on another piece of paper (keep the order paper separate). Students then create change from different options. The teacher can help with different ideas once they have seen the totals. Students calculate the amount of the order if the bill was split evenly amongst the number of people ordering.

Note: Have all money totals, change calculations, split money amounts shown by students in place value Australian notes and coins, such as $32.50 would be shown as 1 x $20 note, 1 x $10 note, 1 x $2 coin, 1 x 50c coin.

The class discuss the task:  what was easy, what was challenging, what tips can the students share to make it easier for others? Students repeat this task for another restaurant. They share their results with the other people/groups and check their calculations. Repeat again.

Students explore bill splitting bill apps  and how people use them to keep track of who owes what (used when people go out or travel) and how they can help pay others their share of the bill in a meal or house sharing situation.

Catalogue madness

Students find an electronics store catalogue to find items that they desire and make a list (only one item per category). Students start to make a running total where they add the items as they go and see who can get their total closest to $10,000 without going over. Students perform these calculations in spreadsheet software. Change the rules along the way by telling students they can get two of one item or their most expensive item went on sale with 10% off, etc.

Counting coins

Students use an Australian Coins Resource Kit to separate the coins into the different categories, checking they recognise the different values.

Students put the different coins into dollar values: how many 10c coins make $1, how many 20c coins make $1, how many are in $2, $5 bundles, etc.

Place sets of random coins around the classroom in stations, to replicate a cash register at the end of a shift that needs totalling. Create a sheet that with a breakdown of each type of coin, how many of each coin are there? What’s the total of each coin? Make a grand total of the register.

Students go around to each station, calculate the breakdown of each note type and calculate the grand total of each register. Students should use a calculator to check their workings out.

Discuss the breakdowns and totals at each station when students have finished.

Counting notes

Use Australian Money Notes Resource Kit to count combinations of:

How much do you have when you have one of each note? How many $5 notes to make $100? How many $10 notes to make $100? How many $20 notes to make $100? How many $50 notes to make $100? How many $100 notes to make $1,000? How many $50 notes to make $1,000?

Place sets of random notes around the classroom in stations, to replicate a cash register till at the end of a shift that needs totalling. Create a sheet that with a breakdown of each type of note, how many of each note are there? What’s the total of each note? Make a grand total of the register. Students to check their totals using a calculator .

Discuss the breakdowns and totals at each station with students when they have finished.

Reconciling cash registers – Counting coins and notes

If students are enjoying or needing more practice with Activities 5 and 6, re-create the task with each station now having a combination of coins and notes. Create the breakdown sheet to now have coins and notes so they can total each coin and note separately before finding the grand total of each register. If students find this task easy, put them under a time limit. Check answers with a calculator.

Patterns in the garden

Students consider where they see real-life examples of patterns or repeated shapes?

Cover some of the image so students guess what the image is showing and ask them to repeat the patterns shown. Ask their opinions, and see what combinations they would choose instead. Examples to look at include, but are not limited to, gardens, road crossings, table cloths and other printed fabrics etc.

Students create an outside garden tile area and cost it. Explore the cost of bigger tiles, smaller tiles, the shapes and varieties that exist. Look at the patterns they would need to lay. Use graph paper to assist if needed.

Look at examples of gardens, does the school have one? Is there a pattern there? Why do we like patterns? Is there an appealing factor to them?

Cost a design for the school’s front garden using a trip to the local nursery or using an online gardening centre. What design pattern would students use? Ask students to look into different designs and maintenance and write a proposal to support their plans.

Stock control

Present examples of stock inventory from the internet and alter it to show trade for a week.

Ask students to highlight the importance of dates of sales, stock quantity, when they re-order, how long it takes for the re-ordering stock to come in, how many items are re-ordered at a time, cost of re-ordering etc.

Present some inventory templates on different tables for students to analyse. Include some mistakes in them, include stock that needs ordering, include stock over-stocked and not moving off the shelves, include dis-continued stock that someone has ordered again etc. Students analyse what is selling well, what isn’t moving and consider what should be taken off their selling list.

This activity could also focus on showing the inventory of one business with the tables dated in order. Students make their way around following this order with another activity running in the room, so students are not waiting around and wasting time.

Overview: Students explore the basic costs of moving out and renting.

Present the idea to students. Use the detailed mind map graphic organiser to organise the class brainstorm about what it means to move out, what costs are associated with it, what responsibilities come with it etc.

To help students understand the task:

- Students create a list of wants and needs when moving out. Discuss this definition before students complete the task. - Students think about an area they would like to live in when they move out. - Students consider how much they will be earning. Settle on an age/time in their life where they can research the wage on Fair Work Australia. - Students consider the bills/utilities that they will need vs want. - Students consider the furniture and other items they will need. - Will the student live on their own or share with someone/a group. Students consider this and write a statement justifying their ideas.

Step 2 – Act on and use the mathematics

Students visit Fair Work Australia first  and find out their wage for this assignment. It could be their first/second/third/final year as an apprentice, or whatever you all agree on. Walk them through the website and find their award wage.

Students start investigating and collecting their information. Use real estate websites for rental properties. Discuss with students what a bond is, and how they need to come up with first month’s rent too. Ask students to create a fortnightly, monthly and yearly total for budgeting.

Explore utility websites.  Some of these websites often have calculators that give estimates. Students create a fortnightly, monthly and yearly total for budgeting.

Students keep a screenshot of all information found as evidence. Students put all totals into spreadsheeting software for calculations.

Students look into creating a simple list of furniture and items needed to move out and cost this.

Students factor in their transport option for the year: will they have a car, what are the costs of a car,  Will they use public transport? If so, estimate their needs. Ask students to create a fortnightly, monthly and yearly total for budgeting.

Students look at their fortnightly, monthly and yearly total. Does it exceed their salary? Will they need a side-hustle? Will they need to alter their budget?

Are the costs reasonable? Do students consider it good use of their money?

Students check their formulas in the spreadsheets. Are items in the correct columns and do they add up properly?

How do students best present this?

Would a poster, presentation or a budget printed out,  communicate this best?

Students produce a draft and ask for feedback again from the teacher/peer.

Exemplar 1 – Module 3: Health and Recreational Numeracy

Focus areas: shape and quantity & measures, focus area: shape.

• common two-dimensional shapes such as circles, triangles, quadrilaterals
• simple three-dimensional objects such as cube, cylinder, simple prisms
• common properties and language of two-dimensional shapes and three-dimensional objects (such as edges, faces, corners) and making connections between nets and three-dimensional objects; e.g. matching solids and nets.
• recognise and name common two-dimensional shapes and simple three-dimensional objects
• construct common two-dimensional shapes and simple three-dimensional objects
• categorise common two-dimensional shapes and simple three-dimensional objects and shapes according to different common classifications
• match common and familiar three-dimensional solids and their nets.

Focus Area: Quantity & measures

• common metric distance and length measurements and quantities
• simple perimeter and area measurements such as measuring area by squares
• simple conversions between common and familiar metric units or common measures such as one teaspoon is 5 ml, one cup is 250 ml
• common units of quantities, such as mass (g, Kg) and volume (ml, L) and temperature in degrees Celsius
• analogue and digital times, including 12-hour time in hours (AM and PM), minutes and seconds on digital clocks, and hours, quarters, and halves, 10 and 5 to/from on analogue clocks
• digital and analogue calendars.
• estimate, measure and compare distance and length, mass (g, Kg) and volume (ml, L) of familiar items and quantities
• estimate, measure and compare simple quantity and measures such as perimeter, area and temperatures in degrees Celsius
• make simple conversions between commonly used units, e.g. one cup is 250 ml
• read and interpret common and familiar dates and times using digital and analogue clocks and calendars.

Health and Recreational Numeracy : The contexts explored in this unit include: hands-on, visually-based activities and technology for planning and scheduling. This incorporates events, food, holidays and provides opportunities for students to practice scheduling with real-life contexts.

Students explore health and recreational numeracy with the focus areas: shape and quantity & measures. For the focus area shape, students work with, recognise and categorise two-dimensional shapes and simple three-dimensional objects. For the focus area quantity & measures, students work with familiar and everyday measurements and measuring tools found within the home, school, workplace and community.

The contexts used in these activities include, but are not limited to, the Yulunga Traditional Indigenous Games, taking a walk around the local neighbourhood, creating a three-dimensional village using nets, incorporating family favourite meals into a cook book, working with 24-hour time with games, and using calendars to create countdowns.

For the assessment task, students create a small trip away exploring Country Victoria, where teachers and students negotiate planning parameters beyond transport and accommodation (with other suggestions including a meals budget and activities budget).

Measuring equipment will be used throughout and may include but not be limited to: tape measures, long rulers, trundle wheels, measuring cups, measuring spoons, measuring jug, medicine cup, digital scales, clocks and yearly calendars.

Technologies include, but are not limited to:

• Yulunga Traditional Indigenous Games website
• Cooking / Recipe websites/apps
• Supermarket websites/apps
• Camera for taking photos
• VicRoads website
• Date/Time/Calendar found on a laptop/computer
• Laptop calculator to simulate phone calculator
• Transport websites/apps (such as flights and trains)
• Accommodation booking websites/apps

Unit 4: Module 3: Health and Recreational Numeracy Focus areas: Shape and Quantity & measures

Please note: These activities must not be taught in isolation from the Problem-solving cycle or the Mathematical toolkit.

Yulunga Traditional Indigenous Games ( https://www.sportaus.gov.au/yulunga )

Students look through the many Yulunga Traditional games. In groups of two to three students choose a game and print out the information about the game. Students read the instructions, gather the equipment, set up the course including the perimeter of the playing field required and look for videos online to help understand the game before instructing the class how to play the game.

After the games have been played, students examine each playing field. They estimate, then measure, each playing field and sketch the dimensions as a record. They return to the classroom to calculate the area of each playing field.

Students then estimate and measure other playing fields in the school, such as tennis courts, hockey pitches, basketball courts etc., and write up a comparison between the fields of the Yulunga Traditional Indigenous Games courts to those in other sports.

Using a map of Australia, students calculate the distance from their capital city or suburb to where that traditional game came from.

Street signs

Students go on a walk and take photos of all signs used in the community for drivers, such as stop signs, roundabouts, speed limits, etc. Examine the message being given and the shape being used. Look at VicRoads for their definitions.

Using the VicRoads website, complete an ‘L’ or ‘P’ test with the class which includes a practical component involving the street signs.

Food to a village

Bring in different examples of food packaging and try to get different examples to cover the simple three-dimensional objects of cubes, cylinder and simple prisms.

Students identify the different names of the objects being used for packaging. Students look at the different shapes used within the packaging and examine how the packaging works (strong base, easy to re-fold top to secure after use, falls apart after opening etc.). Unfold the packaging into its net and get students to draw the nets onto graph paper. Students examine the shapes and area being used and annotate the shape being used.

Students fill out a customised table about each piece of packaging, e.g. how many different shapes are used, how many edges, faces, corners etc. to explore the object in detail.

For practice, present some printable net diagram worksheets for students to build the cube, cylinder and prisms.

Students think about constructing a small village, each student contributing two different net diagrams. Students work together to brainstorm the essentials for a village lifestyle. Each student signs up to create at least two different three-dimensional objects to contribute to the village. Keep the printable net diagrams and food packaging as examples for the students to refer to when designing their own net diagrams. Before securing their pieces together, they must colour or decorate it.

Explore simple measuring tools in the kitchen, such as measuring spoons and cups. Take students into the kitchens or bring in a tub with water/rice, other measuring devices, such as medicine cups, measuring jug and scales. Students estimate how much each measure might hold. Record responses or enter them online and get a calculation of the average. Students carefully measure and create a table that can be put up in the kitchens and taken home.

Students think about a recipe that their family loves to make and bring it in to make a class book. Each student writes out something about their family recipe;where it comes from, who handed it down, or just why they enjoy it etc. Write out the recipe.

Students include:

• Ingredients list.
• Cooking instructions.
• Cooking time.

Students can turn this into a cooking day where they all cook their recipe and sample each other’s meal before producing their cookbook. Students need to produce the ingredients list, work within the budget and time set (so perhaps review their recipe) and cook it on the day. Students can write reviews that are entered into the cookbook.

24-hour time

Present examples of 24-hour time to students, such as airport departure boards and work rosters. Ask students for more examples where they have seen it used and discuss the strengths of using that as a timing system.

Bring in a clock where you can label the 24-hour times next to the 12-hour times. Complete some worksheets to show how 12-hour to 24-hour conversion works. Relate the 24-hour conversion to night time. Develop a timetable of daily activities using 24-hour scheduling .

Play some games with the students as they get comfortable with the 12-hour to 24-hour time conversions:

• 12-hour to 24-hour dominoes:  students play the traditional dominoes game using 12-hour to 24-hour cut-out pieces instead.
• 12-hour and 24-hour memory game: complete this traditional memory game where students need to match the 12-hour time with the 24-hour time.

Students look ahead to Schoolies and how they might celebrate it. Have them plan the ultimate trip that includes flights and accommodation.

Students estimate how many days they are at school for the school year and how many weeks that would be. They use their student organisers/planners to calculate the correct response, remembering to not count the school holidays.

Students estimate how many days and weeks before another event, such as the next school holidays or end of the year, and use their organisers to work it out.

Show students the calendar function on their laptops and how they can count weeks using this function.

Students estimate how old they are in days, weeks, months.  Students find an online calendar to calculate it for them. If you feel brave, have them guess your age and then find the difference!

Students list events they are looking forward to their next big birthday (18th or 21st) starting their apprenticeship, a holiday etc. Make a countdown via days and weeks.

Visit Country Victoria

Assessment Task : Students are planning a small interstate holiday on a budget.

Brainstorm how you plan for an interstate holiday.

Students consider transport, food budgets and what activities they would schedule for their few days away. Consider what budget you would set with the class and the other parameters. Students also explore the time of year they are travelling and the average monthly weather conditions. Students provide a suitable packing list.

Students start to investigate their holiday away and collect evidence of their budget.

Students consider their budget and check they have kept within their budget. Students check their calculations and ask themselves ‘does this seem right?’, by using a calculator or spreadsheet.

Students look at different itineraries to give them ideas on how to compile their piece for submission. Create a checklist to support students to ensure they have completed all aspects as agreed upon at the start, with the scheduling and budgeting requirements.

Exemplar 2 – Module 4: Civic Numeracy

Focus area: data and likelihood.

• simple data collection methods including use of tables, spreadsheets and tallies
• display of data with commonly used tables and graphs with scale of 1’s, 5’s or 10’s including familiar and simple cases of data, graphs and infographics.
• collect, collate, sort and order data sets, e.g. use survey to collect data, use tallies to collate data and insert set of data into a table/spreadsheet, sort from lowest to highest
• construct simple charts or graphs using familiar data with simple scales, e.g. in 1’s, 5’s or 10’s
• read, identify and interpret familiar information and facts from simple tables, graphs and infographics
• make simple comparisons and interpretations between provided simple data sets and their representations.
• likelihood of familiar events or occurrences happening using everyday language of chance
• common likelihoods and chance events such as weather predictions, dice or spinner success rates
• language and relative magnitude of the risk of common or familiar events of chance.
• order and compare simple familiar likelihood events and statements such as evens, for sure, Buckley’s chance, impossible
• read, interpret and make decisions about likelihood statements based on their chance of occurrence or success/failure
• order and compare the relative magnitude of the risk of common and familiar events of chance
• use the language of likelihood such as chance, possibility, highly likely, certain, risk, success/failure, predict.

Civic Numeracy : The focus of the contexts explored in this unit relate to the weather, the occurrence of familiar and slightly unfamiliar events, data collection in the classroom, and exploring data found in the media.

Students explore civic numeracy with the focus areas: data and likelihood. For the focus area data, students explore how to collect data, analyse it and present it, as well as viewing data found in the media and everyday life. For the focus area likelihood, students explore and develop different language to use when considering or interpreting situations and their chance of occurring. They also develop the ability to make decisions based on whether a familiar event would occur or not.

The contexts used in these activities include, but are not limited to, looking at the weather and the language used when forecasting, understanding common and familiar events occurring and how we describe them and exploring more unfamiliar events such as one in one hundred-year floods and the chance of winning TattsLotto. Contexts also include considering how the media can misinterpret data and how we need to develop analytical eyes, exploring two data sets that have a commonality, and collecting data in the classroom and exploring how to use spreadsheeting programs to analyse and display it.

For the assessment task students consider an issue that is important to them, complete a survey, analyse data and produce an infographic to be displayed in the school.

• Spreadsheet software, such as Microsoft Excel, Google Sheets etc.
• Spinner app
• Weather apps or websites

Unit 4: Module 4: Civic Numeracy Focus areas: Data and Likelihood

Testing students weather memories.

Ask students what they predict the monthly average temperature to be each month in Melbourne and ask them for the conditions; the chance of sun and chance of rain. Students can record ‘the chance’ in any way they wish, e.g. percentage, selecting words they associate with chance etc.

Reveal the monthly average temperatures and as a class discuss how students’ predictions went. Discuss how season temperatures and weather patterns are changing and are less predictable (e.g. summer temperatures now appear in March, which is traditionally autumn).

Discuss the language students can use when describing the chance of events occurring:. possible, impossible, highly likely, not likely, will, won’t, certain, Buckley’s chance, one in a million, risky, half-a-chance etc. Ask students to examine the language they used for chance of sun and chance of rain and see if they were happy with their choices. Contrast their language choices and predictions with the weather now revealed on the whiteboard.

Students research the monthly average temperatures in another part of the world and create a table with this data compared to data for Melbourne. Students compare the two sets of weather data. Write a small paragraph to describe these to include in a travel guide promoting the best times for travel and why, and the times to avoid and why. 

Create a spinner with pre-programmed options using a pickerwheel.

In this activity, students express their view using the language associated with the risk of events occurring.

Spinner options include, but are not limited to:

• All students will be in correct uniform today.
• The grass will be green.
• The canteen line will be quicker at the half-way bell.
• There will be snow in Victoria this year.
• (Enter your AFL team here) will win the AFL premiership season this year.
• A female will be the Prime Minister of Australia within the next ten years.
• Victorians will feel another Earthquake this year.

Students create their own spinners and participate by displaying them on the whiteboard and having the class respond.

Exploring events

Explore the events in life where we hear event language being used, such as:

• The chance it takes to win TattsLotto.
• One in one hundred-year floods.
• Volcano eruptions.
• Earthquake eruptions.
• Comparing election forecasts to election results.
• Comparing weather forecasts to the actual weather.

Students explore several of these events to see how often they occur, what monitoring occurs, the implications of it happening, what warning systems are in place etc.

Students connect their research to the language they have developed with risk (see Activity 1), how common occurrences are, causes and effects and current risk status. Students present a view of how the risk might affect them, if at all.

See it, analyse It

Students are given a series of graphs and tables of data and assess what the data is communicating. List key components on the board for students to reference: heading, labels on axes, legend, etc. Analyse a few of the graphs and tables together so the students develop confidence. Then allow students to complete analyses either individually or in pairs.

When the students are confident in their analyses, present some graphs that show media mistakes and examine them together to show that they have to be careful where they get their data from. Show examples such as percentages that do not add up to 100%, poor use of scales on the axes, etc.

Students find their own graph and interpret it. Students can send the graph to a common place on the network or email it to the teacher with two questions to build a group kahoot.

See it, check it

Present students with two data sets that have slight variations, such as the year the data was taken, different age groups or genders, different states or continents, different months etc. Explore different issues such as, but not limited to: opinion polls, climate change data, youth issues data, wage growth/mortgages/cost of living over the years, data from school opinion surveys etc.

Present the data in both table and graph form.

Students look at the data and try to work out how it differs. How are the two data sets similar? How are the two data sets different? Work with the students and help them to develop criteria for writing a small summary.

Students collect and collate data that is the same but with a difference, perhaps the same questions but staff vs students’ opinions, or students’ vs parents’ opinions etc. Explore the differences in data and opinions.

Stop – collect and list It

Students practise collecting data and collating it through small activities in the classroom. Students work in pairs to undertake activities (see below) or create their own activities to gather the data. Work together with students to plan how to gather the data for each activity. Set up different stations and have a session where all data is collected at the same time. Possible activities could include, but are not limited to:

• Student height.
• Favourite number.
• How long it takes for each student to sing the alphabet.
• How many basketball hoops they can score in 30 secs etc.

Students:

• Enter the data into a spreadsheet.
• Sort the data into ascending order.
• Use spreadsheeting conventions to display the data, such as bold headings, borders, etc.

Practise using spreadsheet functions, such as different charts and graphs, to display the results and write small summaries.

Let me tell you …

Students conduct a survey about an issue that is important to them and produce an infographic with the results.

Students are introduced to the idea of making a survey and gathering data. Discuss what surveys they have completed before: what made them easy to fill out, what made them difficult and present samples to direct students.

Students brainstorm issues of important to them, such as youth health, animal welfare, nutrition, sport, etc. Students create five questions, with each question having different answering options (not yes/no questions, but using a scale between 1 and 5 and then explaining the scale, giving tick boxes with options etc.). Be wary of open questions because compiling data can be tricky.

Give students time to work on their surveys and present a draft to you for proofreading.

Students should aim to have at least 10 – 20 people fill out their survey to gain enough data to calculate. Students could complete the survey in class or set up stations in high traffic areas of the school, such as near the canteen, library or sporting fields. Remind students of the ethics and etiquette of surveying people. Remind students that rejection will happen and should be accepted with a ‘thank-you’.

When they have collected their data, discuss with students their experiences and how to collate their data into manageable figures. Use a spreadsheet to analyse the data and produce some graphs to visually explore and summarise the data.

Show students a variety of different infographics or display them around the room. Students place post-it notes on them with ‘what makes this infographic interesting’. Do this at the beginning of the class , so students have some design features in mind to apply to their data.

Find an infographic website to help students produce their infographic and then print a draft.

Students view their draft infographic and make their own annotations for review. They swap it with another student who will give feedback. Students review the work considering:

• Do the results seem reasonable?
• Do they seem realistic?
• Do they seem correct?
• Am I communicating my data strongly?

Teacher has a one-on-one interview with each student where students show their infographic after applying their own and peer feedback and talk through their infographic. By allowing the students to talk through their infographic, it will help them check if they have communicated their ideas correctly onto paper.

Complete all final changes and submit final infographic. Print on A3 for display in school.

Note: Remember, with the Problem-solving cycle, students can move between the four stages and go back and forth regularly. If they find a problem with a calculation during the evaluation stage, they return to the first stage of ‘identify the mathematics’ or second stage of ‘act on and use the mathematics’ to see what they can improve on. This is why the Problem-solving cycle has been created in a circle format showing it can keep moving until the job is done.

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Integrating rich tasks - Activity 1.4

To go back to the introduction to this series of professional development activities, click here

How do higher-order thinking skills (HOTS) relate to rich tasks and problem solving?

This task aims to identify how rich tasks and problem solving fit together.

You will need the following resources:

(The Problem-Solving Cycle.doc is a larger version of the above which might be easier to read.)

How do rich tasks, the problem-solving cycle and higher-order thinking skills fit together?

• Cut out the problem-solving cycle cards ( ProblemSolvingCycleCards.doc ) and lay them out.
• Link them with the rich task description cards ( RichTaskCards.doc ) and with the different aspects of Bloom's taxonomy ( Bloom-descriptors.doc ).

Return to the introduction

Go back to activity 1.3, move on to activity 1.5.

A cyclic sum is a summation that cycles through all the values of a function and takes their sum, so to speak.

Rigorous definition

Note that not all permutations of the variables are used; they are just cycled through.

• Symmetric sum
• PaperMath’s sum

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• Ks2 Maths Polya Problem Solving

KS2 maths – Polya’s problem-solving

From groans and panic to calm and curiosity – how Claire Coolin made systematic maths lessons less scary…

Are you met with groans when you utter the term ‘problem-solving’ to your class?

We all agree that it’s is an important aspect of the curriculum and indeed a life skill, but yet remains a divisive topic of conversation with pupils, parents and colleagues.

So, what is the problem with problem-solving? 

My Year 5 class were high-achieving. In general, they enjoyed maths but when it came to problem-solving many of them froze.

They showed disinterest, nervousness and confusion, saying things like, “The information is hard to process,” and “I couldn’t understand the information as it was just a block of words”.

Interestingly, pupils found the lessons boring when the problems that they solved followed the same routine and pattern, and didn’t provide a level of challenge, interest or excitement.

These routine problems ‘took the fun’ out of maths for them, and while using skills such as addition, subtraction , and so on are all part and parcel of problem-solving, is this what we are trying to achieve here? 

Routine and non-routine problems

Problem-solving can show an in-depth understanding of mathematical concepts where pupils are required to manipulate numbers in order to get to an answer.

However, while routine problems might be easier for the children to solve, they are not aligned to maths in real life .

We don’t live our adult lives practising budgeting for the weekly supermarket shop from Monday to Thursday, with the real deal happening on Friday. Rather, real-life problems can often be spontaneous, unexpected and at first have no obvious solution.

Should we therefore be teaching pupils what to do with the unexpected? Should we be equipping them with the skills to solve non-routine problems? I think the answer is, yes, most definitely. 

Polya’s problem-solving

So what can be done? Is it enough to tell the children in your class to highlight the important words in the problem ?

While this might have helped a few of my pupils to see a way forward, for many this just became a colouring exercise with an impressive array of fancy highlighters on show.

And so I began an action research project to find answers. After poring over many academic articles, I came across Polya’s problem-solving cycle : a cyclical four-step process that could be used to solve any problem, in maths or otherwise. The steps are:  

• Understanding the problem 
• Devising a plan 
• Carrying out the plan 
• Looking back and reflecting 

Polya’s process was something I wanted to introduce to my class, and so I taught this way of thinking about a problem over a term. The results were surprising and long-lasting, with four key takeaway points. 

I found that the cycle: 

Developed time management skills and enhanced focus 

Polya’s cycle automatically forced students to work through each phase in turn, slowing them down and therefore helping them explore the problem with a more thoughtful and connected approach.

Instead of jumping straight into the doing, the children took time to think about their understanding of what was being asked, and would often get out a dictionary to look up a word they didn’t know the meaning of, or on one occasion, a child even dusted off the ‘never-really-used-before-but-every-class-has-one’ maths dictionary to look up the meaning of a mathematical term.  

Equipped all students with a strategy for tackling any maths problem 

When talking to a group of pupils in my class about problem-solving, I recall them saying, “I can’t work out if I should multiply or add or subtract or whatever.”

This is true for many students and can lead to them freezing, or in some cases, frantically adding and subtracting numbers – essentially jumping straight into phase three.

They needed a strategy. Using the problem-solving steps gave my class a framework, allowing them to think in a logical way.

Straight away, students unfroze, the manic scribbling stopped, and they started from a phase one and worked through the problem more systematically.  

Allowed pupils to see the link between maths and the real world  

When I asked my class why they thought we were learning about problem-solving, I was met with answers about becoming better at maths.

But after using Polya’s steps, the children began to see how the skills in solving maths problems are the same ones used in solving real-world problems.

For example, they realised that if a maths problem can be divided into chunks, then so could any problem.

One sunny afternoon, a couple of students came to me to help solve their playground argument. You can imagine my utter surprise when one of the students said, “I think first we need to understand why we have fallen out then we can come up with a plan to put it right.” Real life problem-solving in action! 

Equipped students with the necessary tools to engage with challenge 

Having challenge in any subject is important, but having the right level of challenge is even more so.

Routine problems can remove that challenge for many, and one of my pupils commented that, “You shouldn’t do lots of the same as it gets too easy. You don’t want to spoil it by knowing all the answers.”

Using Polya’s problem-solving steps with non-routine problems gives the more able pupils enhanced scope and freedom to try out and manipulate numbers in different ways, while giving that scaffolding to support the less able in the class. 

So, what now?

What have I learned from this experience? Well, with more than 10 years teaching under my belt, this action research brought me back to my roots.

I started from scratch with something and didn’t rely on my tried-and-tested resources. I had a chance to listen to my pupils and in doing so stumbled onto a way of thinking that works not just for a maths problem, but for all problems.

The steps have since become a classroom philosophy, and I use the vocabulary of the process daily. Trying something new can pay off!  

How to introduce Polya’s process 

• Talk about the cycle and what each phase means. Take note that it’s like a roundabout, which means you can get off at any stop and go back if necessary.  E.g. If you’re carrying out your plan (phase three) but it’s not working, you can go back to phase one: understanding the problem.  
• Use subheadings. I wanted the students to really spend time thinking about each phase, so I made a very simple Performa with each phase sub-headed. The students were required to write within each subheading. 
• Model how to do it. Using different non-routine problems, I modelled how I would use the problem-solving phases. I then built up to working on problems as a class, and finally asked pupils to work in pairs to solve problems using the steps.  
• Make it part of your classroom vocabulary. To embed the process in my class, the language didn’t just come out when doing maths problems, but was used in other subjects and day-to-day school life.   

Claire Coolin conducted this research project while teaching Year 5 at Oxford High Prep School, GDST with the Global Action Research Collaborative, ICGS. From September 2022 she will be a maths specialist teacher and PSHE head of department at Summer Fields School, Oxford.

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Problem solving is an important skill which students should be encouraged to develop over a period of time. With careful planning it is possible to move from a classroom environment where individual lessons are devoted to problem-solving to one where the majority of the teaching and learning is done through problem solving. In this way students experience problem-solving on a daily basis, get into the habit of thinking about their maths and to applying knowledge they already have to tackle something new. This also helps them to see how different topics in maths are connected.

The documents below focus on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterised by the teacher ‘helping’ students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying.

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Mathematics proficiencies

Problem-solving, foundation to year 10, portfolio summary.

In F–2, students solve problems when they use mathematics to represent unfamiliar or meaningful situations.

In Years 3–6, students solve problems when they use mathematics to represent unfamiliar or meaningful situations and plan their approaches.

In Years 7–8, students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

In Years 9–10, students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable. Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively.

Work samples

Number and algebra: My thinkboard

Measurement and geometry: The combined area of spaces

Number and algebra: Fractions and decimals and percentages

Measurement and geometry: Dimensions of a 3D object

Measurement and geometry: Equal areas

Number and algebra: Archers in the plane

Statistics and probability: Baffling box plots

View the proficiencies across f–10, foundation – year 2.

Foundation  includes using materials to model authentic problems, sorting objects, using familiar counting sequences to solve unfamiliar problems and discussing the reasonableness of the answer.

Year 1  includes using materials to model authentic problems, giving and receiving directions to unfamiliar places, using familiar counting sequences to solve unfamiliar problems and discussing the reasonableness of the answer.

Year 2  includes formulating problems from authentic situations, making models and using number sentences that represent problem situations, and matching transformations with their original shape.

Year 3 – Year 6

Year 3   includes formulating and modelling authentic situations involving planning methods of data collection and representation, making models of three-dimensional objects and using number properties to continue number patterns.

Year 4   includes formulating, modelling and recording authentic situations involving operations, comparing large numbers with each other, comparing time durations and using properties of numbers to continue patterns.

Year 5   includes formulating and solving authentic problems using whole numbers and measurements and creating financial plans.

Year 6   includes formulating and solving authentic problems using fractions, decimals, percentages and measurements; interpreting secondary data displays; and finding the size of unknown angles.

Year 7 – Year 8

Year 7   includes formulating and solving authentic problems using numbers and measurements, working with transformations and identifying symmetry, calculating angles and interpreting sets of data collected through chance experiments.

Year 8   includes formulating and modelling practical situations involving ratios, profit and loss, areas and perimeters of common shapes and using two-way tables and Venn diagrams to calculate probabilities.

Year 9 – Year 10

Year 9   includes formulating and modelling practical situations involving surface areas and volumes of right prisms, applying ratio and scale factors to similar figures, solving problems involving right-angle trigonometry and collecting data from secondary sources to investigate an issue.

Year 10   includes calculating the surface area and volume of a diverse range of prisms to solve practical problems, finding unknown lengths and angles using applications of trigonometry, using algebraic and graphical techniques to find solutions to simultaneous equations and inequalities and investigating independence of events.

Related portfolios

Understanding