ThirtyOneTask 86 ... Years 4  8SummaryWhat begins as a game for two becomes a search for a strategy to win. There are 'secret numbers' that control the game and once discovered it becomes clear that they are a pattern of numbers. Why? How do these secret numbers relate to the rules of the game? What happens if we change the rules? Are there still secret numbers? Can we find a way to force a win in all such games? Is there a counterstrategy?This cameo has a From The Classroom section which includes three contributions  two primary and one secondary  that illustrate how tasks have broad application across age and ability ranges. . 
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Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
The card basically provides the rules of the game and leaves the rest to the students. If they choose to play without reflection then you might ask:

Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 
You need one card pack per pair to investigate this task as a class. Perhaps the easiest way to get these is to ask students to bring them. Begin the lesson by gathering around a central table and asking a pair of students to follow your instructions to start the game. From their, pairs move off to explore further, thoughts are shared via the whiteboard and class discussion and as students think they have 'got it' you can test their hypotheses by organising challenges, including challenges against yourself. The investigation proceeds along the lines above. For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 27, Game of 31, which also includes an Investigation Guide and companion software. 
Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The ThirtyOne task is an integral part of:
The Game of 31 lesson is an integral part of:

Ashburton Primary SchoolKate GeddusYear 4 
Two girls were working on the task ThirtyOne. As the period drew to a close, they hadn't had much success yet with finding the 'secret numbers' in the game. So, rather than tell them too much, or have them write trivia like We played a card game and it was fun., when it came to recording time Kate asked How many ways can you make the total 31 when the cards are laid out like that?. As Georgia's work shows, the girls responded well to that question.
Responses like this also open the door to multiplication. For example, consider the first equation:

Regency Park Primary SchoolSimon BlakeYear 5 
Simon writes about using the whole class investigation life of this task (see Three Lives of a Task) through its companion Maths300 lesson titled Game of 31. The lesson ran for two hours  then had to stop because it was home time.
I approached the task with a mindset that I would have to lead the investigation to obtain the outcomes I desired. After about 5 minutes I realised that the students were totally engaged in their learning and actively seeking solutions independently. 
Rosebud Secondary CollegeGreg LeeYear 7 
Greg Lee reports that this investigation could 'do your head in'. He has been trying to explore the variations on the task in an organised way and has offered his work to us all in the hope it will be useful in other classes.
My computer (formatting) skills aren't great but I came up with this table (I like tables) as a sort of summary of the game. The patterns going across are pretty straight forward but going down on the Starter card for a given Game of... seem a bit all over the place. Not sure if its of any value to anyone but I was after some sort of summary for the kids. 