# Thirty-One

### Task 86 ... Years 4 - 8

#### Summary

What 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 counter-strategy?

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.

#### Materials

• Cards 1 - 6 in each of the four suits

#### Content

• basic arithmetic practice as mental calculation
• recognition and interpretation of number patterns
• generalisation and symbolic representation of number patterns
• problem solving

#### Iceberg

A 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:

• Do you only know when someone wins if they get to 31? Can you sometimes tell before that?
• Is there a number you can be on before 31 and know that you will win?
You may need to discuss with the students that it is only sensible to analyse a game if you assume the opponent plays the best possible game. What's the point of winning against someone who is weak? Given that consideration, the 'secret number' is 24. If you are on 24, whichever card your opponent chooses will be less than the 7 needed to make 31 and you will be able to select the winning card.
• But if 24 is the number you need to be on, is there a number before 24 that puts you in a certain position to reach 24?
Using this working backwards strategy, leads to the special numbers 3, 10, 17, 24, 31. So to be sure of winning you go first and choose 3.
• How does the pattern of secret numbers relate to the cards used? Why?
Having found a winning strategy, consider asking if there is a counter-strategy. That is, even if Player A uses the 'secret number' strategy can Player B select cards in response so that when Player A comes to make the last move to 31, the card they need is not available?
• What happens if the winning number is 28 instead of 31?
• What happens if we change the range of cards used?
• What happens if we change the winning total?
• For a given range of cards, what is the highest possible (lowest possible) winning total?
• If I give you any range of cards 1 to N and any (allowable) winning total can you tell me how to play to win?
• Can you tell me which combinations of range and winning total will allow a counter-strategy?
Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

#### Whole Class Investigation

Tasks 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 there, 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 (as in and invitation to 'Beat The Teacher'). 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 hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Thirty-One task is an integral part of:

• MWA Pattern & Algebra Years 5 & 6

The Game of 31 lesson is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Pattern & Algebra Years 7 & 8
This task is also included in the Task Centre Kit for Aboriginal Students and the Primary Library Kit. Solutions for tasks in the latter kit can be found here.

## From The Classroom

#### Ashburton Primary School

Kate Geddus
Year 4
Two girls were working on the task Thirty-One. 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:

• How many times did you write 6?
• How many times did you write 2?
• Do you know the calculator symbol for times?
• Can you write your equation another way?
• Imagine you chose exactly the same cards, but in a different order. Write an equation that shows the different order.
• Can you write this new equation another way?
You can also ask children to check their equations with a calculator. When only additions are involved, all is well with this technology. But the rewritten equations involve mixed operations so, unless you have calculators programmed with the order of operations, new skills have to be learned to account for multiplication taking preference over addition.

#### Regency Park Primary School

Simon Blake
Year 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.

They were able to explore the problem at various levels, with some students applying their addition, data collection and graphing skills whilst other students were investigating algebraic equations. This is the beauty of this task, it has something for everyone.

Prior to the session, my students believed that algebra was something that you undertook in High School, not Grade 5. Another misconception was that A = 1, B = 2 and C = 3, and that those values were set in concrete. At the end of the session, students were able to apply their equations to other situations and have a real understanding of algebra.

It was fantastic to observe the students develop their own ideas and work mathematically throughout the session. I found that the journal logbook was beneficial as it further consolidated my observations of the students.

Jess's Journal - a sample from Year 5, Regency Park Primary School

#### Rosebud Secondary College

Greg Lee
Year 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.