Squound

Task 139 ... Years 2 - 12

Summary

The Squound board has two intersecting shapes - a square and a circle (round). The intersection is called the Squound. The initial challenge involves finding the number of counters in the Squound if you know the number of counters altogether and the number in each of the square and circle. This leads to a generalisation and the generalisation leads to more questions which lead, surprisingly, to the Triangle Numbers and combination theory.

This cameo has a From The Classroom section with three contributions. One shows teachers' interest in the problem in a workshop in Sweden. Another describes how one teacher adapted the task to introduce Venn Diagrams, and more, to a Year 10 class being prepared for academic mathematics in Year 11. A third confirms that this is a fabulously engaging challenge for Year 8 learning about the inclusive and exclusive use of 'or' when working with Venn Diagrams ... and also that Professional Development workshops can be a great learning experience for teachers.

On our Cube Tube page you will find a video of two teachers who are captivated by the problem and in the middle of their investigation into it.

 

Materials

Content

  • sorting and classifying
  • basic operations - addition & subtraction
  • counting every possible case
  • seeking and seeing patterns
  • triangle numbers
  • partioning a collection into two parts in all possible ways
  • generalisation
  • Venn diagrams
Squound

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

One of the nice things about this task is that although it has a considerable number component, it is not necessary to 'know your numbers' in order to do it. At the least threatening level it is only about shifting counters into spaces and looking for a given outcome. Just finding an answer to one of the questions is a success, but the task encourages more by providing a recording sheet. This encourages students to keep data about the 'interesting problem' just like a mathematician would. Then the data encourages looking for a pattern or a connection and before we know it, there is more.

The answers to Questions 1 - 4 on the card are, respectively, 5, 4, 3 and 6. The fifth position on the recording sheet encourages students to invent their own Squound problem using a total of 12 counters.

One key to being able to generalise the problem is to realise that if the sum of the number in the square and the number in the round is larger than the number of counters being used, then something is being counted twice. That something must be what is in the Squound. So, one way to work out a Squound problem is:

Add the number in the square and the number in the circle. Then take away the number of counters. That will be the number in the Squound.
Having written an explanation, most students will be able to readily write it symbolically as: Q = S + C - T, because this 'algebra' will make sense. (These are the symbols used on the Recording sheet.)
  • Applying this rule to the Challenge gives: Q = 67 + 59 - 100 = 26.
  • One other way to check this would be to use 100 counters.

Extensions

The challenge implies that Squound problems could be created for totals of counters other than 12. The answers for 12 suggest that there can be more than one Squound problem for a given total. So:
  • Suppose we only have 3 counters. How many Squound problems can be created?
One way to find that answer is to break the problem into parts and find all the ways that the three counters can be placed in the three sections. The sections are shown by the colours here and are called:

A = square without circle, Q = Squound and B = circle without square

Squound Board

Three or two or one or zero counters could be placed in the Squound. In each case the remaining counters have to be placed in two spaces (A & B). Then, as soon as a choice is made for A, the remainder must go in B. So the problem of counting the number of Squound questions has two parts. Deciding the number to place in the Squound, then deciding the number to go in A. Making these choices in an organised way gives:

A Q B
0 3 0
1 2 0
0 2 1
2 1 0
1 1 1
0 1 2
3 0 0
2 0 1
1 0 2
0 0 3

Ten arrangements in all and they are determined first by the choice for Q, then by the choice for A. The B number is always a consequence of these choices. Also making these choices ensures that every possibility is counted.

Mmm, there seems to be a pattern in the table now:

  • 1 way with three in the Squound
  • 2 ways with two in the Squound
  • 3 ways with one in the Squound
  • 4 ways with zero in the Squound.
The corresponding Squound problems would be:

S Q C
3 ? 3
3 ? 2
2 ? 3
3 ? 1
1 ? 3
2 ? 2
3 ? 0
0 ? 3
2 ? 1
1 ? 2
  • Suppose we ask for the number of Squound problems for 2 counters, 1 counter or even 0 counters?
    • How many solutions are there?
    • How do we know when we have found them all?
Similar reasoning reveals there are 6, 3, & 1 respectively. (If there are no counters, there is only one way to arrange them on the board and that is with zero in each section!) So now the total of Squound problems for 3, 2, 1, 0 counters respectively is 10, 6, 3, 1. There is a hint of Triangle Numbers here. Could it be that:
  • 4 counters produces 15 arrangements?
  • 5 counters produces 21 arrangements?
  • ...
  • and if so, why???
The organised table above can be used to explore the problem for any number, n, and consequently address the why question.

A Q B
0 n 0
1 n - 1 0
0 n - 1 1
2 n - 2 0
1 n - 2 1
0 n - 2 2

... Continuing until 0 are placed in the Squound and n are partioned between A and B ...

n 0 0
n - 1 0 1
n - 2 0 2
n - 3 0 3
... 0 ...
3 0 n - 3
2 0 n - 2
1 0 n - 1
0 0 n

The table clearly shows that there are (n + 1) possiblities if 0 is in the Squound and that the total of possiblities is:

1 + 2 + 3 + 4 + ... + (n +1)
a series of numbers that grows like this until the last tower is (n + 1) high. See Task 51, Staircase, for information about summing this series.

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.
   

This investigation is easy to run as a whole class lesson. All you need is a dozen counters for each pair and your own version of the Squound board. If you have Poly Plug, the red plugs work very well as counters. The development of the lesson can follow the sequence of challenges above.

At this stage, Squound does not have a matching lesson on Maths300.

Visit Squound on Poly Plug & Tasks.

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.
   

Squound is not in any MWA kit. However it can be used to enrich the Number & Computation kit at Years 3/4 and the Pattern & Algebra kit at Years 7/8.

Green Line

From The Classroom

Methodist Ladies' College
Perth

Katherine Stulpner
Year 8
From the email:
Dear Doug
Once again, thank you for an excellent workshop on Tuesday. My colleague and I both used the Squound task in our classrooms this week and it worked really well. The students were fully engaged in the process and it was brilliant to witness every single one of them actively participating. We are studying probability at the moment in Year 8 and we are working with Venn Diagrams. One of the curriculum outcomes is making the distinction between the inclusive and exclusive 'or' and this activity tied into this concept perfectly.

I will be using the activity again and will encourage some of the students to continue with the challenge questions (total number of combinations) during another lesson.

Kind Regards
Kate

Workshop
Ludvika
Sweden

Squound was used as a whole class investigation in this Working Like A Mathematician workshop morning for teachers from Years 6 to 9. The challenge was so involving that when invited to take morning coffee, no one did. In Sweden this is unheard of - 'fika' with coffee and a sandwich is a tradition.


Marlin Lennholm began her organised search for all the Squound questions
that could be asked for twelve counters.

Various methods were being used around the room.

All searches led to a number pattern and and the correct answer of 91...

... however, Marlin's solution not only shows the answer, but reveals why
the answer to this and all other Squound questions is a triangle number.

Marian College, Ararat

Claire Maroske
Year 10
Claire discovered Squound at the annual conference of the Mathematical Association of Victoria in a workshop entitled Beyond the Tip of the Iceberg.
In my first lesson of the year with my Year 10 CAS class I decided to introduce my number theory unit using the Squound problem. This Year 10 class are being prepared for the harder Mathematics stream in Year 11 which is called Mathematical Methods. The task appealed to me because it introduces the concept of a Venn diagram in a practical way. The students worked in pairs and easily found the pattern for calculating the number of discs in the Squound. That is:
Circle + Square - Total = Squound.
I did structure the collection of results by drawing a labelled table on the board.

I posed the challenge of investigating how many possible problems there were using 12 counters and suggested the approach of investigating how many problems there would be for 3 counters. I scaffolded their learning by listing the elements {0,1,2,3} which introduced further set notation. We easily found 8 combinations and worked as a class to find the total of 10.

The students were amazed at their power and the neatness of this task. A lot of students felt pretty good about themselves too.   From this point I posed the more abstract problems of listing the number of combinations for 1, 2 and 0 counters. At this stage the students could see the value in listing the elements for each task. Again I organised the results on a table on the board ... Number of Counters/Number of Combos.

I simply asked if their was a pattern which would save us from investigating 4 counters. I was amazed how quickly and confidently the students found the pattern using the 'difference' method.

I had students explain the pattern to the class, then I asked the class to continue the pattern to find out how many combos their were for 12 counters. They were all pretty glad we found a pattern and did not have to list the combos!

I wrote the number pattern on the board and we discussed the nature of the pattern. We drew triangle dot patterns and staircase block patterns. The students were amazed at their power and the neatness of this task. A lot of students felt pretty good about themselves too.

In the following lesson, I introduced Venn diagrams and Set notation. I took an exercise from the Year 11 text. It contained some probability applications of Venn diagrams. One question stumped a few students, I was able to explain it on the board using the Squound problem. Here is the problem:

Of the 100 students at the Secondary College, 85 are active participants in hockey OR basketball. Basketball is popular with 75 players, but the hockey coach is happy with the squad of 15. Draw a Venn diagram of the students sport participation. Find the probability that a student chosen at random is a) a hockey player b) a participant in both sports, c) not a basketballer d) not a participant in either sport.
Now I might have had some difficulty explaining how to set out the Venn Diagram had I not introduced this unit with the Squound problem. I was able to say:
Hockey + Basketball - Total = Overlap (intersection) ... or 75 + 15 - 85 = 5 (5 students in the overlap)

Now it was my turn to feel pretty pleased with myself! The students said, 'Ahhhh' and I knew it came together for them.

Well done to Douglas Williams and his team! I am so glad I went to his session at the Maths Conference!

Green Line
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