Chains

Task 213 ... Years 4 - 10

Summary

In this game students score points by placing blocks and scores for each move make a progressive total. Each block placed scores at least one point, but if placed to make a chain of more than one block side by side, the score is the number of blocks in the chain. The investigation looks at different total scores that can be made and then extends to look at what happens if the length of the board is changed. The tasks involves significant spatial reasoning which leads to interesting and challenging number patterns.

Materials

• 12 wooden cubes
• 2 playing boards each with 6 cells

Content

• concept of proof
• numbers, odd & even
• numbers, triangle
• patterns, number
• patterns, visual
• reasoning
• recording mathematics
• tree diagrams

Iceberg

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

Students will need to play around for a while to be clear about how the scoring system works. However, as they play they are also beginning to realise that certain placement sequences give the same score even though they may look different. For example these two partially played games are really the same. Turning the board 180º turns one sequence into the other.

Both plays score 1 + 1 + 2 = 4 to this stage.

The clue to finding the highest possible score, as students soon realise, is to make the longest chain possible on each move. These moves then lead to a total of 21, the numbers to the right showing each development of the partial sum.

This form of recording shows so beautifully why the highest score is a triangle number.

The clue to finding the lowest score must be the opposite strategy: score as many 1s as possible then score the lowest possible on each remaining move. Or, in short, score the lowest possible on each move. One way to do this is:

Is there any other way that isn't just a transformation of this?

When it comes to finding a way to play to make every score between 14 and 21, it might be possible to work backwards with a tree diagram. The score must be 6 on the last move, but what might the situation be one move before that? This almost complete tree diagram shows one person's journal record. If this was completed (which might take a bit of planning to fit into a page) working from the bottom up along every possible line would show all possible ways to play and their respective totals.

Alternatively, we could try to find a way to play that produced each of 15, 16, 17, 18, 19 and 20.

The last move has to score 6, so to get a total of 15 we need five scores that will sum to 9; and they have to be scores that can represent moves, for example it isn't possible to score 4 ones and then a five. The context matters; this is not just playing with numbers.

But can your students find a way to make 18 or 20. We can't. And if no one can find a way to play to make these totals, can your students prove that it can't be done.

• Perhaps the tree diagram could be useful after all?
• Perhaps they only need to continue the reasoning of subtracting 6 from 18 or 20, then subtracting another 1 because that must be the first score, then trying to make what's left in a way that represents 4 possible moves.

Challenge

Trying other length boards, as suggested on the card is easy because one board can underlap another to make board lengths from 7 to 12. Shorter boards can be made by placing a piece of paper over the necessary cells of one board. The structure of the card suggests investigating highest and lowest scores for each board length. How would a mathematician tackle this investigation? Perhaps by making a table like this:

 Cells High Low 1 1 1 2 3 3 3 6 5 4 10 8 5 15 11 6 21 14 7 28 17 8 36 21 9 45 25 10 55 29 11 __ __ 12 __ __

The pattern in the highest scores is not surprising given the strategy needed to make the highest. But is there a pattern in the lowest scores? Certainly not one that is straight forward. Perhaps drawing a graph would help.

• Definitely not a straight line, but tantalisingly it does appear that the dots lie on a curve. A reasonably good test that the data is correct. A graphics calculator might help here.
• But perhaps the data is not correct. Can your students find a lower total for any of the board lengths? Let us know if they can ... and of course we want to see the sequence of moves that produces it.

In playing the game students might sense that odd and even length boards are little bit different to play. Therefore it might be worth taking a second look at the table in terms of the odd and even boards.

 Cells High Low 1 1 1 2 3 3 3 6 5 4 10 8 5 15 11 6 21 14 7 28 17 8 36 21 9 45 25 10 55 29 11 __ __ 12 __ __

Now there are more obvious patterns. Predict and check for 11 and 12.

But is there one formula to predict the lowest for any board length n?? Over to you.

One more thought

• Every cell is either occupied or not - a 1 position or a 0 position. How does this problem look in a binary world?

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.

Preparation for this whole class lesson is simply preparing boards to fit the blocks or tiles you are using. Blocks are better because they are easier to pick up and move. This is also a perfect investigation for the yellow/blue Poly Plugs.

To catch student interest, place 6 chairs in the line and explore the rules of the game using people as the 'blocks'. Make sure everyone knows how to score, then ask the class to make the highest score. Students return to their tables to record how the game is played and how the class found the highest score.

Using the table top material, continue with the lowest score challenge and the investigation of finding ways of making the scores in between highest and lowest. The 6 chairs and volunteers can be used at any stage to gather the class for explanation, or demonstration of student discoveries.

Continue the lesson guided by the iceberg above.

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

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.

Chains is not in any MWA kit. However it can be used to enrich the Number & Computation or Space & Logic kits at Years 5/6, 7/8 or 9/10.