Detached Dots

This island is not so much about covering curriculum content but building culture. It encourages creativity so it’s a great entry point to problem-solving strategies.

Let’s start with a square of numbered dots. Draw a line joining two dots.

Then join another two dots.

Draw another line making sure that no line intersects more than one other line. You might end up with something like this:

Here is a non-example that isn’t allowed as the line from 12 to 8 is intersected twice.

This also is not allowed as an end point as the line from 3 to 8 is not intersected once.

More examples obviously help!

Connect as many dots as you can following the rule that each line must be intersected once. What is the smallest total that can be achieved by adding up the unjoined dots?

As I always mention in various tasks, a trial and improvement approach is aided by using dry erase pockets. You can have one grid being their best attempt and one for their experimental attempt.

Initially, there will be plenty to explore using the 4×4 grid. They might come up with something like this as the optimal solution:

A score of 18. But…

What if we had more than one line going from the same dot?

This is the kind of creativity I always hope to promote and so I allow it. This is an example where I might prioritise creativity over exact rules. I would rather allow intersections at the end points as it makes the task better. Now a score of 10 is possible.

Moving forwards, we can explore in so many directions:

  • What if we changed the dimensions of the grid?
  • What if we changed the start position of the number 1?
  • What if we started from 10?
  • What if each line had to be intersected twice?

This is definitely a good island to promote the idea of different children going in different directions and for that reason it makes it a good one to establish a culture of asking ‘what if…?’ questions when they don’t have much experience of it.

There are also easy ways to change the rules to explore even further. Children can definitely come up with their own ideas with this one.

  • What if we used a pentagonal or triangular grid?
  • What if we joined the dots using two lines that form an angle together?
  • What if we tried to make the highest total but had to keep adding lines until it was impossible to add any more?
Search
Search inquiry reasoning skill in maths

Obviously, there is a lot of Seeking going on to achieve lower totals. However, try and use opportunities to comb by considering the subtle changes they can make in their strategies.

Organise

A leader board of totals for different grids can focus them on trying to beat scores and also to focus on grids of certain sizes that don’t have many results. I often use magnetic dry erase labels for leader boards.

Discover

They can consider the strategies and shapes that they are making in their conjectures as well as the totals that they believe are possible. It might be more in the form of Noticing the similar shapes of lines across multiple grids rather than concrete conjectures. From looking at the patterns between them, interesting discussions can be provoked.

Investigate
Investigate inquiry reasoning skill in maths

This is an opportunity to emphasise the use of Expanding to try and beat existing totals. It may well be that the score of 18 is created first and it is through creating peculiar data that it is beaten (by joining more than one line to a dot). Even if they do this from the start, you can focus on how it was a creative approach that wasn’t immediately obvious.

Argue

They can use proof by exhaustion to explain why they can’t create any more lines on a particular example. It’s good to steer them in this direction as it is through emphasising strategies like this that ensures they don’t just create data randomly.

Explore

So many opportunities for the children to take this further! They can really think about this one a lot in terms of the directions they could take it.

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