This is quite a popular puzzle or investigation amongst maths teachers. The reason for this is that it is a great example of a High Ceiling Low Threshold (HCLT) task. You only need to be able to find the difference between two whole numbers to make it worth doing and even with simple whole numbers, you can do lots of investigation.
It’s useful to start the challenge by using a grid like in the challenge picture. Find the positive difference (subtract the smallest from the largest in each case) of each pair of adjacent numbers and write the result at the midpoint of the two numbers. Then draw a line between all of the midpoints and start again. Download the template here.
See how far this one gets until you get 4 results that are all zeros. We can imagine each square as a level. Starting with level 1 as the outermost square, and each square inside as a new level, how many levels do you get with these numbers?
To begin with, This is a good exercise in simple subtraction. Once you’ve done that though, you can start challenging yourself to find numbers that get you to a particular number of levels and then look for patterns in groups of numbers. For example, start with a simple question – How would you produce a level a diffy that has only 1 level? What about 2 levels? Both of these questions would be good to check that students understand the idea of levels. Finding numbers to produce a 3 level diffy might take a bit more investigating. Finding level 4 diffys and on might require a few more attempts.
At this stage, it’s worth using some help with the subtractions. This is not because they are difficult but because the calculations get in the way of the actual investigation. Now you are starting to look for patterns in numbers, spending 10 minutes doing all the subtractions is not helpful. You can use a spreadsheet to automate the process so that children can focus on thinking about what numbers to choose and why they are getting the result that they are getting.
Download the spreadsheet above to help you with calculations. Let us know what level you’ve reached by commenting on this post or tweeting your result using the hashtag #momentofmaths.