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We started work on this problem by looking at 'Button-up' first. Before investigating the 3 buttons in that scenario, we agreed on how 1 and 2 buttons would work, with 1 button obviously giving 1 possible order, and 2 buttons giving 2 possible orders. The children then began to explore the number of possibilities with 3 buttons. Some chose to do this with physical 'buttons' which they reordered, some preferred to record the buttons as written digits (ie. top button = 1, bottom button = 3). One pupil, Caitlin, devised a systematic method of finding all the possibilities beginning with 1, then all the possibilities beginning with 2, and so forth. This yielded 6 possible orders, which the other groups in the class ultimately agreed with. I then recorded the results so far in a simple table of 'Number of buttons'-'Number of possible orders' (see image). To investigate 4 buttons, we tried to apply Caitlin's ordered method again, finding the possible orders beginning 1-2, then 1-3, then 1-4, etc. This yielded 24 possibilities. At this stage, we tried to identify a pattern in the results that would enable us to predict the result for 5 buttons. Predictions were wide-ranging, from 30 to 125, but Liam recognised that there was a mathematical principle behind the results: multiplying the number of buttons by the previous number of possibilities (eg. 3x2, 4x6) produces the number of possibilities for the current number of buttons. Liam used this process to calculate the possibilities for 5 buttons, which he found to be 120. As a class, we then put this to the test by recording actual possible combinations. After finding that the number of possible orders beginning with 1 would be 24, we confirmed that Liam was correct (as 24 x 5 = 120). This then enabled the rest of the class to calculate the number of possible orders for 6 buttons as 720. We enjoyed the challenge!