Sunday, 14 September 2014

$20 Commutative ('Switch up') Challenge

So this one is pretty simple but very fun to run. It's all about the commutative property of multiplication, i.e. 4 x 5 is the same as 5 x 4.

In front of my grade three class, I modelled (after our frenzy warm-up): if I have 4 groups of 6 (4 x 6), what does it equal? We drew the rows and columns as an array, and it made 24. (Also discussed double double, or 12 + 12 for partitioning of the array). What is 6 groups of 4? Again, we drew the rows and columns as an array and, again, it was 24.

Then I announced, $20 note ceremoniously in hand, after pulling it out from my back pocket: The first person to find an example of a multiplication where you switch the position of the two numbers and the answer is different, will receive this as their reward.

One student then asked: Is it possible? I responded, "Maybe" with a smile and slight nod indicating it could be. Another student asked: "Do you know one that works?" I replied, "I might," again, with a slight, sly smile. You need to kind of mislead the group a little, particularly if you have some wise ones. Although this may seem counter-intuitive, if you don't when you get questions like this, the students will not have the opportunity to independently form the view that it is impossible by themselves. To help my cause, at certain parts during the lesson some students thought they'd struck gold and muttered under their breath (though very audibly), "I've found one," thereby sending everyone else into a calculation frenzy on their current attempt.

Most students used arrays to experiment with numbers. Extension students used either traditional or lattice method double-digit and larger multiplications to test it, which they'd been taught earlier, e.g. is 24 x 5689 the same answer as 5689 x 24. Some extension students tried decimals too and found commutativity even works for those equations.

Best thing about this lesson: the students who are getting it wrong are the ones coming to you thinking they have achieved the challenge, so you have no trouble identifying who needs extra assistance :).

By the end, most students have independently concluded that it is entirely impossible. We discussed this as the 'switch up' or commutative property of multiplications and then I gave examples like, 'If I know that ... x ... = ..., what will ... x ... = : every student said 'the same', even the traditional strugglers had fully grasped the concept no matter how I worded or challenged the conception.

Only catch, the students will not trust you ever again - unless you are an even more convincing trickster next time. Very enjoyable! 

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