So It's Times!
Zach sent in a wonderful solution which you can read about at length here: So Its Times - Zach T.pdf
It starts off as follows:
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Saksham from Hounslow Town Primary School wrote:
So first I wrote out all the possible multiplication facts: 1 x 1, 1 x 6, 1 x 31, 1 x 36
Then I decide which one was the biggest and then I did that to all of the facts. After I tallied them and found out which one had the most tallies therefore would give the largest answer.
I have attached a replica of what I did but in excel as I couldn't attach the original file here: So its Times .xlsx
Allison from the The Glasgow Academy in Scotland wrote;
ANSWER : The highest total of flipping over would be to just place the second sheet over the top of the first one in exactly the same place. The lowest total of flipping over would be to place the sheet of plastic down over the first one 180 degrees turned around so that the number 1 is on top of the number 36 (for example).
WORKING : If you think about it, the highest numbers that you can reach are the ones that are multiplied by themselves. Your highest number is 36. So the highest number that you can get is 36 x 36= 1,296. As soon as you flip the second sheet around, the highest number gets much smaller. The largest number you can get now is 36 x 31=1,116. Therefore, the largest numbers are the ones that are multiplied by themselves. The moment you take that away, most of the numbers get much smaller.
Then, what would the smallest number be? Well we know that it can't be just placing the second sheet on top of the first one in the same way because that is how we get the largest number. So we could try multiplying 36 by the smallest number on the sheet which is 1. To do this we would need to turn the second sheet around so that 1 is on top of 36. Then the number 2 would be on top of 35 and so on. This means that it will have the smallest multiplications within the square so that the overall number will be the smallest number possible.
Ayobami from Woolwich Polytechnic School wrote:
Because we were told not to do the number calculations, we considered how we might use algebra to help. Firstly, Ayobami decided to use the nth term to solve the question and find a solution. He started seeing patterns in the nth term calculations, and realised that some of the answers were the same, before deciding a different approach. All the children started
using algebra on the grid by replacing the four corners with a,b,c and d, here is the total file:
Woolwich 2.doc or woolwich.pdf
Meghna from the The British School Kathmandu in Nepal said:
The way to get the highest total is by pairing each number with itself, i.e. 1 with 1, 2 with 2 and so on. This will give us the highest possible total as squaring a number will give a larger product than when we multiply the number by a number smaller than itself. For example, 5 x 5 = 25 whereas 5 x 4 = 20.
The way to get the lowest total is by pairing the largest number with the smallest, 36 with 1, 35 with 2 and so on. This will give us the lowest total, because as I mentioned above, pairing numerals with a number smaller than itself will give us a lower total. So to get the highest total, square each number and to get the lowest total, pair the biggest number with
the smallest, second biggest with the second smallest etc.
Thank you for all these wonderful solutions. It proved it be an activity that really made use of your multiplication facts.
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