Compare the Calculations

Well done to everyone who shared their ideas with our team.

The first part of the problem asked you to order four multiplication calculations. Looking at your answers, there's more than one possible answer depending on which types of calculations you enjoy doing the most.

One of the most popular answers was putting the calculation 70 x 100 first in your list. Harry, Amelia and Marina from Twyford School; Harry and Nancy from Drakes Church of England School; Anika from Boothstown School; Minha from St Michael's International School in Kobe, Japan and Yaochen, from Little Hill School all choose that approach. Yaochen shared this explanation:

I put these four calculations in this order from the easiest to the hardest because 70 x 100 is just 7 x 1 but with two extra zeros at the end of 100 and one extra zero at the end of 70. 

I put 70 x 40 in the 2^nd place because it's just the same as 7 x 4 just with one extra zero at the end of both numbers.

I put 70 x 21 in the 3rd place because you can just calculate it in your head because the ending number is 1 so it's 1 x 7 and anything with one is very easy to calculate.

I put 70 x 57 in 4th place because when you try to calculate 57 x 7 in your brain just like 21 x 7 you can't do that with 57 x 7 cause 7 x 7 = 49 then you do 5 x 7 = 35 then you need to do this calculation 5 + 4 = 9. I find numbers that have a number from 1-9 at the start then zeros easier than two-digit numbers because it's the same as a one digit number but more zeros.        

Yaochen clearly explains the reasoning behind their answer. Do you agree with their ordering? Would you put them in the same order? 

Lachlan, from Full Spectrum Education in Australia, suggested putting the four calculations in a different order:

70 x 40

70 x 100

70 x 21

70 x 57

The reason I placed the numbers in this order is because 70 x 40 and 70 x 100 have zeroes and therefore are much easier to figure out.

70 x 21 is harder because you don't have as many zeroes, so you have to do two lots of working out.

The same reason for 70 x 57, but they use larger numbers.

The second part of the problem asked you to order four division calculations. Again, there's more than one possible answer. 

Yaochen suggested the following solution:

350 Ã· 1   

350 Ã· 3  

350 Ã· 7  

350 Ã· 25

I put these four calculations in this order from the easiest to the hardest because 350 ÷ 1 is just 350 if you were calculating it anyway if you see a 1 then it is definitely the easiest one.

I put 350 Ã· 7 in the 2nd place because you can calculate 35 Ã· 7 in your head then add the zero on to the end.

I put 350 Ã· 3 in the 3rd place because you can't calculate 35 Ã· 3 in your head. You will need a piece of paper and a pen.

I put 350 Ã· 25 in 4th place because dividing a two-digit number is way much harder than dividing a one-digit number. 

Anika, Amelia and Marina suggested a different ordering:

350 ÷ 1

350 ÷ 7

350 ÷ 25

350 ÷ 3

Amelia and Marina shared the reaosning behind their answer:

350 ÷ 1 is the easiest, because by a number 1 ÷ is always going to equal the same number. An example of this is 1 ÷ 2 would be 2. The same so the answer is 350.

350 ÷ 7 is second as 35 ÷ 7 is 5 so then you just have to times it by ten so the answer 50.

350 ÷ 25 is the third easiest because if you double 25 it is 50 so you could do 50 divided by 350 which is 7 so then you would just have to do 7 x 2 to get your answer.

350/3 is the hardest because 3 is not a multiple of 35 so it is not going to go easily into 350.

Anika also used the divisibility rule to check that 350 was not divisible by 3.

Amelia and Marina shared the number of steps they needed to make for each of the four calculations:

This is interesting because on the first question you had to do 1 step and on second you have to do 2 steps and on the third is 3 steps and so on.

Did you put them in the same order as Yaochen or Amelia and Marina? If not, can you explain your reasoning for having the calculations in a  different order? Harry and Nancy suggested this ordering:

350 ÷ 1

350 ÷ 7

350  ÷ 3

350 ÷ 25

In their solution, they explained that they would use the 'bus stop' written method to calculate 350 Ã· 3 but they would choose long division to calculate 350  ÷ 25. I wonder if counting up in steps of 25 might work too?

The final part of the question challenged you to design your own sets of calculations. We'd love to see some of your ideas and we may be able to publish some of them here!