Compare the calculations
Problem
Compare the Calculations printable sheet
There are four multiplication calculations hidden below.
Your challenge is to put them in order, from easiest to hardest. Try to do this without actually calculating each answer if you can.
Click on 'Show' to see them.
70 x 40
70 x 57
70 x 21
70 x 100
How did you decide the order?
You might like to do exactly the same with the set of four division calculations hidden below.
350 $\div$ 7
350 $\div$ 1
350 $\div$ 25
350 $\div$ 3
Create a set of four multiplications or four divisions yourself, which you think could be put in order from easiest to hardest.
Give them to someone else to order.
Do they agree with your final order? Why or why not?
You may find it useful to print off a sheet of the two sets of calculations. You could cut them up into two sets of four cards.
Getting Started
How would you go about answering each one?
Which one would you say is the easiest? Why?
Student Solutions
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 2nd 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 ÷ 1
350 ÷ 3
350 ÷ 7
350 ÷ 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 ÷ 7 in the 2nd place because you can calculate 35 ÷ 7 in your head then add the zero on to the end.
I put 350 ÷ 3 in the 3rd place because you can't calculate 35 ÷ 3 in your head. You will need a piece of paper and a pen.
I put 350 ÷ 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 ÷ 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!
Teachers' Resources
Why do this problem?
This activity is designed to raise learners' awareness of different calculation methods and to help them recognise the value of choosing a method to suit a particular situation. If learners are encouraged to have a flexible approach to calculation, they are freed from feeling that they have to remember the 'right' method to use, and can therefore take greater ownership for their mathematics. This task focuses on multiplication and division, whereas Arranging Additions and Sorting Subtractions offers addition and subtraction examples.
Read more about the benefits of having a flexible approach to calculation in our Let's Get Flexible! article.
Possible approach
Explain that you are going to show the class four calculations and, rather than being interested in the answers, you would like learners to order the calculations from easiest to hardest. Emphasise that you will be wanting them to be able to articulate how they decided on the order.
Reveal the four calculations (it doesn't matter whether you choose to use the set of multiplications or the set of divisions, or whether you use one before the other, or just one set). Give learners a few minutes to look at the whole set on their own to start with before asking them to work with a partner to agree an order. At this point, you may like to give out a set of the calculations, each calculation on a separate card, printed from this sheet (one set per pair). This will enable learners to physically move the calculations around as they discuss the ordering.
As the class works, listen out for pairs who are paying attention to the numbers involved and thinking carefully about how they would solve each one. You may like to stop everyone after five minutes or so to invite them to share some of their thoughts so far. How are they making decisions? Draw out the idea that just because the four calculations all involve the same operation, it doesn't mean we would do them all in the same way. We might be able to apply our knowledge of multiplication facts and/or place value; we might use compensation, or apply an algorithm for example.
Give everyone more time to come to a decision in their pairs before another whole class discussion. You might like to invite a few pairs to share their solution and reasoning, perhaps deliberately picking those who have not reached the same conclusion. It might be that you can reach a concensus on the method you would use to answer each calculation, in which case you could give each one a 'label' so that the whole class has a shared experience and you can refer back to these particular examples in the future.
As a follow-up activity, you could give each pair a piece of A4 paper and ask them to split it into four boxes (for example by folding). In one box, they could write one of the four calculations. In another box, they could work out the answer to that particular calculation (including a description of how they did this). In another, they could show how they would check their answer, using a different method. Finally, in the fourth box they could create a word problem that would be solved using that calculation. These would make a lovely classroom, or school corridor, display.
Of course you could do the same activity but with your own set of four calculations, to suit the experience and needs of your learners.
Key questions
How would you do that calculation?
Why do you think that one is harder/easier than that one?
Possible support
All children should have access to a range of materials to help them calculate, should they find it difficult not to actually work out the answers! This might include concrete objects as well as anything that facilitates jottings.
Possible extension
Challenge learners to create their own set of four calculations, deliberately including a range of difficulty. Having decided on the order from easiest to hardest, they could swap sets with a partner.