Compare the calculations
Can you put these four calculations into order of difficulty? How did you decide?
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 everybody who explained their answers to this question. We received a lot of very similar solutions, as most of the solutions chose the same order!
Multiplication
Illi and Luc from CHPS in Australia said:
We went off of the times tables that are easier to remember and solve, like the tens times tables. That is why we chose 70 × 100 as the easiest. After that we went off from there!
70 × 100
70 × 40
70 × 21
70 × 57
Yuichiro from St. Mary's International School in Tokyo, Japan had this way of thinking about the problem:
The difficulty depends on whether the numbers are multiples of ten and is based on how many steps are involved when solving the calculation.
I like the idea of considering how many steps are involved, when working out the difficulty of a calculation. I wonder why it makes a difference if the numbers are multiples of ten?
We received a lot of different ideas about why multiplying by multiples of ten is easy. Theo from Stourport Primary Academy in the UK sent in this explanation:
We think 70 × 100 is the easiest because you add two zeros
We think 70 × 40 is the second easiest because you add 7 × 4 then add two zeros
We think 70 × 21 is the third easiest because you do 70 × 20 then add a 70 because there is 1 70 left to add on.
We think 70 × 57 is the hardest because you could do 70 × 60 then take away 210 because there is 57 70s not 60 and 3 70s is 210 so you take away 210
These are interesting ideas, Theo. I like the way you've used related facts such as 70 × 60 to help you with your calculations!
Arryan from Eastcourt Independent School in England had a different method of multiplying by 100:
For this question, I put the levels of difficulty as:
70 × 100
70 × 40
70 × 21
70 × 57
I arranged this in this order as 70 × 100 is straightforward to calculate, you just move the decimal point to the right twice which is 7000. 70 × 40 is also straight forward than 70 × 100, but it is still slightly trickier. To do this, you could just do 7 × 4 (which is 28), then move the decimal point right twice so the answer is 2800. For 70 × 21 you could just do 70 × 20 (which is 1400), then add 70 to it, revealing that the answer to this is 1470. 70 × 57 could also be solved with this logic, 70 × 50= 3500 and 70 × 7 = 490 so it is 3990.
Emily and Grace from Stourport Primary Academy had another way of multiplying 70 by 100:
We thought 70 × 100 was the easiest because all you need to do is take the 2 zeros off the 100 and put them on the 70.
Then we thought 70 × 40 was the second easiest because you just take the zeros of 70 and 40 and do 7 × 4 then add the 2 zeros back on.
After that we thought 70 × 21 was the next easiest because the digits of 21 are small so it would be easier to calculate.
Then we thought 70 × 57 was the hardest to calculate because 57 has bigger digits so it would be harder to calculate.
Alex, Freya, Darcie and Aimee from Stourport Primary Academy had very similar ideas to Emily and Grace, but described their multiplication of 70 by 100 in a different way:
70 x 100 is the easiest because all you need to do is add 2 place holders.
The children in Mrs Nelson's Class in Pennyman Primary Academy in England had yet another way of thinking about multiplying by 10 and 100. Their teacher said:
Lots of discussion took place for this one, especially about the strategies used.
We worked together and decided multiplying by 10 and 100 was easiest because you can move them on the place value chart. Then, we chose 70 × 21 because you can multiply by 10 double it, and add 70. We decided you would need to use a written method for the final one!
Adam, Bill, Jason, Masa, Minjun from St. Mary's International School in Tokyo, Japan explained how they decided the order of difficulty. They also explained how they would answer the most difficult calculation using partitioning:
We ordered on how many steps we took to answer and how difficult the multiplication was. For example, 7×1 is easier than 7×5, which is easier than 7×7.
The second easiest one is 70×40 because you do 7×4, then times the answer by 100.
The easiest is 70×100 because you do 7×1 and then times the answer by 100.
The third easiest one is 70×21 because you can just do 70×20 which is 1400 then add 70
(because that’s 70×1)
The hardest one is 70×57 you have partition 57 into 50 and 7. So you do 70×7 (7×7 =49 and then
you times 49 by 10 which is 490.) and then do 70×50 (7×5=35, then times 100 is 3500).
Now you add 3500 and 490 together which equals 3990.
Division
A lot of the solutions for ordering the division calculations were very similar, but we received a few different suggestions about ways to divide 350 by 25. Alex and Freya from Stourport Primary Academy said:
350 / 1 Is the easiest because when you divide by 1 the dividend stays the same.
350 / 7 Is next because 35 / 7 is 5 then do 5 × 10 because there is a placeholder left from the 350.
350 / 25 Is next because 50 / 25 is 2 and there are 7 50s in 350 so then you need to do 2 × 7 because 2 25s fit into 50. But 25 fits into 100 4 times and because it is 350 / 25 and 25 fits in 300 12 times and 25 fits into 50 2 times you do 12 add 2 and then you get the proper quotient.Which is 14.
350 / 3 Is the hardest because there will be a decimal point in the answer because if you use the short division method you will get a remainder 2 but that is not the actual answer because 1 third is 0.33 and 2 thirds is 0.66 and 3 thirds is 1 so the remainder is 2 we need to use the 2 thirds rule so r2 is 0.66 so it will be the hardest. Which is 116.66.
Theo from Stourport Primary Academy sent in this explanation:
Division
We think 350 / 1 is the easiest because you keep the dividend the same.
We think 350 / 7 is the second easiest because 35 / 7 = 5 then you add a place holder from the 0 of the 50.
We think 350 / 25 is the third easiest because you can partition 350 to 250 then divide that by 10 then you have 100 left which you do 100 / 25 which is 4 then you add the answers together which is 14.
We think 350 / 3 is the hardest because it will be a decimal because 3 does not go into 50 because 5 and 0 do not add up to a multiple of 3 so it can't be divided by 3
I wonder how Theo knew that 50 not being divisible by 3 meant that 350 wouldn't be divisible by 3?
Etienne from QUEST Tokyo in Japan used a different method:
350 divided by 1 was the easiest to answer because you need to keep the same answer which is 350.
350 divided by 7 was the next easiest because you can use your knowledge of the 7 times tables. I know 35 divided by 7 = 5. So 350 divided by 7 = 50. It is 10 times bigger.
350 divided by 25 is next because you can count up in 25s. There is a pattern. The numbers end in 25, 50, 75 and 100.
350 divided by 3 is next because we have to use bus stop method. The answer is 116 with a remainder of 2.
It looks like there's a lot of agreement about the calculations which are easiest and hardest to solve! Thank you as well to Oleg from Pechersk School International in Ukraine and Yuna from QUEST Tokyo in Japan, who sent in similar solutions to this problem.
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.