Arranging additions and sorting subtractions
Problem
Arranging Additions and Sorting Subtractions printable sheet
There are four addition 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.
42 + 47
42 + 18
42 + 39
42 + 3
How did you decide the order?
We would love to hear the reasons for your final order.
You might like to do exactly the same with the set of four subtraction calculations hidden below.
Click on 'Show' to see them.
26 - 18
26 - 7
26 - 3
26 - 10
Create a set of four additions or four subtractions 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 this sheet, which contains 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 ordered their calculations and thought carefully about the reasoning behind their answers.
In the first part of the problem we asked you to order four addition calculations. Jasmine, from Meavy Church of England School, sent in this list:
42 + 3 = 45 (Easiest because you are only adding one digit)
42 + 18 = 60 (Second easiest because you can add up the tens 40 + 10 +10)
42 + 47 = 89 (Third easiest because you can add up tens and ones easily, 40 + 40 + 9)
42 + 39 = 81 (Hardest because 9 + 2 = 11 which goes over the tens, 40 + 30 + 11)
Look careful at Jasmine's reasoning for her answers. Do you agree with her ordering, or would you have ordered the calculations another way?
Minha, who is from St Michael's International School in Kobe, Japan sent in her list too. What's the same about Minha's and Jasmine's lists? What's different about them?
My answer is 42+39, 42+18, 42+47, 42+3.
42 + 39 is the hardest because when I calculate 9+2, It is 11. So you have to carry the 10.
And the second hardest is 42+18 is 60, for 2+8 is 10 so you have to carry the 10, but it is easier as there are no ones to remember.
And the next hardest is 42+47 because it is a little bit easier to do, because 7+2 is 9 and 40 plus 40 is eighty so it is a little bit easier to calculate it.
Finally, 42 plus 3 is easiest to calculate because it doesn't need to put in a tens box. And we need to just add on 3 ones. And it is 45.
Simran, from Maurice Hawk School in the USA, sent in her list and shared her reasoning too:
42 + 3 is easiest to add because you are only adding digits in the ones place and they add up to less than 10, so there is no regrouping.
42 + 47 is easy because 2 + 7 in the ones place makes 9. 40 and 40 in the tens place makes 80. Then you add 9 and 80, the total is 89. Regrouping in not required.
42 + 18 is also easy to add. 2 and 8 in the ones place of both numbers add up to 10. 40 and 10 in the tens place add up to 50. Adding 10 to any number is easy. So, the total is 60.
42 + 39 is the hardest because of regrouping. First, I added 2 and 9 in the ones place which is 11. Then, I added 40 + 30 which is 70. Finally, I added 70 + 11 which gives a total of 81.
Simran also shared another way to calculate 42 + 18:
There is another way to solve this: First I take 2 from 42 and add it to 18 to make 20. Then add 40 to 20 and the total is 60.
The second part of this problem asks you to order four subtraction calculations. Both Jasmine and Simran shared their ideas for this part of the problem too.
Let's begin by looking at Jasmine's solution:
26 - 3 is the 2nd easiest because 3 is a unit so you can do this on your fingers count backwards from 26 three times and you get 23.
26 - 7 this is the 3rd easiest because if you count backward from 26 seven times you will get 19.
26 - 18 is the hardest because if I had 26 apples and I ate 18 I would be left with 8 apples.
Thank you, Jasmine. It was really helpful to read your reasons for your ordering.
Now let's look at Simran's list. What do you notice about her solution?
26 - 3 is the easiest because I know that 6 is a double of 3. So, when I subtract 3 from 26, the difference is 23.
26 - 10 is also very easy. You can think about the 100s chart. The number above is 10 less and the number below is 10 more on the hundreds chart. Think about the square right above 26 and you get 16.
26 - 7 is not hard. First, I subtracted 6 from 20 which gives the difference of 20, then I subtracted 1 more and I get 19.
26 - 18 is the hardest because you have to do regrouping. First, I subtracted 8 from 16 which gives a difference of 8, then I had 10 left in the tens place. Then, I subtracted 10 from 10 (in the tens place), so the difference was 8. Another way : First I add 2 to 18 to make it 20 because it is very easy to subtract 20. Subtracting 20 from 26 gives a difference of 6 but because I subtracted two extra numbers, we have to add them back to 6. So, the difference of 26 - 18 is 8.
Many thanks to Jasmine, Simran and Minha for sharing their reasoning behind their answers.
Having read through their answers, I'm wondering if you could write your own lists of calculations and share the reasons behind their ordering?
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 addition and subtraction whereas Compare the Calculations offers multiplication and division 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 additions or the set of subtractions, 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 number bonds and/or place value; we might count on or back, 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.