Subtraction Surprise
1 out of 3
Try out these calculations. Are you surprised by the results?
Problem
Subtraction Surprise printable sheet
In the video below, Alison chooses some three-digit numbers and carries out some calculations which lead to a surprising result!
Watch the video. What do you notice?
Can you figure out the steps that Alison carries out in each calculation?
This video has no sound.
If you can't see the video, click below to see the steps and the examples Alison tried.
Pick any three digit number.
Reverse the digits, so write your number back to front.
Subtract the smaller of your two numbers from the larger one.
Now reverse the digits of the answer you get.
Add the answer to its reverse.
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Choose some three-digit numbers of your own.
(Make sure the first and third digits are different)
Is there a pattern to all the answers?
Now watch the videos again. This time, all three subtractions are carried out at the same time.
You may wish to pause the video at certain points, or watch it several times.
What is the same in each example?
What is different?
Does every example lead to the same answer?
Can you use what you noticed in the video to prove it?
You may be asking yourself...
If we take any four-digit number, reverse the digits, subtract the smaller number from the larger one, reverse the digits of the answer and add, will we always get the same answer?
If not, what answers can you get? Can you find the conditions required to give different answers?
Can you prove your results?
Getting Started
Take a look at the images below, taken from the video.
What do you notice?
What is the same? What is different?
What do you notice?
What is the same? What is different?
What do you notice?
What is the same? What is different?
Now can you mirror or recreate what you are seeing in these calculations but using $abc$ as your starting number?
Student Solutions
Elizabeth and Serena from Withington Girl’s School in the UK described what happens in the video:
In the examples in the video, you subtract the number you think of by the reversed order of your number. Afterwards, you use the answer to add to the reversed order of the answer and get 1089, if the first and the last digits are different. The number that you start with [can be] different.
Surya, Na'ima and Srinika, Abdulla and Issa from British School Al Khubairat in the UAE tried out some more numbers. Surya wrote:
I have done a few calculations and all the answers add up to 1089, for example,
875 - 578 = 297 + 792 = 1089 and 863 - 368 = 495 + 594 = 1089.
Issa added:
You cannot use a number with the same first and 3rd digit. This is because reversed it will equal the same (101 reversed is 101).
Anirudh, Ishbel, Abigail, Ethan, Gemma, Zaina, Ishaan and Luke from Cambourne Village College made some interesting observations about what happens during the process. Here is Abigail's work (click on the image to open a larger version):
Ishaan from Cambourne Village College used an example to show why this leads to a final answer of 1089:
Anh (Alex) from British Vietnamese International School in Vietnam used algebra and some systematic trials to show why the answer is always 1089 (click on the image to open a larger version):
Ci Hui Minh Ngoc Ong from Kelvin Grove State College Brisbane in Australia used similar notation, but showed exactly how the column subtraction and addition works algebraically, in particular borrowing and carrying (click on the image to open a larger version):
Alina, Gemma and Ishbel from Cambourne Village College wondered what would happen for 2 and 4 digit numbers. Here is Alina's work (click to enlarge):
Elizabeth and Serena also experimented with negative numbers. They wrote:
All the examples in the video worked out, however they didn’t use any negatives or use any 0 digits in the 3 digit number. No matter if you use a zero in the 3 digit number, make the first digit smaller than the last digit (which they didn't do in the video) or use a negative 3 digit number, the answer always comes to 1089. For example, if you use the number -293, the reverse would be -392. -293 - -392 comes to -293 +392 which equals 099, and 099 + 990 is 1089. In addition, we noticed that with the examples with 0 as a digit in the number, the first subtraction
usually came to 099 as well.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem offers students an opportunity to apply their understanding of place value to explain a surprising result! We hope that students will be curious about the unexpected outcome and wish to explore and explain why it happens.
Possible approach
"Choose a three-digit number.
Reverse the digits, so write your number back to front.
Subtract the smaller of your two numbers from the larger one.
If your answer isn't a three-digit number, put a leading zero in the hundreds column.
Take your three-digit answer and reverse the digits.
Add these two numbers together."
Once students have had a chance to work out the total, invite a few of them to share their answers. Enjoy the moment of surprise when everyone realises they ALL got the answer 1089.
To investigate why this happens, you may wish to show students the video from the problem and invite them to comment on what is the same and what is different in each example.
This problem offers a chance to challenge students to use very specific and precise language to describe the reasoning underpinning column arithmetic.
In the video, Alison used column subtraction for all three calculations, even though there was a quicker mental method for the third one. This could provoke a useful discussion about choice of methods!
Instead of showing the video, you could invite three students who have used a similar column arithmetic approach to present their examples at the board simultaneously, recreating what happens in the video. Choose a fourth student to work alongside them and mirror what they are doing, but using $abc$ as their starting point. Ask the first three students to articulate their reasoning at each stage in a way that can help the fourth student produce a generalised approach, like that shown in the image below:
Students might use a variety of methods to perform the subtraction, and can be challenged to create a generalisation based on whichever method they choose.
Key questions
What is the same in each calculation?
Why do you always get a 9 in the tens column when you perform the subtraction?
Is there a general way of representing the subtraction that helps to explain why we always get the same answer?
Possible support
You may like to start with students exploring what happens with two digit numbers, taking a similar approach to the one described above for three digit numbers.
"Pick any two digit number.
Reverse the digits, so write your number back to front.
Subtract the smaller of your two numbers from the larger one.
Now reverse the digits of the answer you get.
Add the answer to its reverse."
Does every example lead to the same answer? Can you explain why?
Possible extension
Always a Multiple offers students a chance to appreciate the power of expressing their understanding of place value using algebraic expressions.