Reversing Numbers

The Problem

Take any 2-digit number. Reverse the digits to make another 2-digit number. Add the two numbers together.
How many answers do you get which are still 2-digit numbers?
What do the answers have in common?

Teaching Sequence

1. Introduce the problem – you could do this by writing 2 reversed 2-digit numbers eg 14 and 41. Ask the students what they can tell you about the 2 numbers. If they identify that they are reversed numbers then introduce the problem.
2. It is important that they are clear about how to reverse numbers and that they understand the difference between 2 and 3-digit numbers.
3. You may decide to do an example with the class.
4. Discuss with the students the strategies that they could use to add 2-digit numbers.
5. Let the students work on the problem individually before putting them in small groups. Some students are quicker than others when computing and it is important that all students have the opportunity to "play" with the problem before getting them to share their findings. If all students  have some work to bring to the group they are more likely to be involved in the solution.
6. As you circulate, encourage the students to explain how they are getting the answers.
7. Ask the students how they could organise their reversed numbers so that they could look for patterns in the answers. A good starting point would be to sort the 2 and 3-digit answers into lists or they may decide to identify the reversed numbers that give a 2-digit answer on the hundred's board.
8. Share patterns.

Extension to the problem

Is there a pattern in the numbers that give 3-digit sums?

Solution

There are many patterns that can be found in this problem.  Let's try a few numbers and see what we get:
13 + 31 = 44
26 + 62 = 88
47 + 74 = 121
54 + 45 = 99
68 + 86 = 154
Now we can see that if the sum of the digits in the 2-digit number is less than 10 then the sum of the reversed numbers is less than 100.
27 + 72 = 99
The sum of the digits in the 2-digit number determines the sum of the reversed numbers in the following way:
If the sum is 6 the answer is 66 (24 + 42 = 66; 15 + 51 = 66 etc)
If the sum is 8 then the sum of the reversed numbers is 88.

For the development band students, you might notice that the sum in every case above is a multiple of 11.

Solution to the Extension:

Once again the 3-digit sums are all multiples of 11.  To see this notive that 68 + 86 gives the same answer as 66 + 88.  Now both 66 and 88 are multiples of 11, so the sum is too.