This page continues a discussion begun on a middleweb.com post. Some fourth grade students and I were investigating what happens when you reverse the order of a number's digits and subtract it from the original number. We called this process "backtraction." We began with a list of expressions. Students' initial observations and questions and their early discoveries are included in the post. Here, we pick up where we left off in the article. Toward the end, you will find suggestions for using the project with older and younger students.
Why are the top ones crossed off?
The student who asked this question noticed that (when he used the traditional algorithm) he had to regroup from the tens to the ones and from the hundreds to the tens columns for every "backtraction" calculation. He wondered why this was happening. After examining and discussing multiple calculations, the class saw that:
- The hundreds digit in the minuend was always greater than its ones digit (in order to get a positive number for the difference).
- Because you write the numeral backwards to get the subtrahend, its ones digit is always greater than the ones digit of the minuend. This causes you to regroup from the tens to the ones column.
- Both tens digits always start out the same. However, when you regroup, the tens digit in the minuend becomes 1 less than the tens digit of the subtrahend. This causes you to regroup from the hundreds to the tens column.
- An exception occurs when the hundreds and one digits are equal! In this case, the difference is 0.
Why is 9 always in the tens place?
The observations from the previous question are helpful. The calculations in the tens column will always be:
10 – 1, 11 – 2, 12 – 3, 13, – 4, 14 – 5, 15 – 6, 16 – 7, 17 – 8, or 18 – 9.
Asking new questions
Once students discovered that the differences were always multiples of 99, they became interested in the patterns for these multiples:
0, 99, 198, 297, 396, 495, 594, 693, 792, 891, 990
Soon they made connections to a mental math subtraction method. Instead of adding 99 directly, you can add 100, then subtract 1. This increases the hundreds digit by 1, and decreases the ones digit by 1 (which, by the way, leaves the sum of the two digits the same at 9 –and leaves the tens digit as 9)! The answers also come from a nice subtraction pattern!
100 – 1, 200 – 2, 300 – 3, 400 – 4, 500 – 5 etc.
At this point, students' "What if..." questions began flowing! They were especially curious about what would happen with 2-digit and 4-digit numbers. They were excited to discover that the differences for 2-digits backtraction expressions were multiples of 9!
Based on their experience so far, they predicted that the differences in the 4-digit case would always be multiples of 999. Unfortunately, this did not happen! (It took a while to be convinced of this, because it took some time to recognize multiples of 999 quickly enough. We were getting bogged down in messy calculations.) Eventually, this led to thinking about patterns in multiples of 999!
0, 999, 1998, 2997, 3996, 4995, 5994, 6993, 7992, 8991, 9990, 10989, 11988, 12987, 13986, etc.
These patterns are similar to the patterns for 99, but more complex. Students began making and testing all sorts of conjectures— extending their curiosity out to 5- and 6-digit multiples as well as the multiples of 9999, 99999, etc . I'll let you and your students explore these beautiful patterns yourselves!
More interesting things to think about: Why does 4-digit backtraction not always lead to multiples of 999; and when does it happen? What makes the 4-digit situation different than the 2- and 3-digit cases? (Think about the two middle digits—the tens and hundreds.)
Why are 3-digit "backtraction" differences always multiples of 99?
The students did not suggest this question, but near the end of the investigation, I mentioned it, and I showed them some pictures for 502 – 205 to see what sense they would make of them. Base ten blocks, as shown in the picture below, can help all of us break away from relying solely on traditional ways of thinking about subtraction!
There is a lot to think about here!
- How do "multiples of 99" relate to "groups of 99"?
- What might be the reason for taking 1 away from each "flat" (group of 100) instead of taking 5 away from one of them?
- What happens if you do the steps in a different order?
- What happens if the tens digit is not 0?
- How does this relate to one student's observation (in the original post) that 793 – 397 has the same answer as 783 – 387?
- Is there a quick way to predict the number of groups of 99 without drawing the pictures or going through the whole subtraction algorithm?
- For older students and adults: Can you express (and prove) this idea algebraically? Why are there more backtraction equations having a difference of 396 than 891 (for example)?
Questions that students explored individually
Some students became very curious about negative numbers. They wondered what would happen if they tried "smaller minus larger" calculations. Would there still be patterns in the answers? They often had trouble figuring out how to perform the calculations, but a couple of them discovered that when they swapped the subtrahend and minuend, the got the opposite answer compared to the original expression.
Some students wondered what would happen if they added the possible answers to the backtraction calculations. They discovered that a sum of multiples of 99 is again a multiple of 99! (Later, we realized this is true for other multiples, too.)
A pair of students stepped outside the world of backtraction. They started with a minuend of 99 and investigated what happened when they kept increasing the subtrahend by 1. They were excited to watch the difference decrease by 1 each time. I'm not sure they realized that the same thing would have happened with any minuend!
- Many students predicted that the sum of the digits in each answer would always be a multiple of 9. They tested this conjecture on many numbers and looked for ways to predict which multiple of 9 they would get.
Using the investigation with younger and older students
Many students, especially younger ones, may benefit from beginning with a prompt containing 2-digit versions of backtraction. For example: 72 – 27 31 – 13 54 – 45 83 – 38 etc. It's always possible to extend to the 3-digit case later.
Older students may be able to explore the problem algebraically. For example, by writing and simplifying algebraic expressions that represent the value of each number, they may be able to prove that the 3-digit backtraction expression abc – cba (where a, b, and c are the digits) has a difference of 99(a – c).
100a +90b + c – (100c + 90b + a) =
100a +90b + c – 100c – 90b – a =
(100a – a) + (90b – 90b) + (c – 100c) =
99a –99c =
99(a – c)
This calculation verifies that the tens digit does not affect the difference and that the difference of the hundreds and ones digits determines which multiple of 99 you will get. Notice how this process relates to the pictures of base ten blocks above!
The algebraic reasoning involved here may also help with exploring 4-digit backtraction. Why are the differences not always multiples of 999? Under exactly what circumstances are they multiples of 999? What happens when you have even more digits?