It is my pleasure to introduce my favorite Math Mights characters to you today – Minni and Subbi!
These delightful twins were born in Mathville with an adjoining tail, which makes their lives very complicated. Minni (short for Minuend) represents the first number in subtraction, and she is the rough, tough sister who likes to get muddy and play all kinds of sports. Subbi (short for Subtrahend) represents the second number in subtraction, and is the complete opposite of her sister. Subbi is into cheerleading and dancing, and she likes to make sure things are nice and neat. This includes their tail, which serves a special purpose in making life more manageable for the twins.
As siblings do, especially twins, Minni and Subbi fight all the time. Their parents suggested that, if the twins stay a tail’s width apart, they might be able to keep themselves out of arguments. This works really well, until one of them starts moving. One of Subbi’s biggest pet peeves is when her tail gets dirty. She hates to have her tail dragging in the mud. So, if Minni backs up one, then Subbi has to move one in the same direction as well to keep the distance between them the same and keep their tail clean.
We try not to say that Minni is the larger number in a problem because, when students get to middle school and start talking about integers, the value of the first number in a problem might be a negative number so it wouldn’t necessarily be larger.
Compensation and Shifting the Number Line
Minni and Subbi are helping to show the strategy of compensation as it applies in Subtraction World, and their tail represents the concept of shifting the number line. Many kids think about subtraction as “taking away” something, when really subtraction is all about the distance between two numbers. In Kindergarten, the concept of taking away might look like “I have five M&Ms and then someone ate two. How many are left?” This is simple and it works pretty easily. But when you have larger numbers like 150 – 67, the “take away” perspective of subtraction becomes much more difficult. It would be hard to take away 67 from 150 in your mind or even to start at 150 and count backwards 67 to come up with an answer. This is where Minni and Subbi come in to help shift the number line and make things a little bit easier.
In subtraction, compensation has really great benefits for students, especially as they are trying to go across the zeros. Many times, students try to “go next door” and borrow numbers to regroup the place value as they’re subtracting from neighboring columns. This is a great place to introduce Minni and Subbi.
Take the problem 100-36. Traditionally, we would think we need to regroup, because you can’t take 6 from 0, so we go into the 10s column to “borrow.” Since there’s a 0 there too, they have to go the hundreds to “borrow” one and take it back through the place values. This is the T-Pops strategy, the traditional way. It’s challenging for kids to mentally process the idea of borrowing and keep the place values straight as they change numbers around, and they typically don’t really understand what they’re actually doing
Minni and Subbi dislike regrouping, so they like to shift the number line. Always keeping the distance between them, they like to pick a direction and shift the whole number line so they can avoid a regroup situation. In the problem above, it’s a double regroup that is especially unfavorable to Minni and Subbi, so they use their mathematical understanding to take a shortcut.
On the number line, Minni takes the value of 100 and Subbi has the value of 36. When Minni sees the zeros in her number, she immediately tries to think of 9s, so she moves back one on the number line to 99. As Minni backs up one on the number line, her sister moves one as well to make sure their tail stays clean, so she backs up to 35. Now, when we rewrite the problem to be 99-35, it’s much easier for kids to process mentally because there’s no regrouping involved.
I like to use string to represent their tail. I’ll open the string to represent the distance between the minuend and the subtrahend in the problem and hold it taut so kids can see how the distance doesn’t change, even though the numbers do look a little different as Minni and Subbi move on the number line.
Sometimes, with Minni and Subbi when you’re doing larger numbers, you might shift that number line and some kids need to think of the actual number line, but some kids can transition to doing it in their heads. I have the subtraction problems written more traditionally, however, they are able to see that we are shifting or backing up the minuend three and I need to back up the subtrahend three too.
Compensation doesn’t need to just be done going across the zeros. It can shift anytime you want to make a number a little bit more friendly. A lot of times we’re trying to make the minuend be larger so that when you go to subtract the subtrahend as those numbers line up, you don’t have to regroup.
Compensation: Addition vs. Subtraction
While both Abracus and Minni and Subbi all practice the same strategy of compensation, Abracus lives in the Addition World, and sometimes confusion can occur when we start looking at Minni and Subbi in Subtraction World and their flavor of compensation.
Abracus has the “give one, get one” approach to addition. He zaps a number, maybe adds two, and then he’ll take away two to get his answer. In Subtraction World, Abracus’ wand loses a bit of its power, so this version of compensation doesn’t always work or is very conditional if it does work.
For example, If I had 62-36 and I wanted to use Abracus, I might start by shifting the minuend back to 60 (which really isn’t that friendly because you still have to regroup in your head or count up with another strategy). This gives me 60-36, and I come up with 24. But I have to add two back because I took two away earlier (this is like the “give one, get one” strategy) and so our answer is 26.
Abracus’ strategy does not work the same for the subtrahend. If I were to add 4 to 36 to make it 40, I would need to take away 4 at the end of the day. My problem would be 62-40, which is 22. Taking away 4 would leave me with a wrong answer.
A Strategy that Lasts
We want to be really careful not to teach children conditions or rules in math. Things like “when the minuend is a certain way and you shift it, then it would work to give one/get one” are really just confusing. The better way is to give kids a strategy that’s not only fool-proof, meaning it will always work, but that will also extend with them. Minni and Subbi’s strategy can be done with decimals, money, elapsed time, fractions, and even integers in middle school.
Compensation in Subtraction World is a harder strategy to grasp. Even though we might introduce this strategy in 2nd grade as students are going across as a way that makes more sense, we also have to be careful because some kids’ number sense is still developing. Sometimes they can make the problem more complicated. Students can meet Minni and Subbi in 2nd grade, but their strategy is better developed in 3rd grade, especially to apply it to other problems.