Finishing out our subtraction Math Mights characters, today we introduce…D.C. and T-Pops! They aren’t new faces in Mathville, but today you’ll see how these Math Mights function in Subtraction World!
A few weeks ago, you met Springling, and saw how she used the open number line which helps kids learn how to count up or back on the number line. This strategy is developmentally appropriate to start in 1st grade and will continue all the way up to 5th grade, and even into integers.
Then, you met Minni and Subbi, who demonstrated compensation and how to shift the number line, which is a really great strategy to start in 2nd grade, when kids are starting to learn to subtract from 100 or 200. We want them to learn how to shift that number line instead of just trying to do it with the traditional method. This strategy also extends all the way into 5th grade and into middle school.
D.C. – Decomposing/Composing
In Addition World, D.C. appears as the first character, however, D.C. is developmentally a better fit right before T-Pops in the subtraction line up. Some might think he becomes irrelevant after problems get too hard because kids will eventually fuse into solving with the traditional method, but D.C. really helps them to have an understanding of what’s happening in the traditional method when they get there.
D.C. still stands for Decomposing/Composing, and in Subtraction World, he is still a perfectionist who carries his mallet and his hard hat. He still likes to decompose numbers and break them apart in different ways, but instead of just making friendly 10s, D.C. is trying to prevent regrouping. He renames the numbers to make subtraction easier.
Students can practice renaming numbers in different ways as a foundational skill for this strategy without even having an algorithm. So if I have the number 62, it could be six tens and two ones, but it could also be 50 and 12, or 40 and 22, 30 and 32, etc. This could also be done on an abacus to show the different ways they could build the number. This helps students understand that, when you rename the number in the traditional method, the number doesn’t change. In fact, the quantity stays the same, D.C. is just decomposing it in different ways.
D.C. is very strongly opposed to regrouping, so the first question he asks when he’s looking at a subtraction problem is: Does this need a regroup needed or not?
Lots of kids need help with this because as they start to learn the regrouping process, they immediately start to do it with everything. I vividly remember my son learning how to subtract, and he proudly demonstrated, “I have 38 minus 21 and I’m going to go next door and borrow one,” not even realizing that he didn’t need to regroup!
One way to help students practice answering D.C.’s first question is sorting activities. Give kids a collection of different subtraction algorithms and have them sort by whether a regroup is needed or not. They don’t even have to solve them!
In subtraction, the idea of partial differences is very similar to partial sums, which we represented in Addition World with Value Pak, who can decompose a problem by place value and then subtract easily. If it was a problem that didn’t require a regroup, this works out really well! If I have 85-22, I could decompose the 85 into 80 and 5, and the 22 into 20 and 2, and very simply subtract the tens and then subtract the ones. This is similar to how you would use the Value Pak to add with partial sums.
Some books will teach the idea of partial differences, however, it isn’t a fool-proof strategy. Sometimes when you’re doing a regroup, it doesn’t work. If you have 62-36 and you decompose it into 60 and 2, and 30 and 6, it works out great to subtract your tens but when you go to the ones, you can’t subtract six from two or you’ll end up with a negative number.
I struggled with whether or not to bring Value Pak into Subtraction world and have students use them when they don’t have to do a regroup and use D.C. when you do need a regroup. However, I don’t believe in teaching kids conditions in math. So, we wrapped the idea of partial differences into D.C.’s subtraction strategy.
Kids really need to be able to decide when to regroup. If you don’t need a regroup on a problem, D.C. is just going to smash the number and decompose it by place value to create an easy subtraction problem. But if he needs to do any regrouping, D.C. is going to smash the number and typically back up a decade to to rename it before he subtracts. This is where D.C. can actually help kids understand the traditional method more thoroughly. If you were to stack the numbers 62-36, like you would in the traditional, D.C. could decompose 62, but instead of saying six tens and two ones, you’re making it five tens and 12 ones, which is the same as 50 and 12.
This year, in our 3rd grade classes in some of our project schools, kids didn’t really understand the traditional method. The teachers were trying to do T-Pops, but I actually had them back up to really focus on D.C. (“go slow to go fast” I always say!). D.C. helped the kids make a really solid connection to be able to examine the problem and ask if they needed to regroup before they did anything else. They were able to go through the thought process: If yes, I need to regroup, then I’m going to decompose it and back up a decade. If no, I don’t need to regroup, I’m going to decompose it and subtract by place value.
The beginning side of the poster will show D.C. working with two digit numbers, but when the poster flips over, you will have examples of numbers in the hundreds, decimal numbers. If you’re doing D.C. with decimals, it can be a little tricky. If we have 6.2 – 3.6, you’ll rename it to be five and 12 tenths, which you can’t represent in decimal form. You don’t want to do .12 because that represents 12 hundredths. The values are different, so on the poster, you’ll see that we write out the numbers to help the students keep everything straight.
T-Pops – Traditional Algorithm
Good old T-Pops. He is the oldest citizen in Mathville and the most unique. He enjoys solving problems the traditional way, like your parents learned it. He likes to line up the numbers vertically by place value in order to solve the standard algorithm. Since he’s old, he doesn’t “carry” anything. He prefers to do a regrouping. When he’s looking at a subtraction problem, he uses the non-proportional manipulative of the place value discs to help him understand what’s happening in the regroup.
We don’t want kids to say they’re borrowing when they subtract, as we might have learned to “knock on the door and borrow one from next door.” First of all, you’re not really borrowing it because you’re not giving it back. Secondly, you’re not borrowing one. You might be renaming the number with 10 or 100 or 1000 or a decimal. We want to refrain from that line of thinking and help with T-Pops lining it all up by place value.
When we do rename the number, instead of putting a diagonal slash over it, like you see on the six that has been changed to a 5, we really want kids to attend to place value. We want them to know that they’re changing it from six tens to five tens and that they’re actually taking the two ones and making it twelve ones. The more we can reinforce this idea within T-Pops and the standard algorithm, the better off we are with getting them to understand what’s happening by attending to place value.
The place value discs are really unique in the traditional method because we build the minuend (first number) with the discs and the subtrahend (second number) with strips. You could do it with just the discs, and some books say to do it this way, however, kids get really confused when they’re trying to take away discs and sometimes end up subtracting from the bottom up.
By building the minuend with the discs and the subtrahend with the strips, it helps kids to slowly see how they are going to use the strips as a tray, to check and see if they need to regroup. Let’s take 62-36. We put the two discs on top of six so that it’s off the place value mat. If they leave the two ones on the place value mat, they usually end up just putting eight ones in, which adds up to 10 ones instead of the 12 ones that they actually need. So if the original two ones are off the board as students regroup, they can actually see what’s happening when they rename that 62 into 50 and 12.
After the kids rename, they can the two ones and know they have to take away four more to equal the number of the “tray” they’re using. So they take those discs off the board and can actually see how many are left (six). They follow the same process when they get into the tens column. They see that 60 is now 50, and they know they have to take 30, so they put three 10 discs on top of that tray and are left with 20.
Concretely representing this strategy with the discs and strips is the first step in CPA, and being able to do it abstractly on T-Pops’ dry erase board helps solidify the concept as well. You certainly can do a shorthand pictorial way of looking at this too.