You know how I'm big on strategies? No? Well, you should.
I'm big on them.
My kids know this. We talk a lot about how we can memorize math but if we someday forget it, what will we do? We need a strategy to fall back on. It's just SMART.
Something shocking to me when I moved into third grade was students lack of understanding with addition and subtraction. They seemed to vaguely remember some kind of rules they had been told but they either used them incorrectly or not at all. Every. single. student.
Even students that were "high" math kids and had great understanding would botch computation problems by trying to use some abstract RULE they had been told and simply couldn't recall or understand. When presented another way, these students could get the problem easily.
They had been ingrained with some arbitrary rule of how to do math. A rule that ditched place value, threw out number sense, and was made by one person one time.
And it took me 9 weeks to erase that.
9 week of frustration but my students finally starting valuing getting an answer that made sense over doing "big kid math" (please don't perpetuate this - looking at you 2nd grade teachers).
Here are 2 strategies I guided my students toward and how they ran away with them. Why two? Because I introduced them and using one of these ways, every student in my class could subtract. Every student.
You draw a number line. You know, that line with two arrows on either end (because numbers go on forever). Put the number you want to start with on one end and the number you want to end at on the other.
Now, before you introduce these you need to make sure that your students understand subtraction is the difference between 2 numbers (it's literally called finding the difference). In fact, that little minus symbol means in between. I relate this to book pages. If someone tells you to read pages 35-72 you don't just read 35 and 72, right?
NO. You read those and the pages in BETWEEN. To find the difference, we have to find the numbers between. Start at one number and count up to the second.
There are two ways to put this on your number line. Let's do 74 and 35 as an example.
You can put 35 on one and and 75 on the other and count up.
OR you can put 75 on one end and 35 on the other and count back. Counting up makes sense to me and is how I introduced it at first but changed my mind about that when I had a few students who thought it made more sense the other way. I think because they relate subtraction to getting smaller. Either way works so let students find what makes sense for them.
Once you have your line set up, you can start doing jumps. At first, I have students use 1 jumps. They soon discover this can take forever. So we start circling over ten of the little one jumps and doing 10 jumps. Make sure your students have a firm grasp on adding or subtracting 10 from any number (pull you your hundred charts!). Later, students can do any jumps that make sense to them (5s, 2s, 20s, 100s,).
In our example of 74 and 35, I would start at 35 and do 3 ten jumps to arrive at 65. From there, I could do 9 one jumps and reach my goal - 74. The 3 ten cups and 9 one jumps make 39 jumps. 39 numbers from 35 and 74.
A lot of students stick with ten and one jumps and that is FINE. Tens and ones make sense, they're familiar and students have worked with them before. I wouldn't stress making kids move toward other jumps until they're doing problems in the hundreds or thousands.
Here is an example I show my kids. They help me fill in the boxes under each jump. Then, we see how many jumps it took to find the difference.
The beauty of number lines is if practiced enough, they help kids improve their mental math skills. This is something you should model for your students. Every time we work on a subtraction problem, I show my kids how you can do tens and ones jumps in your head (recording on your hand or a sheet of paper) and arrive at your answer.
Let me paint you picture. You have a student in from of you trying to subtract 53 and 27. She remembers that she absolutely positively has to start with her ones (eye roll) so she looks at 3 and 7. She writes 4. What do you say to her?
Don't tell me.
I got it.
"You CANNOT subtract 3 from 7. If I have 3 dollars could I give you 7? Exactly, no"
WRONG.
Don't believe me? Ask a middle school algebra teacher who is MAD at you right now.
Teach your kids the concept of negative numbers. TRUST ME, it makes way more sense to them than the regrouping thing.
I am not just saying this. I have done it with low kids and high kids. And this is how.
First, we talk about the concept of negative numbers. Like, if you needed 7 dollars but I only had 3 I could give you the 3 NOW and owe you 4 later. Next time I get more money, I will take the 4 out of that and give it to you. This means I owe you 4. Negative 4 (-4)
Together, my students and I build a number line that includes negative numbers to -10 and positive numbers to 10. We play around and we practice. Soon, students realize that any subtraction problem FLIPPED is the negative same number. Like, 5-3 is 2 and 3-5 is negative 2. See how this helps us to get negative answers quickly? But students need to arrive at this conclusion.
Once they've played around, we apply the concept to larger subtraction problems.
Let's go back to our original example. 53 minus 27. First of all, I let students start with the tens. You know, we really might need to borrow or take away so it's good to know what we are working with. Even in standard algorithms this is good. Anyway, we start there. 50 -20 is 30, right? So we write down 30. Then we look at ones. 3-7, students look at the number line and see that this will be -4 (or realize that 7-3 is 4 so the opposite is negative 4 - let them work at their pace). We have our 30 so let's take the 4 from there. That will give us what we need. 30 - 4 is 26. The answer is 26.
I know, I KNOW. This was another mind blower when I thought of it. It makes sense for any subtraction problem as long as you keep the place value. WHY HASN'T SOMEONE SHOWN ME THIS?!
I love this strategy for a lot of reasons. It retains place value so kids aren't thinking about digits only, it introduces them to negative numbers (algebra teachers will praise you - don't tell kids you can't take a bigger from smaller!) and it plain makes sense to kids.
They totally get the idea of negative. Of owing something. It's a real life thing.
At first, I too was skeptical about this method. I used it often with my high kids but wasn't sure about my lows. One day, I decided to just go with it. And guess what? They got it. They write themselves a little negative number line right there and miss way less problems. I saw our computation scores go up big time when I introduced this to them.
If you have students struggling with subtraction, I urge you to try out one of these ways. If you use diligence, you might be surprised how much your students grow with understanding.
Here is a free sheet that you can use to help your kids practice using expanded form and negative numbers to subtract. :) Click to download.
Happy Teaching!
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