When we switched to the Common Core a few years ago, I had to rethink what I thought I knew about teaching addition and subtraction. One of the things I thought was that kids needed to learn the algorithm for addition/subtraction (stack the numbers vertically, carry a one if adding, regroup/borrow if subtracting, voila) and practice it until they mastered it.
I thought they needed to learn it because, well, that's what the other teachers told me; because that's what I learned in my education classes in college; and because that's how I added and subtracted.
So that's what I did, and it mostly worked.
Oh my goodness.
The Common Core came along, and my district created guidelines for teaching (and sample learning tasks, and suggested lessons, and pacing guides, and and and and), and lo and behold, those sacred algorithms were nowhere in the second grade curriculum. Or, as it turned out, the third grade curriculum, but I didn't realize that until my own daughter finally learned them - kind of as an afterthought - in fourth grade. FOURTH grade.
But what was I supposed to do in second grade?
Well, as it turns out, I now do magic in second grade.
Seriously. Those "low," struggling math students who still need their fingers to add 4 + 2? They will be adding 48 + 36 in their heads soon. Those "I don't get it," "I can't do it," whines? Gone. Replaced with mentally solving 55 + 27 and answering confidently.
We began with base ten blocks. I laid them on Elmo so they displayed on the Promethean board, and wrote the matching number sentence on a whiteboard.
(Was I smart enough to take pictures? No. I'm still new enough to this blogging thing that I do NOT take pictures of the things I plan to write about later. Brilliant.)
So I wrote something like "22 + 14" on the whiteboard, modeled it with base ten blocks, and we counted up the total. Yawns all around. "This is so easy."
I spiced it up: "36 + 18," models, quiet and furious counting, shouts of the answer.
Five minutes, tops.
Because that was just the warmup, and now it was time for the meat.
Step one: use hands-on models and manipulatives.
Our warmup only lasted about five minutes or so because I want to get those tens rods and ones cubes into the kids' hands. But I want them to model their answers in a very specific way.
They begin with a partner and a set of red and blue base ten blocks. One partner uses blue to model the first number in the equation on a hundreds chart; the second partner uses red to model the second number on a different hundreds chart. Here's what 45 + 33 looks like:
Then they slide the red number up to join the blue - it's adding time!
But there's a problem:
We talk about how when we did the warmup, we would count all the tens and then count on by the ones. In the blue/red model above, we're counting tens, then ones, then tens, then ones - and we've left a gap. We know 93 is not the right answer - and the kids know that the red tens should go right after the blue tens.
So they slide those five blue ones out of the way and put the tens together first, then add the ones underneath.
Voila. We are adding two-digit numbers.
I LOVE this strategy because 1) we can still see the two original numbers: 45 in blue, 33 in red; and 2) we are grouping the tens together and then grouping the ones together, so essentially the kids are learning to add the tens first - which is very efficient and much easier to do mentally than trying to visualize the algorithm and keep track of all those numbers!
Next, they record their answers on a smaller hundreds chart. This step is crucial because it reinforces the "add the tens first, then add the ones" process.
|(This sweetie was going for speed, not neatness . . .) :)|
Now here's the fun part. I made six pages of these problems. There are eight per page (front and back). But guess what? Those kiddos find the shortcut pretty quickly. :)
Day 1: work with a partner to model the numbers and add the blocks on a shared work space. Record your answers on your own papers.
Day 2: work by yourself to model the numbers and add the blocks on your work space. Record your answers.
Day 3: repeat.
Unless you're one of those kids who asks, "Do I have to use the blocks? Can I just color?"
Teacher (grins to herself) asks the child solemnly, "I don't know. Can you picture what the hundreds chart will look like when you model those two numbers?"
Child nods fervently.
"Can you color all the tens, then all the ones, in the right colors?"
Child nods fervently again.
Teacher says slowly, as if considering this idea for the first time ever, "Well, I don't know. Why don't you do the first two problems and then come show me."
Child leaps away and returns moments later with correct coloring and answers.
Teacher studies the paper very seriously, murmuring, "I can see the red 42 and the blue 37 . . . yes, that looks good, and there's the correct answer in the blank . . . and on this problem I can see where you composed a new ten with all those ones . . ." Teacher looks at the student and says seriously, "It looks to me like you know what to do. Will you do all the rest of the problems the exact same way?"
Child nods excitedly.
"Well, okay, you don't have to use the blocks today . . ." (As the child dashes away, Teacher does a very quiet happy dance. Teacher WANTS the children to be able to visualize how the numbers will be added WITHOUT using the blocks, and now that one student has reached that point, Teacher knows that more will follow. "Hey, he's not using the blocks. Do I have to?" Step one is almost complete.)
Days 4 and 5: repeat.
If, on day 4, I still have children using the tens and ones blocks on their work space, then I'll call them over to work with me and ask, before they begin modeling, "Can you picture what it's going to look like when you color?" The child always responds, "Yes," and I encourage him/her to "go ahead and color before you model." Then we talk about how we can see the two numbers in their different colors - and we can see the final answer - and the child is elated that it took so much less time. :)
Day 6: send the final page home for homework! By now the kids are so quick to color the tens first, and then the ones, and reach the correct answer, that I am confident they can complete this homework by themselves without any parent help.
Here's a preview of the packet:
You can pick up Adding with Base Ten Blocks in Two Colors on Teachers Pay Teachers for only $3.00! What a steal! ;)
Step two: draw representations of the models.
Now that we understand the concept of adding tens and ones, we move to drawing our own tens and ones. We start out on whiteboards and move to paper once I am sure that every child is setting up the problems just so.
And by just so, I mean, MY WAY. Because I have reasons. And if they don't do it MY WAY (which is logical and organized and reasonable), there will be confusion and mistakes later.
I've learned over time, you see.
We start by dividing our whiteboards into spaces tens and ones:
Then we draw the two numbers. The first number (34) goes at the top of the whiteboard, tens on the left, ones on the right.
The second number (22) goes in the MIDDLE of the whiteboard.
We count up what we've got on both sides, and voila, we're adding two digit numbers again.
MY WAY: Oh, you drew your tens too close together? Erase and draw them again with at least a finger or two of space in between. (Important when we get to subtraction later, so we start modeling it the right way now.) Oops, you drew your numbers so big they filled up your whole whiteboard? Erase and draw the first number at the top and the second number in the middle. No, not at the bottom. Erase and draw it in the middle. The bottom should be blank.
(Geesh, this teacher is picky.)
(Yes, I am. Thank you. Now draw it the right way.)
Sometimes we have to compose a ten, right? So again, we draw the numbers at the top and in the middle:
And then we circle ten of those ones, draw an arrow over to the tens side, and draw our newly composed ten UNDER the other two numbers. (Now the children begin to understand why they had to leave space under their second number!)
Last, we write the value of each side and put them together for our final answer.
Sometimes we talk about how the top number is the "red" number and the bottom number is the "blue" number, and about how that new ten is the two-colored tens row on the hundreds chart, because I want them to keep visualizing the tens grouped together and then the ones grouped together. We'll also revisit "red tens" and "red ones" and "blue tens" and "blue ones" with our next strategy, so keeping that fresh in their minds is helpful.
We'll come back to these drawings when we subtract, but that's a post for another week. ;)
Step three: use a number line.
I don't know about you, but most of my second graders are not ready to jump into drawing their own number lines right away. They need some structure and scaffolding first.
So I made them some number lines, and together we figured out how to add using the number line to help.
This student is working on a page from my Adding 2-Digit Numbers Using a Number Line. It has five pages of Teacher's Notes (i.e. lesson plans) that model different ways to add using a number line.
For example, we can:
- Start at the first number and jump the tens and ones of the second number (start at the red number and jump the blue number);
- Start at the second number and jump the tens and ones of the first number (start at the blue number and jump the red):
- Start at the TENS of the first number and jump the TENS of the second number, then jump all the ones of both numbers (start at the red tens, jump the blue tens, now jump the red and blue ones);
- and more.
For some students, this number line idea is a challenging strategy. After some specific teaching and practice, however, it's "easy-peasy," as one student told me earlier this week. :)
I made six front-and-back student pages (in the picture above, the child is working on the back of one of these pages) where the number lines are marked at the ones, fives, and tens. These six pages give the students lots of practice trying out different addition strategies on the number line.
There are also six more front-and-back pages where the number lines are only marked at the tens (a nice segue into open number lines later in the year), which require a little more number sense and math strategy.
Here's the preview for Adding 2-Digit Numbers Using a Number Line . . .
. . . but you might prefer to get the BUNDLE instead, which also includes Subtracting 2-Digit Numbers Using a Number Line AND Adding and Subtracting Two-Digit Numbers on a Number Line and, of course, saves you money! I'll be writing more about how I use those packets in my classroom in future posts.
Step four: use a number line with only the tens marked,
so that students have to use their number sense to add the ones.
Here's one of the Teacher's Notes pages for these six pages of number lines:
As you can see, there are many different ways to use the number lines to solve these addition problems!
An eventual step five: draw your own number lines! We will use open number lines later in the year when working with 3-digit numbers, so sometimes I introduce the concept of open number lines while we're adding 2-digit numbers.
And that's how we add in second grade.
Magic, I'm telling you.
Next week: subtraction!