Adding to 100

I asked some students today to add the numbers from 1 through 100. After fielding some questions and making sure the students knew we were adding 1 + 2 + 3 + … + 99 + 100, they were off!

A few tried the obvious… 1 + 2 + 3..the long way. Luckily I had only given them about 1/8 of a page to work on!

The way I tried to show was using some skills we learned back in elementary school…looking for groups of numbers that add up nicely. For example, add the list: 2 + 7 + 4 + 3 + 6 + 8. Did you get 30? I did…but I did some rearranging (2 + 8) + (3 + 7) + (4 + 6) and said 10 + 10 + 10 is 30. I used that, along with an abbreviated list of numbers, to get the sum:


By adding groups of 100, I was able to get 4900! Then there’s the extra 100 sitting on top, which brings us to 5000. The missing 50 brings us right up to 5050.

A student, though, took a different approach. Same idea, different way of looking at it. I was impressed!

I presented a modified version…he wrote out a bunch more work. I presented only the pattern that he figured out from looking at it.



4. Lattice Multiplication

We’ve made it…we can multiply small numbers using area, we can multiply small numbers pretending to do area, and we can keep pretending as we use bigger and bigger numbers.

Now comes full-blown lattice multiplication. Here we go.

How it looks at the end:


What IS that, you may wonder. It’s a mix of area model, with some old-school place value thrown in. Let’s break it down, and start over.


That looks just like the area model…except no zeroes! Ok, we can deal with that. We’ll just use place value later on. Let’s multiply like area model (but we’ll do it a little off-center).


It looks more or less like area, only with the answers in corners, and no extra zeroes. That’s basically it. Now we just need to add place values just like we used to. Only instead of straight up and down, we need to go at an angle. So…let’s add imaginary lines in red to cut each box in half.


Now we just need to add inside each band…they all correspond to the same place value, and we can carry (in red) to the next line.

lattice-finalFinally, the answer is there and we can read it off: 33,813!

It’s not intuitive, but if you take a little bit from all the other methods, then you end up with lattice.

Good luck!

**note: if you multiply, and the answer to go in a box is only one digit, put a zero in the upper corner:


3. Expanded Area Model for Multiplication

We saw previously that drawing pictures is a good way to think about multiplication.

Multiplying is really just adding a whole bunch of things together, as long as they’re the same size. The little boxes showed that, but it got a little crazy once we used bigger numbers.

So, let’s keep on with the pattern of pretending, by trying out 14 x 7. Since I don’t actually know 14 x 7, I will break it down: 14 is really just 10 and 4. So I’ll split that side length in my picture. Then I have two little rectangles…10×7 and 4×7! I can do that!
(and if you want to get really fancy/technical, we did just demonstrate the distributive law!)

unnamedWow, that was WAY easier than expected! Maybe we can apply that same logic to our original problem?

867 = 800 + 60 + 7
39 = 30 + 9


BAM! Still annoying…but addition is better than multiplication.

Now…how does this all help with the lattice model? That ugly beast of multiplication…

2. Area Model of Multiplication

Another way to do multiplication is to think about it as finding the area of some rectangles.

Now, if we draw pictures of area for multiplication, we can start to visualize it! Let’s pretend, over the next few posts, that I can draw squares and rectangles. And that I can take good pictures.

For example, 3 x 2 = 6 in the area model is…


Not too bad. But when we get to bigger problems, we don’t really want to draw out every box. It gets tedious even for small numbers, and worse as they get bigger. Who’s going to count?!? Instead, we rely on our basic math facts, and pretend that there are little boxes!

Displaying IMG_1063.JPG

It seems like we’re just reviewing our basic multiplication facts, right? Well, yes.

But the interesting part is in the next post, when we see how larger numbers (bigger than 10) mix with the area model to simplify things.

1. Lattice Multiplication: Introduction

Remember the good old days of long multiplication? You multiply, bring down, and carry. There are so many new methods, and lattice is one. It makes complete sense logically once you understand it, but it is one of the least intuitive structures. Check it out:


It’s crazy looking, and definitely not natural. Let’s get there one step at a time. Let’s solve a problem using our old-school math, and then figure it out using a few different methods. Eventually we’ll do it using lattice.


867 x 39

Traditional Solution:

Displaying IMG_1061.JPG

Not too bad…so familiar and comfortable. Next we’ll look at basic multiplication using the area model.

6b: Following Up When Students Aren’t Meeting the Expectations

We set up expectations for lining up, and most students are on top of it. But there are 1 or 2 students who are just keeping the whole class from earning stars. What do you do? Yell at them? Hold them in their spots in line? should follow up with them. Instead of holding back the class if there are a few students, you can adjust the requirement to 80% of students doing it. That will take the pressure off the students who are struggling. There’s nothing worse than having a hard time, and having everyone hate you for it.

T: Johnny, come have lunch with me today. We need to check in.

J: But why? I don’t want to! I hate getting in trouble!

T: We need to check in about lining up. It’s been hard, so I want to work on it. It’s not punishment…you can still go to recess.

[after whatever it takes, Johnny comes to you for lunch]

T: Hi. What’s going on in line?

J: I hate lining up. I don’t want to be quiet and just stand there. And XX keeps poking me, because she knows it makes me upset, so I just yell.

T: I want you to be able to stand in line and help earn those stars. What do we need to do?

J: You can move me next to YY! We’re best friends, and I’ll do great there!
T: I don’t know about that…you may talk there too. I can move you to another spot, next to LL.

J: I don’t like LL. I don’t think that will work.

T: Well, LL won’t bother you though. Let’s do this: if you can help earn stars 5 times near LL, then you can earn a separate reward (break, toy, candy, etc). I know you can do it – you control yourself the rest of the day.

J: Whatever.

It may not sound like it’s going to work (Johnny isn’t that excited). Will he self-sabotage? Maybe. But right before the next transition, check in with Johnny. Remind him that he’s working for his own separate reward too. If it works, keep it up! If not, try a few more times.

6a: Practical Implementation of Positive Reinforcement (with recording sheet)

In a training today (a little while ago now), my supervisor made a point:

Skill development is the best way to get rid of problem behaviors.

That really gets to the heart of what I want to say. Build new skills and reinforce them. Then those will happen more frequently, and problem behaviors will go away on their own (in the general case, excluding kids with specific needs).

For this post, we’re going to do a class-wide behavior, which could be an issue for many teachers: lining up to transition to other classes. This is appropriate more for elementary teachers (probably up through grade 5, though older groups line up too).

Often students fool around, don’t stand in a straight line, change places, etc.

Here, we’re looking for (1) a straight line, (2) in line order, and (3) kids are quiet. Also, it shouldn’t take more than (4) 1 minute (this will definitely change over time)

This represents only one possible way to work with a general classroom of kids. Tweak/adapt for your group!

Step 1: Baseline (where the kids are right now)

One common mistake when doing a baseline for a group behavior is to tell students what you’re looking for, and to do it at a not-quite-natural time. Maybe during morning meeting, you tell them you want to see what they can do, and you have them line up. The problem is that it’s not a real transition: it’s practice lining up. They know what to do. What you’re checking is where they are: what do they actually do.

Now, we need to track it in some way. Get out your stopwatch and clipboard! I would use the sheet here (google docs) – it’s short, to the point, and will work for a week’s worth of lines.

Step 2: Tell the students what you’re looking for!

Now that you’ve seen what they do, check in with the students! We’ve mentioned this a few times. Here’s a sample script for a morning meeting check-in:

T: Good morning!

Ss: Good morning!

T: You may have noticed my clipboard yesterday afternoon when you were lining up…any ideas why I had it out?

S: We’re bad at lining up…you want to get us in trouble!

T: Well, not exactly. But close. We’ve had a lot of trouble transitioning, so I wanted to see how we do. We’re going to look at it, and see how to get better. (take out data sheet – projected, or copied onto a large paper for the class)

S: There’s no way it took us 5 minutes to line up! We were in line fast!

T: You got in the area in a minute, but it took 5 minutes to get into the line we’re supposed to be in. We’ve gone over the expectations, so that’s what I was looking for.

S: So you’re saying that we’re bad at lining up!

T: Not quite. I’m saying I want us to work together on lining up how we’re expected to. I’m going to use this data sheet when we line up. If we meet 2 out of the 3 goals (in the allowed time), we’ll earn a star on this chart. Once we have 20 stars, we’ll earn a reward!

Step 3: Follow Through!

Now that you track data, give them stars! Try to make it as easy as possible for them…you can give reminders, have them practice during the day if a lesson finishes early (tell them it’s practice, so that now is the time to make mistakes). That way, they get it. The more practice they have, the better it is.

When students go to specials (gym, art, music, etc) you can pass the sheet along for those teachers. Not only is it consistent (which will help the kids), but I’m sure the teacher who only sees students once a week will appreciate it too!

Step 4: Shape it Up!

Once they get 2/3 consistently and earn the first reward, make them work harder! Either decrease the time to 45 seconds, or add a fourth behavior requirement (and they have to earn 3/4). Or track how much help they need – if they do it independently, they earn 2 stars. If a teacher has to remind them (but they do it), they earn 1 star.

Adjust according to the class’ needs, and slowly but surely they will do awesome! And they’ll love you too, because you are so nice to them too.

5. Dealing with Negative Behaviors

When something goes wrong, the student knows it. You’re not surprising them when you address behaviors. At first they may seem resistant, argumentative, and like they don’t want to do any better (even though we assume that they do!). What, then, is the next step?

Whenever you set up a follow-up time, you absolutely must follow up at that time. Below are guidelines/steps for following up with behaviors. I’m offering the following suggestions based on many different things: my experience going through CPI training, and a great book I read while at a previous job.
But really, and most importantly, these ideas come from the fact that I have had the privilege of experiencing wonderful teachers who have shaped it all. I have worked with them, and collaborated with them outside of school. Every one of them has taught me something important. This is just my synthesis of what is really their hard work.

Addressing Behaviors – Guidelines

  1. Make the discussion about what happened here. Not about how he always does this, and always does that. Keep it to what happened.
  2. Caveat: if it is part of a larger pattern, you can bring it up appropriately. But keep it focused on this time, and the future. (e.g. “I’ve noticed we’ve been having this conversation a lot lately. This can’t keep happening…” Now you’re bringing up the past, but focusing on the future)
  3. Make it about you, the student, and the class. Abstract moral discussions are for another time.
  4. Make it about you, the student, and the class. The more personal, the less room for argument. And it’s on the list twice because it’s that important.
  5. Focus not on the negative behavior, but on the replacement behaviors. Your expectations for what to do instead.

Addressing Behaviors – Process

Here is a short sample conversation related to addressing behaviors. It keeps in mind the guidelines above, and will structure the conversation without limiting it.


  • Address the concern from your perspective. (Don’t lead with the open-ended, abstract “What happened?” Give your point first, and do it directly)
    • EG: When you banged the desk and swore, that really upset me. I was trying to run a class, and it got derailed. That can’t happen.
  • Ask for the student’s input, and (the hardest part): LISTEN TO IT.
    • EG: I’m assuming that wasn’t how you would normally react. What happened?

Side note: listening to the student’s side does not excuse the behavior. They will probably have a great reason (I was exhausted, and the student next to me had been poking me with scissors all day, and etc etc) and you can sympathize. And you can say that. The reasoning may make perfect sense, but the behavior still can’t happen.

  • Discuss a plan for making it up.
    • Maybe the student apologizes to the class, maybe the student comes during recess or lunch, maybe the student does detention with you. But make it meaningful: it’s not a punishment, it’s a way for the student to make up for a mistake. They can clean, organize, grade papers, help develop a class-wide plan for addressing behaviors that s/he presents to the class. The possibilities are endless.
  • Follow through, and make sure the student follows through.

It’s challenging, logistically and emotionally. But things will get better! By engaging the student in solving the problem, you can often find out what motivates him/her and how to address it. But you do have to be committed to making the change as well!

Coming next: A practical way to set up positive reinforcement in a classroom.

4. Preventing Negative Behaviors

I’ll tell you right now: the title is somewhat of a lie. I am not going to discuss ways to prevent negative behaviors (at least, not directly). Instead, it will continue the expectation/motivation factor. The real goal is to promote positive behaviors (but that wouldn’t be quite as exciting a headline).

I remember thinking, and I have spoken and got the same impression from other teachers, that punishment and reinforcement are two different things. One is used to eliminate bad behaviors, and the other is an abstract notion based on charts and individual plans that take too much work to get going.

I am happy to say that I, and those others, were wrong.

How? Let me explain.

Why Positive Reinforcement

Building a positive reinforcement system is, at first, difficult. You have to decide on and teach the behaviors you want to see, you have to have reward systems for it, and you have to follow through on it. Punishment is easier: you see a bad behavior, and you squash it. Sounds easy enough to me.

But it doesn’t work that way. If I want you to walk down the hall, and you’re always running, then an interaction may go like this:

T: Stop running!
S: OK! [starts skipping]
T: No skipping!
S: OK! [stats hopping]
T: Fine! Come sit in my room at recess!

What’s the problem here? The students was supposed to walk. But in that snippet, the teacher never really said to walk. I know, it’s been said before. And it’s come up before. And we can’t just give every student a prize every single time they walk in the hall. But imagine the rapport between the teacher and student now…things are not good. And the hostility from this may lead to more problems in the future. Who knows.

Positive Reinforcement: A Primer

The title says that you’re reducing negative behaviors, but you’re increasing positive behaviors. So how does it all come together? The idea is to determine appropriate behaviors for certain contexts. Students shouldn’t run in the hall…does that mean running is negative? Absolutely not.

So, what to do then? Here are some rough guidelines:

  1. Find time to check in with students. As a whole class initially, then individually with those who need extra support. This should involve laying out the routines and basic expectations.
  2. Find time to check in with students (part 2). As students make progress, you can revise and edit any expectations. You’re not stuck with it forever – if things come up, add to the list. If things aren’t a problem, let them go.
  3. Reward students – this usually involves some kind of check-in. Whether you give checks, tokens, or just meet about it once in a while – you must reward students. If positive behaviors don’t get them any rewards for being there, then they’ll do things they find rewarding. Like running. Reward: fun.

Ultimately it comes down to communication. The more frequent, the better (though not so frequent that it’s every 5 minutes and you never get to class). The rewards can be simple: students can bring lunch and eat with you. You can bring in treats or a special lunch for them. They can get extra recess minutes, or a break from a long/difficult activity to draw. Be creative – Pinterest is full of great ideas! Don’t reinvent the wheel!


The goal of positive reinforcement is to make doing the right thing worth it. After a while, you fade out rewards. Make them work harder for them. It works. You’re teaching students better behaviors for situations – but still give them a chance to do the crazy things! We can’t always assume they get to do it at home.

Next up: dealing with negative behaviors and consequences. Positive reinforcement reduces negative behaviors, but not 100%. Just like communication is key in positive reinforcement, it’s key in dealing with negative behaviors.

3. Set Up Expectations and Routines, Not Rules

I’ve read in a few books and heard in many classes, as I have mentioned, that limiting rules to a select bunch is good. As you may guess, I disagree. I feel that starting with a list of rules is very limiting.


Also, setting up very specific consequences (if you do X, then Y will hapen [insert call home, office, etc] ) can get out of hand – if you don’t follow through, then you look bad. And if you do, then does the punishment really help the behavior in the long run? My guess: no.

Try this instead:


No rules. Instead, make a list of routines. What are your requirements? How do you want things to go? For example, turning in homework. Where, when, and how should students turn it in? Then teach them the process. For many teachers, the “I thought I turned it in, I gave it to you during class, etc” gets out of hand. Tell students how to do it, and that’s it. Then when there’s an excuse, you can respond simply: “Since you know the routine, I’m sorry but I can’t be responsible for losing it even if you did give it to me in the middle of the lecture as we transitioned to the quiz and 10 students were asking questions, which is when you decided to hand me a half-crumpled and ripped piece of paper.” Hopefully you don’t say all of that every time, but you get the point.

Routines are the background processes. Without them, nothing else gets done. It’s how you file into class and sit, it’s how to transition between classes, it’s how you turn it and pass back homework. Put the onus on students, tell them that it HAS to be done that way. Not because you’re strict and arbitrary, but because any other way will be too distracting and the class won’t function. I’ve heard these called non-negotiables. If you hand me homework mid-class, I will lose it. I don’t negotiate that. I need you to start and finish the warm-up in 10 minutes, since that helps me figure out how to frame the rest of the activity. Must happen.


Expectations are broader. Rules are strict and limiting, and don’t necessarily allow for flexibility in how different students approach and handle things. Expectation 1: Work when it’s time to work. 2. If you finish early, help other students. 3. Walk appropriately in the hallways. 4. Speak at the right volume for the activity.

“But students will try to take advantage of loopholes and play games and it will lead to arguments!”, I hear people screaming. Maybe. At first. But it leads to a broader discussion: what is really appropriate? Students know it’s not appropriate. Don’t argue, don’t discuss it right away. Tell them. “You do know that’s not appropriate, and if you want to talk about it, then see me ____ [lunch/recess/after school/study block].”

I am also going to add a possibly unpopular sentiment here: the middle of class when a behavior happens is NOT a teaching moment. Instead, you keep it in your back pocket. A student got upset and slammed his desk and walked out. You want to have a rational discussion about expectations? Good luck! (but not really, because don’t do it!) What to do: tell the student: “You know that you’re expected to work appropriately, and I feel you didn’t do that. I’ll see you at lunch/recess/after school/Friday advisory to discuss it.” NO argument allowed (you need to move on). What you MUST do is actually follow up with the student. The immediate consequence may be that the student now sits and works, maybe goes to the office to work away from whatever caused the scene, etc.