 Hello, my name is Pat Lohr, and I'm a member of the Edmonton Regional Learning Consortium's math team. Thanks for joining me today. I'm going to share four examples of number strings based on the ideas of Kathy Fosno. They're designed to give you some ideas about how you can use strings like these to encourage your students to develop good addition strategies. You'll be able to download the files so you can adapt them and make use of them in your classroom if you'd like. I've discussed the first two strings, which make use of 10 frame visuals in Part 1. In Part 2 we'll be looking at the last two strings, which make use of an open number line. This professional learning opportunity is made possible by a grant from Alberta Education to support the implementation of the revised mathematics curriculum. The file has a page of teaching notes with information about the math behind the strings and suggestions for presenting the problems in the classroom. I'll be talking about some of these ideas during the webcast. The four examples of strings are definitely meant to be a starting point for you, and I encourage you to make use of the ideas to create some number strings of your own. I think they'll help move your students on to more abstract ways to work with numbers. So now we're going to have a look at the third number string. This one is designed to help students who have used number lines at school already. It'll introduce them to the idea of an open number line, one that doesn't include divisions and labels for all the numbers. In this string, I'm not using a completely open number line. I'm saving that for the fourth string. If you've already watched Part 1 of this webcast, you may notice that I'm using the exact same string of problems, as I used in example 2, adding five facts. This time, instead of a 10 frame model, I'm going to use a number line marked only at intervals of five. A number line is a powerful model for adding and subtracting numbers. And by removing some of the divisions, I'm trying to push my students beyond counting one by one to find the answers. I always want to start any number string at a place where I'm sure my struggling kids can be successful and try to go to a place such challenging for my strongest kids. It's about helping everyone in the class move forward, rather than trying to get everyone to exactly the same place at the same time. For each example, you're going to ask, who knows the answer? How do you know what was your strategy? You write the answer when it's given and do your best to model the student strategies on the number line. If a student suggests starting at five and then counting on five more, model it like this. But then ask, does anyone have a different strategy that doesn't include counting on one by one? You're really trying to help them move beyond that. Likely, a student will suggest that they could count by fives or know that two fives make one ten. Again, I would model that like this. I suggest you do the writing on the board because it really should be a mini lesson and you don't want it to go on too long. The next question is five plus seven. Notice that I'm only presenting one question at a time and leaving the previous questions and answers on the board. I want to encourage them to use the facts they know to help them with facts they don't know. I'm going to use the same number line, though, so let's erase the work from the last problem. Again, model strategies as they're suggested and ask, did anyone think of it in a different way? Acknowledge a counting on strategy if someone suggests it. But by the second or third problem, I might not actually model it. Remember, we're trying to get them to move on beyond that. Someone might suggest that they start with the five, add another five, and then two more to get twelve, or you may have a student that starts with the seven, adds three more to make the ten, and then two more to get to twelve. Both of those are great strategies. The next problem is five plus six. That's a near double. You might have a student suggest that six plus six is twelve, so the answer would be one less. Model that one like this. So we have six, and six more to make twelve, and then we're going to take one away to give us eleven. When a good strategy is shared, consider asking someone else to restate it in their own words, or ask them all to turn to a neighbor and explain the strategy to them. Next is five plus nine. You can see how important it is that students have a strong understanding of five and ten as anchors, as well as how to take apart numbers. As long as students know that nine is five and four more, then adding the two fives and then the four is a possible strategy. If they know that nine and one more is ten, and that fives of one and a four, then that's another great strategy to start with the nine and make a ten, and then add four more. This also requires that they know five plus nine is the same amount as nine plus five, and that's not always as obvious to our students as we think it is. So this is why my last question presents the problem with the five in the second spot. I'm wondering if anyone will still think of taking the eight apart to make a five and three. Of course, that's not the only good strategy for this one. And there's an even more open number line with only the multiples of ten marked. Move to this model when you think your kids are ready. And I've included a couple of sample number lines with the modeling there for you to have a look at. Okay, let's take a look at the fourth number string. In this one, I'm using a completely open number line, and I'm going to move on to multi-digit addition. One of the reasons we want our kids using flexible strategies for addition basic facts, rather than rote memorization, is that those strategies will be just as powerful and meaningful when it comes time to add larger numbers. This number string is similar to many I've used with my grade five and six students, because they often lack strong number sense, and many lessons like these give us a chance to talk about the concepts they might be missing, like those five and ten anchors I keep talking about. This string of problems will focus on adding with nines, because there are some great strategies and I want to make sure my kids can use them. Again, I start with the fact that I think is within reach of all my students, nine plus seven. You can see that this number line is truly open, no markings at all. I ask the students to show me a thumbs up when they have an answer, and when most are ready, I ask, who knows the answer? What was your strategy? Who thought of it in a different way? I know I sound like a broken record. If someone says they know that nine and one more is ten and six more is sixteen, I would model it like this. The nine and one more to make ten and six more to make sixteen. If someone says I move one from the seven to the nine and turn it into ten plus six, you might want to actually record the nine plus seven equals ten plus six. And ask your kids if it works every time. Someone else might say that I know ten plus seven is seventeen, so nine plus seven is one less, and I might model that one like this. So we do the plus seven and then we take away the one. Ask your kids, why are you doing it that way? Why does that work? This string continues with nineteen plus seven, which is moving us into a two-digit example, and you can start to see the power of the open number line. No need to get fancy, no need to be perfectly proportional, and it's very easy to model any number of strategies. Twenty plus seven minus one, nineteen plus one plus six, even seven plus three plus sixteen. When I ask students to suggest other strategies, they think of all kinds of possibilities. It's that kind of flexible thinking that I want us to encourage. Here are some other problems in my string. Nine plus twenty-seven, ninety plus seventy, and seven hundred plus nine hundred. I hope you can see how I'm trying to make connections and help everyone move ahead in their thinking. How does what you know about nine plus seven help you with ninety plus seventy, or seven hundred plus nine hundred? Never settle for an explanation like, I just do nine plus seven and add a zero. I tell my kids that, in my world, sixteen plus zero is still sixteen. The last one is my challenge problem. It's probably out of reach of lots of my kids, but that's okay. Everyone can benefit from the discussion. With any of the questions, let your kids jot down intermediate steps if they need to. But remember that this is all about personal strategies. And that's it for part two. If you haven't had a chance already, please check out part one. Again, here are a few sample modeling strategies for you. I owe a debt to the ideas of Kathy Fosnow, and I do highly recommend that you check out her work if you found this interesting. Remember, keep the many lessons quick and don't expect everyone to master the strategies the first time you present them. We want our kids to see the math behind the facts, build strong anchors of five and ten, and work flexibly with numbers. We definitely do not want them to believe that rope memorization is the key to math power. Thanks for spending the past few minutes with me, and I hope I've given you some food for thought.