 Hello, my name is Pat Lohr and I'm a member of the Edmonton Regional Learning Consortium Math Team. Thanks for joining me. Today I am sharing four examples of number strings based on the ideas of Kathy Fosnoe. They're designed to give you some ideas about how you could use strings like these to encourage your students to develop some good multiplication strategies. You're watching the second of two parts and in part one I shared two sample strings that used an array model for multiplication. In part two I'm going to discuss two more strings but both use variations of an open array model which is something a little different. 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 couple of pages 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 these ideas during the webcast. The four sample 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. And I want to remind you that the lessons are not meant to replace concrete experiences and contextual kinds of problems that you're already using with your students. What they're designed to do is hopefully help your students move on to more flexible and abstract ways to work with numbers. In this sample string I want students to begin to develop an understanding of open arrays and how they could be used to model multiplication, especially with larger numbers. I use strings like these as many lessons, maybe at the beginning of math class or right after lunch, whatever works for me. I like to do the writing and modeling myself in a mini lesson because it really should move along pretty quickly. I'll be using a smart board in this webcast, but you certainly don't need a smart board to use number strings with your kids. It does have some nice features that I'll be making use of, but I won't be doing anything that you couldn't do with an overhead projector or a blackboard. So let's have a look at the strings. In any string of problems I do with my students I always want to leave the previous questions and their answers displayed so my kids can use what they know to help them figure out something they don't know. But with this string, because of the amount of space taken out by the later arrays, not to mention the grid overlay, the previous questions aren't on the next slide. However, it's still really important for students to be able to see the answers to those questions as they move along the string. So I'm going to record each equation on chart paper or on an adjacent whiteboard or blackboard. You won't be able to see that, you'll just have to take my word for it. I've got the grid right here because moving to an open array is a big stretch. And I think that this might provide a nice support for that. For each question in the string you'll ask the same basic questions. What's the answer? What was your strategy? Did anyone think of it in a different way? Each array is proportional so you can drag the grid over the array during the discussion if you want. Use a pen to divide the open array into sections with dimensions labeled to model strategies that the students suggest. This one's pretty easy, but your students will still probably suggest two strategies, seeing it as two groups of four or as four groups of two. I always want to begin a number string with a problem that I think is within reach of even my weakest student. So that student can join the conversation in a meaningful way. Then I'm going to move on to more and more complex questions until I think I'm pushing my strongest student at least a bit. It isn't about having everyone reach the same place at the same time. It's about everyone being able to take one step forward from wherever they are. So here's the next question in my string, four times ten. Don't let a student get away with suggesting just add a zero to the four. I tell them in my world four plus zero is still four. It's four tens, so that would make a good strategy. Someone else might see it as four times five and another four times five. Now maybe that's not a strategy they'd actually use to find the answer to this question, but it's great for your students to realize that. And I might pull the grid over to make that clearer. Here's the next question. And again, I'll show you how I might divide the array to model my kid's strategies. For example, if a student thinks of four times twenty as four times ten and four times ten, divide the array down the middle and maybe label each as a four by ten array, equalling forty each for a total of eighty. This string moves on to eight times twenty, ten times ten, ten times twenty, and my challenge fact, twelve times twenty. Lots of multiples of ten and I'd probably have my students look over the string at the end to see if they notice anything interesting. I'm happy if they notice the connection between eight times two and eight times twenty. I just don't want to add a zero to be some mindless rule that they memorize. The second sample string uses equations only and now it's your job to sketch an open array to model the strategies as they're suggested. This takes some practice so I have included a few sample models on the last page and there they are. Open arrays are a bit like open number lines. There's no need to be particularly accurate in your proportions when you're drawing them. Just do the best you can. But before you present a string like this to students, you need to think about what strategies they might suggest and be ready to model them. I always model the equation as height first times width, like this. Remember for each problem you ask, who knows the answer to this question? What was your strategy? Did anyone think of it in a different way? The questions move from simpler to more difficult and before I made the string I had the last question already in mind. I have to think about which easier questions will support my kids as they move on to the next harder one. I know some of them are going to drop off along the way but that's okay. So for three times ten I'm expecting someone will notice that it's three groups of ten. And the answer is thirty. Let's have a look at the rest of the questions. I'm going to present them one by one and the previous questions with their answers are right there for support. When someone shares a good strategy I might ask someone else to say it again in their own words or maybe I'll ask them all to turn to a neighbor and explain the strategy to them. There's no need to leave the arrays on the boards so I'm going to erase this one to give me a little bit more room. Again I draw the array and label it but now I'm expecting that someone will give me the strategy that they knew three times ten was thirty and they know that three times two is six and they can use those two ideas to give them the answer of thirty-six. The next problem is ten times ten and ten times twelve. Remember I'm looking for more than add a zero to the twelve. So ten times twelve might look like this. A ten times ten and a ten times two for the answer a hundred and twenty. Then we have thirteen times twelve and finally twenty-three times twelve. So you can see again how those are increasing in complexity and ending up with a problem that I think is really going to push my strongest kids. Remember modeling with open arrays is probably a new idea for you and your students so you might want to practice modeling a string before you actually do it with your kids. So that's it for part two. If you missed part one be sure to check it out. And if you found any of this intriguing I highly recommend you check out Kathy Fosnoe's work. Remember to keep the mini lessons quick and don't expect everyone to master the strategies the first time you present them. We want our kids to see the big ideas behind number facts and be able to work flexibly with numbers. We definitely do not want them to believe that rote memorization is the key to math power. And thank you for spending the past few minutes with me.