 Hi, my name's Pat Lohr and I'm a member of the Edmonton Regional Learning Consortium 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 could use strings like these to encourage your students to develop good multiplication strategies. You're watching the first of two parts and in this part I'm going to discuss the first two strings which make use of an actual array model. In part two I'll discuss the last two strings and they use an open array model. That's something that might be new for a lot of you so I encourage you to check out part two if you have the chance. 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 pages of teaching notes with information about the math behind the strings and some suggestions for presenting the problems in the classroom. I'll be talking about the 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 strings of your own. I want to remind you that they are not meant to replace concrete experiences or contextual problems that you're already using with your students. But I do think that they might help your students move on to more abstract ways to work with numbers. In the first string I want students to begin to develop an understanding of arrays because they are a really powerful way to model multiplication. I use strings as a short mini lesson with my students and I like to do the writing or modeling to keep things moving along. I'll be using the 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 do anything that you couldn't do with an overhead projector or a blackboard and some paper or arrays. Now in the first sample you don't see any equations because I'm going to write them on the board as students suggest them during our discussion. I expect you would use this string with students who are just being introduced to multiplication or maybe students who haven't had much experience with arrays. For each picture I'm going to ask my students, how many fruit do you see? How do you know? Can you find the total without counting them one by one? And did anyone think of it in a different way? You can circle groups of fruit and write equations to model the strategies students suggest. You can see the fruit are arranged in arrays so I guarantee students are going to make use of those equal groups and their strategies even if you haven't introduced multiplication yet. So if a student suggests that they see two groups of six apples you can circle it like this and record 6 plus 6 equals 12 or 2 times 6 equals 12 depending on what your kids are ready for or where you want to go with the lesson. Let them suggest as many different ways to find the total number of apples as they can. They may suggest four groups of three or six groups of two or they may have some other ideas as well. You always want to begin the number string in a place where even your lowest student can join in the conversation and then you try to move up in complexity to a problem that's going to stretch your strongest kids. What you're trying to do is help everyone move forward in their understanding from wherever they happen to be. You're never going to end up with everyone in the same place. The next picture has oranges and the 3 by 3 arrays are a little more challenging. And then I have a picture of two 5 by 4 arrays of tomatoes. So the numbers are greater but kids can start to make use of tenness like 4 times 10 or 2 times 20 and right away we're actually moving beyond basic facts. Now the second string is going to use equations with the arrays and in this particular example I'm going to try to nudge my students to think about how the 5 times facts could be used to help you with the 6 times facts. Now for each problem remember those questions. Who knows the answer? What was your strategy? Did anyone think of it in a different way? Model their strategies by circling sections of the array. So here a student might say I counted by 5s 4 times and I would model it like this and label the arrays. Another student might say I know 5 times 2 is 10 and then I just doubled it. So again I can show that on my array. Once you ask that question who thought of it in a different way you'd be surprised by what your kids come up with. Now most of the facts in this string are organized in pairs with the 5 times fact first followed by the related 6 times fact. In each case the extra row for the 6 times array is hiding on top of the bottom row of the 5 times array and you can just pull it down and into place before you discuss the 6 times fact. Again a little smart board trick but you could have the extra row folded back on your paper array. It's a not too subtle way to nudge kids to think about my target strategy but remember that in almost every case there are other good strategies as well. 6 times 4 could be modeled as 5 times 4 and 1 times 4 but it could also be modeled as 6 times 2 and 6 times 2. Both of those are great strategies for that fact. I present the problems one by one and I always leave the previous questions on the board for support. I want my students to use what they know to help them with what they don't know. If someone suggests a particularly good strategy why not ask someone else to restate it in their own words or you might have everyone turn to a neighbor and explain the strategy to them. So 5 times 3 is followed by 6 times 3 5 times 7 followed by 6 times 7 and then the final challenge fact to this sample string is 6 times 8 presented with no helper fact. The 6th row of the array could be pulled down though if you wanted to do that during the discussion. And that's it for part 1. In part 2 I am going to share a couple of sample strings that use that open array model. If you found any of this intriguing I highly recommend you check out Kathy Fosnows' 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 the facts and work flexibly with numbers. We definitely do not want them to believe that rote memorization is the key to math power. Thanks for spending the past few minutes with me and I hope you take some time to check out part 2.