 Hi, my name is Sandy Bacus and today we will be exploring multiplication using Touch Times. Touch Times is a free iPad app that can be found and downloaded from the App Store. It is designed to introduce primary students to multiplication in an alternative way to repeated addition. Repeated addition tends to become firmly entrenched as the dominant perception of multiplicative situations for students with the move to larger whole numbers, fractions, decimals, percentages, ratios, and proportions. Multiplicative reasoning becomes key to many mathematical situations that students encounter in middle school. Rather than rely exclusively on repeated addition, approaches need to be developed and implemented in the early grades that highlight the function aspect of multiplicative reasoning that is so critical to future success with mathematics. Touch Times has two worlds, the graspify world, which is the focus of this video and the zappify world. In graspify, learners use their fingers to create pips and pods. Recent research shows that fingers can be powerful tools for understanding and expressing number. The intent of the Touch Times design is for learners to notice the relation between the number and color of pips and the shape and content of the pods. As this is the basis for the multiplicative operation. Touch Times takes care of the multiplying, both in terms of making sure that the pods are reflective of the number of pips, and in terms of ensuring that the equation at the top of the screen corresponds to the pips and pods that have been created by the user. Let's have a look at additive thinking versus multiplicative thinking. Multiplication includes a variety of skills and abilities that all center around relationships of quantity, while counting and additive reasoning start at zero and focus on a linear accumulation of value, such as adding three four times to get 12. Multiplication involves bundling and manipulating quantities to elicit more complex relationships. A child thinking multiplicatively might propose doubling five to get 10 or partitioning 10 into two groups. A significant multiplicative skill is the idea of unitizing or grouping, which is the ability to consider a set of countable items as a single countable item, such as grouping a set of three and counting it as one as in the diagram. This then becomes three four times. And finally, a single product of 12. Let's have a look at how Touch Times shows this multiplicative model. Using Graspify, students can explore double unitizing first by making a set of countable items such as three and making a single unit of three. This can then be iterated or copied multiple times, and then Touch Times unitizes again by encircling the pods to make a single product of 12. Graspify also enables children to change one unit and see the change spread across other units, which is another significant multiplicative idea. This blue pip has now spread across each and every one of the pods. Let's have a look at some tasks that can be used with Touch Times in your classroom. The first tasks involve doubling and having. So I might say to my students, your challenge today is to double the product so that it is six. What is a product again? Yes, a product is the math name for a multiplication answer. See if you can figure out how to make the product six by changing only the number of pips. Once you've done that, double the product again to make 12, then double it again to make 24. You want to ensure that they've doubled the pips. Let's begin by increasing the number of pods. And we really want to draw attention to the spread effect of the pips within the pods. So in order to double the product from six to 12, I am going to double the number of pips from two to four. I'm going to double the product again. I'm going to double the pips from four to eight. And now I have a product of 24. I often like to get my students to work on this in partners. First of all, it encourages them to discuss the mathematics that they're seeing and experiencing. And it also just gives them more fingers to use to create larger numbers. If you successfully completed this task, I often pull them together and ask questions like what happened when I increased the number of pips? What color is the pit that I just made? Where else do you see pips that are that purple color? Do you notice anything about the shape of the pods? Again, we want students to notice what is happening in the pods. You can point to the array button. If I push this button, it will create an array. Notice how touch times organized two times three is six into an array with three rows and two columns. There are three purple pips in one column and three more blue pips in the second column, which shows three doubled. So when we increase the number of pips to double the product, were we adding more groups? No. What was happening instead that caused the product to double? Asking students to draw what's on the screen also helps to draw attention to how the pips and the pods are related. Let's look at a second task. This time let's have a look at a halving task. Right now, the product is 20. How can you have the product so that it becomes 10? Your challenge is to have the product by only changing the pips. Once you've done that, have the product again to make five. So it's not possible for students to have the product by trashing the pods and to get a product of 10 with these particular numbers. So in order to have 20, they're going to need to have the number of pips from four to two. If they'd like to have the product of 10, they are going to have the number of pips so that there's only one. Now they have a product of five. Early finishers can be challenged to try halving the product of eight times five is 40, or be sure to try an impossible question, like three times seven is 21 and ask them to have that. Once students have had a chance to complete the task, you can bring them back together and ask them questions such as how is doubling different than halving? Did you notice anything about halving that reminded you about what you've learned about doubling? Let's have a look at a final few tasks beginning with how can I make a single pod of five and then moving from there to skip counting by five using touch times. Your task is to figure out how to make a single pod of five. A single pod of five can also be called a five pod. This many countable items to one item seems easy, but children don't initially find this easy to do or easy to understand. They'll often start by making five one pods. So once they've created their five pod, I can bring them together and ask questions like predict what will happen if we put one more pip down? What happened to the shape of the pod? What do you notice about the colors? If I put one pip down and then take one off, what happens to the color and the shape? Right now, I have five times one equals five. If I make another pod, I have 10, 15, 20, 25. I'm showing you how to skip count by five by changing the number of pods. I want you to skip count by five by changing the number of pips. If you're here for students to do that, they need to think about how many pips they're going to put down first and then how many pods they're going to put down. So in this case, I need five pods. Right now, my product is five, 10, 15, 20, 25. How is skip counting by pips different than skip counting by pods? Here is a summary of the key multiplicative ideas that are emphasized by touch times. The first is unitizing, which is the ability to consider a set of countable items as a single countable item. And this occurs when one hand makes a set of pips. Multiplying, which is simultaneously creating multiple versions of the original unit, happens when the other hand makes a set of pods, either all at once or each time additional pods are iterated. Co-varying, multiplication is a varying of two quantities. So when one factor is varied, the product co-varies with respect to the other factor. So co-varying or altering the pips affects each and every pod as well as the product. Spreading, this can be seen when a unit is varied and the variation can be seen to spread across all the other units. So as we've seen when you change the pips, this spreads across all of the other pods. And this idea emphasizes scaling or enlarging. Here are some references if you're interested in some of the research behind the ideas that have been shared here. Thank you for watching my video. I hope that you'll find some of these ideas helpful. If you would like further information about touch times, it can be found on the touch counts website. If you have any questions about touch times or the tasks that have been developed to go along with it, feel free to contact me via email at sbacus at sfu.ca.