The Allure of Large Numbers

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1 The Allure of Large Numbers Daniel Scher Scott Steketee KCP Technologies Children love large numbers. As they count, 2, 3, 4, 5 and higher, it doesn t take long before they take a mighty leap to one hundred, one thousand, or one million, leaving smaller, skipped values in the dust. Pity those small numbers: Drawing 2 dots is easy enough, and as a result, 2 is tangible and perhaps a little boring. Drawing one million dots is another matter all together: one million is intangible, mysterious, and alluring. Given the strong attraction of big numbers, you might think that elementary mathematics curricula would embrace them. But what is to be done with them? One million by itself sizzles, but multiplying,956,584 by 6,460,889 is no one s idea of fun. Similarly, you can t very well ask a student to draw, a miniscule number with a huge denominator. 000 So too bad large numbers! You re lovable as an idea, but unmanageable in practice. This was our thinking at the start of the Dynamic Number Project, an NSF-funded research and development effort whose goal was to create interactive models of numbers and numerical operations using The Geometer s Sketchpad software. We focused on reasonable numbers those that did not yield nasty calculations or out-of-the-ordinary images (like a picture of ). But then something interesting happened: As students 000 experimented with our Sketchpad models, they took our reasonable numbers and replaced them with unreasonable numbers. Sketchpad didn t complain; it simply updated the models to reflect these new numerical values. Our work with students demonstrated that with calculation and illustration tasks offloaded to the software, students are able to focus on what the numbers and pictures are telling them. And rather than a distraction, these large numbers become an aid to students understanding in a variety of ways: Large numbers spur students to find clever ways to think about calculations too unwieldy to compute by hand. Large numbers expose mathematical relationships that are less apparent when restricted to smaller numbers. Large numbers, viewed in a continuum starting with smaller numbers, give students a way to analyze the behavior of mathematical models across time. Large numbers, viewed visually, create indelible images of the mathematics. In this article, we give examples of elementary students interactions with numerical Sketchpad models and focus on what happens when numbers are released from their confines and allowed to roam free.

2 Factor Puzzles Factor Puzzles is a Sketchpad number game that injects an element of logical reasoning into the study of factors. The Sketchpad model shows four letters, a, b, c, and d, each of which has been assigned a secret numerical value by the software. Students pick any two letters, drag them to the right across a vertical line, and Sketchpad displays their product. In the example in figure, a student has dragged b and d and learned that b x d is 30. Students can drag letters back and forth across the divider line and display the product of any two letters. No product appears when one, three, or four letters sit to the right of the divider. By picking various pairs of letters, a student might learn that b d = 30, a b = 5, a d = 8, and b c = 0 and reason that a = 3, b = 5, c = 2, and d = 6. Figure : The product of b and d is 30. Students play the game multiple times, with Sketchpad generating new random values for the four letters. To keep the computations reasonable, we programmed the puzzle so that no letter ever exceeded a value of 4. While students enjoyed solving these challenges, what they really wanted to do was create factor puzzles for each other. We included a make-your-own version of the puzzle that allowed students to work in pairs and choose their own values for a, b, c, and d. In one classroom, Matt and Lisa were partners and as Lisa looked away, Matt chose values for a through d. Not content with the wimpy small numbers that Sketchpad had chosen, Matt approached the problem with gusto and entered values like a = 5, b = 296, c = 632, and d = 784. When Matt was done, he hid the numbers and told Lisa, with a devilish grin, that the puzzle was ready to solve. Lisa, of course, stood no chance. Dragging a and b across the divider, she discovered that a x b = 5,256. The other products were equally daunting. If nothing else, Matt and Lisa had learned something valuable about products and factors: While it s straightforward to multiply two numbers together and obtain a large product, it s much more difficult to pull large numbers apart into their factors. We expected Matt and Lisa to call a truce and revert to manageable numbers, but Lisa said she could craft a puzzle with large products that would not be difficult to solve. Lisa asked all of us to turn away while she entered new values for the four letters. With the 2

3 puzzle ready, Matt dragged pairs of letters across the divider and found that a b = 60,000, b c = 80,000, and c d = 200,000. Lisa had kept the products large, but cleverly found a way to make the problem tractable. Matt noticed that each product ended with four zeroes, a strong indicator that the values of a through d were multiples of one hundred. By ignoring the zeroes, Matt was able to reduce the problem to a simpler, related version: a b = 6, b c = 8, and c d = 20. Taking the solution a = 3, b = 2, c = 4, and d = 5, Matt appended two zeroes to each number to obtain the answer to Lisa s original problem: a = 300, b = 200, c = 400, and d = 500. Lisa and Matt s insightful exchange was made possible simply by relaxing our grip on the field of permissible values for a through d. Sketchpad knew nothing about what made for a good or bad puzzle, so it happily allowed Lisa and Matt to enter any numbers they wanted. In the process, both students used their knowledge of small products to reason about much larger products. Multiples and Number Dials It is common for students to take a number like 7 and name some of its smaller multiples, but it s unlikely they could answer either of these questions relating to larger numbers without performing long division: Name the number closest to 5,736 that is a multiple of 7. Is a multiple of 7? Using a Sketchpad model of a number dial, however, students examine such questions without lengthy computation and gain a deeper understanding of number properties in the process. Figure 2 shows three time-lapse pictures of a number dial with 7 tick marks as a pointer moves clockwise from tick mark to tick mark. A student has instructed the pointer to move 7 times. The goal is for the pointer to return to its original upright position, which, when accomplished, opens a virtual lock. The lock does not open after 5 rotations (the middle picture), but after 7 rotations, the pointer is back to where it began and the message Lock Opens! appears. Figure 2: Time-lapse pictures of a pointer moving around a number dial 3

4 Students soon discover that the pointer can move more than once around the dial. Fourteen rotations, 2 rotations, or any multiple of 7 results in the pointer traveling multiple times around the dial and opening the lock. Students enjoy watching the pointer s circular journey and want to know what will happen when they program the pointer to move by a larger value like 5,736. While 5,736 may not seem especially large in the context of this article, it takes nearly half an hour for the pointer to move 5,736 times, making it plenty big from a student s perspective. As the count progresses, students focus on other work and periodically check to see the current value. At the conclusion of 5,736 rotations, the pointer does not end upright and open the lock. But seeing the pointer just 3 tick past a full revolution of 7 (fig. 3) allows students to reason that 5,733 is a multiple of 7 without having to start the counter again at 0 or perform long division. And knowing that 5733 opens the lock makes it a cinch to name the next four numbers that will work: 5740, 5747, 5754, 576. Figure 3: Time-lapse pictures of a pointer rotating 5,736 times around a number dial Similarly, students are able to reason that if,239 opens the lock and 2,737 opens the lock, then so will their sum, A typical explanation is: First go,239 rotations. We know that opens the lock. The point is back where it began and it s like starting at 0 again. Now go another 2,737 rotations. That opens the lock. So if you go the whole way at once, 3,976 rotations, that opens the lock. This line of reasoning is an intuitive way to think about the algebraic statement, If a and b are each evenly divisible by n, then a + b is divisible by n as well. As the examples in the section demonstrate, students are able to use the number dial model to reason about calculations involving large numbers without actually performing the calculations a very powerful habit of mind (Goldenberg, Mark, and Cuoco, 200). Fractions and Their Denominators When textbooks represent fractions pictorially as parts of a circle, they rarely exceed a denominator of 2. While there isn t widespread call for visualization of fractions like /23 or 3/56, we were eager nonetheless to see what would happen when we field tested a Sketchpad area model of fractions that allowed for denominators of any size. 4

5 Figure 4 displays the Sketchpad model in its original state. With students assembled on a rug in front of an interactive whiteboard, we selected the denominator of 2 on screen and steadily pressed the + key on the keyboard to increment its value from 2 to 3 to 4 and higher, all the way up to 20. As the value of the fraction changed, so did its corresponding picture. Figure 5 shows a time-lapse view of the model for denominators of 7 through 20. Figure 4: An area model of 2 Figure 5: A succession of fraction images as the denominator increases by When we asked students to describe what they were seeing, they focused on how the model changed over time rather than mentioning any one particular image: As the denominator gets bigger, the circle is divided into more parts. The more equal parts you have, the smaller each part gets. The shaded part gets smaller because the parts are smaller. It s easiest to see a difference in the fractions when they change from 2 to 3 or 4 to. It s harder to see a change from 5 9 to 20. 5

These astute observations were aided by the dynamic nature of the Sketchpad model and the steady progression of denominator values from 2 through 20.

6 These astute observations were aided by the dynamic nature of the Sketchpad model and the steady progression of denominator values from 2 through 20. And as we expected, stopping our exploration at was simply not an option. Students called out other 20 values of the denominator they wanted to see, including 50, 00, 500, and 000. Once again, we pressed the + key and watched a stream of subtly changing images as the circle divided into more and more parts and the shaded portion became just a sliver of the whole. When we reached, we stopped, double-clicked the denominator, and 00 changed its value from 00 to 500 in the dialog box that appeared. This allowed us to jump immediately from to. Figure 6 shows the fraction model for denominators of 50, 00, and 500. Figure 6: Viewing the fraction model with denominators of 50, 00, and 500 6

7 In conjunction with Figure 6, students discussed the possibility of taking a pizza and dividing it into 50, 00, or even 500 equal slices. They recognized the folly of this proposal, but delighted in the ability of Sketchpad to make the result achievable through a virtual model. By pushing the model to its limits, students created a convincing and memorable demonstration that as the denominator of a fraction increases, the value of the fraction decreases. Fractions on a Number Line Aside from representing fractions as parts of a circle, we can also represent them as lengths along a number line. Our Sketchpad number line model presents students with a list of numbers that can serve as the numerators and denominators of fractions. Using a custom-built fraction tool, students select a numerator and then a denominator. When they do, a labeled segment appears whose length is equal to the fraction. The segment can then be placed on a number line either with its left endpoint at 0 or its left endpoint connected to an existing segment. In Figure 7, a student has just created a segment of length 2 by clicking the 2 and 3 in the 3 list of numbers. By clicking 0 on the number line, the student can attach the labeled segment to it. Figure 7: Constructing a segment of length 2 3 and placing it on a number line Restricting the possible numerators and denominators to the numbers in the list leads to some interesting questions: What fraction can you make this is less than, but as close to as possible? What is the smallest fraction you can make? What is the largest fraction you can make? This last question turns the search for the smallest fraction on its head. While the segment 00 of length is barely detectable on the number line, a segment of length 00 extends way off the visible portion of the Sketchpad window and reliably elicits many oohs and aahs from the class. As students work diligently to scroll to the right in search of the segment s endpoint, they experience the size of the improper fraction in a very tangible and memorable way. 7

8 Conclusion The title of this article, The Allure of Large Numbers, is a nod to the book The Lore of Large Numbers by Philip J. Davis (975). While our title could well have focused on the mathematical insights gained from the study of large numbers, we chose instead to highlight the pedagogical benefits of fueling young students' curiosity. Just as earlier generations of mathematicians found motivation in the seemingly unmanageable and wondrous nature of irrational, transcendental, and imaginary numbers, so too do young learners find inspiration in large numbers. References Davis, Philip J. The Lore of Large Numbers. Washington, DC: Mathematical Association of America, 975. Goldenberg, E. Paul, June Mark, and Al Cuoco. An Algebraic Habits-of-Mind Perspective on Elementary School. Teaching Children Mathematics, 6, no 9 (May 200):

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