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  • Engaging Students through TechnologyAuthor(s): A. KURSAT ERBAS, SARAH LEDFORD, DREW POLLY and CHANDRA H. ORRILLSource: Mathematics Teaching in the Middle School, Vol. 9, No. 6 (FEBRUARY 2004), pp. 300-305Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41181924 .Accessed: 30/09/2013 01:48

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  • A. KURSAT ERBAS, SARAH LE D FORD, DREW POLLY, and CHANDRA H. ORRILL

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  • representations and mathematical situations and rela- tionships (NCTM 2000). Technology empowers stu- dents who may have limited mathematical knowl- edge and limited symbolic and numeric manipulation skills to investigate problem situations. Technology not only frees the students from tedious and repeti- tive computations but actually encourages the use of multiple representations. Students can easily move from a spreadsheet to a graph or geometry software in their quest for solutions to a given problem. When supported by the teacher, these tools of technology provide students with opportunities to investigate and manipulate mathematical situations to observe, ex- periment with, and make conjectures about patterns, relationships, tendencies, and generalizations. Teach- ers should emphasize and encourage the use of mul- tiple representations to support students' thinking and understanding of concepts and problem-solving situations in all areas of mathematics.

    In this article, we will elaborate on NCTM's vision for technology use by highlighting the flexibility and richness that it offers in problem solving through a multiple-representation approach. "When technologi- cal tools are available, students can focus on decision making, reflection, reasoning, and problem solving" (NCTM 2000, p. 25). We offer such use of technology with two problems adapted from the InterMath Web site (www.intermath-uga.gatech.eduA InterMath is designed to support teachers in developing mathe- matical understanding. Some investigations on the Web site are appropriate for middle school class- rooms, and teachers involved in InterMath have adapted other investigations to use with their stu- dents. The intent of InterMath problems, whether for students or teachers, is to support the development of understanding, using, and appreciating mathemat- ics. Consistent with the multiple-representations theme of this article, InterMath problems encourage learners to work with the problems in multiple ways. The vignettes that follow offer ideas for supporting students in working with open-ended problems that lend themselves to multiple representations and help provide concrete examples of what technology- enhanced mathematics can look like.

    Jack's Mealworms Jack and Jill each have two mealworms. Jack's mealworm population will triple each month. Jill's mealworm population will increase by 40 each month. During what month (s) will Jack and Jill once again have the same number of mealworms? How many mealworms will each of them have?

    This kind of problem promotes the use of prob- lem-solving strategies through the investigation of

    Fig. 1 A spreadsheet showing Jack's and Jill's mealworm populations

    mathematical concepts. In this problem, students are required to extend a pattern by using the first term and the rule of the pattern. Although this problem could be solved using pencil and paper to complete a table, it provides an excellent opportu- nity for technology to be integrated into instruction. By using graphing software or a spreadsheet, stu- dents will quickly be able to represent the problem in multiple ways and investigate whether there are multiple solutions to the problem.

    A spreadsheet provides one effective way to model this problem, since it allows a pattern to be extended quickly and easily. In a spreadsheet, stu- dents can create their table by using three columns. The first column is the month, which increases by 1. Let the next column represent Jack's meal- worms, which start at 2 and triple each month. The final column represents Jill's mealworms, which start at 2 and increase by 40 each month. The num- ber of mealworms that Jack has will always be the number of mealworms from the previous month times 3. The number of mealworms that Jill has will always be the number of mealworms from the pre- vious month plus 40.

    As shown in figure 1, Jill has more mealworms than Jack until the end of the fourth month, when they each have 162 mealworms. After the fourth month, Jack has more mealworms than Jill. The use of a spreadsheet allows students to extend the pat- tern without tedious calculations that may prohibit them from looking for multiple solutions or other emerging patterns. For example, in this problem, when the pattern is extended for thirty months, for example, Jack's mealworm population is 2.1 x 1014, whereas Jill's population is 1202. Students can eas- ily see that Jack's number is increasing more rapidly than Jill's. By analyzing the pattern in the table, students should be able to tell that Jill will never again have as many worms as Jack.

    This problem features two patterns - exponen- tial and linear. Once the patterns have been identi- fied and discussed, the spreadsheet allows us to ex-

    VOL. 9, NO. 6 FEBRUARY 2004 301

    Month Jack JHI_ 0 2 2_

    _J 6 42_ 2 18 82_ 3 54 122 4 162 162 5 486 202 6 1458 242

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  • Fig. 2 A table showing the calculation of Jack's and Jill's mealworm populations

    tend the pattern quickly. As with any open-ended in- vestigation, technology-enhanced investigations re- quire the teacher to take an active role in support- ing students to find generalizations and make sense of mathematical ideas. In this situation, the teacher could lead a discussion regarding the two patterns, helping students see that Jack's pattern is exponen- tial and that Jill's pattern is linear.

    The equation for Jack's mealworms is exponen- tial, since it triples each month. This problem leads to a discussion about exponential equations, which can be facilitated by the question, "How are exponen- tial and linear patterns different?" Student responses may include the following kinds of observations:

    The previous term is being multiplied by 3 to cal- culate the next term.

    A linear pattern involves adding a constant amount each term, which is not occurring here.

    The terms in the pattern are growing large quickly, which indicates an exponential pattern.

    To capitalize on helping students understand ex- ponential patterns, the teacher may want to ask them to consider what information repeats in the pattern. Looking at figure 2, students should notice that the number of mealworms for a given month equals the previous month's number times 3. To reinforce the move to symbolic manipulation, you may want to work with your students to explore how exponents can be used to simplify the pattern, as in figure 2. The equation for Jack's mealworm population, if x = months, is 2(3*). Jill's mealworm population, how- ever, can be modeled with a linear equation, since it increases by the same number each month. Jill started with two mealworms, and each month her population increases by 40. The pattern is also ex- tended in figure 2. The graph can be put into slope- intercept form, a topic that most middle school stu- dents would have experienced. If x = months, the number of Jill's mealworms equals 4(k + 2.

    Once the students have identified the equations and nature of the equations, the class discussion should support the students in developing general- izations from the table. For example, after looking at the extended pattern on the spreadsheet, students

    Fig. 3 A graph of Jack's and Jill's mealworm populations -

    short term

    should form the generalization that other than at the beginning, Jack and Jill will only have one month when they have the same number of mealworms.

    Of course, simply stopping with the spreadsheet solution leaves students with an incomplete under- standing of the problem. The question can be an- swered, or more fully understood, by using graphing software or a graphing calculator. By graphing the equations for the patterns, as shown in figures 3 and 4, linear and exponential functions can be explored. To guide the use of the graph of the two equations, the teacher can ask the students to consider the fol- lowing question: "If the number of mealworms con- tinues to grow at the same rate, will Jack and Jill ever have the same number of mealworms again?"

    Students can derive their own equations or if the class has developed an equation, it can be entered into graphing software. The graphs allow students a differ- ent way to develop an understanding of the difference between exponential and linear equations. At this point, ask the students to again explore how linear and exponential patterns are different You may also work with the students to develop an understanding of what it means when the lines of the graphs cross each other. This activity becomes a powerful way for stu- dents to understand intersection, given that they will see it in both numerical and graphical forms.

    302 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

    Month 1 2 3 Jack's extended pattern 2 2x3 2x3x3 2x3x3x3 Jack's exponential pattern 2x3 2x3* 2x32 2x33 Jack's total mealworms 2 6 18 54 Jill's extended pattern 2 2 + 40 2 + 40 + 40 2 + 40 + 40 + 40 Jill's total mealworms 2 42 82 122

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  • Fig. 4 A graph of Jack's and Jill's mealworm populations- long term

    The mealworm problem focuses on reinforcing mathematical knowledge and skills, as it requires students to identify the rule of a pattern, extend pat- terns, and write equations for functions. The use of technology enriches learning by providing multiple representations in the forms of spreadsheets and graphs. Students can see the problem graphically, numerically, and symbolically. The integration of technology also allows students to have the freedom to focus on problem solving and interpretation rather than rote arithmetic if using paper and pencil.

    Polygon Angles What is the sum of the angles of a triangle? Of a quadrilateral? Of a pentagon? Of a hexagon? What is the sum of the angles in any convex poly- gon in terms of the number of sides?

    The problem posed here allows students to in- vestigate and develop the formula for finding the sum of the angles of polygons. Approaching this problem through investigation gives students the chance to better understand it instead of focusing on computational skills to get an answer. There are many different approaches that students could take to solve this problem, ranging from paper-and-

    Fig. 5 Using Sketchpad to find the sum of the angles of a triangle

    Fig. 6 Using Sketchpad to find the sum of the angles of a quadrilateral

    pencil methods to using calculators, spreadsheets, graphing calculators, or such geometry software as The Geometer's Sketchpad (Sketchpad). The use of technology enables students multiple ways to gen- erate solutions, form conjectures, and test those conjectures as they investigate the problem in an in- teractive learning environment.

    With this problem, a good starting point would be to look at triangles using Sketchpad, lbe teacher can ask the students to create a wide variety of triangles and use the "measure angle" function in Sketchpad to find the sum of the angles (see fig. 5) . After work- ing through a few examples, the teacher can ask each student to make a generalization about the total number of degrees in a triangle. Teachers can also encourage students to copy the angles of the triangle to form a straight line to see that the sum of the an- gles is always 180 degrees, information that becomes useful in finding the sums of the angles of other poly- gons. Sketchpad, at this point, is valuable because it allows students to quickly create, measure, and record the sums of angles in multiple shapes.

    Before the students move on to more complex shapes, the teacher may want to go over class find- ings so far. It is likely that the students will have spent one class period on this task and will need to discuss their findings to make sense of them. Typi- cally, student findings will be within about 1 degree of the expected 180 degrees because of rounding within the software. At this point, it will be impor- tant to help students understand why the measure- ment is not exactly "correct" and to develop a con- sensus among your students to call the sum of angles 180 degrees.

    VOL. 9, NO. 6 FEBRUARY 2004 303

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  • Fig. 7 Excerpt from the spreadsheet finding polygon angle sums

    Fig. 8 A spreadsheet plot of data points

    As a next step in solving this problem, ask stu- dents to repeat their exploration using quadrilater- als. As the students work with more complex shapes, the teacher may prompt students to employ the previous technique to find this sum (see fig. 6). To further student understanding, you may also ask students to look for a relationship between the min- imum number of triangles needed to form the quadrilateral and the measure of the angles in the quadrilateral. Once students conclude that two tri- angles form a quadrilateral, they may feel confident in generalizing that the angles of a quadrilateral sum to 2 x 180 = 360. If the teacher is comfortable with the progress being made using Sketchpad, the students may further delve into the study of the pentagon and then the hexagon.

    Once the students have explored a variety of shapes with Sketchpad, you may ask them to move to a spreadsheet and incorporate these headings: type of polygon, number of sides, number of trian- gles, and sum of the angles. The students can use the "fill down" command to find the values for all n- gons (see fig. 7). In so doing, students can make generalizations about all w-gons, when n is the num- ber of sides of the polygon. You may prompt your students to look at the values in the columns involv- ing the number of sides and the number of trian- gles. Students should see the pattern - the number of triangles that make up a polygon is 2 less than the number of sides. The formula (n - 2) x 180 re-

    turns the sum of the angles of any polygon when the number of sides is known.

    To tie this investigation to other mathematical problems, you may also want your students to de- velop a working knowledge of the symbolic manipu- lation that will yield the answers. One way of solving this problem symbolically would be by using a graphing calculator, graphing calculator software, or the "plot function" within the spreadsheet to plot some of the data points found. Students may plot the sum of the angles as it compares with the number of sides of the polygon (see fig. 8). The graph shows that the data points are linear, which may lead to a discussion on finding the equation that is repre- sented. Finding the equation simply involves using any two points to find the slope, then calculating the ^intercept. For the slope-intercept form of a linear equation, y = mx + b. See the following example.

    Using points (3, 180) and (4, 360):

    sloPe = m = ^ =36-180= = 180 x2-x1 4-3 1 Since

    y = mx + ,

    then 360 = 180x4 + 0 360 = 720 + 0

    -360 = b.

    Therefore, y = 180* - 360 is the equation.

    Once the equation has been found, it may be used to predict or check polygon angle sums (what if n = 100?) that may or may not be found on the spreadsheet

    Some extensions to this problem include finding the measure of each interior angle of a regular poly- gon. Students may want to further their investiga- tions by finding these values using their spreadsheet. They may also want to look at concave polygons to see if their conjectures still hold or if other general- izations can be made for this different case.

    Technology is very useful in the study of this problem as students are able to engage in problem solving and discovery of mathematical concepts in a short period of time. It allows students to see multi- ple representations including graphs, spreadsheets, and equations that allow them to make their own generalizations. Students may then continue to use the technology to confirm those generalizations. If students formulate more questions and ideas than can be answered during class, they may even leave class excited and wanting to learn more!

    304 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

    polygon # of sides # of triangles sum of angles triangle 3 1 180

    quadrilateral 4 2 360

    pentagon 5 3 540

    hexagon 6 4 720

    heptagon 7 5 900

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  • Conclusions

    IN SUMMARY, TECHNOLOGY IS NOT just a simple tool to perform some calculations or engage in drill-and- practice exercises. Technology al- lows students to interact with and ex- plore abstract and concrete concepts through multiple representations, which will enable them to be better problem solvers. Through the use of multiple computer-based applica- tions, students can developer richer understandings of the mathematics that they encounter. When used in ways that allow students to engage in mathematical problem solving, tech- nology becomes an important ele- ment in quality mathematics educa- tion, as described in NCTM's Principles and Standards.

    Reference

    National Council of Teachers of Mathemat- ics. Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. D

    VOL. 9, NO. 6 FEBRUARY 2004 305

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    Article Contentsp. 300p. 301p. 302p. 303p. 304p. 305

    Issue Table of ContentsMathematics Teaching in the Middle School, Vol. 9, No. 6 (FEBRUARY 2004), pp. 289-352Front MatterReaders Write [pp. 291-292]Using the Stock Market for Relevance in Teching Number Sense [pp. 294-299]Engaging Students through Technology [pp. 300-305]Inverting a Triangular Array: Involving Students in Mathematical Inquiry [pp. 306-313]Cartoon Corner [pp. 316, 323]The Thinking of Students: Married People [pp. 317-319]FEBRUARY'S: Menu of Problems [pp. 320-323]Food for Thought from Jay's Diner: Over Laps [pp. 324, 323]Reasoning and Working Proportionally with Percent [pp. 326-330]Families Ask: Manipulatives in the Middle School [pp. 332-333]IT'S FRIEZING IN HERE: Tessellations throught Art, Architecture, and Cultural Artifacts [pp. 334-338]TAKE TIME FOR ACTION: Students' Interpretations of Misleading Graphs [pp. 340-343]SPOTLIGHT ON THE PRINCIPLES: Technology Enhances Student Learning across the Curriculum [pp. 344-349]Window on ResourcesCOMPUTER MATERIALSReview: untitled [pp. 350-350]

    BOOKSReview: untitled [pp. 350-351]

    PRODUCTSReview: untitled [pp. 351-352]

    FROM OTHER PUBLISHERSReview: untitled [pp. 352-352]

    Back Matter