Research and Application of Concept Algebra Volume 2


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Both the experimental and control groups used the same text. However, the experimental group, which used a CAS system, covered the material in 5 weeks; in contrast, the control group took 10 weeks to cover the same material. Furthermore, the experimental group significantly outscored the control group on both computational and conceptual exams; however, despite efforts to ensure the same teaching style, the difference in conceptual scores could be explained by teacher variation. Further, the CAS group overall had slightly more confidence in their success in future mathematics courses, and a larger percentage of students in the experimental group indicated that they had learned more in this class than in any other mathematics class.

Ninety-five percent of the experimental group claimed they would sign up for another class using a CAS system. CIA focused more on concepts, employing symbolic manipulators to perform most of the skills. Three of four hypotheses for greater conceptual understanding were supported, with significant gains found in the ability to model functions, interpret functions, and translate functions. No difference was found in manipulative procedures. Hembree and Dessart found that, when calculators are integrated with regular instruction, students at all achievement levels show an improved attitude toward mathematics, improved test scores in basic operations, and improved scores in problem solving.

These results certainly support the use of technology in the classroom. However, no data have been found regarding a tutorial such as that featured in the Casio FX2. Not only will the Casio FX2. It can, in a sense, teach as well as allow practice of symbolic manipulation. A description of each step appears at the top of the screen, and the results of the particular step are shown below the line. By working through examples such as this, users can then try solving equations on their own, selecting the manual option provided on the calculator. The central question of this study sought to determine if the Casio FX2.

The study was conducted at a small 2-year liberal arts college with a focus on preparing students for a 4-year college or university.

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Ninety-four percent of the students enrolled were traditional students, attending college immediately after high school graduation. The sample was taken from the enrollment in two beginning algebra classes. To control for instructor variability, two sections taught by the same person were chosen. One of these was randomly selected as the experimental group and the other as the control group. Each class had an initial enrollment of 25 students. The instructor also detected no difference between the abilities of the two classes, both of which she perceived as weak.

However, she did think that the early start time a. Use of the Casio FX2. Although students were allowed to use the Solve feature on the TI or 83 Plus to check their work, use of this feature was not taught. The section on solving linear equations through symbolic manipulation lasted approximately 3 weeks. Students in the experimental group used the tutorial on a daily basis in class as the teacher lectured on step-by-step manipulation. They were also encouraged to use the tutorial while doing homework. Skill in solving linear equations was measured by a posttest given at the conclusion of the section on solving linear equations.

The posttest consisted of 14 linear equations and two word problems that were solved with pencil and paper only; calculators were not permitted see Appendix A. Scoring of the problems was based on a rubric provided by the Educational Testing Service Network. Possible scores ranged from 0 to 4, with categories of algebraic knowledge, communication of this knowledge, and demonstration of the skills comprising the score. Scores of 3 or 4 represented successful responses, a 2 represented some conceptual understanding with inconsistent proficiency and incomplete explanations, and scores of 1 or 0 were deemed unsatisfactory.

Two group interviews, one with students in the experimental group and one with students in the control group, were conducted near the conclusion of the unit. Six students in each class, 3 students from the top and bottom quartiles as determined by SAT scores, were randomly selected for the interviews. The questions for both groups included attitudes toward calculators and solving linear equations and general feelings toward mathematics. The experimental group had additional questions concerning the use of the Casio FX 2.

See Appendix B for the interview protocol. Excerpts from the questionnaire used for this study are provided in Appendix C. Finally, an anonymous homework questionnaire was administered to the experimental group on five different days during the linear equation unit to determine the extent of use and the degree to which students found the tutorial helpful. Overall, mean scores for the two groups were 2. Posthoc statistics were performed to determine on which questions there was a significant difference between the two groups.

The experimental group outperformed the control group on six questions; the control group did not outperform the experimental group on any questions. Interestingly, of the six questions on which the experimental group bested their counterparts, four contained fractions numbers 2, 4, 8, and 11 , with the experimental group average score above 3 on the first three of these. On Problem 11, the experimental group averaged 1. On the test, only five problems contained fractions; in fact, the experimental group showed a significant gain on every problem that contained a fraction on both sides of the equation.

Further, the experimental group outperformed the control group on multistep problems. Whether or not these differences occurred for all levels of students in the classes was also investigated. Students were sorted into low, medium, and high groups based on their SAT scores; only students in the medium and high groups had gains on individual questions.

The high attrition and failure rate in the lowest performing group might suggest that these students were either unprepared or unwilling to put forth the necessary effort for success. Consequently, the results concerning the increase in attitudes are tenuous. Post hoc analysis showed that gains in attitude occurred for students in the medium group, but not for students in the low or high groups. Although the time students in the experimental class spent on homework varied daily, there were some detectable patterns. For the first assignment of the unit, most students spent the majority of their time with the tutorial; all but one student reported it was useful.

As the unit progressed, those who perceived the assignments as being difficult spent more time with the tutorial than did those who found the assignments relatively easy. Some students found that the tutorial was difficult to learn, but once learned, it was helpful. The tutorial clearly helped students master the process of solving linear equations. This confirms an important and fifth potential use of graphing calculators in the mathematics classroom, that of developing algebraic skills. The interviews and observations suggest that this additional benefit may have occurred because, to use the tutorial, students had to understand and distinguish coefficients.

In regard to calculator usage, higher-level students took advantage of the calculator, while students in the lower group may have used the calculator as a crutch. It gives us an opportunity to check our answers so we have extra confidence. It is also very useful because it shows us the steps if we forget. Although the results concerning improved attitudes are more tentative, students in the experimental group showed a statistically significant improvement in attitudes between the pre-intervention survey and the post-intervention survey, but the control group did not.


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Ferrara Sci. Theory Appl. Special linearly unrelated sequences, J. Kyoto Univ. On the irrationality of factorial series, Acta Arith. In: W. We are going to be looking at mixing solutions of different percentages to get a new percentage. The solution will consist of a secondary liquid mixed in with water.

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The secondary liquid can be alcohol or acid for instance. Note as well that the percentage needs to be a decimal. However, this is exactly what we want. The basic equation tells us to look at how much of the secondary liquid is in the water. So, this is the correct wording. So, the new word equation is,. Do not get excited about the zero in the first term. This is okay and will not be a problem. So, we need to add in Notes Quick Nav Download. Notes Practice Problems Assignment Problems.


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You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Okay, this may be a little bit of overkill here. However, the point of these first two steps is that you must read the problem.

You need to read the problem very carefully and as many times as it takes. You are only done with this step when you have completely understood what the problem is asking you to do. This includes identifying all the given information and identifying what you are being asked to find. Sometimes a single word can completely change how the problem is worked.


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  5. If you just skim the problem you may well miss that very important word. Represent one of the unknown quantities with a variable and try to relate all the other unknown quantities if there are any of course to this variable.

    Applied mathematics

    If applicable, sketch a figure illustrating the situation. This may seem like a silly step, but it can be incredibly helpful with the next step on occasion. Form an equation that will relate known quantities to the unknown quantities. To do this make use of known formulas and often the figure sketched in the previous step can be used to determine the equation. Solve the equation formed in the previous step and write down the answer to all the questions.

    It is important to answer all the questions that you were asked. The problem asked us to find the dimensions. This means that we also need the width of the set of shelves. So the dimensions will need to be 4x16 feet. This means that 0. In other words, we have the following equation. Note that since we are dealing with money we rounded the answer down to two decimal places. This problem is pretty much the opposite of the previous example. Therefore, the equation and solution is,. These are some of the standard problems that most people think about when they think about Algebra word problems.

    The standard formula that we will be using here is. Now, we need to sketch a figure for this one. From this figure we can see that the Distance Car A travels plus the Distance Car B travels must equal the total distance separating the two cars, miles.

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    We used the standard formula here twice, once for each car. We know that the distance a car travels is the rate of the car times the time traveled by the car. Plugging these into the word equation and solving gives us,. For this problem we are going to need to be careful with the time traveled by each car. The only difference is what we substitute for the time traveled for the faster car. In this case the slower car will travel for 3. Here is the figure for this situation. Plugging these into the work equation and solving gives,.

    The standard equation that will be needed for these problems is,.

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    The word equation for this problem is,. We can use the following equation to get these rates. Plugging these quantities into the main equation above gives the following equation that we need to solve. So, it looks like it will take the two machines, working together, 1. The basic word equation for this problem is,. This time we know that the time spent working together is 3. We now need to find the work rates for each person. We are going to be looking at mixing solutions of different percentages to get a new percentage.

    The solution will consist of a secondary liquid mixed in with water. The secondary liquid can be alcohol or acid for instance. Note as well that the percentage needs to be a decimal. However, this is exactly what we want. The basic equation tells us to look at how much of the secondary liquid is in the water. So, this is the correct wording. So, the new word equation is,. Do not get excited about the zero in the first term. This is okay and will not be a problem.

    Research and Application of Concept Algebra Volume 2 Research and Application of Concept Algebra Volume 2
    Research and Application of Concept Algebra Volume 2 Research and Application of Concept Algebra Volume 2
    Research and Application of Concept Algebra Volume 2 Research and Application of Concept Algebra Volume 2
    Research and Application of Concept Algebra Volume 2 Research and Application of Concept Algebra Volume 2
    Research and Application of Concept Algebra Volume 2 Research and Application of Concept Algebra Volume 2
    Research and Application of Concept Algebra Volume 2 Research and Application of Concept Algebra Volume 2
    Research and Application of Concept Algebra Volume 2 Research and Application of Concept Algebra Volume 2

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