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Domain D: Impact on Student LearningDimension 10:Impact on Student LearningThe sequence of lessons focused on The Real Number System. Students must comprehend that the real number system is the foundation for mathematical disciplines. They need to master the skills to prove that a number is rational because it can be expressed as a fraction. They need to become familiar with the functions on a calculator that can support the notions of repeating and non-repeating decimals and decimal expansions. Students should hone their approximations skills in order to determine the upper and lower boundaries of an irrational number. The following Common Core Learning Standards and Mathematical Practices were followed:The Number System 8.NSKnow that there are numbers that are not rational, and approximate them by rational numbers.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.Mathematical Practices 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. 7. Look for and make use of structureThey also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. In order for students to develop problem-solving strategies and improve mathematical fluency and conceptual mastery, it is essential that they have a firm foundation and understanding of the Real Number system and how it connects to the mathematical disciplines of Algebra and Geometry. This includes being able to identify the sub-categories within the real number system, such as rational and irrational numbers. They are required to use mathematical tools, such as a number line and calculator, to support their claims. Additionally, they will use theorems and algorithms to identify rational and irrational numbers. Students need to become familiar with radical symbolism and terminology in order to prepare them for the units on rational exponents and logarithms. Students require mastery of real numbers in order to help prepare them to extend their knowledge to complex and imaginary numbers.The specific learning objectives (SLOs) are:a. Students will be able to identify rational and irrational numbers b. Students will be able to use algebraic expressions to define rational numbers c. Students will be able to use calculator functions to support procedures and calculations d. Students will be able to make connections between geometric models and real numbers In addition to direct instruction, the students engage in hands-on, kinesthetic activities that are learner-centered. Learner-centered practices help students build academic self-confidence, become more engaged and invested in their learning. Additionally, learner-centered activities are challenging, but flexible, so that each individual finds a way to participate. The teacher becomes the facilitator of learning rather than the source of all knowledge (Allen, 2009.) These activities include constructing perfect squares from colored tiles and drawing conclusions from their constructions. Rigor was introduced through the inclusion of theorems and the development of procedures to prove these theorems. Students utilize calculators to represent rational and irrational numbers and to become familiar with repeating, non-repeating decimals and decimal expansions. Students are encouraged to use multiple representations to extend their conceptual knowledge. Differentiation was incorporated to adapt the content, level, pace and products of instruction to accommodate the diverse student population.The assessment method was an end of unit test. The test included traditional questions and open-ended responses. Artifacts I and II represent the work of two students, whose responses exemplify the majority of responses from the other students.The following table is used to provide simple summary statistics reflecting the students’ ability to meet the specific learning objectives. A rating of “0” indicates that the student did not display any conceptual understanding of concepts and procedures. A rating of “1” indicates that the student displayed some conceptual understanding of concepts and procedures. A rating of “2” indicates the student displayed mastery of conceptual understanding of concepts and procedures. Student 1SKILL012Identifying rational/irrational numbersMaking connections within mathematical disciplinesUtilizing multiple representations to express abstract conceptsUsing calculator functions and/or other appropriate tools to support procedures and calculationsStudent 2SKILL012Identifying rational/irrational numbersMaking connections within mathematical disciplinesUtilizing multiple representations to express abstract conceptsUsing calculator functions and/or other appropriate tools to support procedures and calculationsDomain D - ArtifactsArtifact IThe following represents the work of Student I. Her construction of a number line with accurate intervals along the horizontal axis indicates an understanding and appreciation for precision and acknowledges her appreciation of the number line as an accurate measuring tool. She exhibited correct use of mathematical language by indicating that a perfect square is the product of a factor multiplied by itself. The student produced an accurate illustration of a perfect square, indicating the ability to connect geometric concepts with real number concepts. She supports her claim that √5 is an irrational number by explaining the sequence of calculator keys she uses to display the decimal expansion and convert the decimal to a fraction. She explains the results by stating that an irrational number cannot be converted to a decimal. Her response to the Extra Credit question is concise, accurate, and procedurally and conceptually sound. Her answer is organized and she provides more than one example. She displays mastery of the specific learning objectives and mathematical fluency.-167640194310right244159Artifact IIright4139248Student 2’s responses are not as precise or procedurally accurate as Student I’s.. She does not expand on her knowledge of perfect squares. The response could have been enhanced if the student recalled the lab activity, which required the students to construct perfect squares from colored tiles. This would have added depth and substance to her response. Additionally, she does not include any calculator functions to support her claim that 29 is irrational. She does include information about the square root of 29, which is irrational and alludes to the fact that “perfect squares” can provide clues to finding irrational numbers. She struggles to convey her mastery of conceptual and procedural fluency in identifying irrational numbers and their connection to perfect squares. She incorrectly responds that √5 is a repeating decimal. She does not provide visual or numerical data to support her claim. Although she attempted to answer the Extra Credit problem, she merely writes repeating numbers without providing procedural documentation to prove they are rational. -409575282575 ................
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