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Transcript
SUBJECT GRADE LEARNING UNIT MATHEMATICS 11th OPERATING IN THE REAL NUMBERS (R) SET. TITLE OF LEARNING OBJECT RECOGNITION OF THE ORDER RELATION BETWEEN REAL NUMBERS CURRICULAR AXIS STANDARD COMPETENCIES Numerical thinking, and Systems of Numeration. Recognize the density and incompleteness of the rational numbers through numerical, geometrical, and algebraic methods. To determine the set of real numbers through its properties. SCO: Identify the order properties in real numbers. SKILL 1. Interpret the definition of the order relation. SKILL 2. Recognize the symbology associated to the order relation. SKILL 3. Establish the properties of order in real numbers. SCO: Order real numbers SKILL 4. Make comparisons between real numbers. SKILL 5. Order real numbers from lowest to highest and vice versa starting from their location in the number line. SKILL 6. Order real numbers according to their symbolic representation. SKILL 7. Order real numbers in the number line by approximation or geometric methods. (Pythagoras and Thales). SKILL 8. Find real numbers between two given real numbers. SKILL 9. Recognize the property of density in real numbers. LEARNING OBJECTIVES SKILLS/ KNOWLEDGE LEARNING FLOW ASSESSMENT GUIDELINE Introduction Objectives. The teacher presents the objectives and may establish others if desired. Development Activity 1. Relation of order of real numbers (Skills: 1, 2, 3). Activity 2. Order of real numbers (Skills: 4, 5, 6, 7). Activity 3. Density of real numbers (Skills: 8, 9). Abstract Homework Evaluation Glossary Vocabulary box Bibliography English review topic Students are expected to recognize the definition of the order relation, and its properties, which he/she will be able to use to order different numbers form the subsets contained in the real numbers and to locate them in the number line, making use, if necessary, of the geometric methods of Thales and Pythagoras. Stage Learning flow Introduction Introduction Teaching/Learning Activities Recommended Resources Using the animation, the teacher will Interactive resource. present the relation of order in real Animation. numbers, and the properties numbers must meet to keep such relation; the teacher will also introduce the symbols utilized to establish numerical relations, namely: greater than (>), less than (<), and equal (=) The location of numbers in the number line is also applied, which implies the construction of rational numbers and non-exact square roots; to this end, the geometric methods of Thales and Pythagoras are used, accordingly. Development Activity 1 Finally, the property of density of real numbers, and the way to find a number in-between two given real numbers are presented. Activity 1 RELATION OF ORDER OF REAL NUMBERS Using the animation, the teacher introduces the definition of the order relation in real numbers, the symbology utilized to organize numbers, and the properties numbers should meet to keep such relation. The teacher uses the Did you know that...? tool to introduce the existence of two kinds of order relation. There are two kinds of order relation: partial order relation, which meets the reflexive, antisymmetric, and transitive properties; and the relation of Total Order Interactive resource animaction which meets the condition of being partial and all of its elements are comparable Learning Activity The student develops the learning activity Interactive resource making use of the True or False tool in the platform to answer 5 proposed situations. Recreational resource Activity 2 Activity 2 ORDER OF REAL NUMBERS The teacher used the infographic to explain students how to compare real numbers in order to know which number is greater or less than other number among a given set; besides he/she presents the way of constructing and locating rational numbers and roots in which answers are not exact, with the purpose of locating numbers in the number line. The teacher used the animation to present the methods of Thales and Pythagoras for the construction of rational numbers, and non-exact roots. The teacher gives practical examples of the comparison and location of rational and irrational numbers corresponding to non-exact roots in the number line., The teacher introduces the information making use of the fun fact Remember that… Constructing the square root of some numbers such as 3 and 7, among others, depends on the location of other roots previously created in the number line. Learning Activity in the Classroom In groups, the students solve the proposed exercises. 1. Order from lowest to highest and locate in the number line, use ruler and compass to make auxiliary constructions if necessary. 1 1 ● −3.5, −2, 4, − 2 , 2 ● −5,3,5,−3 ● √5,− √2, 2, − 3 ● −2,5, 1.3, −√3, − , 2.3 9 4 2 1 1 True or false game. 2 3 Learning Activity The student develops the learning activity making use of the True or False tool in the platform to answer: 1. From lowest to highest: −3.5, 1 2 , , 2 3 √2 , 3 2 _____________True 2. From lowest to highest: −√2, −√3, −1.2, 1.2, 4 3 ______________False 3. From highest to lowest: √7, 3, √5, 1.3, −2, −√7 _____________False 4. From highest to lowest: 2.3, √5, 1.5, Activity 3 −1.4, 7 , −√5 2 1 , 3 _____True Activity 3 DENSITY OF REAL NUMBERS Using a dialogue infographic, the teacher presents the formula used to find a real number in-between two given numbers, as well as the steps that need to be followed; which helps the student recognize the density property of real numbers. Example The teacher presents an example in which two real numbers should be found 2 4 between fractions 3 and 5, resulting in their location in the number line. Recreational resource The teacher uses the fun fact Did you know that...? to present important information. The property of density of real numbers is applied to the rational numbers set, and consequently, the result will always be a True or false game. rational number. Learning Activity The student develops the learning activity making use of the True or False tool in the platform to answer the proposed situations. 1 3 5 1. From these numbers , , the number 2 4 9 located in between the two other 3 numbers is 4. FALSE 2. The correct way of finding a real number between two other numbers is to add them and divide the result in 2. TRUE 3. The property of density is the one that indicates that between any two whole numbers there will always be another whole number. FALSE 8 4. The midpoint between numbers 2, − 3 1 3 is − .TRUE Abstract Assignment/ homework The teacher presents through an Conceptual map. infographic the most relevant information about the order relation of real numbers, the properties that numbers should meet to belong to such relation, how to represent different numbers in the number line taking into account the geometric methods of Thales and Pythagoras; as well as the property of density of real numbers, and how to find a number between two other given numbers. In group, answer the following exercises. 1. Indicate in each case what are the highest, the lowest, and the equal Student material elements, if there are any, in each case 2 1 4 ● −√5, −3, 5 , √2, − 4 , 3 ● ● −4.25, √3, 5 12 ,− 5 , 2 8 , 7 9 1.35, − 2 , − 17 4 1 −√7, 1.42, −2.4 , − 5 2. Locate the following numbers from lowest to highest in the number line. ● −√3, 1, − 1, √2, √3 ● 1.5, 4 , 5 3 ● 3 , 10 1.25, −√7 , − 2.5, − 4 , − 2, 7 4 6 5 3. Find three numbers between the two given numbers, and locate them in the number line. 5 ● 3 and 2 ● − ● 3 4 1 4 and and −3 4 1 2 4. Sara, Camila, Andrea, and Sonia went shopping; Sara spent less than Camila, but more than Andrea. Sonia Spent more than Sara, but less than Andrea. Organize the expenses of each girl in the line from lowest to highest. Indicate who spent less and who spent more 5. Santiago and Carlos are heavier than Juan, while Andrés is less fat than Santiago, but, more than Carlos. After representing them in the number line, indicate who the least heavy boy is. Evaluation Answer true or false in questions 1 to 4, as appropriate. 1 1 1. If 0 < 2 and 0 < 3 then 0 < 2 ∙ 3 TRUE Feedback: Whenever two positive numbers are multiplied, the result will be positive. 1 2. If − 4 < 0 then −1 ∙ 4 > 0 FALSE Feedback: Whenever a positive number is multiplied by a negative number the result will be negative 4 2 3. If − 5 < 5 then 4 5 2 < −5 FALSE Feedback: If each number of inequality is multiplied by −1, inequality is reversed. 1 2 the the 1 2 4. If −3 < − then −3 + 4 < − + 4 TRUE Feedback: If the same number is added to both sides of an inequality, the inequality will not change. Select the correct answer in questions 5 to 9. 5. The geometric method of Thales allows us to construct one of the following numbers in the number line: a) Natural numbers. b) Whole numbers. c) Rational numbers. d) Irrational numbers. Answer key:/ c Feedback: Thales’ method allows dividing the unit in as many segments as the denominator of a fraction indicates. 6. The most adequate method to locate the number √7 in the number line is: a) Pythagoras’ Method. b) Approximation Method. c) Thales’ Method. d) Location of exact numbers. Answer key:/ a Feedback: Pythagoras’ method is used to construct non exact square roots. 7. The simplest way to find a number in-between two real numbers is to find their midpoint. Consequently, it can be affirmed that the midpoint between 2 5 and 6 is: 3 𝑎 1 4 𝑏. 3 4 𝑐. 2 6 d. 4 9 Answer key:/ b Feedback: When comparing the given numbers it with becomes evident that it is less than greater than 3 4 5 6 and 2 3 8. In the following sets of numbers, we can state that number √2 is the highest in: 1 3 ,−5 2 3 2 1 1.4, 2 , √2, − 7 , 3 2 7 1 2 − 5 , √2, 6 , − 2 , 3 8 9 , , −1.6, 1.33, √2 9 4 a) √2, 2.5, b) c) d) Answer key:/c Feedback: After converting all the numbers to decimals and comparing them, √2 is the highest only in answer c. 9. The correct ascending order for 3 these numbers −2, 2 , 1.8, √2, −√3 is: 3 a) −√3, −2, 1.8, 2 , √2 3 b) −2, −√3, √2 , 2 , 1.8 c) 1.8, 3 , √2, −√3, −2 2 d) −√3, 1.8, −2, Glossary Answer key:/b Feedback: When comparing the numbers in pairs, answer b is the only one organized from lowest to highest. Relation: correspondence or Interactive resource connection between two things. Vocabulary box 3 , √2 2 Reflection: Relation of an element to itself. Antisymmetry: Relationship of two elements in which both of them are equal. Transitivity: Relationship between three elements in such a way that if 𝑎𝑅𝑏 and 𝑏𝑅𝑐 then 𝑎𝑅𝑐. Thales’ Method: Based on the Thales’ theorem in which correspondences between triangles can be found by creating congruent segments. Pythagoras’ Method: Based on the Pythagorean Theorem, in which the legs of a right triangle are related with the measure of the hypotenuse. To allow: To give permission for someone to do something, or to not prevent something from happening. Aid: help. Take into account: to consider. Therefore: For that reason. Tool: an instrument for doing work, especially by hand Thus: (referring to something mentioned before or after) in this or that way. Whenever: at any time that, at ● ● ● ● ● ● ● Bibliography English review topic every time that. Rodríguez, J., & Astorga, A. Orden en el conjunto de los números reales. Retrieved on March the 30th, 2016 from https://tecdigital.tec.ac.cr/revista matematica/cursoslinea/MATEGENERAL/t1-realesexpresionesalgebraicas/T1-1numeros-realesjulioetall/node12.html Ferrari, G., & Tenembaum, S. (2015). Resumen de Relaciones y Funciones. Retrieved on March the 31st, 2016 from http://www.x.edu.uy/ y http://www.x.edu.uy/inet/RELACIO NES_FUNCIONES.pdf OCWUPM, UPM. División de segmentos en n segmentos iguales. Retrieved on March the 31st, 2016 from http://ocw.upm.es/ y http://ocw.upm.es/apoyo-para-lapreparacion-de-los-estudios-deingenieria-y-arquitectura/dibujopreparacion-para-launiversidad/contenidos/archivos/pr oporcionalidad/04-division-de-unsegmento-en-n-partes-iguales.pdf Ordinal numbers Comparatives and superlatives Passive voice ● ● ●