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SUBJECT MATH TITLE OF LEARNING OBJECT CURRICULAR AXIS STANDARD COMPETENCIES LEARNING OBJECTIVES SKILLS/ KNOWLEDGE GRADE MATH 11th GRADE LEARNING UNIT OPERATING IN THE SETS OF REAL NUMBERS Characterization of real numbers Numerical thinking and systems of numeration Analyze decimal representations of real numbers to differentiate between rational and irrational numbers. Recognize the density and incompleteness of rational numbers through numeric, geometric and algebraic methods. Compare and contrast the numeric properties (natural, whole, rational and real) and their relations and operations to properly create, handle and use the different number systems. Based on their properties, establish the set of Real numbers. SCO: Recognizes the set of real numbers SKILL 1: Research the history and importance of real numbers in mathematical processes. SKILL 2: Research the origin of imaginary numbers and their difference with real numbers. SKILL 3: Find characteristics in the Cauchy, Weierstrass and Cantor’s theorems regarding real numbers. SKILL 4: Establish features of the Dedekind cut to define real numbers. SKILL 5: Build a definition of real numbers. SCO: Represent ray intervals SKILL 6: Understand the definition of real interval. SKILL 7: Identify the types of intervals (finite, infinite, semi-infinite) and their corresponding notation. SKILL 8: Relate the continuity notion to real intervals. SKILL 9: Represent real intervals on a line. LEARNING FLOW ASSESSMENT GUIDELINE SCO: Recognize the properties of real numbers SKILL 10: Identify the number sets to which a certain real number belongs to. SKILL 11: Identify real numbers as the intersection between the set of rational and irrational numbers. SKILL 12: Locate real numbers on the number line, and if possible represent them using a ruler and a compass. SKILL 13: Do usual operations between real numbers. SKILL 14: Use technological tools to calculate using real numbers. SKILL 15: Recognize the addition and multiplication properties among real numbers. SKILL 16: Understand the distributive property as a property that relates sums and multiplication between real numbers. Introduction Objectives. The teacher presents the objectives and may establish other ones if desired. Development Activity 1. History of real numbers (Skills: 1, 2, 5) Activity 2. Real number interval (Skills: 6, 7, 8, 9) Activity 3. Rational and irrational numbers (Skills: 10, 11, 12) Activity 4. Addition and multiplication properties of real numbers (Skills: 13, 14, 15, 16) Activity 5. Conceptions of real numbers (Skills: 3, 4) Summary Homework With this assignment we hope the student identifies the sets that make up the real numbers, apply the addition and multiplication properties for problem solving, expresses real numbers on the number line, has the ability to work using real number intervals in order to present and solve everyday life problems, and finally that they may define real and imaginary numbers Stage Introduction Learning flow Introduction Teaching/Learning Activities Recommended Resources Using the animation, the teacher will present the historical processes that Animation contributed to creating real numbers (from the Egyptian time until the “Numbers” theories that formed the concept of real numbers proposed during the XVIII century). On the other hand, we show the importance of representing real numbers using intervals that describe the sets, which may be: open, closed, halfclosed or half-open and infinite, as well as the graphical representation of the number line. Some real numbers cannot only be placed on the number line, but can also be represented geometrically using a ruler and a compass by forming triangles and rectangles (this allows to see the real dimensions of certain numbers which are hard to measure). Content The teacher presents the topic Finally, the addition and multiplication properties of real numbers are formalized, presenting different methods and tools to work with the set of real numbers. Activity 1 SKILL 1: Research the history and importance of real numbers in mathematical processes. SKILL 2: Research the origin of imaginary numbers and their difference with real numbers. SKILL 5: Build a definition of real numbers. History of Real Numbers Using the computer graphics, the teacher will present the history of real numbers, keeping in mind the time period when each subset of real numbers is acknowledged: natural, whole, rational and irrational numbers, Map Inphograps as well as, sets that are not part of real numbers, such as imaginary ones. “History Of The Numbers” Learning activity Activity 1.1. In groups, and based on the information given, create the subsets that Exercise 1. make up the real numbers. Give an example for each case. The teacher uses the Did you know that? tool to teach a basic concept of real numbers. Interactive resource Did you Real numbers are made up of rational and irrational numbers. Meaning know that? that all natural and whole numbers may be written as a fraction. The teacher presents the topic Activity 2 SKILL 6: Understand the definition of real interval. SKILL 7: Identify the types of intervals (finite, infinite, semi-infinite) and their corresponding notation. SKILL 8: Relate the continuity notion to real intervals. SKILL 9: Represent real intervals on a line. Real number intervals Using an animation, the teacher presents the type of intervals of real numbers (closed, open, half-open and infinite intervals), as well as their representation on the number line and their symbolic expression. Animation “ Real Numbers Intervals” Furthermore, the Very important resource shows information about the Interactive continuity of functions in a real number interval. resource Very Important that? Starting from a problematic situation, present the applicability of intervals and how they may vary depending of the characteristics of each situation. Furthermore, develop an example of continuity of intervals using real number graphics. Learning Activity Students solve the activity through an interactive game. Activity 2.3. Match the data on the left to the type of interval on the right. exercise 1 f(x) = {x/2 ≤ x ≤ 9} f(x) = {x⁄−3 < 2𝑥 < 10} f(x) = {x⁄4 < 𝑥 ≤ 15} f(x) = {x/x ≤ 5} Closed interval Open interval Hal-open interval Infinite interval Activity Exercise 2Activity Exercise 2. 2.3. 2.3. Classroom learning activity Students do the suggested exercises. 1. In groups: Match each of the situations to the corresponding type of interval Represent it algebraically Represent it on the number line. Situations: Real numbers between 0 and 5. Real Real Real Real Real numbers numbers numbers numbers numbers from 1 to 10 greater than -4 less than 3 from -3 and less than 12 greater than -5 and up to 6. 2. Find the intervals where the functions are continuous. f(x) = x 3 + 3x + 1 1 f(x) = x x2, 𝑥 < 2 f(x) = { 4, 𝑥 ≥ 2 −x, 𝑥 < −3 f(x) = { x, 𝑥 > −3 f(x) = √x 2 − 1 Activity 3 Skill SKILL 10: Identify the number sets to which a certain real number belongs to. SKILL 11: Identify real numbers as the intersection between the set of rational and irrational numbers SKILL 12: Locate real numbers on the number line, and if possible represent them using a ruler and a compass Rational and Irrational numbers. Activity 3.1. and Using computer graphics of a dialogue, the teacher presents different activity 3.2. Map subsets that are part of the rational numbers, meaning natural and whole Inphograps and numbers, fractions, periodical decimals and irrational numbers. Image, text. The teacher presents the representation of number √2 on the number line, by using a ruler and a compass. Learning activity Along with your classmates, develop the following exercises on your notebook. 1. Indicate to which number set or sets (natural, whole, rational irrational) the following events correspond to. a. Counting how many students have shoes size 38 in the classroom. b. Dividing 8 portions of pizza among 6 people. c. The diagonal of a room with 2m * 3m sides. d. The minimum temperature of New York is -3°. Activity Exercise 1. 3.1. 2. Complete the chart using the symbols “it belongs” ( ∈ ) or “it does not Activity belong” ( ∉ ) depending on the case. If a number belongs to more than Exercise 2 one number set, mark it as such. ℕ ℤ ℚ I 3.1. −8 3 4 20 √9 √7 π − 7 12 3. Express the following irrational numbers. a. √3 b. √5 c. √7 d. √10 Activity 4 Skill SKILL 13: Do usual operations between real numbers. SKILL 14: Use technological tools to calculate using real numbers. SKILL 15: Recognize the addition and multiplication properties among real numbers. Activity Exercise 1. 3.1. SKILL 16: Understand the distributive property as a property that relates sums and multiplication between real numbers Properties of Addition and Multiplication of Real Numbers Using animation, the teacher presents the addition and multiplication properties for real numbers, as well as their symbolic expression and usefulness. Animation “properties of Addition and Multiplication Of Real Numbers” Technological tools for performing operations Activity 4.2. Table Different technological tools which can be used to do operations with real Inphograps and numbers as well as graph them on the number line and Cartesian plane, audio. are presented. Some examples like Geogegra, Cabri, Derive and the page http://www.wolframalpha.com, among other, are presented. Question Activity 4.1. Exercise 1 Learning Activity The student matches each of the operations to the corresponding property. 1 −3 ∗ (4 + ) 5 = −3 ∗ 4 + (−3) 1 ∗ 5 1 −12 ∗ =1 −12 √2 + (−√2) = 0 Distributive Inverse for multiplication Inverse for addition 1 −4 ∗ (3 ∗ ) 2 1 2 1 −9 + (−5) = 2 2 √3 ∗ 1 = √3 Associative for multiplication = (−4 ∗ 3) ∗ Internal for addition Neutral element for multiplication Activity 5 SKILL 3: Find characteristics in the Cauchy, Weierstrass and Cantor’s theorems regarding real numbers. SKILL 4: Establish features of the Dedekind cut to define real numbers. Conceptions of Real Numbers The teacher presents information about the conceptions that different mathematicians such as, Cauchy, Cantor and Weierstrass, have about real numbers. Dedekind cut were also presented. Activity 5.1. Map Inphograps The teacher uses the Did you know that? resource to present data about Activity 5.2. the conception of infinity. Image and Audio Abstract Abstract Using computer graphics, the teacher presents the most relevant Map Inphograps information regarding subsets that are part of the real numbers, the operations that can be done and the addition and multiplication properties. The way to represent real number subsets by using the different types of intervals, or their location on the number line, is also taught. Finally, the conceptions that have defined real numbers are taught. Homework Homework 1. On your own, represent the following algebraic expressions as an interval in the number line. 5 a. {x⁄−2 < 𝑥 < } Homework 1 Table 4 b. {x⁄0 ≤ x < √2} 3 6 c. {x⁄ ≤ x ≤ } 2 7 d. {x⁄−2 > 𝑥} e. {x⁄x < √5} 2. The following problems can be solved using the different addition and multiplication properties of real numbers, indicate which property Homework 2 Text and Image applies for each problem and solve. a. In a park there were 5 firs and 4 pines. The gardener planted another 6 pines. How many trees are there in total now in the park? b. At a bakery they made 5 boxes of French bread and 4 boxes of churros, each box weighs 3kg. How many kilos of bread are there? c. If Camila has 20 packages of 12 candies each and Samuel has 12 packages of 20 candies each, how many candies does Camila have and how many does Samuel have? d. At a soccer tournament the winning team gets 3 points, the loosing team 0 and the team that ties 1. If team A ties 8 games and team B loses 7, how many points does each team have? Evaluation Evaluation 1. The set of real numbers is made up of two number subsets that are: a. Natural and Irrationals b. Rational and Irrational c. Rational and Imaginary d. Irrational and Whole Evaluation Answer key: b Feedback: The set of real numbers is made up of rational and irrational numbers 2. In a Company they are offering three positions. You must be between 20 and 30 years old to position A, for position B from 25 to 35 years old, and for position C from 24 to 32. The age range for people can who apply to the three positions is: a. [25,32] b. [25,30] c. [20,35] d. [24,30] Answer key: b Feedback: It is the only range where the three intervals of age meet 3. Match Number set that goes from 2 to 5. A number set which elements are between -2 and 5. Number sets which elements go from -2 and are less than 5. [−2,5] (−2,5) [−2,5) 4. Complete: a. Examples of ________ numbers are: 2, 5, 12. b. _______ numbers consist of fractions and finite and periodical decimals. c. Roots that have an infinite and non-periodical decimal result are part of the _______numbers. d. Numbers -3, -5, 2, 10, are part of the set of _________numbers. e. _________ numbers are not part of the set of real numbers. f. Answer key: 1. Natural 2. Rational 3. Irrational 4. Whole 5. Imaginary For the following cases, indicate the property that applies for solving each case: 5. Andrea and her daughter take a certain amount of money to the mall 2 1 in order to do some shopping; they spend 5 on clothes and 4 on toys. In order to know how much money they used, they have to add the expenses using which property? a. Associative for addition b. Neutral element for multiplication c. Inverse addition d. Internal addition e. Answer key: Feedback: The right answer is d: since adding to real numbers results in another real number. 6. After shopping where they spent 2 5 of their money on clothes and 1 4 on toys, they decided to eat something, reason why they used 0.3 of their money. At the end, to know how much they spent in total, you have to use which property? a. Distributive b. Associative for addition c. Commutative for multiplication d. Inverse addition Answer key: Feedback: The right answer is b: when adding three real numbers they have to be associated by two terms, added and then the remaining term must be added. 7. Ana builds trash cans using plastic bottles she recycles. To build small 1 trash cans she uses 5 out of 150 bottles, and to build big trash cans she uses 11 2 Glossary Glossary ● ● ● ● ● 3 5 out of the 150 bottles. If Ana builds 21 2 of a small trash can and of a large one, and I wish to know how many bottles she used in total, I must apply which property: a. Commutative for multiplication. b. Associative for multiplication c. Distributive d. Internal for addition e. Answer key: Feedback: The right answer is c: since the final result is obtained by adding the amount of bottles used for a big trash can and the amount of bottles used for a small trash can; and then the resulting amount is multiplied by the amount of both trash cans built by Ana Rational numbers: number set made up of all the numbers that can be Table Inphograps expressed as fractions or finite or periodical decimals. Irrational numbers: number set made up of radicals and infinite and non-periodical decimals. Complex numbers: reference to adding a real and an imaginary number. Interval: Measurable set of real numbers. Periodical decimal: a decimal point with one or more decimals that repeat themselves. Bibliography Bibliography ● ● ● ● Vocabulary Box Vocabulary Box Stewart. J, Redlin. L, & Watson. S. (2012) Precálculo, Matemáticas para el cálculo (6a. ed.) México: Cengage Learning. http://www.profesorenlinea.cl/ (s.f.) Operaciones aritméticas y propiedades con números reales. Retrieved on March 16 from http://www.profesorenlinea.cl/matematica/Numeros_reales_propieda des.html http://tecdigital.tec.ac.cr/ (s.f.) Los números reales según Cantor u Dedeking. Una propuesta didáctica. Retrieved on March 17 from http://tecdigital.tec.ac.cr/revistamatematica/propuestas-didacticasem/v6n1-may-2005/ http://www.pedagogica.edu.co/ (s.f.) Concepciones históricas asociadas al concepto de límite e implicaciones didácticas. Retrieved on March 17 from http://www.pedagogica.edu.co/storage/ted/articulos/ted09_08arti.pdf ● Cathetus: In general, a cathetus is a line falling perpendicularly on a surface or another line. Retrieved on April 20, 2016 from: http://mathworld.wolfram.com/ ● Field: A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra… Examples include the complex numbers ( ), rational numbers ( ), and real numbers ( ). Retrieved on April 20, 2016 from: http://mathworld.wolfram.com/ ● Fir: a tall evergreen tree (= one that never loses its leaves) that grows in cold countries and has leaves that are like needles. Retrieved on April 20, 2016 from http://dictionary.cambridge.org/dictionary/ ● Pythagoras: c582–c500 b.c, Greek philosopher, mathematician, and religious reformer. Retrieved on April 20, 2016 from http://www.dictionary.com/ Table Inphograps Glossary Inphograps ● Theorem: (especially in mathematics) a formal statement that can be shown to be true by logic. Retrieved on April 20, 2016 from http://dictionary.cambridge.org/dictionary/ ● English Review topic English Review topic 4. Simple present, Imperatives, Compound sentences Index