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MATHEMATICS SECONDARY I - IV ALBUM Created by Darin M. Bicknell 2013 Montessori Observation Big Picture Montessori This work is licensed under a Creative Commons Attribution Non Commercial - Share Alike 3.0 Unported License. Math Secondary I - IV Contents 1. Overview 1.1. 1.2. 1.3. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. Aims The Elementary Mathematics Curriculum* Presentation: The Great Story of Mathematics Continues Number, Set Notation and Language Squares and Cubes Directed Numbers Vulgar and Decimal Fractions and Percentages Ordering Standard Form The Four Rules Estimation Limits of Accuracy Ratio, Proportion, Rate Percentages Use of an Electronic Calculator Measures Time Money Personal and Household Finance Graphs in Practical Situations Graphs of Functions Straight Line Graphs Algebraic Representation & Formulae Algebraic Manipulation Functions Indices Solutions of Equations & Inequalities Linear Programming Geometrical Terms & Relationships 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. Geometrical Constructions Geometrical Constructions Symmetry Angle Properties Locus Mensuration Trigonometry Statistics Probability Vectors in Two Dimensions Matrices Transformations 1. Overview 1.1. Aims The aims of the curriculum are the same for all candidates. The aims are set out below and describe the educational purposes of a course in Mathematics. They are not listed in order of priority. The aims are to enable candidates to: o Develop their mathematical knowledge and oral, written and practical skills in a way which encourages confidence and provides satisfaction and enjoyment; o Read mathematics, and write and talk about the subject in a variety of ways; o Develop a feel for number, carry out calculations and understand the significance of the results obtained; o Apply mathematics in everyday situations and develop an understanding of the part which mathematics plays in the world around them; o Solve problems, present the solutions clearly, check and interpret the results; o Develop an understanding of mathematical principles; o Recognise when and how a situation may be represented mathematically, identify and interpret relevant factors and, where necessary, select an appropriate mathematical method to solve the problem; o Use mathematics as a means of communication with emphasis on the use of clear expression; o Develop an ability to apply mathematics in other subjects, particularly science and technology; o Develop the abilities to reason logically, to classify, to generalise and to prove; o Appreciate patterns and relationships in mathematics; o Produce and appreciate imaginative and creative work arising from mathematical ideas; o Develop their mathematical abilities by considering problems and conducting individual and co-operative enquiry and experiment, including extended pieces of work of a practical and investigative kind; o Appreciate the interdependence of different branches of mathematics; o Acquire a foundation appropriate to their further study of mathematics and of other disciplines. 1.2. The Elementary Mathematics Curriculum* 1.2.1. 1.2.2. 1.2.3. 1.2.4. 1.2.5. 1.2.6. 1.2.7. 1.2.8. 1.2.9. 1.2.10. 1.2.11. 1.2.12. 1.2.13. 1.2.14. The Story of Numbers Numeration Multiplication Division Fractions Decimal Fractions Squaring and Cubing Square Root and Cube Root Powers of Numbers Negative Numbers Non-decimal Bases Word Problems Ratio and Proportion Algebra * The Elementary Math Album licensed under a Creative Commons Attribution Non Commercial - Share Alike 3.0 Unported License by Jonathan Feagle can be found at http:/www.freemontessori.org 1.3 Presentation: The Great Story of Mathematics Continues 2. Number, Set Notation and Language 2.1. Natural numbers Definition: Numbers used for counting i.e., numbers beginning from 1 to infinity. 1, 2, 3, 4 Proficiency & Practice with Natural Numbers Student uses BODMAS (BEDMAS) Student practice putting numbers in order Students practice rounding to the nearest ten, hundred, and thousand. Student practice solving simple problems, using the four operations of arithmetic: addition, subtraction, multiplication and division Presentation 1: ORDER OF OPERATIONS We have used natural numbers in many ways in the classroom. We often use the numbers with operators like addition, subtraction, multiplication, and division and at times we have also used two operators in one problem. Like when we used them for binomial work (2+3) x 4. Suppose you have to work out the answer to 4 + 5 × 2. What do you think the correct answer is? The student may say the answer is 18, but the correct answer is 14. There is an order of operations which you must follow when working out calculations like this. Multiplication × is always done before addition +. In 6 + 5 × 3 this gives 6 + 15 = 21. What would be the answer if you have to work out (4 + 3) × (7 –2). Give the students the correct answer of 35. What did you realise? Did you see additions and subtraction within the brackets have to be done first, giving 7 × 5 = 35? Have the students work on their own from questions the make up for each one another. Presentation 2: BODMAS (BEDMAS) So, how do you work out a problem such as 64 ÷ 8 + 12 × 3? To answer questions like this, there is the simple BODMAS (BEDMAS) rule. This tells you the order in which you do the operations. B O D M A S Brackets Order Division Multiplication Addition Subtraction B E D M A S Brackets Exponents (powers) Division Multiplication Addition Subtraction For example, to work out 64 ÷ 8 + 12 × 3: First divide: 64 ÷ 8 = 8; giving 8 + 12 × 3 Then multiply: 12 × 3 = 36 giving 8 + 36 Then add: 8 + 36 = 44. Give another problem 34 – 5 + 4 × 3 ÷ 2. Have students identify the correct order. First 12 ÷ 2 = 6; giving 34 – 5 + 4 x 6 Second 4 x 6 = 24; giving 34 – 5 + 24 Third 5 + 24 = 29; giving 34 – 29 Finally 34 – 9; giving 5 Hand the lesson off to the student(s) once you see they understand the order and let them create their own equations. Extension From state, provincial, local or national exams provide questions with answers in a timed fashion to allow the students to challenge their skills. The teacher or students can mock up speed test cards with 10 questions on one side and answers. The students should be provided with a stop watch or egg timer to count down trying to improve their accuracy and time taken to complete the test. Presentation 3: Hierarchy of Numbers Revision The teacher writes out the following numbers using the green, red, blue and green for each hierarchy. Have the students arrange the numbers in order from the smallest first to the largest. 8045 4085 48058504 8540 5840 Look at the thousands first and then each of the other hierarchies in turn. The correct order is 8504, 8504, 8045, 5840, 4805, and 4085. Presentation 4: Hierarchy of Numbers Revision Write each of the following numbers in natural numbers using the colour code for the hierarchies, green for units, blue for tens, and red for hundreds. Eight million, two hundred thousand, fifty-eight Nine million, four hundred and six thousand, one hundred, seven One million, five hundred, two Two million, seventy-six thousand, forty Presentation 5: Approximation, Rounding with Natural Numbers Prepare newspaper articles with estimates of sizes, also include statements about your approximate time to travel to the school written down as well as getting directions for cooking, baking times from a box of frozen food or pre-packaged food. “It takes me about 15 minutes to get to school without traffic and if it is raining it could take me as much as 30 minutes or more.” “Cook in the microwave for 5 minutes on high” “Boil hot water and pour it into in the container of noodles for 5 minutes or until soft.” “The demonstration in Jakarta had over 5000 people.” We use rounded information all the time. Look at these examples. All of these statements use rounded information. Each actual figure is either above or below the approximation shown here. When rounding is done correctly, you can find out what the maximum and the minimum. For example, if you know that the amount of time people demonstration in Jakarta is rounded to the nearest 1000. The smallest figure to be rounded up to 5000 is 4500, and the largest figure to be rounded down to 5000 is 5400 (because 5500 would be rounded up to 6000). So there could actually be anywhere from 4500 to 5400 people at the demonstration. Examples: Round each of these numbers to the nearest 10. 57 34 12 21 58 107 20 60 110 Answers rounded to the nearest 10. 60 30 10 Give these cooking times to the nearest 5 minutes. Note that smallest figure to be rounded up to 5 minutes is 2 minutes and 30 seconds, and the largest figure to be rounded down to 5 minutes is 7 minutes and 29 seconds (because 7 minutes and 30 seconds would be rounded up to 10 minutes). 34 minutes, 57 minutes, 13 minutes, 7 minutes and 50 seconds Answers rounded to the nearest 5 minutes. 35 minutes, 55 minutes, 15 minutes, 10 minutes Extension From state, provincial, local or national exams provide questions with answers in a timed fashion to allow the students to challenge their skills. The teacher or students can mock up speed test cards with 10 questions on one side and answers. The students should be provided with a stop watch or egg timer to count down trying to improve their accuracy and time taken to complete the test. 2.2. Integers Definition: Integers are whole numbers i.e., they can be positive negative or zero. -3, -2, -1, 0, 1, 2, 3 Positive integers: 1, 2, 3, 4 Negative Integers: -1, -2, -3, -4 Proficiency & Practice with Integers Students will use Integers in real life applications Students will explore the meaning of the Integer Students will practice using Integers within inequalities Students will practice how to do arithmetic with Integers Presentation 1: The Integer Number Line Prepare an Integer number line either as a single bead on a line that can be slid left or right with the integers clearly marked behind the fishing line. Or you can create from thick cardboard a vertical / horizontal number line with beads placed in holes. We have presented the Integer number line material before so this is a review for some and maybe new to others. Look at the number line. Notice that the negative numbers are to the left of 0, and the positive numbers are to the right of 0. Numbers to the right of any number on the number line are always bigger than that number. Numbers to the left of any number on the number line are always smaller than that number. –3 Negative –2 –1 0 1 2 3 Positive The Integers can also be represented vertically on a number line. Look at the number line. Notice that the negative numbers are below of 0, and the positive numbers are above 0. Numbers above any number on the number line are always bigger than that number. Numbers below any number on the number line are always smaller than that number. 5 Positive 4 3 2 1 0 -1 -2 -3 -4 -5 Negative 2.3. Prime numbers A number which has two unique factors i.e., 1 and the number itself. 2, 3, 5, 7, 11, 13, 17 e.g., 3 = 3 × 1, 7= 1 × 7. Note that: 1 is not a prime number as it has two factors but both are same. 2.4. Square numbers 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49 These number obtained by squaring integers are called square numbers. Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81 2.5 Factors Factor is a whole number which divides exactly into a whole number, leaving no remainder. e.g., 1) 2) Factors of 4 1×4=4 2 × 2 = 4, so 1, 2, 4 are factors of 4. Factors of 12 1 × 12 = 12 3 × 6 = 12 3 × 4 = 12, so 1, 2, 3, 4, 6, 12 are factors of 12. 2.6 Prime Factors Factors that are Prime. e.g., Prime Factors of 252: 22 × 32 × 7 2.7 Common Factors Common factors of numbers. e.g., 1) 2.8 Common factor of 15 and 12 Factors of 12 = 1, 2, 3, 4, 6, 12 Factors of 18 = 1, 2, 3, 6, 9, 18 Common Factor of 12 & 18 are 1, 2, 3, and 6 Highest common Factor: 6 Multiples A multiple of a number is what you get when you multiply that number by some other whole number. e.g., 1) Multiplying 3 by 5 gives 15 so 15 is a multiple of 3. 2) Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81 2.9 Common Multiple Common multiples of numbers. e.g., 1) 2.10 Common multiples of 2 and 4 Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 Common multiples of 2 & 4 are 4, 8, 12, 16, 18, 20, and 24 Lowest Common Multiple: 4 Rational numbers A rational number is a number which can be expressed in the form of p/q where p and q are whole numbers. Recurring decimals are rational. e.g., 0.666666666 i.e., 0.6' Terminating decimals (that end) are rational. e.g., 0.8896 is a rational number. More Examples: -3/5, -2, 6/9, 12, 0.9', Square root of 4/9 = 2/3 2.11 Irrational numbers An irrational number is a number which cannot be represented in the form of p/q where p and q are integers. 2.12 Real numbers All the numbers that can be represented on number line are called Real numbers. i.e., all the integers, rational and irrational numbers. 2.13 Sequences A set of numbers which follows a certain pattern are said to be in sequence. e.g., 2, 4, 6, 8, 10, and 12. This is a sequence of numbers which have a common difference of 2 between every consecutive number. Each number of a sequence is said to be the term i.e., term 1(t1) = 2 t1 = 2(1) = 2 t2 = 2(2) = 4 t3 = 2(3) = 6 So, tn = 2(n) is the formula for nth term. Different types of sequence: Sequence with same difference between terms: The nth term (tn) of sequences with same common difference is: tn = an + b; Where a = common difference, b = preceding number of the sequence e.g., find the formula of nth term for the sequence 9, 16, 23, 30. Here common difference (t2 – t1 = t3 – t2 = t4 – t3) a =7 Preceding number of the first term of sequence i.e., (9 – 7), b = 2, Hence formula for nth term is tn = 7n + 2. Sequence with consecutive terms having same ratio between them e.g., 1, 4, 16, 64, 256. In this sequence each term is four times the previous term. As, t1 = 1, t2 = 4, So, t2 = 4 (1) gives t2 = 4 t1 Which can now be used to define formula for nth term: tn = 4 t(n-1) , since the nth term is four times the (n-1)th term. Known Sequences: Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23 (In this sequence you won't find any specific difference between the consecutive terms) Square Numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81 (Here it can be seen that there is a difference of 2 increasing with every consecutive term) Triangular Numbers: 1, 3, 6, 10, 15 (Here the difference between the terms increases by 1 each time) Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 is the Fibonacci sequence. (Here the first two terms are both 1 and then every consecutive term is obtained by adding the two previous terms) 2.14 Sets A set is a collection of well-defined objects. Defining a set: E.g. i) Number of Montessori schools in Indonesia ii) Number of the best schools in Indonesia Statement 1 is a set: Number of Montessori schools can be listed easily so it is well-defined. Statement 2 is not a set: This statement is not well defined because it can be argued, what is meant by "best schools", it open to interpretation. These well-defined objects are often called as "elements" of the set. Also a set is represented by listing all these elements within ‘curly’ brackets or braces { }. E.g., A set of vowels can be written as {a, e, i, o, u}. Other form of set representation: {x : x is a vowel of English alphabets}, Read as a set consisting of all elements of x, such that x is a vowel of the English alphabet. In this representation, "x" stands for all the elements that can be written according to the definition of the set which here is the vowel from the English alphabet. Usually a set is identified by naming it with an alphabet written in capital. Some Important Definitions: A = {x : x is a natural number} = {1, 2, 3, 4, 5, 6.........} B = {x : x is an even number } = {2, 4, 6, 8, 10 ..........} C = {x : x is a prime number } = {2, 3, 5, 7, 11 ..........} D = {x : 2 < x < 8} = {3,4,5,6,7} E = {(x,y): y=2x + 1, x and y are natural number} = {(1,3), (2,5), (3,7)...} Method: As x and y are natural numbers Taking x=1, substituting it in the formula y = 2x + 1 gives Y = 2(1) + 1 = 3 and hence the first element is (1,3), Similar can be done to find out taking other natural numbers as value of x or y. 2.15 Venn Diagram A Venn Diagram is another easy way to represent sets. A Venn Diagram is a large rectangle with circles or ovals inside to represent different sets and their properties. In a Venn Diagram, the circles may or may not overlap depending on the similarity of elements in those sets. i.e., If the sets have some of the elements same then the circles overlap and the same elements are written within that overlapped part. As shown in the figure the three circles are the sets and the overlapping parts show the common elements in them. A Universal set is the set consisting of all the other sub-sets. As shown in figure, the set consisting of all elements of the sets A, B and C. If an element is not in the given sets represented as a circle then it is lies inside the rectangle but outside the circles representing the sets. Set Notation: E.g; Let set A be set of Prime numbers less than 11 i.e., A = {2, 3, 5, 7} Venn Diagram for A':