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SJO PW - Język angielski ogólnotechniczny, Poziom B2 Opracowanie: Teresa Olechowska DLA LEKTORA Lekcja 2 B ______________________________________________________________________________________________ MATHEMATICS INTRODUCTION (in the class): Explain that in this part they are going to tackle the mathematical concepts of powers, roots and matrices. The introduction and TASK 1 should not take more than 5 minutes. Think and try to match the answers to the following questions: 1. What does it mean, when we say x squared? d 2. What does it mean when you see the following: 16 b 3. 4. 5. 6. a. b. c. d. e. What is a matrix? f What do matrices consist of? c How can matrices be added? a How can matrices be multiplied? e By adding corresponding elements - only if they have the same order. It means the square root of sixteen and it equals 4. Elements (numbers/letters), rows and columns. Then the 2 is called the power or index. If the number of elements in the columns of one matrix equals the number of elements in the rows of the other matrix. f. It is an array of numbers or letters in the shape of a rectangle. Key: 1d; 2b; 3f; 4c; 5a; 6e INPUT – HOMEWORK TASK 2 When checking this task, just point out that there are several ways of reading the powers, i.e. “squared, cubed, but - to the power of [number], or to the nth power”. Knowing that: a x a² = a³ (a times a squared is equal to a cubed) a³ x a² = a (a cubed times a squared is equal to a to the power of five, or a to the fifth power ) a. practice reading out the following: 1. a² + b² a squared plus b squared 2. x² + y³ x squared plus y cubed 3. p p to the power of four / to the fourth power 4. xⁿ x to the power of n / to the nth power b. Can you solve the following equations: 1. 4a² + 2a² = ?6a² 2. 4a² - 2a² = ?2a² 3. 4a² x 2a² = ? 8a 4. 4a² ÷ 2a² = ? 2 TASK 3 Point out the plural of the word index > indices. Complete the following statements about procedure: 1. When multiplying two numbers which have been raised to certain powers e.g. n³ (n cubed) and n (n to the fifth), we add the indices . 2. When we wish to raise a power to a power, the indices must be multiplied . In the above case the answer is n¹. 1 SJO PW - Język angielski ogólnotechniczny, Poziom B2 Opracowanie: Teresa Olechowska DLA LEKTORA Lekcja 2 B ______________________________________________________________________________________________ To learn more you can visit the YouTube site: http://www.youtube.com/watch?v=qOn-NW_lsvE&feature=related Key: 1. cubed; to the; indices; 2. raise; to; indices; multiplied. TASK 4 Before checking the answers, again underline that there are different ways of reading the roots, i. e. a square root, a cube root, but the nth root. However, some people also say the second root and the third root. Why "Root" ... ? When you see "root" think: "I know the tree, but what is the root that produced it?" Example: in √9 = 3 the "tree" is 9, and the root is 3. http://www.mathsisfun.com/numbers/nth-root.html Knowing that: is called the radical sign; 64 means the square root of 64; 27 means the cube root of 27; x means the fifth root of x. If we want to find the root in the following example, we must divide the index by the root: Practice reading out the following, and then express them in more simple terms: 1. √x²; 2. √4x; 3. √a²b² 4. ∜mn 5. b³ 6. a³ 7. √16b² If you want to learn and practice more, check the YouTube sites: http://www.youtube.com/watch?v=PlZZ6ZZJBg4 Key: 1. x; 2.2 x²; 3. ab; 4. mn; 5. b; 6. 2a²; 7. 4b. 2 SJO PW - Język angielski ogólnotechniczny, Poziom B2 DLA LEKTORA Opracowanie: Teresa Olechowska Lekcja 2 B ______________________________________________________________________________________________ TASK 5 While checking in class - it is worth mentioning that in Polish language we have the same rules. Words which retain their original Greek or Latin forms make their plurals according to the rules of Greek and Latin. -us > -i -is > -es -um -ix -a -on -ices -ex Supply the proper plurals of the words in the brackets. 1. The new (syllabus) syllabi will be drawn up according to different (criterion) criteria. 2. These (index) indices must be multiplied. 3. (Matrix) matrices can be added if they have the same order. 4. He agreed that these were strange (phenomenon) phenomena . 5. Television and newspapers are the mass (medium) media . Key: 1. syllabi; 2. indices; 3. matrices; 4. phenomena; 5. media TASK 6 1. Now add two matrices: a 4 1 7 5 2 8 3 6 9 + b 3 5 6 1 4 9 7 2 8 = 2. And multiply the following two matrices: b 3 1 a 2 0 3 x 2 1 - 1 3 1 1 0 = If you want to learn more - visit: http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php http://www.youtube.com/watch?v=sYlOjyPyX3g Key: 1. addition: 7 6 10 6 6 8 13 17 17 2. multiplication: (2 x 3 + 0 x 2 + 3 x 1) (2 x 1 + 0 x 1 + 3 x 0) 9 2 = (- 1 x 3 + 3 x 2 + 1 x 1) (- 1 x 1 + 3 x 1 + 1 x 0) 4 3 2 SJO PW - Język angielski ogólnotechniczny, Poziom B2 Opracowanie: Teresa Olechowska DLA LEKTORA Lekcja 2 B ______________________________________________________________________________________________ CLASSWORK CHECKING TASK 7 When you check their homework – which should take only a few minutes, you can control how they have got the result of addition or multiplication. The questions in TASK 7 are for some discussion about mathematical concepts in general. The discussion can be serious or a humorous one – it depends on the group. TASK 7 Discuss the following questions: 1. Can numbers or other mathematical concepts be said to exist? 2. Are numbers ‘discovered’ or ‘invented’ (Leopold Kronecker said that “God made the integers. All else is the work of men”)? 3. Taking negative numbers into consideration – has anyone seen ‘minus one cow’? 1. 2. 3. 4. 5. 6. 7. BIBLIOGRAPHY A Mathematical Dictionary, Jason, R.E., Pergamon Press, 1979 Alex’s Adventures in Numberland, Bellos, Alex, Bloomsbury, 2010 Computer Adventures in Lingua Land, Olechowska, Teresa, Rouba, Wojciech, WSiP, 1992 English for Basic Maths, Blackie, David, Nelson, 1978 Lingua Land Kappa Land, Olechowska, Teresa, WPW, 1986 Mathematics and the Imagination, Kasner, Edward, Newman, James, Penguin Books, reprinted 1979 MINI anglojęzyczne – EAP & ESP, Olechowska, Teresa, wydanie własne, 2012 http://dictionary.reference.com http://www.purplemath.com/modules/radicals.htm http://www.mathsisfun.com/numbers/nth-root.html http://en.wikibooks.org/wiki/Primary_Mathematics/Powers,_roots,_and_exponents http://www.youtube.com/watch?v=qOn-NW_lsvE&feature=related http://www.youtube.com/watch?v=PlZZ6ZZJBg4 http://en.wikipedia.org/wiki/Matrix_multiplication http://www.mathwarehouse.com/algebra/matrix/multiply-matrix.php http://www.youtube.com/watch?v=sYlOjyPyX3g 4