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Calculating the height of a vertical object from its shadow We use the following method to find out the height of the tree AB We hammer a stake vertically into the ground AB We measure the length of the stake, AB . Then we measure the length of the shadows, AC and AC , of the tree and the stake, that are made by the Sun at the same moment. Triangles ABC and A'B'C' are similar because they have two equal angles: A A because both are right angles. C C because the Sun’s rays shine on the tree and the stake at the same angle. Since the triangles are similar, their sides are proportional: AB AC
Suma Piezas is an educational project by Anaya for Secondary Education.
The following people have worked on this book: Editorial team: Lucía de la Rosa (adaptation of material from the book Matematicas 2.o ESO and creation of the fact files and activities in this book) DESIGN, TECHNICAL DRAWINGS AND MAPS: Miguel Ángel Castillejos, Miguel Ángel Díaz-Rullo, Marta Gómez, Patricia G. Serrano
ILLUSTRATIONS: Celia López
Layout: Elvira López and Celia López
Translation: Robin Munby
PICTURE EDITING: Olga Sayans
Photographs: Age Fotostock (Dawna Moore), Archivo Anaya (Cosano, P.; Osuna, J.), iStock/Gettyimages (DrPAS, JackF, Nerthuz), 123rf (Olga Yastremska)
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Index
1 The set of natural numbers • 2 The relation of divisibility • 3 Prime and composite numbers • 4 Lowest common multiple of two or more numbers • 5 Greatest common divisor of two or more numbers • 6 The º set of integers • 7 Operations with integers • 8 Powers of integers • 9 Roots of integers
1 Decimal numbers • 2 Operations with decimal numbers • 3 Decimal numbers and sexagesimal numbers • 4 The square root of a decimal number • 5 Fractions • 6 Fractions and decimal numbers
3. OPERATIONS WITH FRACTIONS.................................25 1 Adding and subtracting fractions • 2 Multiplying and dividing fractions • 3 Problems with fractions • 4 Powers and fractions
4. PROPORTIONALITY..............................................................33 1 Ratios and proportions • 2 Directly proportional quantities • 3 Inversely proportional quantities • 4 Problems of compound proportionality • 5 Problems of proportional distribution
5. PERCENTAGES......................................................................... 41 1 Percentages. Concept • 2 Problems with percentages • 3 Bank interest • 4 Other arithmetic problems
6. ALGEBRA .................................................................................... 49 1 Why do we use algebra? • 2 Algebraic expressions • 3 Polynomials • 4 Notable products
7. EQUATIONS ................................................................................57 1 Equations: meaning and use • 2 Equations: elements and names • 3 Transposing terms • 4 Solving simple equations • 5 Equations with denominators • 6 The general method for solving first-degree equations • 7 Solving problems with equations • 8 Second-degree equations • 9 Solving second-degree equations 8. SYSTEMS OF EQUATIONS ............................................65 1 First-degree equations with two unknowns • 2 Systems of linear equations • 3 Methods for solving linear system • 4 Solving problems with systems of equations
19. PYTHAGORAS’ THEOREM...............................................73 1 Pythagoras’ theorem • 2 Calculating a side when two are known • 3 Applications of Pythagoras’ theorem
10. SIMILARITY............................................................................79 1 Similar shapes • 2 Plans, maps and models • 3 How to build similar figures • 4 Thales’ theorem • 5 Similarity between right-angled triangles • 6 Applications of the similarity of triangles
11. Three-dimensional geometric shapes 87
1 Prisms • 2 Pyramids • 3 Truncated pyramids • 4 Regular polyhedra • 5 Cross-sections of polyhedra • 6 Cylinders • 7 Cones • 8 Truncated cones • 9 Spheres • 10 Sections of spheres, cylinders and cones
12. Measuring volume.....................................................97 1 Units of volume • 2 Cavalieri’s principle • 3 Volume of a prism and a cylinder • 4 Volume of a pyramid and a truncated pyramid • 5 Volume of a cone and a truncated cone • 6 Volume of a sphere
13. Functions ..........................................................................105 1 The concept of a function • 2 Increases, decreases, maximums and minimums • 3 Functions given as tables of values • 4 Functions given as equations • 5 Proportional functions: y = mx • 6 The slope of a line • 7 Linear functions: y = mx + n • 8 Constant functions: y = k
14. statistics ........................................................................... 113 1 Drawing a table and its graph • 2 Location parameters • 3 Dispersion parameters • 4 Measures of position • 5 Two-way tables
