Contents
Review 1 Calculating with numbers &expressions ··········································· 4
Review 2 Linear equations, Simultaneousequations ·············································· 56
Review 3 Coordinates and functions ········ 76
Review 4 Figures ················································· 110
Review 5 Organizing and making use ofdata; Probability ···························· 196of square roots
3 Calculating expressions containing radicalsigns ·········································································· 48
Addition and subtraction / Four arithmetic
calculations / Using the expansion formulas /Evaluating expressions / Using square roots
Study seminar Rationalizing denominators thatare in the form of a ± b

Comprehension test for Chapter 2,
End-of-chapter problems ···································· 54Chapter

1Calculating with expressions

1 Calculating with polynomials ······················· 7Multiplying a monomial and a polynomial / Dividing apolynomial by a monomial / Four arithmetic operations
with monomials and polynomials / Multiplying twopolynomials / Formula for expanding expressions of
the form (x＋a)(x＋b) / Expanding expressions of theform (a＋b)2 and (a−b)2 / Expanding expressions of
the form (a＋b)(a−b) / Various calculations
2 Factoring ······························································· 19
Prime numbers and prime factorization / Extractingcommon factors / Factoring x 2＋(a＋b)x＋ab /Factoring a2＋2ab＋b2 and a2−2ab＋b2 / Factoring
a2−b2 / Various ways of factoring / Factoring byreplacement
3 Using calculations of expressions ·········· 27
Evaluating expressions / Using formulas whencalculating numbers / Using expressions to set up aproof / Using calculations of expressions for figures

Comprehension test for Chapter 1,
End-of-chapter problems ···································· 32
Complements
Complex factoring ························· 34
ComplementsGreatest common factors and least
common multiples ····················································· 35

Chapter

2Square roots

1 Square roots ························································· 37
Meaning of square roots and how to represent them /Relative size of square roots / Size of square roots /Rational and irrational numbers, Recurring decimals
2 Multiplying and dividing expressions
containing radical signs ································ 42
Products and quotients of square roots / Converting
between a and b c / Rationalizing denominators /Multiplication and division / Approximating values

Chapter

3Quadratic equations

1 How to solve quadratic equations ··········· 59
How to solve ax2−b＝0 / How to solve (x＋m)2＝n /How to solve x2＋px＋q＝0 / How to use the quadratic
formula to solve quadratic equations / How to use
factoring to solve quadratic equations / Various
quadratic equations
2 Using quadratic equations ··························· 66
Problems about values of variables / Problems about
numbers / Problems about figures / Problems aboutgraphs of functions / Problems about ratios / Solving
problems using the quadratic formulaComprehension test for Chapter 3,
End-of-chapter problems ···································· 74

Chapter

4Functions

1 The function y＝ax2 ·········································· 81
The function y＝ax2 / Expressing the relationshipbetween two quantities as y＝ax2 / Graphing thefunction y＝ax2 / Properties of the function y＝ax2
2 Properties of the function y＝ax2 ············ 87
Domains / Rate of change / Using the rate of change/ Points where a parabola and a line intersect / Areaof triangles
3 Using graphs of functions ····························· 95
Graphs of functions and segments / Graphs offunctions and squares / Graphs of functions andparallelograms / Bisecting areas / Changing shapes
without changing area / Changing shapes withoutchanging area using quadrilaterals / Moving pointsand functions / Moving figures and functions /Various functions

9th grade

Comprehension test for Chapter 4,
End-of-chapter problems ·································· 108

Chapter

5Similar figures

1 Similar figures ·················································· 115
Similarity between figures / Properties of ratios /Properties of similar figures, Similarity ratios /Conditions for similar triangles / Ratios of thelengths of sides in similar triangles / Proving thesimilarity of two triangles / Using reductions
2 Parallel lines and ratios of segments ·· 125Triangles and ratios / Parallel lines and ratios /Using the relationship of parallel lines and ratios /Midpoint theorem / Using the midpoint theorem /Angle bisectors and ratios
Study seminar The centroid of a triangle
3 Ratios of area, Ratios of volume ············ 134
Ratios of area / Ratios of the areas of similar
figures / Ratio of the surface areas and ratio of thevolumes of similar figures
4 Using similarity ················································· 138
Ratios of segment / Using ratios of area / Usingratios of volume / Using ratios of segment on
coordinate planes
Study seminar
Ceva’s theorem, Menelaus’theorem

Comprehension test for Chapter 5,
End-of-chapter problems ·································· 144

Chapter6

Properties of circles

1 Circles and lines ·············································· 147
Cir cles and lines / Lengths of tangents /Circumscribed circles and inscribed circles
2 Inscribed angle theorem ····························· 151
Inscribed angle theorem / The converse of theinscribed angle theorem / Proofs using the inscribed
angle theorem / A circle and similar triangles /Proving similarity using a circle / Proofs using theconverse of the inscribed angle theorem
Study seminar
Quadrangle inscribed in acircle / Alternate segment theorem / Intersecting
chords theorem

Chapter

7The Pythagorean theorem

1 The Pythagorean theorem ·························· 167
The Pythagorean theorem / Proving that thePythagorean theorem is true / The converse of thePythagorean theorem
2 The Pythagorean theorem and plane
figures ···································································· 171
Special right triangles / Drawing an auxiliary line
in an isosceles triangle or a trapezoid / Varioustriangles / Distance between two points / Chords
in a circle / Tangent to a circle
3 The Pythagorean theorem and solids ·· 180
Cuboids and cubes / Pyramids / Cones / Spheres
Study seminar Height and volume of a regulartetrahedron
4 Solving various problems by applying
the Pythagorean theorem ··························· 186
Problems about plane figures / Similarity and thePythagorean theorem / Problems about circles /The shortest distance on the surface of a solid /Polygon formed inside a solid / Problems aboutsolids

Comprehension test for Chapter 7,
End-of-chapter problems ·································· 194

Chapter

8Sample surveys

1 Sample surveys ················································· 199
Censuses and sample sur veys / Using samplesurveys

Comprehension test for Chapter 8,End-of-chapter problems ·································· 201
Complements
Complements

Inequalities ··································· 202
Base-n number system ··············· 205

Key points learned during junior highschool ··········································································· 206

Comprehension test for Chapter 6,
End-of-chapter problems ·································· 164
Note: “Selections of entrance examination and Final test (placed from page 206 to page 223 in the original book)” are omitted in this book.