Let’s review what you have learned at elementary school

3 Ratios, Graphs, Proportions and inverse proportions
contents

1Percentages and Buai＊

2

Calculating percentages and Buai for various amounts

3

Proportions and graphs

4Ratios

5

Proportions and inverse proportions

1 Percentages and Buai Change the following decimals into equivalent percentages and Buai.
□⑴

□⑵

0.2

□⑶

0.62

□⑷

3.2

2 Calculating percentages and Buai for various amounts Fill in□⑴

60 yen is□⑶

60％ of 300 m2 is□⑸

The payment for an 800-yen article at a 40％ discount is0.164

with a suitable number.

wari of 200 yen.

□⑵

42 meters is％ of 600 meters.

m2.

□⑷

5 wari 4 bu of 600 g isg.

yen.

3 Proportions and graphs The circle graph on the right shows the land areas used for growing different
fruits in an orchard. Answer the following questions.

(%)□⑴ What percentage of the land area is used for growing strawberries,grapes and pears, respectively?

OthersPears

If the whole area of the orchard is 7,200 m2, how many□⑵

square meters of land is used for growing strawberries, grapesStrawberries

Grapesand pears, respectively?

4 Ratios Answer the following questions.
□⑴

Simplify the following ratios.
①□⑵
①

②

8：24

Fill in③35：49

④

1.2：3

4
2：
3
5with a suitable number.
②

2：5＝6：：12＝7：4③：1.6＝5：4

5 Proportions and inverse proportions y is proportional to x in the table ⑴, while y is inversely
proportional to x in the table ⑵. Fill in blanks ① to ④.
□⑴＊Proportion

1 wari isx

2

4②

8④y8①24③ 40□⑵

Inverse proportion

x2

3②

6④y18

①

9③4111or 10％, 1 bu isor 1％, 1 rin isor 0.1％. These expressions are used in Japan and101001,000

are called buai.
7Chapter 2

Algebraic expressions

Algebraic expressions

Chapter 2●Key points of study●

1

How to express products☞P 40〜

Expressions containing multiplication with variables＊ are written without × symbols. Numbers are⑴

Chapter 2

always written in front of variables, and variables are usually arranged in alphabetic order.
⑵ a×1 and 1×a are written as a, instead of 1a . (−1)×a is written as −a.
⑶2

The product of a variable and itself is written using exponents: a×a×a=a3
How to express quotients☞P 41〜⑴

Expressions containing division with variables are written in fraction form without ÷ symbols.

⑵

a1
1 1
a
can also be written as a: a÷2=a× = a=2
2
2 2
2⑶

When the numerator or the denominator has a negative sign, it is put in front of the fraction:

3

How to express quantities☞P 45〜−3
3=−xx

When writing an expression that contains variables, first explain the relationships between the quantities
in words. Then replace the words with variables and numbers to form a mathematical expression.

4

Evaluating expressions☞P 49〜⑴

Plugging in a number for a variable in an expression is called substitution.

⑵

Calculating the result of an expression by substituting a number for a variable is called evaluating theexpression.

5Terms and coefficients, Combining terms☞P 51〜⑴

In an expression containing addition, each item joined by a + sign is called a term.

⑵

In a term containing one or more variables, the number in front of the variable(s）is called thecoefficient of the term.

⑶

Terms containing the same variables are called “like terms” and can be combined into one term. In thisway, an expression can be simplified.

6Linear expressions☞P 51〜

3x−5 consists of two terms: 3x and −5. Terms like 3x that contain only one variable are called terms of
the first degree. An expression that contains only terms of the first degree or that indicates the sum ofterms of the first degree and a number is called a linear expression.

7

Calculation with linear expressions⑴☞P 53〜

To add or subtract two linear expressions, first remove the parentheses, then combine the terms having
the same variable portion and those consisting of only numbers.

⑵

To multiply a linear expression by a number, multiply each term of theexpression by the number, using the distributive law as shown on the right.

⑶

a（b+c）=ab+ac

To divide a linear expression by a number, divide each term of the expression by the number. You canalso multiply each term by the reciprocal of the divisor.

8How to express the relations of quantities⑴☞P 59〜

An expression that uses an equal sign (=) to indicate an equivalent relationship between two quantities
is called an equality.

⑵

An expression that uses an inequality sign（＜， ＞， ≦， ≧）to indicate the relative size of twoquantities is called an inequality.

＊

Variables are uppercase or lowercase letters used to represent unknown quantities in mathematic

expressions.

38

Chapter 5

Plane figures

Plane figures

Chapter 5●Key points of study●

1Lines and angles⑴☞P 119〜Line ABA

A straight line that passes through two points A and B is called line AB.

⑵

A portion of line AB that has two endpoints A and B is called segment AB.

⑶

A portion of line AB that starts at A and goes off in the direction of B to infinity

ABRay ABAis called ray AB.

BSegment ABB⑷

The halfway point of a segment is called the midpoint.

⑸

Triangle ABC is written as △ABC, using the symbol △.

⑹

The angle formed by ray OA and ray OB is called angle AOB, which is written as ∠AOB.

⑺

When lines AB and CD are parallel, the relationship is expressed as AB™CD. When they areperpendicular, the relationship is expressed as AB⊥CD.

2

Circle arcs and chords, Tangents☞P 121〜

Aarc AB(⑴

When points A and B are on the circumference of a circle,

Chapter 5

the part from A to B is called arc AB, which is written as A
⌒
B.

(AB)

chord ABB

The segment that links A and B is called chord AB.
⑵

When line ¬ touches circle O at a single point A, line ¬ is called a tangent of circle O and point A iscalled the tangent point. Line ¬ is perpendicular to OA (¬⊥OA).

3Figure transformation☞P 124〜

Translation4Rotation

Figure construction⑴

Reflection☞P 128〜

Figure construction involves drawing figures with only a straightedge and compass. The straightedge
is used to draw straight lines, and the compass is used to draw circles and replicate the length ofsegments.

⑵

Since it is important to show the process of constructing figures, do not erase the lines or parts ofcircles that are drawn when solving construction problems.
Perpendicularbisector5

Angle bisector

Sector arc length and area

Perpendicular line thatpasses through point O☞P 139〜⑴

A portion of a circleʼs area bounded by two radii is called a sector.

⑵

The angle made by the two radii in the sector is called the central angle.

⑶

For a sector having radius r, central angle a°, arc length ¬, and area S,
you can set up the following formulas.
①118¬＝2∏r×a360② S＝∏r 2×

a1，S＝ ¬r
360
2

Perpendicular line thatpasses through point P

Chapter 5

Plane figures

Comprehension test for Chapter 5
1 Parallel and perpendicular lines Trapezoid ABCD is shown on the right.
Points E, F, G, and H are the points where each side is tangent to circle O.
Answer the following questions.
□⑴

Write an expression to indicate that segments OG and OH have the same length.

□⑵Side AD is parallel to side BC. Express this relationship using a symbol.

□⑶

Segment OG is perpendicular to side DC. Express this relationship using a symbol.

2 Figure construction Construct the following figures.
□⑴

Perpendicular bisector of segment AB (Diagram 1)□⑶

Straight line that passes through point P and is perpendicular to line ¬ (Diagram 3)
Diagram 1□⑵

Angle bisector of ∠AOB (Diagram 2)Diagram 2

Diagram 33 Figure transformation As shown on the right, △DEF is theresult when △ABC is reflected over the axis of line ¬ and thenrotated 30° clockwise around the center of rotation F. Answer thefollowing questions about △DEF and △ABC.
□⑴

Which point corresponds to point A?

□⑵

Which side corresponds to side AB?

□⑶

Which angle corresponds to ∠DEF?

□⑷

When ∠ACB＝58°, find the size of ∠A'FE.

4 Sectors Find the arc length and area of each sector shown below.
□⑴144□⑵