Everything You Need to Ace Math in One Big Fat Notebook 読書ノート3/3【働学併進#017】

正式な書名は「Everything You Need to Ace Math in one Big Fat Notebook: The Complete Middle School School Guide」。"数学で評価Aを取るために必要な全て: 中学教育の完全な手引書"、和訳されたものは「14歳からの数学」。

Everything You Need to Ace Math in One Big Fat Notebook: The Complete Middle School Study Guide (Big Fat Notebooks)

前編は第1章のnumber system とfractionと、第2章のratio, rate, proportion などについての語句をまとめた。中編は第3章のexpressionやequationと、第5章のprobabilityとstatisticsの語句をまとめた。
今回の後編は4章のgeometry 幾何と、6章のcoordinate plane 座標系, function 関数の語句をまとめる。

Geometry

Geometry 幾何学 is the branch of mathematics that deals with lines, shapes, and space.

Key Concepts

• Point 点

• Line segment 線分
  a part of a line that has two endpoints. $${\overline{AB}}$$

• Line 直線
  a line that continues forever in both directions. $${\overleftrightarrow{AB}}$$

• Ray 半直線
  a line with only one endpoint $${\overrightarrow{AB}}$$

• Parallel lines 平行線
  lines that are always the same distance apart. they never intersect. $${m \| n}$$ (\|)

• Angle 角度
  formed by two rays with the same endpoint. $${\angle A}$$ (\angle)
  we use degrees(°) to measure the size of an angle.

• Vertex 頂点
  the point of intersection of rays or lines that form an angle. $${\angle A \rm{\ or\ } \angle BAC}$$

• Right angle 直角
  a 90-degree angle

• Perpendicular lines 垂線
  two lines that form a right angle. $${P \perp Q}$$ (\perp)

• Congruent lengths/angles
  the shapes, lines, or angles are equal in size. $${\cong}$$ (\cong)

Angles

• Right triangle 直角
  Measures exactly 90°

• Acute triangle 鋭角
  Measures less than 90°

• Obtuse triangle 鈍角
  measures greater than 90°

• Complementary angle 余角
  Two angles whose sum is 90°

• Supplementary angle 補角
  Two angles whose sum is 180°

• Adjacent angles 隣角
  angles that share a vertex and a common side

• Vertical angles 対角
  angles formed by two intersecting lines that are opposite each other

Geometry Tools

• Ruler 定規

• Protractor 分度器

• Set square (or triangle) 三角定規

  • 60-30 set square

  • 45 set square

• Drawing compass コンパス

Polygons

Plane geometry 平面幾何 deals with 2-dimensional (2D) shapes (or flat shapes).
a polygon 多角形 is a closed plane figure with at least three straight sides.

Quadrilaterals and Area

a quadrilateral 四辺形 is a polygon with four(quadri) sides(lateral) 辺.

• Parallelogram 平行四辺形
  Opposite sides are parallel and equal in length

• Rectangle 長方形
  A parallelogram where all four sides form right angles

• Rhombus 菱形ひしがた
  A parallelogram where all sides are equal in length

• Square 正方形
  A parallelogram where all sides are equal in length and all sides form right angles

• Trapezoid 台形だいけい
  Has exactly two parallel sides, which are called base1 and base2. Sides do not have to be equal in length

the perimeter 外周 is the distance around a two-dimensional object.
the area 面積 is the size of a surface or is the amount of space inside a two-dimensional object.
To calculate the area of a parallelogram, multiply the base by the height
($${A = bh}$$).

To calculate the are of trapezoid, use the formula 公式
$${A = \frac{1}{2}h(b_1+b_2)}$$

Triangles and Area

a triangle 三角形 has three sides and three angles. 
the symbol for a triangle is $${\triangle}$$. (\triangle)

• Equilateral triangle 正三角形
  3 equal sides, 3 equal angles, always 60°

• Isosceles triangle 二等辺三角形
  2 equal sides, 2 equal angles

• Scalene triangle 三角形
  No equal sides, no equal angles

• Right Triangle 直角三角形
  Has a right angle.

• Obtuse triangle
  Has an angle more than 90°

• Acute triangle
  all angles are less than 90°

To calculate the area of a triangle, multiply the base 底辺 by the height 高さ, then multiply that amount by half.
$${A = \frac{1}{2}bh}$$

因みに、三角形の内角はinterior angle、外角はexterior angleと言う。

The Pythagorean Theorem

Pythagorean theorem by Wikimedia

the pythagorean theorem ピタゴラスの定理/三平方の定理 is a fundamental relation in Euclidean geometry ユークリッド幾何 among the three sides of a right triangle.
the pythagorean theorem is used to find the length of hypotenuse 斜辺.

$$
a^2 + b^2 = c^2
$$

• $${3^2 + 4^2 = 5^2}$$

• $${5^2 + 12^2 = 13^2}$$

• $${8^2 + 15^2 = 17^2}$$

Circles, Circumference, and Area

• Circumference (C) 円周
  the distance around the circle

• Chord 弦
  a line segment whose endpoints are on the circle

• Diameter (d) 直径
  a chord that passes through the center of the circle

• Radius (r) 半径
  a line segment that has one endpoint at the center and the other on the circle.

• Pi (π) 円周率
  the ratio of a circle's circumference to its diameter

• Arc 孤

To find the circumference, use the following formula
$${C = 2 \pi r = \pi d}$$

To find the area of a circle, use the following formula
$${A = \pi r^2}$$

Similar Figures

similar figures 相似形 are figures that have the same shape, but not necessarily the same size.
similar figures have corresponding angles that are congruent and corresponding sides that are proportional in size.
the symbol for similar figures is $${\sim}$$(\sim)

a scale drawing 縮尺図 is a drawing that is similar to an actual object just made bigger or smaller.
the scale 縮尺 is the ratio of the length in the drawing to the actual length.

Polyhedra

Solid geometry 空間幾何 deals with 3-Dimensional (3D) shapes.
3D shapes is also called a "space figure" or a "solid".
Polyhedron 多面体 is a 3D figure that is made up of regions that are in the shape of polygons, and plural of polyhedron is polyhedra

Regular polyhedron

a regular polyhedron 正多面体 is a polyhedron where all the faces are identical polygons.

Pagesのテンプレートを使わせてもらった

• Tetrahedron 正四面体

• Cube

• Octahedron 正八面体

• Dodecahedron 正十面体

• Icosahedron 正二十面体

Rectangular prism

a prism 角柱 is a 3D figure that has two polygon bases that are parallel and congruent as well as lateral faces that are parallelograms.

これだけ不細工な画像で申し訳ない

• Triangular prism 三角柱

• Rectangular prism 四角柱

• Pentagonal prism 五角柱

• Hexagonal prism 六角柱

Other polyhedra

• Sphere 球

• Pyramid 錐

  • triangular pyramid 三角錐

  • rectangular pyramid 四角錐

  • pentagonal pyramid 五角錐

  • hexagonal pyramid 六角錐

• Cylinder 円柱

Volume

the volume of a 3D figure refers to the number of cubic units needed to fill the figure.

• Volume of prisms: $${V = Bh}$$

  • Volume of rectangular prism: $${V = lwh}$$
    l: length, w: width: h: height

  • Volume of triangular prism: $${V = \frac{1}{2}bh_th_p}$$

    • $${h_t}$$: height of triangle

    • $${h_p}$$: height of prism

• Volume of a cone: $${V = \frac{1}{3}\pi r^2 h}$$

• Volume of a pyramid: $${V = \frac{1}{3}Bh}$$

• Volume of a sphere: $${V = \frac{4}{3}\pi r^3}$$

  • Surface area 表面積: $${A = 4 \pi r^2}$$

ちなみに、断面図は英語でcross sectionと言う。

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The Coordinate Plane & Functions

The Coordinate Plane

a coordinate plane 座標平面 is a flat surface formed by the intersection of two lines or axes 軸: the horizontal line known as the x-axis x軸, and the vertical line, known as the y-axis y軸.
the x- and y-axes intersect (cross) at the origin 原点.
an ordered pair gives the coordinates (exact location) of a point. like $${(x, y)}$$

the coordinate plane is divided into four quadrants 四象限(quadrants Ⅰ, quadrants Ⅱ, quadrants Ⅲ and quadrants Ⅳ).

Cartesian plane (デカルト平面) by Wikimedia

to find the distance between two point, use this distance (d) formula..
$${d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}$$

Relation and Functions

a relation is a set of ordered pairs. and in a relation, all of the x-coordinates are called the domain 定義域, and all of the y-coordinates are called the range 値域.
e.g. (-6, 0), (-3, 1), (0,2), (3, 3), (6, 4)
domain: -6, -3, 0, 3, 6
range: 0, 1, 2, 3, 4

functions 関数 are a kind of relationship where there is only one y-value for each x-value.
the input 入力 refers to all of the domain that can be substituted in the formula for $${x}$$. the output 出力 refers to all of the range that result after we input the x-values and simplify

$${y = \frac{1}{3}+2}$$

the slope (m) is a number that is a ratio that describes the tilt of a line.
$${\rm{slope} = m = \frac{\rm{rise}}{\rm{run}} = \frac{y_2-y_1}{x_2-x_1}}$$
rise yの変化量 is how much a line goes up or down.
run xの変化量 is how much a line moves left or right.

なぜ傾きを表すアルファベットがmなのかは諸説あるらしい。
最も有名なのは、幾何学を発展させたことで有名なフランス人の数学者Descartesデカルトが「上昇」を表すフランス語 montant の頭文字から取ったと言う説である。
しかし、現在のフランスでは英語のslopeからy=sx+bとして教えられているとか。

• positive slope: 正の傾き

• negative slope: 負の傾き

• zero slope: xに平行な傾き

• undefined slope: yに平行な傾き
  vertical line because the run is 0, and any number divided by 0 is undefined.

Linear Equations

a linear equation is an equation whose graph is a line.
a linear equation always has the form:
$${y = mx + b}$$
b represents the y-intercept 切片 (where a line crosses the y-axis).

Simultaneous Linear Equations

simultaneous linear equations or system of linear equation is a collection of one or more linear equations involving the same variables.

Nonlinear Functions

Quadratic Function
parabola

Transformation Polygons on the Coordinate Plane