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Everything You Need to Ace Math in One Big Fat Notebook 読書ノート 2/3 【働学併進#016】

nariXcode

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」。

前回は第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。
今回は第3章の数式 expression や方程式 equation と、第5章の確率 probability と統計 statistics の語句をまとめる。
第4章と第6章は幾何 geometryの内容なので明日の後編にてまとめる。

Expressions

An expression 数式 is a mathematical phrase that contains numbers, variables 変数 (letters or symbols used in place of a quantity we don't know yet), and/or operators 演算子. and it allow us to do calculations to find out what quantity the variable is.
a constant 定数 on the other hand, is a number that stays fixed in an expression.

A term 項 is a single mathematical expression. It may be a single number (positive or negative), a single variable, several variables multiplied but never added or subtracted. Some terms contain variables with a number in front of them. The number in front of a term is called a coefficient 係数.

We collect/combine like terms 項をまとめる to rewrite the expression so that it contains fewer numbers, variables and operations.

Operator

  • + sum

    • Keyword: "greater than", "more than", "plus", "added to", "increased by"

      • "21 increased by b" → 21 + b

  • - difference

    • Keyword: "less than", "decreased by", "subtracted from", "fewer"

      • "5 less than h" → h - 5

  • × product

    • Keyword: "times", "multiplied by", "of"

      • "The product of 11 and y" → 11y

      • "15% of x" → 0.15x

  • ÷ quotient

    • Keyword: "divided by", "per"

      • "The quotient of 7 and r" → $${\frac{7}{r}}$$

Properties

Properties 性質 are like a set of math rules that are always true. There are four basic properties of numbers: commutative, associative, distributive, and identity

  1. Identity property 恒等法則

    • of addition: $${a + 0 = a}$$

    • of multiplication: $${a \times 1 = a}$$

  2. Commutative property 交換法則

    • of addition: $${a + b = b + a}$$

    • of multiplication: $${a \times b = b \times a}$$

  3. Associative property 結合法則

    • of addition: $${(a + b) + c = a + (b+c)}$$ 

    • of multiplication: $${(a \times b) \times c = a \times (b \times c)}$$

  4. Distributive property 分配法則 of multiplication over addition/subtraction

    • multiplication over addition: $${a(b + c) = ab + ac}$$

    • subtraction over subtraction: $${a(b - c) = ab - ac}$$

    • distribute (the number across the terms inside the parentheses) 展開する

    • factoring/factorization 因数分解: the reverse of the distributive property.

      • factor 因数分解する

因みに、property 性質とlaw 法則の違いは曖昧である。そのためcommutative property をcommutative law と呼ぶ人もいる。
ただ、語感の違いだけに注目すると「人間が発見した/誰にも強制されていないようなものはproperty」であり、「人間が計算しやすいように都合よく定義した基本的な規則のようなものがlaw」とも取れる。

Exponents

an exponent 指数 is the number of times the base number is multiplied by itself.
$${2^8 = 256}$$ "2 to the 8th power is 256" or "2 to the power of 8"

Laws of Exponents

前述の通りlawとpropertyの違いは曖昧なため、Properties of exponentsとも言える。
英語では指数の整理の仕方それぞれに名前がついているが、日本語では全てをまとめて指数法則と呼んでいる。この本でも、一つひとつ以下のような名前で区別して紹介されることはなかった。

  • Law of Product: $${a^m × a^n = a^{m+n}}$$

  • Law of Quotient: $${a^m \div a^n = a^{m-n}}$$

  • Law of Zero Exponent: $${a^0 = 1}$$

  • Law of Negative Exponent: $${a^{-m} = \frac{1}{a^m}}$$
    Negative exponent 負の指数 shows how many times we have to multiply the reciprocal 逆数 of the base.

  • Law of Power of a Power: $${(a^m)^n = a^{mn}}$$

  • Law of Power of a Product: $${(ab)^m = a^mb^m}$$

  • Law of Power of a Quotient: $${(\frac{a}{b})^m = \frac{a^m}{b^m}}$$

Power of a Power: Multiply Exponents を覚えるために、"Powerful Orangutans Propelled Multiple Elephants (屈強なオラウータンが複数の象を駆り立てる)"と言う略名 mnemonic があるらしい。

Scientific Notation

Scientific notation 科学的表記法 is a shorthand way of writing numbers that are often very small or large by using power of 10.
e.g. $${6.02214076 \times 10^{23}, 8.854187 \times 10^{-12}}$$

累乗を使わない表記法のことは英語ではStandard notation, もしくはStarndard formと言うらしいが、「標準表記法」などと言う言葉は日本では見かけない。

Square & Cube Roots

a square 平方 is the result of multiplying a number by itself. 
e.g. "square of 3 is 9", "18 squared is 324"

the opposite of squaring a number is to take a number's square root 平方根.
e.g. "square root of 16 ($${\sqrt{16}}$$) is 4"
The square root of a number is indicated by putting it inside a radical sign 根号 (√ )

Perfect square 完全平方 is a number that is the square of an integers.
e.g. 16, 25, 36, 
※ if a number under the radical sign is NOT a perfect square, it is an irrational number.

a cube 立方 is the result of multiplying three instances of numbers together.
the opposite of cubing a number is to take a number's cube root 立方根.
The cube root of a number is indicated by putting it inside a radial sign with a 3 on top ($${\sqrt[3]{}}$$).
Perfect cube 完全立方 is a number that is the cube of an integers.  e.g. 8, -8, 27, -27, 64, -64

Equations

an equation 方程式 is a mathematical sentence with an equal sign (=). 
the solution 解 is the missing number or variable that makes the sentence true.

There are 2 different types of variables that can appear in an equation. 
independent variable and dependent variable.

  • Independent variable 独立変数
    the variable you are substituting for.

  • Dependent variable 従属変数
    the other variable that you solve for.

Wikiのページには、"A symbol that stands for an arbitrary任意の、恣意的な input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable."と書かれていた。

Evaluation 評価 is the process of simplifying a mathematical expression by first substituting/replacing a variable with a number, and then solving the expression using order of operations.

Solving for the unknown or x 方程式を解く is about finding the missing information that you're looking for.
In order to do so, we must isolate the variable on one side of the equal sign

Inequalities

an inequality 不等式 is a mathematical sentence that contains a sign indicating that the values on each side of it are NOT equal.

  • =: "is"

  • >: "is greater than"

  • <: "is less than"

  • $${\ge}$$: "at least", "is greater than or equal to"

  • $${\le}$$: "at most", "is less than or equal to"

Statistics

statistics 統計 is the study of data.
data is a collection of facts. and there are 2 types of data.

  • quantitive data 定量的データ
    information give in numbers. (information that you can count or measure)

  • qualitative data 定性的 データ
    information given that describes something. (information that you can observe)

a statistical question is a question that anticipates予測する many different responses. answers that differ have variability/dispersion 統計的ばらつき(describes how spread out or closely clustered a set of data is).

sampling 標本調査 is when we take a small part of a larger group to estimate characteristics about the whole group.

Measures of Central Tendency

measures of central tendency 代表値 is a single number that is a summary of all of a data set's values.

  • mean or arithmetic average 平均値
    a calculated central value of a set of numbers

  • median 中央値
    the middle number of a data set when all of the items are written in order from least to greatest

  • mode 最頻値
    the item in a data set that occurs most often.

Measures of Variation

  • range 幅
    the difference between the minimum 最小値 and maximum 最大値 unit in a data set.

  • variance 分散

  • standard deviation 標準偏差

Displaying Data

データの表示法は、日本と共通しているものが多い。

  • (2-way) table 表

  • line plot 

  • histogram ヒストグラム

  • scatter plot 散布図

    • line of best fit 近似直線

    • correlation 相関関係

      • positive correlation

      • negative correlation

      • no correlation

(少なくとも私の世代の)日本の数学教育では習わなかったかもしれないが、この本ではbox (and whisker) plot 箱ひげ図も扱っていた。
box plot displays data along a number line and splits the data into quartiles(quarters) 分位数. the different boxes show the different quarters-25% of the data is in each of the four quarters. the median splits the data into two halves.
the median of the lower half is known as the lower quartile 第一四分位数 and is represented by "Q1". the median of the upper half is known as the upper quartile 第三分位数 and is represented by "Q3".
the interquartile range (IQR) 四分位範囲 is the difference between the 75th and 25th percentiles of the data.


box plot from Wikimedia

Outlier 外れ値 is a data value that is significantly lower or higher than the other values.
Malcom Gladwell氏の"天才"と和訳された本"Outliers"、その意味こそが「外れ値」であり、一般大衆からかけ離れた存在と言う意味を表している。

Probability

probability 確率 is the likelihood that something will happen. It is a number between 0 and 1.

the action 試行 is what is happening.
the outcomes 根元事象/単一事象 are all of the possible results of an action.
an event 事象 is any outcome, or group of outcomes.
we use a ratio to find out how likely it is that an event will happen (probability of an event).

$$
\rm{Probability}(\rm{Event}) = \frac{\rm{number\ of\ favorable\ outcomes}}{\rm{number\ of\ possible\ outcomes}}
$$

the complement of an event/complementary event 余事象 is the opposite of the event happening

やはり(アメリカの)中学数学までで習う確率はこの程度が限界なのだろうか。
確率をより詳しく勉強する予定もあるため、これ以上の単語は後ほど紹介することにする。

明日は残りの幾何の部分をまとめる。平面図系や立体図形の英語名もひと目見てわかるようにまとめるため期待してほしい。

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