Superb Algebra 1 2018, Student Edition
第 1 單元，第 1 課

Variables and Expressions代數式

這課一眼看
Definitions定義
  • variable變數: A letter that stands for a number.
  • constant常數: A number whose value does not change.
  • algebraic expression代數式: Variables, constants and operations combined; no equal sign.
  • term項: A number, a variable, or their product; terms are separated by + and −.
  • coefficient係數: The number multiplied by the variable in a term.
  • factor因數: A number or variable that is multiplied.
Formulas and Rules公式與性質
  • verbal expression文字敘述: A quantity described in words.
  • substitute代入: Replace a variable with a number.
Problem Types題型範例
    1Definitions定義

    AVariables and algebraic expressions變數與代數式

    1. A variable is a letter that stands for a number that can change or is not yet known, such as \(x\), \(n\) or \(t\). A constant is a number whose value does not change, such as \(5\) or \(-\frac{1}{2}\). An algebraic expression combines variables, constants and operation symbols: \(3x + 7\), \(\frac{n}{4}\) and \(a^2 - 2ab\) are algebraic expressions. Like a numerical expression, it has no equal sign.變數是代表會變動或還不知道的數的字母，例如 \(x\)、\(n\)、\(t\)。常數是值不會改變的數，例如 \(5\) 或 \(-\frac{1}{2}\)。代數式把變數、常數和運算符號結合起來：\(3x + 7\)、\(\frac{n}{4}\)、\(a^2 - 2ab\) 都是代數式。和數值算式一樣，它沒有等號。
    2. In algebra, multiplication is usually written without a symbol. \(3x\) means \(3 \times x\), \(ab\) means \(a \times b\), and \(2(x + 1)\) means \(2 \times (x + 1)\). Division is usually written as a fraction: \(\frac{n}{4}\) means \(n \div 4\). The symbol \(\times\) is avoided because it looks like the variable \(x\).代數裡的乘法通常不寫符號。\(3x\) 表示 \(3 \times x\)，\(ab\) 表示 \(a \times b\)，\(2(x + 1)\) 表示 \(2 \times (x + 1)\)。除法通常寫成分數：\(\frac{n}{4}\) 表示 \(n \div 4\)。不用 \(\times\) 是因為它看起來像變數 \(x\)。
    Numerical or algebraic?數值算式還是代數式？
    Expression式子Has a variable?有變數嗎？Kind類別Note說明
    \(12 \div 3 + 5\)\(12 \div 3 + 5\)no沒有numerical expression數值算式can be evaluated to one number, \(9\)可以算出一個數 \(9\)
    \(4t - 1\)\(4t - 1\)yes有algebraic expression代數式value depends on \(t\)值取決於 \(t\)
    \(\frac{p}{2} + p^2\)\(\frac{p}{2} + p^2\)yes有algebraic expression代數式the same variable can appear more than once同一個變數可以出現不只一次
    \(4t - 1 = 11\)\(4t - 1 = 11\)yes有equation, not an expression是方程式，不是代數式has an equal sign; you solve it有等號；它是要「解」的
    練習 1

    BTerms, coefficients and factors項、係數與因數

    1. A term is a single number, a single variable, or numbers and variables multiplied together. Terms are separated by \(+\) and \(-\) signs. The expression \(4x^2 - 7x + 9\) has three terms: \(4x^2\), \(-7x\) and \(9\). The sign in front of a term belongs to it.項是一個數、一個變數，或是數與變數相乘的結果。項與項之間用 \(+\) 和 \(-\) 隔開。\(4x^2 - 7x + 9\) 有三項：\(4x^2\)、\(-7x\) 和 \(9\)。項前面的符號屬於那一項。
    2. The coefficient of a term is the number multiplied by the variable. In \(4x^2\) the coefficient is \(4\); in \(-7x\) it is \(-7\). A variable written alone, like \(y\), has coefficient \(1\), and \(-y\) has coefficient \(-1\). A term with no variable, like \(9\), is a constant term. Numbers or variables that are multiplied together are factors: in \(5mn\), the factors are \(5\), \(m\) and \(n\).一項的係數是乘在變數前面的數。\(4x^2\) 的係數是 \(4\)；\(-7x\) 的係數是 \(-7\)。單獨一個變數（如 \(y\)）的係數是 \(1\)，\(-y\) 的係數是 \(-1\)。沒有變數的項（如 \(9\)）叫常數項。相乘的數或變數叫做因數：\(5mn\) 的因數是 \(5\)、\(m\) 和 \(n\)。
    Parts of \(5x^2 - x + \frac{2}{3}y - 12\)\(5x^2 - x + \frac{2}{3}y - 12\) 的各部分
    Term項Coefficient係數Variable part變數部分Type種類
    \(5x^2\)\(5x^2\)\(5\)\(5\)\(x^2\)\(x^2\)variable term變數項
    \(-x\)\(-x\)\(-1\)\(-1\)\(x\)\(x\)variable term變數項
    \(\frac{2}{3}y\)\(\frac{2}{3}y\)\(\frac{2}{3}\)\(\frac{2}{3}\)\(y\)\(y\)variable term變數項
    \(-12\)\(-12\)(none)（無）(none)（無）constant term常數項
    注意
    Terms are separated by addition and subtraction, not by multiplication. \(5mn\) is one term with three factors, while \(5 + m + n\) is three terms.項是用加減分開的，不是用乘法。\(5mn\) 是一項，有三個因數；\(5 + m + n\) 則是三項。
    練習 2
    2Formulas and Rules公式與性質

    ATranslating words into symbols把文字翻成符號

    1. A verbal expression describes a quantity in words, such as "six less than twice a number". To translate it, choose a variable for the unknown number, then match each key word to an operation. Watch the order in subtraction and division: "six less than \(n\)" is \(n - 6\), not \(6 - n\).文字敘述是用語言描述一個量，例如「一個數的兩倍少 6」。翻譯時先用一個變數代表未知數，再把每個關鍵字對應到一種運算。減法和除法要注意順序：「比 \(n\) 少 6」是 \(n - 6\)，不是 \(6 - n\)。
    Key words for operations運算的關鍵字
    Operation運算Key words關鍵字Phrase句子Expression代數式
    Addition加法sum, plus, more than, increased by, total和、加、比…多、增加、總共eight more than a number比一個數多 8\(n + 8\)\(n + 8\)
    Subtraction減法difference, minus, less than, decreased by, fewer than差、減、比…少、減少eight less than a number比一個數少 8\(n - 8\)\(n - 8\)
    Multiplication乘法product, times, twice, of, multiplied by積、倍、兩倍、乘以twice a number一個數的兩倍\(2n\)\(2n\)
    Division除法quotient, divided by, ratio of, per, half of商、除以、比、每、一半the quotient of a number and six一個數與 6 的商\(\frac{n}{6}\)\(\frac{n}{6}\)
    Powers次方squared, cubed, to the power of平方、立方、…次方a number squared一個數的平方\(n^2\)\(n^2\)
    Grouping分組the sum of…, the difference of… (then another operation)「…的和」「…的差」後面再接另一個運算three times the sum of a number and one一個數與 1 的和的三倍\(3(n + 1)\)\(3(n + 1)\)
    注意
    "The sum of \(x\) and \(3\), times \(5\)" needs parentheses: \(5(x + 3)\). The comma shows that the whole sum is multiplied. Without the comma, "the sum of \(x\) and \(3\) times \(5\)" is \(x + 15\).「\(x\) 與 3 的和，乘以 5」需要括號：\(5(x + 3)\)。逗號表示整個和都要乘。沒有逗號的「\(x\) 與 3 乘以 5 的和」則是 \(x + 15\)。
    練習 3

    BEvaluating an algebraic expression代數式的求值

    1. To evaluate an algebraic expression, substitute the given number for each variable, then follow the order of operations. Put the number in parentheses when you substitute, especially if it is negative: for \(x = -3\), \(x^2\) becomes \((-3)^2 = 9\), not \(-3^2 = -9\).要對代數式求值，先把給定的數代入每個變數，再照運算順序計算。代入時把數放進括號，負數尤其要這樣做：\(x = -3\) 時，\(x^2\) 變成 \((-3)^2 = 9\)，而不是 \(-3^2 = -9\)。
    Steps to evaluate an algebraic expression代數式求值的步驟
    Step步驟What to do做什麼Example: \(x^2 - 3x\) when \(x = -2\)例子：\(x = -2\) 時的 \(x^2 - 3x\)
    1. Substitute1. 代入Replace each variable with its value, in parentheses.把每個變數換成它的值，放在括號裡。\((-2)^2 - 3(-2)\)\((-2)^2 - 3(-2)\)
    2. Powers2. 次方Evaluate exponents.算出指數。\(4 - 3(-2)\)\(4 - 3(-2)\)
    3. Multiply and divide3. 乘除Left to right.由左到右。\(4 - (-6)\)\(4 - (-6)\)
    4. Add and subtract4. 加減Left to right.由左到右。\(4 + 6 = 10\)\(4 + 6 = 10\)
    練習 4
    3Problem Types題型範例

    AIdentify terms, coefficients and constants找出項、係數與常數

    1. Use this type when you are asked to name the parts of an expression. First split the expression at each \(+\) or \(-\) sign, keeping the sign with the term that follows it.當題目要你說出代數式的各部分時，就是這種題型。先在每個 \(+\) 或 \(-\) 處把式子切開，符號跟著後面那一項。
    例題
    For the expression \(6a^2 - a + 3b - 8\), list the terms, give the coefficient of each variable term, and name the constant term.對代數式 \(6a^2 - a + 3b - 8\)，列出所有的項，寫出每個變數項的係數，並說出常數項。
    觀念提示
    Each minus sign starts a new term and belongs to it. A variable with no number in front has an invisible coefficient of \(1\).每個減號都開始一個新的項，而且屬於那一項。前面沒有數字的變數，係數是看不見的 \(1\)。
    關鍵算式
    \(6a^2 \;\; {-a} \;\; 3b \;\; {-8}\)\(6a^2 \;\; {-a} \;\; 3b \;\; {-8}\)
    完整詳解
    • Terms: \(6a^2\), \(-a\), \(3b\), \(-8\). There are four terms.
    • Coefficients: \(6a^2\) has coefficient \(6\); \(-a\) has coefficient \(-1\); \(3b\) has coefficient \(3\).
    • The constant term is \(-8\).
    • 項：\(6a^2\)、\(-a\)、\(3b\)、\(-8\)，共四項。
    • 係數：\(6a^2\) 的係數是 \(6\)；\(-a\) 的係數是 \(-1\)；\(3b\) 的係數是 \(3\)。
    • 常數項是 \(-8\)。
    自我檢查
    How many terms does \(2xy + \frac{x}{3} - 5\) have? What is the coefficient of \(xy\)?\(2xy + \frac{x}{3} - 5\) 有幾項？\(xy\) 的係數是多少？
    看答案
    • Three terms: \(2xy\), \(\frac{x}{3}\) and \(-5\).
    • The coefficient of \(xy\) is \(2\). (The coefficient of the second term is \(\frac{1}{3}\).)
    • 三項：\(2xy\)、\(\frac{x}{3}\)、\(-5\)。
    • \(xy\) 的係數是 \(2\)。（第二項的係數是 \(\frac{1}{3}\)。）
    練習 5

    BTranslate words into an algebraic expression把文字敘述翻成代數式

    1. Use this type when a phrase describes a number with words. Underline the key words, decide the operation for each, and check the order for "less than" and "divided by".當一個句子用文字描述一個數時，就是這種題型。畫出關鍵字，決定每個關鍵字的運算，並檢查「少」和「除以」的順序。
    例題
    Write an algebraic expression for "nine less than three times a number".把「一個數的三倍少 9」寫成代數式。
    觀念提示
    "Less than" reverses the order: nine less than something is that something minus nine.「少」會把順序反過來：比某個量少 9，就是那個量減 9。
    關鍵算式
    \(3n - 9\)\(3n - 9\)
    完整詳解
    • Let the number be \(n\).
    • "Three times a number" is \(3n\).
    • "Nine less than" that amount means subtract \(9\) from it: \(3n - 9\).
    • The expression is \(3n - 9\).
    • 設這個數為 \(n\)。
    • 「一個數的三倍」是 \(3n\)。
    • 「少 9」表示從它減掉 9：\(3n - 9\)。
    • 代數式是 \(3n - 9\)。
    例題
    Write an algebraic expression for "the quotient of a number and four, increased by the square of the number".把「一個數除以 4 的商，加上這個數的平方」寫成代數式。
    觀念提示
    Translate one phrase at a time. "Quotient of A and B" is \(A \div B\) in that order.一句一句翻。「A 與 B 的商」是照順序的 \(A \div B\)。
    關鍵算式
    \(\dfrac{x}{4} + x^2\)\(\dfrac{x}{4} + x^2\)
    完整詳解
    • Let the number be \(x\).
    • "The quotient of a number and four" is \(\frac{x}{4}\).
    • "The square of the number" is \(x^2\).
    • "Increased by" means add: \(\frac{x}{4} + x^2\).
    • 設這個數為 \(x\)。
    • 「一個數除以 4 的商」是 \(\frac{x}{4}\)。
    • 「這個數的平方」是 \(x^2\)。
    • 「加上」表示相加：\(\frac{x}{4} + x^2\)。
    自我檢查
    Write an algebraic expression for "seven times the sum of a number and two".把「一個數與 2 的和的七倍」寫成代數式。
    看答案
    • The sum is \((n + 2)\); seven times the whole sum is \(7(n + 2)\).
    • 和是 \((n + 2)\)；整個和的七倍是 \(7(n + 2)\)。
    練習 6

    CTranslate an algebraic expression into words把代數式翻成文字敘述

    1. Use this type when you must describe an expression in words. Name the operation that is done last as the main phrase, and describe the parts inside it.當題目要你用文字描述一個代數式時，就是這種題型。把最後做的運算當作主要句型，再描述裡面的各部分。
    例題
    Write a verbal expression for \(2(m - 5)\).把 \(2(m - 5)\) 寫成文字敘述。
    觀念提示
    The parentheses tell you the subtraction happens first, so the multiplication is the outer, main idea.括號表示先做減法，所以乘法是外層、主要的意思。
    關鍵算式
    twice (the difference of \(m\) and \(5\))（\(m\) 與 5 的差）的兩倍
    完整詳解
    • The last operation is multiplying by \(2\), so the main phrase is "twice" or "two times".
    • Inside the parentheses is "the difference of \(m\) and five".
    • Verbal expression: "twice the difference of a number and five".
    • 最後做的運算是乘以 2，所以主要句型是「兩倍」。
    • 括號裡是「\(m\) 與 5 的差」。
    • 文字敘述：「一個數與 5 的差的兩倍」。
    自我檢查
    Write a verbal expression for \(x^3 + 10\).把 \(x^3 + 10\) 寫成文字敘述。
    看答案
    "The cube of a number increased by ten" (or "ten more than the cube of a number").「一個數的立方加 10」（或「比一個數的立方多 10」）。
    練習 7

    DEvaluate for given values代入給定的值求值

    1. Use this type when the problem gives a value for each variable. Substitute carefully with parentheses, then follow the order of operations exactly as for a numerical expression.當題目給每個變數一個值時，就是這種題型。小心地用括號代入，然後完全照數值算式的運算順序計算。
    例題
    Evaluate \(2x^2 - 5x + 1\) when \(x = 4\).當 \(x = 4\) 時，求 \(2x^2 - 5x + 1\) 的值。
    觀念提示
    Replace every \(x\) with \((4)\). The exponent applies only to the \(4\), not to the \(2\) in front.把每個 \(x\) 都換成 \((4)\)。指數只作用在 4 上，不作用在前面的 2。
    關鍵算式
    \(2(4)^2 - 5(4) + 1\)\(2(4)^2 - 5(4) + 1\)
    完整詳解
    • Substitute: \(2(4)^2 - 5(4) + 1\).
    • Power: \(4^2 = 16\), so \(2(16) - 5(4) + 1\).
    • Multiply: \(32 - 20 + 1\).
    • Add and subtract left to right: \(12 + 1 = 13\).
    • The value is \(13\).
    • 代入：\(2(4)^2 - 5(4) + 1\)。
    • 次方：\(4^2 = 16\)，得 \(2(16) - 5(4) + 1\)。
    • 相乘：\(32 - 20 + 1\)。
    • 由左到右加減：\(12 + 1 = 13\)。
    • 值是 \(13\)。
    例題
    Evaluate \(\dfrac{a + 3b}{b - a}\) when \(a = -2\) and \(b = 6\).當 \(a = -2\)、\(b = 6\) 時，求 \(\dfrac{a + 3b}{b - a}\) 的值。
    觀念提示
    Two variables, two substitutions. Keep the negative number in parentheses so that \(6 - (-2)\) is read correctly as \(6 + 2\).兩個變數就代入兩次。把負數放在括號裡，\(6 - (-2)\) 才會正確看成 \(6 + 2\)。
    關鍵算式
    \(\dfrac{(-2) + 3(6)}{(6) - (-2)}\)\(\dfrac{(-2) + 3(6)}{(6) - (-2)}\)
    完整詳解
    • Substitute: \(\dfrac{(-2) + 3(6)}{(6) - (-2)}\).
    • Numerator: \(-2 + 18 = 16\).
    • Denominator: \(6 - (-2) = 6 + 2 = 8\).
    • Divide: \(16 \div 8 = 2\).
    • The value is \(2\).
    • 代入：\(\dfrac{(-2) + 3(6)}{(6) - (-2)}\)。
    • 分子：\(-2 + 18 = 16\)。
    • 分母：\(6 - (-2) = 6 + 2 = 8\)。
    • 相除：\(16 \div 8 = 2\)。
    • 值是 \(2\)。
    自我檢查
    Evaluate \(3(y + 2)^2\) when \(y = -5\).當 \(y = -5\) 時，求 \(3(y + 2)^2\) 的值。
    看答案
    • \(3((-5) + 2)^2 = 3(-3)^2 = 3(9) = 27\).
    • \(3((-5) + 2)^2 = 3(-3)^2 = 3(9) = 27\)。
    練習 8

    EWrite an expression for a situation依情境寫出代數式

    1. Use this type when a real situation has one unknown amount. Choose a variable, write what each part of the story does to it, and then evaluate the expression if a value is given.當生活情境中有一個未知量時，就是這種題型。選一個變數，寫出情境中每個部分對它做了什麼，若題目有給值，再代入求值。
    例題
    A gym charges a \(\$25\) sign-up fee plus \(\$30\) per month. Write an expression for the total cost for \(m\) months, and find the cost for 6 months.一間健身房收 25 元入會費，之後每月 30 元。寫出 \(m\) 個月總費用的代數式，並求 6 個月的費用。
    觀念提示
    "Per month" means multiply by the number of months. A fee paid once is a constant term.「每月」表示乘以月數。只付一次的費用是常數項。
    關鍵算式
    \(30m + 25\)\(30m + 25\)
    完整詳解
    • Monthly charges for \(m\) months: \(30m\).
    • Add the one-time fee: \(30m + 25\).
    • For 6 months, substitute \(m = 6\): \(30(6) + 25 = 180 + 25 = 205\).
    • The expression is \(30m + 25\); 6 months cost \(\$205\).
    • \(m\) 個月的月費：\(30m\)。
    • 加上一次性的入會費：\(30m + 25\)。
    • 6 個月時代入 \(m = 6\)：\(30(6) + 25 = 180 + 25 = 205\)。
    • 代數式是 \(30m + 25\)；6 個月的費用是 205 元。
    自我檢查
    A rectangle is \(w\) cm wide and its length is 3 cm more than its width. Write an expression for its perimeter.一個長方形寬 \(w\) 公分，長比寬多 3 公分。寫出它周長的代數式。
    看答案
    • Length: \(w + 3\). Perimeter: \(2w + 2(w + 3)\), which can also be written \(w + w + (w + 3) + (w + 3)\).
    • 長：\(w + 3\)。周長：\(2w + 2(w + 3)\)，也可以寫成 \(w + w + (w + 3) + (w + 3)\)。
    練習 9

    這課的重點字

    • variable變數
    • constant常數
    • algebraic expression代數式
    • term項
    • coefficient係數
    • constant term常數項
    • factor因數
    • verbal expression文字敘述
    背這課的 9 個字

    申請試用一個月