Superb Algebra 1 2018, Student Edition
第 1 單元，第 2 課

Order of Operations數值算式

這課一眼看
Definitions定義
  • numerical expression數值算式: Numbers joined by operation symbols; no equal sign.
  • evaluate求值: Carry out the operations to find the value of an expression.
  • power次方: A base with an exponent; repeated multiplication of the base.
  • base底數: The number that is multiplied repeatedly in a power.
  • exponent指數: Tells how many times the base is used as a factor.
Formulas and Rules公式與性質
  • order of operations運算順序: Grouping symbols, powers, multiply/divide left to right, add/subtract left to right.
  • grouping symbols分組符號: Parentheses, brackets, braces and the fraction bar; they mark what to evaluate first.
Problem Types題型範例
    1Definitions定義

    ANumerical expressions and evaluating數值算式與求值

    1. A numerical expression is a group of numbers joined by operation symbols, such as \(18 - 4 \times 3\) or \(\frac{30}{5} + 2\). It has no equal sign and no letters. To evaluate an expression means to carry out all the operations and find its single value. For example, evaluating \(18 - 4 \times 3\) gives \(18 - 12 = 6\).數值算式是由數字和運算符號組成的式子，例如 \(18 - 4 \times 3\) 或 \(\frac{30}{5} + 2\)。它沒有等號，也沒有字母。求值就是把所有運算做完，算出式子唯一的值。例如 \(18 - 4 \times 3\) 求值得到 \(18 - 12 = 6\)。
    Operation symbols運算符號
    Operation運算Symbols符號Example例子Read as讀法
    Addition加法\(+\)\(+\)\(9 + 4\)\(9 + 4\)9 plus 4; the sum of 9 and 49 加 4；9 與 4 的和
    Subtraction減法\(-\)\(-\)\(9 - 4\)\(9 - 4\)9 minus 4; the difference of 9 and 49 減 4；9 與 4 的差
    Multiplication乘法\(\times\), \(\cdot\), parentheses\(\times\)、\(\cdot\)、括號\(9 \times 4\), \(9 \cdot 4\), \(9(4)\)\(9 \times 4\)、\(9 \cdot 4\)、\(9(4)\)9 times 4; the product of 9 and 49 乘 4；9 與 4 的積
    Division除法\(\div\), \(/\), fraction bar\(\div\)、\(/\)、分數線\(36 \div 4\), \(36/4\), \(\frac{36}{4}\)\(36 \div 4\)、\(36/4\)、\(\frac{36}{4}\)36 divided by 4; the quotient of 36 and 436 除以 4；36 與 4 的商
    注意
    An expression is not an equation. \(7 + 5\) is an expression; \(7 + 5 = 12\) is an equation, because it has an equal sign. You evaluate an expression, but you solve an equation.算式不是方程式。\(7 + 5\) 是算式；\(7 + 5 = 12\) 是方程式，因為它有等號。算式是「求值」，方程式才是「解」。
    練習 1

    BExponents and powers指數與次方

    1. A power is a short way to write repeated multiplication. In \(5^3\), the base is \(5\) and the exponent is \(3\). The exponent tells how many times the base is used as a factor: \(5^3 = 5 \times 5 \times 5 = 125\). We read \(5^2\) as "five squared", \(5^3\) as "five cubed", and \(5^4\) as "five to the fourth power".次方是重複相乘的簡寫。在 \(5^3\) 中，底數是 \(5\)，指數是 \(3\)。指數表示底數當作因數用了幾次：\(5^3 = 5 \times 5 \times 5 = 125\)。\(5^2\) 讀作「5 的平方」，\(5^3\) 讀作「5 的立方」，\(5^4\) 讀作「5 的四次方」。
    Reading and evaluating powers次方的讀法與值
    Power次方Base底數Exponent指數Meaning意思Value值
    \(4^2\)\(4^2\)\(4\)\(4\)\(2\)\(2\)\(4 \times 4\) (four squared)\(4 \times 4\)（4 的平方）\(16\)\(16\)
    \(4^3\)\(4^3\)\(4\)\(4\)\(3\)\(3\)\(4 \times 4 \times 4\) (four cubed)\(4 \times 4 \times 4\)（4 的立方）\(64\)\(64\)
    \(10^4\)\(10^4\)\(10\)\(10\)\(4\)\(4\)\(10 \times 10 \times 10 \times 10\) (ten to the fourth power)\(10 \times 10 \times 10 \times 10\)（10 的四次方）\(10000\)\(10000\)
    \(\left(\frac{1}{2}\right)^3\)\(\left(\frac{1}{2}\right)^3\)\(\frac{1}{2}\)\(\frac{1}{2}\)\(3\)\(3\)\(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\)\(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\)\(\frac{1}{8}\)\(\frac{1}{8}\)
    \(7^1\)\(7^1\)\(7\)\(7\)\(1\)\(1\)\(7\) used once\(7\) 只用一次\(7\)\(7\)
    注意
    \(2^5\) means \(2 \times 2 \times 2 \times 2 \times 2 = 32\), not \(2 \times 5 = 10\). The exponent counts factors; it is not a factor itself.\(2^5\) 是 \(2 \times 2 \times 2 \times 2 \times 2 = 32\)，不是 \(2 \times 5 = 10\)。指數是在數因數的個數，它本身不是因數。
    練習 2
    2Formulas and Rules公式與性質

    AOrder of operations運算順序

    1. The order of operations is the agreed order for evaluating an expression, so that everyone gets the same answer. Work inside grouping symbols first, then powers, then multiplication and division from left to right, and finally addition and subtraction from left to right.運算順序是大家約定好的求值順序，這樣每個人算出來的答案才會一樣。先算括號等分組符號裡面，再算次方，接著由左到右做乘除，最後由左到右做加減。
    Order of operations運算順序
    Step步驟What to do做什麼Example: \(8 + 2(7 - 4)^2 \div 6\)例子：\(8 + 2(7 - 4)^2 \div 6\)
    1. Grouping symbols1. 分組符號Evaluate inside parentheses, brackets, braces and fraction bars, innermost first.先算小括號、中括號、大括號和分數線裡面，最裡面的優先。\(7 - 4 = 3 \;\to\; 8 + 2(3)^2 \div 6\)\(7 - 4 = 3 \;\to\; 8 + 2(3)^2 \div 6\)
    2. Powers2. 次方Evaluate all exponents.算出所有的指數。\(3^2 = 9 \;\to\; 8 + 2 \cdot 9 \div 6\)\(3^2 = 9 \;\to\; 8 + 2 \cdot 9 \div 6\)
    3. Multiply and divide3. 乘法與除法From left to right, whichever comes first.由左到右，哪個先出現就先算。\(2 \cdot 9 = 18\), then \(18 \div 6 = 3 \;\to\; 8 + 3\)\(2 \cdot 9 = 18\)，再 \(18 \div 6 = 3 \;\to\; 8 + 3\)
    4. Add and subtract4. 加法與減法From left to right, whichever comes first.由左到右，哪個先出現就先算。\(8 + 3 = 11\)\(8 + 3 = 11\)
    注意
    Multiplication does not come before division, and addition does not come before subtraction. Each pair is done from left to right. \(24 \div 6 \times 2 = 4 \times 2 = 8\), not \(24 \div 12 = 2\).乘法不比除法優先，加法也不比減法優先。每一組都是由左到右做。\(24 \div 6 \times 2 = 4 \times 2 = 8\)，不是 \(24 \div 12 = 2\)。
    練習 3

    BGrouping symbols分組符號

    1. Grouping symbols tell you which part of an expression to treat as a single quantity. Parentheses \(( \;)\), brackets \([ \;]\) and braces \(\{ \;\}\) all do the same job; different shapes just make nested groups easier to read. A fraction bar is also a grouping symbol: evaluate the whole numerator and the whole denominator before dividing. When groups are nested, start with the innermost one.分組符號告訴你算式裡的哪一部分要當成一個整體。小括號 \(( \;)\)、中括號 \([ \;]\)、大括號 \(\{ \;\}\) 的作用都一樣，不同形狀只是讓一層套一層的分組容易看清楚。分數線也是分組符號：先把整個分子和整個分母算完，再相除。有多層分組時，從最裡面那一層開始算。
    Grouping symbols分組符號
    Symbol符號Name名稱Example例子Value值
    \(( \;)\)\(( \;)\)parentheses小括號\(3(10 - 4)\)\(3(10 - 4)\)\(3 \cdot 6 = 18\)\(3 \cdot 6 = 18\)
    \([ \;]\)\([ \;]\)brackets中括號\(2[5 + (9 - 7)]\)\(2[5 + (9 - 7)]\)\(2[5 + 2] = 14\)\(2[5 + 2] = 14\)
    \(\{ \;\}\)\(\{ \;\}\)braces大括號\(\{12 - [6 - (1 + 2)]\}\)\(\{12 - [6 - (1 + 2)]\}\)\(12 - [6 - 3] = 9\)\(12 - [6 - 3] = 9\)
    fraction bar分數線groups numerator and denominator把分子、分母各分成一組\(\frac{9 + 6}{5 - 2}\)\(\frac{9 + 6}{5 - 2}\)\(\frac{15}{3} = 5\)\(\frac{15}{3} = 5\)
    練習 4
    3Problem Types題型範例

    AEvaluate expressions with powers求含次方算式的值

    1. Use this type when an expression mixes a power with other operations. Find the value of the power first, then continue with the order of operations.當算式裡有次方又有其他運算時，就是這種題型。先算出次方的值，再照運算順序繼續。
    例題
    Evaluate \(4 + 3^2 \times 2\).求 \(4 + 3^2 \times 2\) 的值。
    觀念提示
    There are no grouping symbols, so the power is the first thing to evaluate. Do not add \(4 + 3\) first.沒有分組符號，所以第一步算次方。不要先算 \(4 + 3\)。
    關鍵算式
    \(3^2 = 9\), then \(4 + 9 \times 2\)\(3^2 = 9\)，再算 \(4 + 9 \times 2\)
    完整詳解
    • Power first: \(3^2 = 9\), so the expression becomes \(4 + 9 \times 2\).
    • Multiply: \(9 \times 2 = 18\).
    • Add: \(4 + 18 = 22\).
    • The value is \(22\).
    • 先算次方：\(3^2 = 9\)，式子變成 \(4 + 9 \times 2\)。
    • 相乘：\(9 \times 2 = 18\)。
    • 相加：\(4 + 18 = 22\)。
    • 值是 \(22\)。
    例題
    Evaluate \(50 - 2^4 \div 8\).求 \(50 - 2^4 \div 8\) 的值。
    觀念提示
    Write out the power as repeated multiplication so you do not confuse \(2^4\) with \(2 \times 4\). Division comes before subtraction.把次方寫成重複相乘，就不會把 \(2^4\) 和 \(2 \times 4\) 搞混。除法在減法之前。
    關鍵算式
    \(2^4 = 16\), then \(50 - 16 \div 8\)\(2^4 = 16\)，再算 \(50 - 16 \div 8\)
    完整詳解
    • Power first: \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).
    • The expression is now \(50 - 16 \div 8\).
    • Divide: \(16 \div 8 = 2\).
    • Subtract: \(50 - 2 = 48\).
    • The value is \(48\).
    • 先算次方：\(2^4 = 2 \times 2 \times 2 \times 2 = 16\)。
    • 式子變成 \(50 - 16 \div 8\)。
    • 相除：\(16 \div 8 = 2\)。
    • 相減：\(50 - 2 = 48\)。
    • 值是 \(48\)。
    自我檢查
    Evaluate \(6^2 - 5 \times 4\).求 \(6^2 - 5 \times 4\) 的值。
    看答案
    • \(6^2 = 36\), then \(5 \times 4 = 20\).
    • \(36 - 20 = 16\).
    • \(6^2 = 36\)，\(5 \times 4 = 20\)。
    • \(36 - 20 = 16\)。
    練習 5

    BEvaluate expressions with grouping symbols求含分組符號算式的值

    1. Use this type when the expression has parentheses, brackets or braces. Work from the innermost group outward, and treat each group as one number once it is evaluated.當算式裡有小括號、中括號或大括號時，就是這種題型。從最裡面的分組往外算，每個分組算完後就當成一個數。
    例題
    Evaluate \(3[20 - (4 + 2)^2 \div 4]\).求 \(3[20 - (4 + 2)^2 \div 4]\) 的值。
    觀念提示
    The parentheses are inside the brackets, so they come first. Inside the brackets you still follow the order of operations: power, then division, then subtraction.小括號在中括號裡面，所以先算。中括號裡面仍然照運算順序：先次方、再除法、最後減法。
    關鍵算式
    \((4 + 2) = 6\), then \(3[20 - 36 \div 4]\)\((4 + 2) = 6\)，再算 \(3[20 - 36 \div 4]\)
    完整詳解
    • Innermost group: \(4 + 2 = 6\). The expression is \(3[20 - 6^2 \div 4]\).
    • Power: \(6^2 = 36\). Now \(3[20 - 36 \div 4]\).
    • Inside the brackets, divide first: \(36 \div 4 = 9\), then subtract: \(20 - 9 = 11\).
    • Multiply: \(3 \times 11 = 33\).
    • The value is \(33\).
    • 最裡面的分組：\(4 + 2 = 6\)。式子變成 \(3[20 - 6^2 \div 4]\)。
    • 次方：\(6^2 = 36\)。現在是 \(3[20 - 36 \div 4]\)。
    • 中括號裡先除：\(36 \div 4 = 9\)，再減：\(20 - 9 = 11\)。
    • 相乘：\(3 \times 11 = 33\)。
    • 值是 \(33\)。
    例題
    Evaluate \(2\{18 - 2[7 - (9 - 5)]\}\).求 \(2\{18 - 2[7 - (9 - 5)]\}\) 的值。
    觀念提示
    Three layers of grouping: parentheses inside brackets inside braces. Peel them off one layer at a time, from the inside.三層分組：小括號在中括號裡，中括號在大括號裡。從最裡面一層一層往外剝。
    關鍵算式
    \((9 - 5) = 4\), then \([7 - 4] = 3\), then \(\{18 - 6\}\)\((9 - 5) = 4\)，再 \([7 - 4] = 3\)，再 \(\{18 - 6\}\)
    完整詳解
    • Innermost: \(9 - 5 = 4\). Now \(2\{18 - 2[7 - 4]\}\).
    • Brackets: \(7 - 4 = 3\). Now \(2\{18 - 2 \times 3\}\).
    • Braces: \(2 \times 3 = 6\), then \(18 - 6 = 12\).
    • Multiply: \(2 \times 12 = 24\).
    • The value is \(24\).
    • 最裡面：\(9 - 5 = 4\)。現在是 \(2\{18 - 2[7 - 4]\}\)。
    • 中括號：\(7 - 4 = 3\)。現在是 \(2\{18 - 2 \times 3\}\)。
    • 大括號：\(2 \times 3 = 6\)，再 \(18 - 6 = 12\)。
    • 相乘：\(2 \times 12 = 24\)。
    • 值是 \(24\)。
    自我檢查
    Evaluate \(5(12 - 8)^2 - 30\).求 \(5(12 - 8)^2 - 30\) 的值。
    看答案
    • \(12 - 8 = 4\), then \(4^2 = 16\).
    • \(5 \times 16 = 80\), then \(80 - 30 = 50\).
    • \(12 - 8 = 4\)，\(4^2 = 16\)。
    • \(5 \times 16 = 80\)，\(80 - 30 = 50\)。
    練習 6

    CExpressions with a fraction bar含分數線的算式

    1. Use this type when an expression is written as a fraction with operations above or below the bar. The bar groups the numerator and groups the denominator; evaluate each completely, then divide.當算式寫成分數，分數線上下又有運算時，就是這種題型。分數線把分子分成一組、分母分成一組；各自算完後再相除。
    例題
    Evaluate \(\dfrac{5^2 + 11}{2 \times 9 - 6}\).求 \(\dfrac{5^2 + 11}{2 \times 9 - 6}\) 的值。
    觀念提示
    Think of the fraction as \((5^2 + 11) \div (2 \times 9 - 6)\). Finish the top and the bottom separately before dividing.把分數想成 \((5^2 + 11) \div (2 \times 9 - 6)\)。上下分別算完，再相除。
    關鍵算式
    \(\dfrac{36}{12}\)\(\dfrac{36}{12}\)
    完整詳解
    • Numerator: \(5^2 + 11 = 25 + 11 = 36\).
    • Denominator: \(2 \times 9 - 6 = 18 - 6 = 12\).
    • Divide: \(36 \div 12 = 3\).
    • The value is \(3\).
    • 分子：\(5^2 + 11 = 25 + 11 = 36\)。
    • 分母：\(2 \times 9 - 6 = 18 - 6 = 12\)。
    • 相除：\(36 \div 12 = 3\)。
    • 值是 \(3\)。
    自我檢查
    Evaluate \(\dfrac{40 - 4 \times 7}{3 + 1}\).求 \(\dfrac{40 - 4 \times 7}{3 + 1}\) 的值。
    看答案
    • Numerator: \(40 - 28 = 12\). Denominator: \(3 + 1 = 4\).
    • \(12 \div 4 = 3\).
    • 分子：\(40 - 28 = 12\)。分母：\(3 + 1 = 4\)。
    • \(12 \div 4 = 3\)。
    練習 7

    DWrite and evaluate an expression for a situation依情境列式並求值

    1. Use this type when a short story gives several numbers and asks for a total or a leftover amount. Write one numerical expression that describes the whole situation, using grouping symbols where needed, and then evaluate it.當題目用一小段情境給出好幾個數，要求總數或剩下多少時，就是這種題型。先寫一個描述整個情境的數值算式，需要的地方加分組符號，再求值。
    例題
    A bakery packs 4 boxes with 6 muffins in each box and 3 boxes with 8 muffins in each box. Write a numerical expression for the total number of muffins and evaluate it.一家麵包店裝了 4 盒每盒 6 個的瑪芬，和 3 盒每盒 8 個的瑪芬。寫出表示瑪芬總數的數值算式並求值。
    觀念提示
    Each kind of box is a multiplication. Joining the two kinds is an addition. The order of operations already multiplies before adding, so no parentheses are needed.每一種盒子都是一個乘法，把兩種合起來是加法。運算順序本來就是先乘再加，所以不需要括號。
    關鍵算式
    \(4 \times 6 + 3 \times 8\)\(4 \times 6 + 3 \times 8\)
    完整詳解
    • Muffins in the first kind of box: \(4 \times 6\). In the second kind: \(3 \times 8\).
    • Total: \(4 \times 6 + 3 \times 8\).
    • Evaluate: \(24 + 24 = 48\).
    • There are \(48\) muffins.
    • 第一種盒子裡的瑪芬：\(4 \times 6\)。第二種：\(3 \times 8\)。
    • 總數：\(4 \times 6 + 3 \times 8\)。
    • 求值：\(24 + 24 = 48\)。
    • 共有 \(48\) 個瑪芬。
    例題
    Mei had \(\$60\). She bought 3 notebooks for \(\$4\) each and a pen for \(\$2\). Write an expression for the money she has left and evaluate it.小美有 60 元。她買了 3 本每本 4 元的筆記本和一枝 2 元的筆。寫出表示她剩下多少錢的算式並求值。
    觀念提示
    Without parentheses, \(60 - 3 \times 4 + 2\) would add the \(2\) back instead of subtracting it. Group the whole amount spent before subtracting.如果不加括號，\(60 - 3 \times 4 + 2\) 會把 2 加回去而不是減掉。要先把花掉的全部圈起來再減。
    關鍵算式
    \(60 - (3 \times 4 + 2)\)\(60 - (3 \times 4 + 2)\)
    完整詳解
    • Amount spent: \(3 \times 4 + 2\).
    • Money left: \(60 - (3 \times 4 + 2)\). The parentheses group everything she spent.
    • Inside: \(12 + 2 = 14\).
    • Subtract: \(60 - 14 = 46\).
    • She has \(\$46\) left.
    • 花掉的錢：\(3 \times 4 + 2\)。
    • 剩下的錢：\(60 - (3 \times 4 + 2)\)。括號把她花掉的全部圈成一組。
    • 括號內：\(12 + 2 = 14\)。
    • 相減：\(60 - 14 = 46\)。
    • 她剩下 46 元。
    自我檢查
    A train car has 12 rows of 4 seats. 7 seats are empty. Write an expression for the number of people in the car and evaluate it.一節車廂有 12 排、每排 4 個座位，其中 7 個座位是空的。寫出表示車廂裡人數的算式並求值。
    看答案
    • \(12 \times 4 - 7\).
    • \(48 - 7 = 41\) people.
    • \(12 \times 4 - 7\)。
    • \(48 - 7 = 41\) 人。
    練習 8

    這課的重點字

    • numerical expression數值算式
    • evaluate求值
    • operation運算
    • power次方
    • base底數
    • exponent指數
    • order of operations運算順序
    • grouping symbols分組符號
    背這課的 9 個字

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