第 1 單元，第 1 課
A Plan for Problem Solving解題的計畫
這課一眼看
Definitions定義
- four-step plan解題四步驟: Understand, Plan, Solve, Check: a plan for solving problems.
- strategy策略: A method you choose to solve a problem.
- multi-step problem多步驟問題: A problem that needs two or more operations.
- estimate估算: To find an answer close to the exact one.
1Definitions定義
AThe four-step plan and strategies解題四步驟與策略
- The four-step plan is a way to solve any word problem in order: Understand, Plan, Solve and Check. A strategy is a method you choose in the Plan step. Drawing a picture, making a list and choosing an operation are all strategies.解題四步驟是依序解任何文字題的方法：了解、計畫、解題、驗算。策略是在「計畫」步驟中選用的方法，例如畫圖、列清單、決定要用哪一種運算，都是策略。
BMulti-step problems and estimates多步驟問題與估算
- A multi-step problem needs two or more operations. For example, to find the cost of 3 drinks at $2 each and one $5 sandwich, you multiply and then add. To estimate means to find an answer close to the exact one by using rounded numbers. For example, \(48 + 31\) is about \(50 + 30 = 80\).多步驟問題需要用到兩種以上的運算。例如：買 3 杯每杯 2 元的飲料和一個 5 元的三明治，要先乘再加。估算是用四捨五入後的數，找出接近正確答案的值。例如 \(48 + 31\) 大約是 \(50 + 30 = 80\)。
2Formulas and Rules公式與性質
AThe four-step plan and choosing an operation解題四步驟與選擇運算
The four-step plan解題四步驟
| Step步驟 | Questions to ask要問自己的問題 |
|---|---|
| 1. Understand1. 了解 | What facts do I know? What do I need to find?我知道哪些條件？要求的是什麼？ |
| 2. Plan2. 計畫 | How are the facts related? Which strategy or operation should I use?條件之間有什麼關係？要用哪一個策略或運算？ |
| 3. Solve3. 解題 | Carry out the plan. Do the computation carefully.照著計畫做，仔細計算。 |
| 4. Check4. 驗算 | Does the answer make sense? Did I answer the question?答案合理嗎？我有回答題目問的問題嗎？ |
Choosing an operation選擇運算
| Question in the problem題目在問 | Operation運算 | Example例子 |
|---|---|---|
| How many in all? (joining amounts)總共有多少？（合併數量） | addition \(+\)加法 \(+\) | 12 red pens and 9 blue pens: \(12 + 9\)紅筆 12 枝、藍筆 9 枝：\(12 + 9\) |
| How many more? How many are left?多多少？還剩多少？ | subtraction \(-\)減法 \(-\) | 20 cm is how much longer than 14 cm: \(20 - 14\)20 公分比 14 公分長多少：\(20 - 14\) |
| How many in all? (equal groups)總共有多少？（每組一樣多） | multiplication \(\times\)乘法 \(\times\) | 5 bags with 6 apples each: \(5 \times 6\)5 袋、每袋 6 顆蘋果：\(5 \times 6\) |
| How many in each group? How many times as many?每組有多少？是幾倍？ | division \(\div\)除法 \(\div\) | 36 cards shared by 4 friends: \(36 \div 4\); 24 is how many times 8: \(24 \div 8\)36 張卡片平分給 4 人：\(36 \div 4\)；24 是 8 的幾倍：\(24 \div 8\) |
注意
A division answer may need to be rounded up. If 64 students ride in vans that hold 7 students each, \(64 \div 7 = 9 \text{ R } 1\). You need 10 vans, not 9, because one student still needs a seat. Always ask what the remainder means.除法的答案有時要無條件進位。64 位學生坐每台可載 7 人的廂型車，\(64 \div 7 = 9 \cdots 1\)（商 9 餘 1），需要 10 台車而不是 9 台，因為還有 1 位學生需要座位。一定要想一想餘數代表什麼。
3Problem Types題型範例
AMulti-step word problems多步驟文字題
- Look for more than one action in the story, such as buying several boxes and adding what you already have, or saving the same amount each week toward a goal.注意題目裡是否有不只一個動作，例如買了好幾盒再加上原有的數量，或每週存一樣多的錢來達成目標。
例題
A club buys 6 boxes of markers with 12 markers in each box. It already has 18 markers. How many markers does it have now?社團買了 6 盒麥克筆，每盒 12 枝，原本已經有 18 枝。現在共有幾枝？
觀念提示
Use the four-step plan. Equal boxes suggest multiplication, and joining the old markers with the new ones suggests addition.使用解題四步驟。每盒一樣多要想到乘法，把原有的和新買的合起來要想到加法。
關鍵算式
\(6 \times 12\), then add \(18\)\(6 \times 12\)，再加上 \(18\)
完整詳解
- Understand: 6 boxes, 12 markers per box, 18 markers already. Find the total.
- Plan: multiply to find the new markers, then add 18.
- Solve: \(6 \times 12 = 72\) and \(72 + 18 = 90\).
- Check: about \(6 \times 10 = 60\), plus about 20, is about 80. 90 is close, so it makes sense.
- The club has 90 markers.
- 了解：6 盒、每盒 12 枝、原有 18 枝，要求總數。
- 計畫：先用乘法算新買的枝數，再加上 18。
- 解題：\(6 \times 12 = 72\)，\(72 + 18 = 90\)。
- 驗算：大約 \(6 \times 10 = 60\)，再加約 20，約 80。90 很接近，答案合理。
- 社團共有 90 枝麥克筆。
例題
Lin has $40 and saves $15 each week. She wants a bike that costs $205. How many weeks must she save?小琳有 40 元，每週存 15 元。她想買一台 205 元的腳踏車，要存幾週？
觀念提示
First find how much money is still missing. Then think: how many groups of $15 make that amount?先找出還差多少錢，再想一想：要幾個 15 元才湊得到這個金額？
關鍵算式
\(205 - 40\), then divide the result by \(15\)\(205 - 40\)，再把結果除以 \(15\)
完整詳解
- Money still needed: \(205 - 40 = 165\) dollars.
- Weeks: \(165 \div 15 = 11\).
- Check: \(40 + 11 \times 15 = 40 + 165 = 205\). ✓
- She must save for 11 weeks.
- 還差的錢：\(205 - 40 = 165\) 元。
- 週數：\(165 \div 15 = 11\)。
- 驗算：\(40 + 11 \times 15 = 40 + 165 = 205\)。✓
- 她要存 11 週。
自我檢查
A hall has 24 rows with 35 seats in each row. 650 tickets were sold. How many seats are empty?禮堂有 24 排，每排 35 個座位，賣出 650 張票。還有幾個空位？
看答案
- Total seats: \(24 \times 35 = 840\).
- Empty seats: \(840 - 650 = 190\).
- 座位總數：\(24 \times 35 = 840\)。
- 空位：\(840 - 650 = 190\) 個。
BOverlapping groups重疊的群組（文氏圖）
- When some people belong to two groups at the same time, draw two overlapping circles (a Venn diagram). Write the number that belongs to both groups in the middle first.當有些人同時屬於兩個群組時，畫兩個重疊的圓（文氏圖），先把同時屬於兩組的人數寫在中間重疊的部分。
例題
In a class of 30 students, 18 play soccer, 15 play basketball, and 7 play both. How many play only soccer? How many play neither sport?一班有 30 位學生，18 人踢足球，15 人打籃球，其中 7 人兩種都有。只踢足球的有幾人？兩種都沒有的有幾人？
觀念提示
Draw a Venn diagram. The 7 students who play both are counted inside each circle, so take them out before you count 'only'.畫文氏圖。兩種都有的 7 人在兩個圓裡都被算到，所以算「只有」的人數前要先扣掉他們。
關鍵算式
only soccer: \(18 - 7\); only basketball: \(15 - 7\); neither: \(30 -\) (everyone in the circles)只踢足球：\(18 - 7\)；只打籃球：\(15 - 7\)；都沒有：\(30 -\)（圓裡的總人數）
完整詳解
- Only soccer: \(18 - 7 = 11\).
- Only basketball: \(15 - 7 = 8\).
- In at least one circle: \(11 + 8 + 7 = 26\).
- Neither: \(30 - 26 = 4\).
- Check: \(11 + 7 = 18\) and \(8 + 7 = 15\). ✓
- 11 students play only soccer, and 4 play neither.
- 只踢足球：\(18 - 7 = 11\)。
- 只打籃球：\(15 - 7 = 8\)。
- 至少參加一種：\(11 + 8 + 7 = 26\)。
- 兩種都沒有：\(30 - 26 = 4\)。
- 驗算：\(11 + 7 = 18\)，\(8 + 7 = 15\)。✓
- 只踢足球的有 11 人，兩種都沒有的有 4 人。
自我檢查
50 teens were asked about pets. 32 have a cat, 21 have a dog, and 12 have both. How many have only a dog? How many have neither?調查 50 位青少年養寵物的情形：32 人養貓，21 人養狗，其中 12 人兩種都養。只養狗的有幾人？兩種都沒養的有幾人？
看答案
- Only a dog: \(21 - 12 = 9\). Only a cat: \(32 - 12 = 20\).
- With a pet: \(20 + 9 + 12 = 41\). Neither: \(50 - 41 = 9\).
- 只養狗：\(21 - 12 = 9\) 人。只養貓：\(32 - 12 = 20\) 人。
- 有養寵物：\(20 + 9 + 12 = 41\) 人。兩種都沒養：\(50 - 41 = 9\) 人。
CIs the answer reasonable?判斷答案是否合理
- Use this when you must decide if an answer makes sense. Round the numbers, compute quickly in your head, and compare your estimate with the answer.當題目要你判斷答案是否合理時使用：把數四捨五入、心算出估計值，再和答案比較。
例題
Ravi says that \(19 \times 31 = 389\). Use estimation to decide if his answer is reasonable.小瑞說 \(19 \times 31 = 389\)。用估算判斷他的答案是否合理。
觀念提示
Round each factor to the nearest ten so you can multiply in your head. Then compare the estimate with Ravi's answer.把每個因數四捨五入到十位，就能心算相乘，再把估計值和小瑞的答案比較。
關鍵算式
\(19 \times 31 \approx 20 \times 30\)\(19 \times 31 \approx 20 \times 30\)
完整詳解
- Estimate: \(20 \times 30 = 600\).
- 389 is far from 600, so the answer is not reasonable.
- Exact answer: \(19 \times 31 = 19 \times 30 + 19 = 570 + 19 = 589\), which is close to 600.
- 估算：\(20 \times 30 = 600\)。
- 389 離 600 很遠，所以答案不合理。
- 正確答案：\(19 \times 31 = 19 \times 30 + 19 = 570 + 19 = 589\)，和 600 很接近。
自我檢查
A store sold 198 tickets at $12 each. Is $2,376 a reasonable total? Use estimation.商店賣出 198 張票，每張 12 元。總收入 2,376 元合理嗎？用估算判斷。
看答案
- Estimate: \(200 \times 12 = 2400\).
- 2,376 is close to 2,400, so it is reasonable. (Exact: \(198 \times 12 = 2376\).)
- 估算：\(200 \times 12 = 2400\)。
- 2,376 和 2,400 很接近，所以合理。（正確值：\(198 \times 12 = 2376\)。）
DDivision problems with remainders有餘數的除法問題
- In a division word problem, decide what the remainder means. Round up if every item must fit somewhere. Drop the remainder if a leftover part is too small to use.除法文字題要先想清楚餘數的意思：如果每一個都要有地方放，就無條件進位；如果剩下的部分太少不能用，就把餘數捨去。
例題
64 students go on a trip in vans. Each van holds 7 students. How many vans are needed?64 位學生搭廂型車去校外教學，每台車可載 7 人。需要幾台車？
觀念提示
Every student needs a seat, so a van that is not full is still needed.每位學生都要有座位，所以沒坐滿的車也還是需要。
關鍵算式
\(64 \div 7\)\(64 \div 7\)
完整詳解
- Divide: \(64 \div 7 = 9 \text{ R } 1\), because \(7 \times 9 = 63\).
- 9 vans carry 63 students, and 1 student is left.
- That student needs one more van: \(9 + 1 = 10\).
- 10 vans are needed.
- 相除：\(64 \div 7 = 9 \cdots 1\)，因為 \(7 \times 9 = 63\)。
- 9 台車載 63 人，還剩 1 人。
- 這位學生還需要一台車：\(9 + 1 = 10\)。
- 需要 10 台車。
例題
A ribbon is 50 m long. Each bow uses 3 m of ribbon. How many bows can be made?一條緞帶長 50 公尺，每個蝴蝶結要用 3 公尺。可以做幾個蝴蝶結？
觀念提示
Each bow needs a full 3 m. A leftover piece shorter than 3 m cannot make another bow.每個蝴蝶結都要完整的 3 公尺，剩下不到 3 公尺的部分就不能再做一個。
關鍵算式
\(50 \div 3\)\(50 \div 3\)
完整詳解
- Divide: \(50 \div 3 = 16 \text{ R } 2\), because \(3 \times 16 = 48\).
- The 2 m left over is too short for a bow, so drop it.
- 16 bows can be made.
- 相除：\(50 \div 3 = 16 \cdots 2\)，因為 \(3 \times 16 = 48\)。
- 剩下的 2 公尺不夠做蝴蝶結，捨去。
- 可以做 16 個蝴蝶結。
自我檢查
A farm packs 150 eggs into cartons that hold 12 eggs each. How many cartons are needed to pack all the eggs?農場要把 150 顆蛋裝進每盒 12 顆的蛋盒。要裝完所有的蛋需要幾盒？
看答案
- \(150 \div 12 = 12 \text{ R } 6\), because \(12 \times 12 = 144\).
- The 6 extra eggs need one more carton, so 13 cartons are needed.
- \(150 \div 12 = 12 \cdots 6\)，因為 \(12 \times 12 = 144\)。
- 多出的 6 顆蛋還要再一盒，所以需要 13 盒。
這課的重點字
- four-step plan解題四步驟
- strategy策略
- multi-step problem多步驟問題
- estimate估算