CONNECT THE IDEAS
Chapter 1, in your own words.
Use these reminders to explain a method to a classmate. Open a misconception to see what needs correcting.
試下向同學解釋每個方法。遇到唔肯定嘅地方，可以直接返去嗰課睇圖解。
“Per one” → divide to find a rate.
Whole journey → total distance ÷ total time.
Total or difference → write amounts using k.
Two linked ratios → match the common quantity.
Missing value → write fractions, then cross multiply.
Drawing and actual size → same units first; area needs two dimensions.
01 · Rates & unit conversion
速率與單位轉換
- A rate tells you how much for one unit of another quantity.
- Convert units before dividing. For speed, check both distance and time.
- Write the unit in your answer and use multiplication to check it.
Spot the mistake
“72 km/h = 72,000 m/s.”
Think first, then open the explanation.Why does it need correcting?
The distance changed units, but the time did not. 1 h = 3,600 s, so 72,000 ÷ 3,600 = 20 m/s.
02 · Journeys & average speed
兩段路程與平均速度
- Average speed = total distance ÷ total elapsed time.
- Two-part journeys: first time x, remaining time T − x.
- Use distance = speed × time, solve, and answer in the requested unit.
Spot the mistake
“The average of 40 km/h and 60 km/h is always 50 km/h.”
Think first, then open the explanation.Why does it need correcting?
Only if equal time is spent at the two speeds. Use total distance ÷ total elapsed time.
03 · Ratios & simplest form
比與最簡整數比
- Keep the order requested in the question.
- Use the same units before cancelling them in a ratio.
- Clear fractions/decimals, then reduce to the simplest whole-number ratio.
Spot the mistake
“0.6 kg : 450 g = 0.6 : 450.”
Think first, then open the explanation.Why does it need correcting?
Convert both amounts to grams first: 600 : 450 = 4 : 3.
04 · Ratio equations
比例方程
- Write each ratio as a fraction in the same order.
- Multiply both sides by the denominators to understand cross multiplication.
- Solve and check in the original ratio.
Spot the mistake
“x : 12 = 3 : 4 means x/12 = 4/3.”
Think first, then open the explanation.Why does it need correcting?
Keep the order on both sides: x/12 = 3/4. Then 4x = 36, so x = 9.
05 · Ratios with algebra
含代數式的比例方程
- Write a fraction first; cross multiply with brackets.
- Expand every term, solve, and check in the original ratio.
- From pa = qb, derive a/b by division; do not copy the coefficients in order.
Spot the mistake
“3(x + 2) = 3x + 2.”
Think first, then open the explanation.Why does it need correcting?
The 3 multiplies both terms: 3x + 6.
06 · Sharing in a ratio
按比以總數或相差分配
- Let every amount use the same k.
- Given a total: add the parts. Given a difference: subtract the smaller from the larger.
- For worded three-way relationships, choose one common reference before forming the ratio.
Spot the mistake
“For 3 : 5, a difference of 24 means 8k = 24.”
Think first, then open the explanation.Why does it need correcting?
The difference is 5k − 3k = 2k. So 2k = 24 and k = 12.
07 · Combining three quantities
合併兩個比成三項比
- Identify the letter common to both ratios.
- Use a common multiple and scale each whole ratio.
- Combine in the requested order, then check both original ratios.
Spot the mistake
“a : b = 2 : 3 and b : c = 4 : 5 give 2 : 3 : 5.”
Think first, then open the explanation.Why does it need correcting?
The two values of b must match. Use 8 : 12 and 12 : 15 to get 8 : 12 : 15.
08 · Ratios of expressions
代數式的比
- Match the common variable before combining two ratios.
- Use one shared non-zero k for every quantity.
- Substitute into a fraction, simplify, then write the final ratio.
Spot the mistake
“x : y = 2 : 3 and x : z = 4 : 5 give y : z = 3 : 5.”
Think first, then open the explanation.Why does it need correcting?
Match x at 4 first. Then y = 6 parts and z = 5 parts, so y : z = 6 : 5.
09 · Maps & scale drawings
地圖與比例尺
- Scale means drawing length : actual length, in the same units.
- For 1 : n, multiply drawing length by n; divide actual length by n.
- Check whether the answer should be an enlargement or a reduction.
Spot the mistake
“Scale 1 : 50,000 means 1 cm represents 50,000 m.”
Think first, then open the explanation.Why does it need correcting?
Both lengths use the same unit: 1 cm represents 50,000 cm, which is 500 m.
10 · Scale drawings & area
比例尺、面積與每平方米費用
- Find actual length AND actual width before calculating area.
- A length multiplier n gives an area multiplier n².
- Divide the total cost by actual area for cost per square unit.
Spot the mistake
“A length multiplier of 200 gives an area multiplier of 200.”
Think first, then open the explanation.Why does it need correcting?
Both length and width multiply by 200, so area multiplies by 200² = 40,000.
Can you explain why your method works? Have you kept the order, checked the units, and checked your answer against the original information?
識講點解，比只係記住答案更重要。