🧠 WHY BOX — Why learn this?
Understanding this topic helps you solve real-life math problems and prepares you for the SSPA exam.
學好這個課題能幫助你解決生活數學問題，為 SSPA 考試做好準備。
📖 Story Context / 故事情境
Imagine you are shopping and need to calculate totals, discounts, or split bills. Math is everywhere in daily life!
想像你在購物時需要計算總額、折扣或分攤帳單。數學無處不在！
Lam Fung Academy· LF Academy
Primary 5 · Lesson 31 · Student Handout
Advanced algebraic expressions + two-step equations
Unit 3 · Algebra (T3 · T7) · 65 minutes · 1-to-3 Online Lesson
Corresponding textbooks:"New Thinking in Primary School Mathematics (Second Edition)" Volume 5C Unit 9 + Modern Education 5C Unit 21-22
Core traps:🔴 High frequency
The sub-test must be taken every year, accounting for about 12-15% of Paper 1Prerequisite knowledge:Class 14 (Algebraic expression recognition + simple equations) · Class 15 (Equation problems) · Class 27 (Decimal division)
Objective of this class:❶ Simplification of algebraic expressions (including brackets + distributive law) ❷ ax+b=c type two-step equation ❸ a(x+b)=c type equation ❹ Equation word problems
Our goals:❶ Simplification of algebraic expressions (including brackets + distributive law) ❷ ax+b=c-type two-step equation ❸ a(x+b)=c-type equation ❹ Equation word problems
Student Name:
Class:
Date:
Time Spent:
I.Warm-Up Questions（total 5 question，5 minutes）
🧮 方程解謎·計時挑戰
解出方程才能過關！每題限時90秒。小心陷阱：移項錯、符號錯、忘記驗算！
🔢 開始解謎 →
| # | Question | Difficulty | Working Space (write out complete process) |
|---|
| 1 | Simplify: 3a + 5a = ? | Basic | |
| 2 | Simplify: 7b − 2b = ? | Basic | |
| 3 | Find the value of 2x + 5 when x = 3. | Basic | |
| 4 | Solve the equation: x + 7 = 15 (one-step equation review) | Basic | |
| 5 | Solve the equation: 3y = 18 (one-step equation review) | Basic | |
II.Core Knowledge ＋ Worked Examples
Knowledge point 1: Advanced algebraic expressions - including parenthetical simplification and distributive law 🔴 SSPA
① Merge similar items (review):Only items with the same letters and the same exponent can be merged. For example, 3a + 2a = 5a, but 3a + 2b cannot be combined
② Distributive law (new learning):a(b + c) = ab + ac；a(b − c) = ab − ac
③ Rules for removing brackets:The word "+" before the bracket → remove the bracket directly; the word before the bracket is "−" → change the sign of each item after removing the bracket
④ Simplification steps:① Remove the brackets (use the distributive law) → ② Combine similar terms → ③ Write the simplest form
⑤ Common traps:3(x + 2) Many people only multiply the first term and write it as 3x + 2 (forget that 3 × 2 = 6!)
🪤 Example of trap detonation (the most important demonstration in this class)
Simplify: 3(x + 2) + 2(x − 1)
❌ Common mistakes (65% students)
3x + 2 + 2x − 1 = 5x + 1
Forget 3×2=6; forget 2×(−1)=−2. Correct: 3x+6+2x−2 = 5x+4
✅ Correct solution
5x + 4
① 3(x+2) = 3x+6 ② 2(x−1) = 2x−2 ③ (3x+6)+(2x−2) = 5x+4
🧠 Tip: "Multiply outside the parentheses to avoid missing items when multiplying item by item; when combining similar items, the letter indices must be the same."
⚠️ The most frequent error: the distributive law only multiplies the first term and misses the second term. 3(x+2) → 3x+2 (wrong!) → correct: 3x+6
⚠️ The second most frequent error: forgetting to change the sign when there is a minus sign before the bracket. −(x−3) → −x−3 (wrong!) → correct: −x+3
Knowledge Point - Worked Examples
| # | Question | Difficulty | Working Space |
|---|
| Example 1 | Simplify: 2(x + 3) = ? | 🌱 | |
| Example 2 | Simplify: 4(2a − 1) = ? | 🌱 | |
| 6 | Simplify: 5(y + 2) + 3y = ? | 🌱 | |
| 7 | Simplify: 3(m − 4) + 2(m + 1) = ? | 🌿 | |
| 8 | Simplify: 6 + 2(3x − 2) = ? | 🌿 | |
| 9 | Simplify: 4(2a + 3) − 3(a − 1) = ? | 🌳 | |
| 10 | Simplify: 2(x + y) + 3(x − y) − (x + 2y) = ? | 🌳 | |
Knowledge point 2: Two-step equation — ax + b = c type🔴 SSPA must take the exam
Problem-solving tips: first process addition and subtraction, then multiplication and division (reverse operation)
① Step 1: Eliminate constant terms(−b or +b on both sides of the equation at the same time) → ax = c − b
② Step 2: Eliminate coefficients(÷a on both sides of the equation) → x = (c − b) ÷ a
③ Verification:Substitute the answer into the original equation, the left side must be equal to the right side
④ Key principles:Both sides of the equation must do the same operation! "You do what, I do what"
①
elimination constant
−b on both sides leaves only ax on the left
②
elimination coefficient
Find x by ÷a on both sides
③
verify
Substitute into the original equation to check that the left and right sides are equal.
④
answer
x = Don’t miss the value x=
Example (two-step equation)
Example 3: Solve 3x + 5 = 20. Write out each step.
Example (two-step equation)
Example 4: Solve 4y − 7 = 13.
❌ Common mistakes
3x + 5 = 20 → x = 20 − 3 − 5 = 12
Move items randomly! There is no "simultaneous calculation of both sides of the equation". 3x is a whole and cannot be taken apart.
✅ Correct solution
x = 5
① 3x+5=20 → Both sides −5: 3x=15 ② Both sides ÷3: x=5 ③ Verification: 3(5)+5=20 ✓
🧠 Tip: "First do additions and subtractions, then multiplications and divisions, and move both sides of the equal sign simultaneously; after completing the substitution test, it's just the reverse."
Knowledge Point 2 Synchronization practice
| # | Question | Difficulty | Working Space |
|---|
| 11 | Solve the equation: 2x + 3 = 11 | 🌱 | |
| 12 | Solve the equation: 5y − 4 = 16 | 🌱 | |
| 13 | Solve the equation: 7m + 8 = 29 | 🌱 | |
| 14 | Solve the equation: 6p − 10 = 14 | 🌿 | |
| 15 | Solve the equation: 3x + 12 = 3 | 🌿 | |
Knowledge point 3: Two-step equation — a(x + b) = c type (bracket equation) 🔴 SSPA Advanced
Two solutions (recommended method one):
Method one (cancel coefficients first):Both sides of the equation simultaneously ÷a → x + b = c ÷ a → both sides −b → x = (c ÷ a) − b
Method two (remove the brackets first):Use the distributive law to expand → ax + ab = c → becomes ax + b = c type re-solution
① Both methods are correct, method one usually has fewer steps
② If c ÷ a is not an integer, use method two (the fraction is left for final processing)
example
Example 5: Solution 2(x + 3) = 14 (use method 1: cancel the coefficients first)
example
Example 6: Solve 3(x − 4) = 15 (use method 2: remove the brackets first and compare which one is faster)
⚠️ High-frequency trap: type a(x+b)=c, students often write x+b=c−a directly (wrong!). Correct: ÷a first, not −a first!
Knowledge Point 3 Synchronization practice
| # | Question | Difficulty | Working Space |
|---|
| 16 | Solve the equation: 2(x + 1) = 8 | 🌱 | |
| 17 | Solve the equation: 3(y − 2) = 12 | 🌿 | |
| 18 | Solve the equation: 5(m + 3) = 25 | 🌿 | |
| 19 | Solve the equation: 4(x − 1) = 10. Tip: 10÷4 = 2.5, you can first use the distributive law to expand. | 🌳 | |
| 20 | Solve the equation: 2(3x + 1) = 20 (process the coefficient 3 in the brackets first, then use method 1 or 2) | 🌳 | |
III. Lesson Layered Synchronization Practice
Basic layer (total 5 questions, everyone must do)
| # | Question | Difficulty | Working Space |
|---|
| 21 | Simplify: 3(a + 2) = ? | 🌱 | |
| 22 | Simplify: 4x + 2x − 3x = ? | 🌱 | |
| 23 | Solve the equation: 2x + 1 = 9 | 🌱 | |
| 24 | Solve the equation: 3(x + 1) = 15 | 🌱 | |
| 25 | Solve the equation: 6y − 5 = 19 | 🌱 | |
Advanced layer (total 5 questions, 🚶🚀 choose do)
| # | Question | Difficulty | Working Space |
|---|
| 26 | Simplify: 5(2m − 3) + 4(m + 2) = ? | 🌿 | |
| 27 | Solve the equation: 4x + 9 = 2x + 15 (hint: move the terms first, putting aside all those containing x) | 🌳 | |
| 28 | Solve the equation: 2(3x − 1) = 16 | 🌿 | |
| 29 | Solve the equation: 5(x + 2) − 3 = 17 | 🌳 | |
| 30 | Solve the equation: 3x + 7 = 4x − 2 (hint: have x on both sides) | 🌳 | |
🌳 challenge layer (total 3 questions, 🚀 choose do, SSPAKiller Questions)
| # | Question | Difficulty | Working Space |
|---|
| 31 | Solve the equation: 2(x + 3) = 3(x − 2) + 1
Tips: First use the distributive law to expand both sides, and then transfer the terms to merge. | 🏔️ | |
| 32 | When x = 4, 3(x + a) = 30. Find the value of a. | 🌳 | |
| 33 | Solve the equation:x2+ 3 = 7 (equations containing fractions: eliminate the constant first, then the denominator) | 🌳 | |
IV.applicationquestionspecial topic (equation + solution equation + answer sentence)
Knowledge Point 4: Two-Step Equation Application Questions 🔴 SSPA Required
Four-step method for solving problems (must follow!):① Let the unknowns("Let x be...") ②Make an equation(Convert it into an equation according to the meaning of the question) ③Solve the equation(Write the complete steps) ④Write an answer sentence(Answer + unit, otherwise step points will be deducted!)
example
Example 7: Xiao Ming has a certain amount of yuan. After he spent 8 yuan, he divided the remaining money equally among his three younger brothers, and each brother received 4 yuan. How much yuan does Xiao Ming originally have?
| # | Question | Difficulty | Working Space (let→column→solution→answer) |
|---|
| 34 | Xiao Ming has x yuan. He spent 15 yuan, and then spent three times the remaining money on books, for a total of 45 yuan. Find x.
Tip: 3(x − 15) = 45 | 🌿 | |
| 35 | The length of a rectangle is 2 times its breadth plus 3 cm. If the perimeter is 42 cm, find the width.
Tips: Let width = x, length = 2x+3, perimeter = 2(length+width) = 42 | 🌳 | |
| 36 | Four times a number plus 7 is equal to six times that number minus 5. Find this number.
Tips: Let the number be x, and write the equation 4x+7=6x−5 | 🌳 | |
| 37 | The older sister is 5 years older than the younger sister. After 5 years, the older sister will be twice as old as the younger sister. Please tell me your sister's current age.
Tips: Suppose the younger sister is x years old now, older sister = x+5. 5 years later: younger sister x+5, older sister x+10. Sequence equation x+10 = 2(x+5) | 🏔️ | |
V.🏔️ Ultimate challenge area
| # | Question | Difficulty | Working Space |
|---|
| 🏔️1 | Solve the equation: 3(2x + 1) − 2(x − 3) = 4x + 11
Tips: Remove the brackets first → merge similar terms → move terms → solve. Notice there are x's on both sides. | 🏔️ | |
| 🏔️2 | Simplify and evaluate: When a = 2, b = −1, find the value of 3(2a + b) − 2(a − b) + 4b.
Tips: Simplify the algebraic expression first, and then substitute the numerical values. Be careful with the minus sign! | 🏔️ | |
| 🏔️3 | The lengths of the three sides of a triangle are (x+2) cm, (2x−1) cm and (3x−3) cm respectively. If the perimeter is 22 cm, find the value of x. | 🌳 | |
VI. Mix comprehensive practice (mixed algebraic expression + equation)
| # | Question | Difficulty | Working Space |
|---|
| 38 | Simplify: 2(x + 3) − (x − 4) = ? | 🌿 | |
| 39 | Solve the equation: 8x − 3 = 5x + 9 | 🌿 | |
| 40 | If 3(x + 2) = 2x + 11, find x. | 🌳 | |
| 41 | Solve the equation: 4(2x − 1) = 3(x + 3) + 2 | 🌳 | |
🧠 General tips: "For algebraic simplification, break down the brackets first, and you will not go wrong by multiplying item by item; do the two-step equation in reverse, adding and subtracting first, then multiplying and dividing; set the unknown number for application problems, solve the problem in columns, and then answer the sentence."
VII. Lesson afterhomework
Basic must-do questions (total 5 questions, mustwrite out complete steps)
| # | Question | Difficulty | Working Space |
|---|
| H1 | Simplify: 4(2x + 1) = ? | 🌱 | |
| H2 | Simplify: 3(a − 2) + 5a = ? | 🌱 | |
| H3 | Solve the equation: 3x + 4 = 19 | 🌱 | |
| H4 | Solve the equation: 2(x + 5) = 18 | 🌿 | |
| H5 | Solve the equation: 7y − 6 = 15 | 🌿 | |
Advanced choose doquestion (total 3 questions, 🚀 choose do)
| # | Question | Difficulty | Working Space |
|---|
| H6 | Solve the equation: 5(x − 2) + 3 = 2x + 4 | 🌳 | |
| H7 | Solve the equation: 3(2x + 1) = 4(x + 3) − 5 | 🌳 | |
| H8 | Xiao Ming has some stickers. After giving 8 stickers to his sister, he put 4 times the remaining stickers into the collection book, for a total of 36 stickers. How many stickers does Xiao Ming originally have? (Suppose the equation is solved) | 🌳 | |
VIII.The Lessoncorecommon errorsummary
✅ Self-examination in this hall (tick after completion)
☐ I know the pitfalls of solving each knowledge point
☐ I can complete 🌱basic questions independently
☐ I can challenge 🌿advanced questions
☐ I remember the formula
🎯 Review of Learning Objectives - After completing this lesson you should be able to:
☐ Identify all trap types in our hall
☐ Solve 🌱basic questions independently (100% correct)
☐ Challenge🌿Advanced questions (80%+ correct)
☐ Explain the lesson formula to classmates
| # | common error | Correct Approach |
|---|
| 1 | The distributive law only multiplies the first term：3(x+2) → 3x+2 | The number outside the brackets must be multiplied by each item inside the brackets |||SEP|||every item：3(x+2) = 3x+6 |
| 2 | There is a minus sign before the parentheses and I forgot to change the sign.：−(x−3) → −x−3 | After removing the parentheses of the minus sign, the sign of each item in the parentheses must be changed: −(x−3) = −x+3 |
| 3 | The two-step equation first eliminates the coefficients (wrong!)：3x+5=20 → x+5=20÷3 | First cancel the constants (addition and subtraction), then cancel the coefficients (multiplication and division): −5 → 3x=15 → x=5 on both sides |
| 4 | a(x+b)=c type first subtract a：2(x+3)=14 → x+3=14−2 | The brackets are a whole, first ÷a: 2(x+3)=14 → x+3=7 → x=4 |
| 5 | Forgot to change the number when moving the item：3x+7=4x → 3x+4x=7 | When moving the terms to the other side of the equal sign, + becomes − and − becomes +: 3x−4x=−7 → −x=−7 → x=7 |
| 6 | The same operations are not performed on both sides of the equation | "You do what, I do what" - both sides of the equation must do exactly the same operation simultaneously |
| 7 | Missing unknowns/missing sentences in application questions | The four-step method must be complete: ① Assume x ② Make an equation ③ Solve the equation ④ Answer the sentence + unit |
Lam Fung Academy · LF Academy · We don't teach math. We teach trap avoidance.
📚Related topics: L14 Understanding algebraic expressions · L15 Equation word problems · L31 Advanced algebraic expressions