LF-P5-S2-L28 Comprehensive—Volume + Area + Fraction

Lam Fung Academy· LF Academy

Primary 5 · Lesson 28 · Student Handout

Comprehensive—Volume + Area + Fraction

Cross-Topic integration (T1 · T5 · T9) · 65 minutes · 1-to-3 Online Lesson

Corresponding textbooks:"New Thinking in Primary School Mathematics (Second Edition)" Volume 5B Units 7-8 + Modern Education 5B Unit 18-20
Core traps:🔴 High Frequency
Sub-test cross-topic comprehensive questions, accounting for about 18-22% of Paper 1Prerequisite knowledge:Class 21 (Circle + Section + Origami) · Class 22-23 (Fundamentals of Volume + Application) · Class 24-26 (Multiplication and Division of Fractions) · Class 27 (Division of Decimals)
Objectives of this class:❶ Comprehensive application of volume and area formulas ❷ Application of fractions in volume ❸ Identification and avoidance of cross-topic traps
Our goals:❶ Comprehensive application of volume and area formulas ❷ Application of fractions in volume ❸ Identification and avoidance of cross-topic traps

Student Name: Class: Date: Time Spent:

I.Warm-Up Questions（total 5 question，5 minutes）

🎯 總複習·陷阱獵人

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#	Question	Difficulty
1	The cuboid is 6 cm long, 4 cm wide and 5 cm high. Volume = ?	Basic
2	Triangular base 8 cm, height 6 cm. Area = ?	Basic
3	The parallelogram has a base of 12 cm and a height of 5 cm. Area = ?	Basic
4	The base area of a cuboid is 24 cm² and the height is 3 cm. Volume = ?	Advanced
5	The volume of a cuboid is 120 cm³, its length is 6 cm, and its width is 5 cm. Seek high.	Advanced

II.Core Knowledge ＋ Worked Examples

Knowledge point 1: Comprehensive application of volume + area formula🔴 SSPA

① Volume formula (cuboid/cube):V = length × width × height; V = base area × height
② Area formula review:Triangle A = (base × height) ÷ 2 · Parallelogram A = base × height · Trapezoid A = (upper base + lower base) × height ÷ 2
③ Key distinction:The area is the size of the "surface" (cm²/m²), and the volume is the size of the "space" (cm³/m³)
④ Composite graphics strategy:Divide into basic graphics → Find area/volume separately → Add or subtract
⑤ Unit trap:The base and height units must be consistent; the volume unit is cubic (³), not square (²)!

🪤 Example of trap detonation (the most important demonstration in this class)

A rectangular container has a trapezoidal base (upper base 4 cm, lower base 6 cm, height 5 cm), and the height of the container is 10 cm. Find the volume of the container.

❌ Common mistakes (70% students)

V = (4+6) × 5 × 10 = 500 cm³

Forget about the area of a trapezoid ÷2! First find the base area = (4+6)×5÷2 = 25 cm², then × 10 = 250 cm³

✅ Correct solution

V = 250 cm³

① Base area = (4+6)×5÷2 = 25 cm² ② V = 25×10 = 250 cm³

🧠 Tip: "First determine the surface and determine the body, and distinguish the basis of the formula; divide the area of a trapezoid by two, and then multiply the volume by the height; the unit is cubed, and the area is raised to the second power."

⚠️ The most frequent error: Forget the area of the trapezoid ÷2, use (upper base + lower base) × height directly as the area. Remember that the trapezoidal formula must be divided by 2!

⚠️ The second most common error: the unit of volume is written as cm² instead of cm³ - area is square and volume is cubic!

Knowledge Point - Worked Examples (write out complete steps: ①findbasearea ②findvolume ③mark unit)

#	Question	Difficulty
Example 1	The side length of a cube is 5 cm. Find: (a) the area of a face (b) the volume	🌱
Example 2	The cuboid is 8 cm long, 6 cm wide, and 4 cm high. Find the area and volume of the base.	🌱
6	A rectangular parallelepiped with a square base (side length 7 cm) and height 10 cm. Find the volume.	🌿
7	A rectangular container has a parallelogram base (base 8 cm, height 5 cm) and the height of the container is 12 cm. Find the volume.	🌿
8	A rectangular parallelepiped has a base area of 36 cm² and a height of 9 cm. If the height is increased by 3 cm, how much will the volume increase?	🌳
9	The picture below consists of two cuboids (stacked one above the other). Upper layer: 3×3×4 cm; lower layer: 5×5×6 cm. Find the total volume. Tip: Calculate separately and then sum.	🌿
10	The side length of one cube is 2 times the length of the other cube. How many times the volume of the large cube is the volume of the small cube?	🌳

Knowledge Point 2: Application of Fractions in Volume 🔴 SSPA Required Exam

Four ways to consider fractions in volume:
① Side length contains fractions:Length = 212cm → Convert to false fraction and multiply: V =52Container volume
② Fraction of volume:The container is packed34of water → volume of water = volume of container ×34
③ Find the volume of the parts and find the whole:known25The volume is 40 cm³ → whole = 40 ÷25 = 100 cm³
④ Scaling:Length enlargement 112times → volume enlargement (32)³ = 278times

①

Side length includes fraction

Mixed fractions→improper fractions and then multiply to find the volume

②

Find the fraction

Volume × Fraction = Partial Volume

③

Rounding the known parts

Part ÷ Fraction = Whole Volume

④

scaling

Side length × kVolume × k³

example

Example 3: A cuboid is 212cm long, 4 cm wide, and 3 cm high. Find the volume.

example

Example 4: A water tank has a volume of 80 L and is filled with35water. What is the volume of water in L?

❌ Common mistakes

212 × 4 × 3 = 2.5 × 12 = 30

After converting mixed numbers to improper fractions, the numerator and denominator are processed separately: 52×4×3 = 30 ✓ (This question happens to be correct), but it is easy to get confused in complicated situations!

✅ Correct approach

V = 52 × 4 × 3 = 30 cm³

① 212 = 52 ② 52×4×3 = 602 = 30

Knowledge Point 2 Synchronization practice

#	Question	Difficulty
11	The side length of the cube is 112cm. Find the volume. (Answer in fractions)	🌿
12	A water tank has a volume of 60 L and contains23water. How many liters of water is there?	🌿
13	A fish tank34has 45 L of water. What is the volume of the entire fish tank in L?	🌳
14	The length of the cuboid is32m, the width23m, and the height is 2 m. Find the volume. (Answer with m³)	🌳
15	After the side length of the cube is enlarged by 113times, how many times is the new volume the original? (Answer in fractions)	🏔️

Knowledge Point 3: Comprehensive Application—Volume + Area + Fraction Three-in-One 🔴 SSPA Advanced

Strategies for solving cross-topic comprehensive questions (three-step method for reading questions):
① Circle all the numbers and units in the question- Clarify which are areas, which are volumes, and which are fractions
② Judge the order of operations- First find the area (base area) → then find the volume → finally deal with the fraction ratio
③ Check the consistency of the units——m and cm cannot be multiplied directly! Convert to the same unit first

example

Example 5: A rectangular water tank with a rectangular base (1 m long, 50 cm wide) and 80 cm high. The water tank contains35water. Find the volume of water (answer in L, 1 L = 1000 cm³).

example

Example 6: The side length of cube A is 6 cm, and the side length of cube B is23of A. What fraction of the volume of B is A?

🧠 Tip: "Unify the units first, then the area and then the volume; multiply the fractions last, and avoid all traps."

⚠️ High frequency trap: m and cm are mixed up - 1 m = 100 cm, but 1 m² = 10000 cm², 1 m³ = 1,000,000 cm³!

Knowledge Point 3 Synchronization practice

#	Question	Difficulty
16	The base area of a cuboid is 25 cm² and the height is 8 cm. What is the volume34in cm³?	🌿
17	A fish tank is 50 cm long, 40 cm wide, and 30 cm high. (a) Find the volume of the fish tank (b) If water is filled to the height of \|\|\|SEP\|\|\|, what is the volume of water in L?45Height, what is the volume of water in L?	🌳
18	The side length of a cube water tank is 20 cm. Existing water 4000 cm³. What fraction of the tank's volume does water occupy? (Answer with the simplest fraction)	🌳
19	The cuboid is 1.5 m long, 0.8 m wide, and 0.6 m high. (a) Volume = ? m³ (b) If the space of58has been used, how much m³ is the remaining space?	🌳
20	A cylinder has a base area of 78.5 cm² and a height of 10 cm. If the height is reduced by \|\|\|SEP\|\|\|, what is the new volume? (Hint: new high = 10 × (1 −15, what is the new volume? (Hint: new high = 10 × (1 −15)）	🏔️

III. Lesson Layered Synchronization Practice

Basic layer (total 5 questions, everyone must do)

#	Question	Difficulty
21	The cuboid is 10 cm long, 5 cm wide, and 4 cm high. Find the volume.	🌱
22	The side length of the cube is 8 cm. Find the volume.	🌱
23	Triangular base 12 cm, height 9 cm. Find the area.	🌱
24	The upper base of the trapezoid is 4 cm, the lower base is 8 cm, and the height is 5 cm. Find the area.	🌱
25	The volume of a cuboid is 72 cm³ and the base area is 12 cm². Seek high.	🌱

Advanced layer (total 5 questions, 🚶🚀 choose do)

#	Question	Difficulty
26	A cuboid has a square base (sides 6 cm) and a height 15 cm. Find the volume.	🌿
27	Water tank volume 120 L, used \|\|\|SEP\|\|\|. How much L space is left?56. How much L space is left?	🌿
28	The side length of the cube is 113cm. Find the volume. (Answer with improper fractions)	🌿
29	Cuboid A: length 6 cm, width 4 cm, height 5 cm. Cuboid B: All sides are twice as long as A. How many times the volume of B is A?	🌳
30	A combined solid: the lower cuboid is 10×8×5 cm, and the upper cube with side length 4 cm is placed in the center. Find the total volume.	🌳

🌳 challenge layer (total 3 questions, 🚀 choose do, SSPAKiller Questions)

#	Question	Difficulty
31	A rectangular container has an interior length of 30 cm, a width of 20 cm, and a height of 15 cm. After placing a stone, the water level rises by 2 cm. What is the volume of the stone in cm³? Tips: Drainage method - stone volume = bottom area × water level rise height	🌳
32	The side length of the cube is34m. Find (a) the area of a surface (answer in m²) (b) the volume (answer in m³) (c) how many times the volume is the area of the base?	🌳
33	A trapezoidal cylinder: the base is trapezoidal (upper base 5 cm, lower base 9 cm, height 4 cm), and the height of the cylinder is 10 cm. Find the volume. Tips: V = base area × height = trapezoid area × cylinder height	🏔️

IV.applicationquestionspecial topic training

#	Question	Difficulty
34	A rectangular pool is 8 m long, 5 m wide, and 2 m deep. (a) What is the volume of the pool in m³? (b) If the water depth is only \|\|\|SEP\|\|\|, what is the volume of water in m³?34, what is the volume of water m³?	🌿
35	A rectangular container is 12 m long, 2.5 m wide, and 2.8 m high. (a) What is the volume of the container in m³? (b) The loaded goods occupy \|\|\|SEP\|\|\|, how many m³ of goods can be loaded in the remaining space?57, how many m³ of goods can be loaded in the remaining space?	🌳
36	A cube water tank has a side length of 40 cm. 8000 cm³ of water is injected every minute. (a) What is the total volume of the tank in cm³? (b) How many minutes does it take to fill the water tank? (c) After 5 minutes, what fraction of the water will be in the tank?	🌳
37	A rectangular fish tank is 60 cm long, 30 cm wide, and 40 cm high. After first pouring 36 L of water and adding some stones, the water level rose by 5 cm. (a) 36 L What is the height of the water in the tank in cm? (1 L = 1000 cm³) (b) What is the total volume of the stone in cm³? (c) What is the final water depth in cm?	🏔️

V.🏔️ Ultimate challenge area

#	Question	Difficulty
🏔️1	The length of a cuboid is 2 times the width, and the breadth is 112times the height. If the height is 4 cm, find: (a) What is the width? (b) What is the length? (c) What is the volume? (d) What is the surface area? (Surface area = 2×(length×width + length×height + width×height))	🏔️
🏔️2	A cube has a side length of 10 cm. A small cube with a side length of 2 cm is dug out in the center of each of its six faces (digging through to the opposite side). Find the volume of the remaining solid. Tips: Note that the perforations in the three directions will overlap, be careful not to deduct repeatedly!	🏔️
🏔️3	A trapezoidal cylindrical pool: the bottom is trapezoidal (upper bottom 3 m, lower bottom 7 m, height 4 m), and the pool depth is 2 m. 32 m³ of water available. (a) What is the total volume of the pool in m³? (b) What fraction of the total pool is the existing water? (Answer in simplest fractions) (c) How many m³ of water must be injected to fill it completely?	🏔️

Cross-topic integration: volume + area + fraction, a universal problem-solving framework🔴 essential for score examinations

When encountering comprehensive questions, deal with them in the following order:
① Read the questions:Circle all the numbers, mark the units (cm/m/L/cm³/m³), and find out "a fraction"
② Unify the units:Unify all lengths into the same unit (all cm or all m)
③ Area first, then volume:If you need the base area → use the corresponding formula (rectangle/triangle/trapezoid/parallelogram)
④ Processing fractions:Volume × fraction = volume of part; part ÷ fraction = whole
⑤ Check the answer:Are the units correct? Are fractions reduced? Is the answer reasonable?

VI.Class afterhomework

Basic must-do questions (total 5 questions, mustShow full working)

#	Question	Difficulty
H1	The cuboid is 7 cm long, 5 cm wide, and 3 cm high. Find the volume.	🌱
H2	The side length of the cube is 9 cm. Find the volume.	🌱
H3	A cuboid has a base area of 28 cm² and a height of 6 cm. Find the volume.	🌱
H4	Water tank volume 90 L. Installed25water. How many liters of water is there?	🌿
H5	A cuboid is 1.5 m long, 0.8 m wide, and 0.5 m high. Find the volume (answer in m³).	🌿

Advanced choose doquestion (total 3 questions, 🚀 choose do)

#	Question	Difficulty
H6	The cuboid is 212cm long, 4 cm wide and 3 cm high. Find the volume.	🌿
H7	A cube water tank has a side length of 25 cm. Pour 8 L of water first, then add the stones, and the water level rises by 3 cm. What is the volume of the stone in cm³? (1 L = 1000 cm³)	🌳
H8	Cuboid A: 6×4×5 cm. Cuboid B: All side lengths are 112times that of A. How many times the volume of B is A? (Answer in fractions)	🏔️

VII. 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	Forget the area of the trapezoid ÷2(upper base + lower base) × height, not the area of the trapezoid!	Trapezoidal area = (upper base + lower base) × height ÷ 2, you must also divide 2 when using it to calculate the base area.
2	The volume unit is written incorrectly \|\|\|SEP\|\|\|: cm³ is written as cm²: cm³ is written as cm²	Area → square (²), volume → cubic (³). Check the index before each answer
3	m/cm conversion error \|\|\|SEP\|\|\|: 1m=100cm but 1m³≠100cm³: 1m=100cm but 1m³≠100cm³	1 m³ = 100×100×100 = 1,000,000 cm³. Three-dimensional conversion requires conversion in each dimension
4	The fractions are not reduced after multiplication：612Take it directly as the answer	The last step is to check whether the numerator and denominator have common factors, which must be reduced to the simplest
5	"Part ÷ Fraction = Whole" is reversed \|\|\|SEP\|\|\|: Use multiplication instead of division:Use multiplication instead of division	Given that35is 60 → overall = 60 ÷35= 100 (not 60 ×35）
6	The side length is enlarged k times → the volume is enlarged k times(Wrong!)	The side length is enlarged k times → the volume is enlarged k³ times (cubic!)
7	Misunderstanding of drainage method \|\|\|SEP\|\|\|: Thinking that the rising height of the water level = the height of the stone: Thinking that the rising height of the water level = the height of the stone	The volume of the stone = the bottom area of the container × the rising height of the water level (not the height of the stone)

Lam Fung Academy · LF Academy · We don't teach math. We teach trap avoidance.

📚 Related topics: L03 Area of parallelograms and triangles · L04 Area of trapezoidal polygon · L05 Area trap special project · Related topics: L25 Volume word problems · L26 Volume concept · L27 Composite three-dimensional drainage method