Cubes and Their Prime Factors
45-Minute Lesson Plan
Hook Activity (4 Minutes)
Begin with a visual demonstration of cubes and their volumes. Show a 3D cube and ask students to calculate its volume if each side is 4 units. Then ask: "What if we break down the number 4 into its prime factors? How would that relate to the cube's volume?"
Explicit Teaching/Teacher Modelling (8 Minutes)
Demonstrate the relationship between a number, its cube, and their prime factors using examples from the textbook:
| Number | Prime Factorization | Cube | Prime Factorization of Cube |
|---|---|---|---|
| 4 | 2 × 2 | 64 | 2 × 2 × 2 × 2 × 2 × 2 = 2³ × 2³ |
| 6 | 2 × 3 | 216 | 2 × 2 × 2 × 3 × 3 × 3 = 2³ × 3³ |
| 12 | 2 × 2 × 3 | 1728 | 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2³ × 2³ × 3³ |
Key Observation: Each prime factor appears three times in the prime factorization of a cube.
Group Work (16 Minutes)
Divide students into groups and assign each group a number: 12, 45, 216, or 225. Each group should:
- Calculate the cube of their assigned number
- Find the prime factors of the cube
- Determine if it's a perfect cube
- Present their findings to the class
Independent Work (8 Minutes)
Students work individually on the "Try These" problems from the textbook:
Which of the following are perfect cubes? 400, 3375, 8000, 15625, 9000, 6859, 2025, 10648
Closure (4 Minutes)
Summarize the key concept: A number is a perfect cube if all its prime factors can be grouped into triplets. Quick review with examples: 216 (perfect cube) vs. 500 (not a perfect cube).
3D Cube Visualization
Observe this rotating cube. Each face represents a dimension. When we cube a number, we're essentially multiplying it in three dimensions.
Try calculating the prime factors of different numbers and see if they form perfect cubes!
Prime Factorization Examples
Example 1: Is 243 a perfect cube?
Prime factorization of 243: 3 × 3 × 3 × 3 × 3
We can group: (3 × 3 × 3) × 3 × 3
We have one complete triplet, but two 3's remain ungrouped.
Conclusion: 243 is not a perfect cube.
Example 2: Is 216 a perfect cube?
Prime factorization of 216: 2 × 2 × 2 × 3 × 3 × 3
We can group: (2 × 2 × 2) × (3 × 3 × 3)
All prime factors are in complete triplets.
Conclusion: 216 is a perfect cube (6³).
Try These Questions
Question 1: Is 400 a perfect cube?
Prime factorization of 400: 2 × 2 × 2 × 2 × 5 × 5
We can group: (2 × 2 × 2) × 2 × (5 × 5)
We have extra 2 and two 5's that cannot form a complete triplet.
Answer: No, 400 is not a perfect cube.
Question 2: Is 3375 a perfect cube?
Prime factorization of 3375: 3 × 3 × 3 × 5 × 5 × 5
We can group: (3 × 3 × 3) × (5 × 5 × 5)
All prime factors are in complete triplets.
Answer: Yes, 3375 is a perfect cube (15³).
Question 3: Is 8000 a perfect cube?
Prime factorization of 8000: 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5
We can group: (2 × 2 × 2) × (2 × 2 × 2) × (5 × 5 × 5)
All prime factors are in complete triplets.
Answer: Yes, 8000 is a perfect cube (20³).
Interactive Worksheet
Section A: Is it a Perfect Cube?
1. Is 729 a perfect cube?
2. Is 1000 a perfect cube?
3. Is 15625 a perfect cube?
Section B: Find the Cube Root
4. If 216 is a perfect cube, what is its cube root?
5. If 3375 is a perfect cube, what is its cube root?
