Cubes and Cube Roots
An Interactive Learning Experience for Class 8 Students
Sub-topic 6.1: Introduction to Cubes
Teaching Learning Process
This lesson follows a structured approach to ensure effective learning:
Hook Activity (4 Minutes)
Helps to grab students' attention and activates prior knowledge to set the context.
Explicit Modelling (I Do) - 8M
Teacher demonstrates the skill or concept clearly, setting a strong example.
Group Work (We Do) - 16M
Encourages collaboration, discussion, and peer learning.
Independent Work (You Do) - 8M
Allows students to apply learning individually, reinforcing understanding.
Closure - 4M
Wraps up the lesson with a quick review, reflection, or takeaway to solidify learning.
Hook Activity (4 Minutes)
Let's start with an engaging activity to spark your interest in cubes!
Cubes in Real Life
Cubes aren't just in math class; they're everywhere! Think about ice cubes, boxes, or even some buildings. Why do you think so many things are cube-shaped?
Cube-shaped Objects
- Dice
- Building blocks
- Ice cubes
- Rubik's cube
- Sugar cubes
- Boxes
- Some buildings
Why Cube Shapes?
Cube shapes are common because:
- They are stable and stack easily
- They maximize storage space
- They are easy to manufacture
- They have equal sides, making them strong
Discussion Questions:
1. What cube-shaped objects can you find in your home or school? Describe them.
Sample Answer: In my home, I can find several cube-shaped objects. For example, sugar cubes in the kitchen are small cubes used for sweetening tea. There are also dice that we use for board games - they have numbers on each face. In my room, I have a Rubik's cube puzzle with different colored sides. In the living room, there are cube-shaped storage boxes that help organize items neatly.
2. Can you think of practical reasons why objects are designed in cube shapes? (Hint: Think about stacking and packing!)
Sample Answer: Objects are often designed as cubes for practical reasons. Cubes stack easily without wasting space, which is why boxes and storage containers are often cube-shaped. They're also stable because all sides are equal, making them less likely to tip over. In manufacturing, cubes are easier to produce since all sides are identical. For transportation, cubes pack efficiently into trucks and containers, maximizing space usage.
3D Cube Visualization
Below is a rotating 3D cube. Notice how all six faces are identical squares, and all edges are equal in length.
Explicit Modelling (I Do) - 8 Minutes
Now, let me introduce you to the fascinating story behind Ramanujan's number 1729 and how it relates to cubes.
The Story of 1729
The number 1729 is known as the Hardy-Ramanujan number. It's the smallest number that can be expressed as the sum of two cubes in two different ways:
1729 = 12³ + 1³ = 1728 + 1
1729 = 10³ + 9³ = 1000 + 729
This number was part of a conversation between mathematicians G.H. Hardy and Srinivasa Ramanujan, highlighting Ramanujan's extraordinary mathematical intuition.
What is a Cube Number?
A cube number is obtained when a number is multiplied by itself three times.
Cube Number Examples
Check for Understanding
What is the product of:
2 × 2 = ? (Square)
2 × 2 × 2 = ? (Cube)
Group Work (We Do) - 16 Minutes
Now, let's work together to understand cubes better through interactive activities.
Building Cube Structures
Imagine you have a puzzle cube that is 3 × 3 × 3. How many unit cubes would it contain?
Total cubes: 27
Thinking Challenge: If you wanted to count the boxes in a 10 × 10 × 10 cube in thirty seconds, how would you do it?
Sample Answer: Instead of counting each box individually, I would use the mathematical formula for finding the volume of a cube. Since a cube has equal sides, I would multiply the length × width × height. For a 10 × 10 × 10 cube, that would be 10 × 10 × 10 = 1000 cubes. This method is much faster than counting each box and works for cubes of any size!
Independent Work (You Do) - 8 Minutes
Now it's your turn to practice what you've learned about cubes.
Practice Exercises
-
Represent the numbers in exponential form:
a) 5 × 5 × 5 = =
b) 2 × 2 × 2 = =
c) × × = 7³
-
For calculating a cube, how many times does the number multiply by itself?
Prior Knowledge Check
Before we proceed to cube roots, let's review some important concepts:
- What is prime factorization?
- How do we find the square root of a number?
- What is the relationship between squares and square roots?
Sample Answers:
- Prime factorization is breaking down a number into its prime factors (numbers that can only be divided by 1 and themselves).
- To find the square root of a number, we look for a number that when multiplied by itself gives the original number.
- The relationship between squares and square roots is that they are inverse operations - squaring a number and then taking the square root brings you back to the original number.
Closure - 4 Minutes
Let's wrap up our lesson with a quick review of what we've learned about cubes.
Lesson Summary
In this lesson, we've learned:
- What cube numbers are and how to calculate them
- The relationship between a number and its cube
- How to visualize cubes using physical models
- The fascinating story behind Ramanujan's number 1729
Key Takeaway
Cube numbers grow much faster than square numbers. For example, while 10² = 100, 10³ = 1000. This exponential growth is why cube numbers become large very quickly!
Looking Ahead
In our next lesson, we'll explore cube roots - the inverse operation of cubing a number. Think about:
- If we know the volume of a cube, how can we find its side length?
- How are cube roots related to the prime factorization method we learned for squares?
Homework
Complete the following worksheet to reinforce your understanding of cubes:
Worksheet Problems
1. Find the cube of the first 10 natural numbers.
The cubes of the first 10 natural numbers are:
1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000
2. Identify which of the following are perfect cubes: 64, 100, 125, 216, 300
Let's check each number:
- 64 = 4 × 4 × 4 = 4³ (Perfect cube)
- 100 = 10 × 10, but 100 ÷ 10 = 10, not a perfect cube since 4.64 × 4.64 × 4.64 ≈ 100
- 125 = 5 × 5 × 5 = 5³ (Perfect cube)
- 216 = 6 × 6 × 6 = 6³ (Perfect cube)
- 300 = Not a perfect cube (6.69 × 6.69 × 6.69 ≈ 300)
So the perfect cubes are: 64, 125, and 216.
3. If the volume of a cube is 343 cm³, what is the length of its side?
We know that volume of a cube = side × side × side = side³
Given volume = 343 cm³
So we need to find a number that when multiplied by itself three times gives 343.
Let's try: 7 × 7 × 7 = 49 × 7 = 343
Therefore, the side length of the cube is 7 cm.
4. Find the smallest number by which 256 must be multiplied to obtain a perfect cube.
First, let's find the prime factorization of 256:
256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸
For a number to be a perfect cube, each prime factor must appear in groups of three.
In 2⁸, we have eight 2's. To make groups of three, we need one more 2 to make 2⁹ (since 9 is a multiple of 3).
Therefore, the smallest number we need to multiply 256 by is 2.
256 × 2 = 512, and 512 = 8 × 8 × 8 = 8³, which is a perfect cube.