Patterns in Cubes: Complete Interactive Lesson
Prior Concepts & Skills
- Read and write numbers with multiple digits along with expanded forms
- Basic arithmetic operations: addition, subtraction, multiplication, and division
- Understanding of squares and square roots
- Knowledge of exponents and powers
- Prime factorization of numbers
- Ascending and descending order of numbers along with values
Vocabulary
- Cube: A symmetrical three-dimensional shape
- Patterns: A repeated design, numbers, etc.
- Multiple of Numbers: A number that is the product of a given number and some other natural number
- Consecutive Numbers: Numbers that follow each other in order
- Odd Numbers: Numbers not divisible by 2
- Even Numbers: Numbers divisible by 2
Hook Activity (4 Minutes)
Let's start with an interesting observation about cubes and odd numbers!
Observe the following pattern of sums of odd numbers:
\[1^3 = 1 = 1\]
\[2^3 = 8 = 3 + 5\]
\[3^3 = 27 = 7 + 9 + 11\]
\[4^3 = 64 = 13 + 15 + 17 + 19\]
Interactive Cube Builder
Adjust the cube size to see how many consecutive odd numbers are needed!
Cube of 3 = 27
= 7 + 9 + 11
Lesson 6.3: Patterns of Cubes of Even and Odd Numbers
Learning Objective: Students will be able to observe the pattern of cubes of even numbers and generalize that cube of even numbers are even and that of odd numbers is odd.
Pattern Observation
Let's examine what happens when we cube even and odd numbers:
Even Numbers Cubed
Even × Even × Even = Even
Pattern: The cube of an even number is always even.
Odd Numbers Cubed
Odd × Odd × Odd = Odd
Pattern: The cube of an odd number is always odd.
Try These: Without calculating, determine if these cubes are even or odd:
1. 12³
12 is even, so 12³ is even.
2. 15³
15 is odd, so 15³ is odd.
3. 21³
21 is odd, so 21³ is odd.
4. 30³
30 is even, so 30³ is even.
Experience and Reflection
Exploring Number Sequences
Let's explore the number sequence: 2, 8, 4, 3, 1
What patterns do you notice in this sequence?
Pattern Analysis:
- The sequence contains both even and odd numbers
- 2, 8, and 4 are even numbers
- 3 and 1 are odd numbers
- 8 is 2³ (the cube of 2)
- 1 is 1³ (the cube of 1)
- The sequence shows that cubes can be both even and odd, depending on the original number
Observation
Look for patterns in everyday objects. How many cube-shaped items can you find around you?
Critical Thinking
Why do you think the cube of an even number is always even? Can you prove this mathematically?
Real-World Connection
Where might knowing whether a large number is a cube be useful in real life?
Equation Analysis: Consider the equation 0 × 4 = 6.5
Equation Analysis:
This equation is mathematically incorrect. Let's break it down:
- 0 × 4 = 0 (any number multiplied by 0 is 0)
- 6.5 is not equal to 0
- This might be a common mistake when learning multiplication
- Remember: The product of 0 and any number is always 0
Correct equation: 0 × 4 = 0
Explicit Teaching/Teacher Modelling (I Do) - 8 Minutes
Let's explore two interesting patterns related to cubes:
1. Adding Consecutive Odd Numbers
We can express cubes as the sum of consecutive odd numbers. Notice the pattern:
For \(n^3\), we need \(n\) consecutive odd numbers, starting from \(n(n-1)+1\).
Example: For \(5^3 = 125\), we need 5 consecutive odd numbers starting from \(5×4+1=21\)
So, \(5^3 = 21 + 23 + 25 + 27 + 29 = 125\)
2. Pattern in Difference of Consecutive Cubes
Observe this pattern:
\[2^3 - 1^3 = 1 + 2 \times 1 \times 3\]
\[3^3 - 2^3 = 1 + 3 \times 2 \times 3\]
\[4^3 - 3^3 = 1 + 4 \times 3 \times 3\]
In general: \(n^3 - (n-1)^3 = 1 + n \times (n-1) \times 3\)
Try These: Using the above pattern, find the value of:
(i) \(7^3 - 6^3\)
Using the pattern: \(7^3 - 6^3 = 1 + 7 \times 6 \times 3 = 1 + 126 = 127\)
(ii) \(12^3 - 11^3\)
Using the pattern: \(12^3 - 11^3 = 1 + 12 \times 11 \times 3 = 1 + 396 = 397\)
Group Work (We Do) - 16 Minutes
Divide into groups of 3-4 students and work on these problems:
Activity 1: Express the following numbers as the sum of odd numbers using the pattern:
(a) \(6^3\)
\(6^3 = 216\). We need 6 consecutive odd numbers starting from \(6×5+1=31\)
So, \(6^3 = 31 + 33 + 35 + 37 + 39 + 41 = 216\)
(b) \(8^3\)
\(8^3 = 512\). We need 8 consecutive odd numbers starting from \(8×7+1=57\)
So, \(8^3 = 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 = 512\)
(c) \(7^3\)
\(7^3 = 343\). We need 7 consecutive odd numbers starting from \(7×6+1=43\)
So, \(7^3 = 43 + 45 + 47 + 49 + 51 + 53 + 55 = 343\)
Activity 2: How many consecutive odd numbers will be needed to obtain the sum as \(10^3\)?
For \(10^3 = 1000\), we need 10 consecutive odd numbers.
They start from \(10×9+1=91\)
So the numbers are: 91, 93, 95, 97, 99, 101, 103, 105, 107, 109
Independent Work (You Do) - 8 Minutes
Now try these problems on your own:
1. Express \(9^3\) as the sum of consecutive odd numbers.
\(9^3 = 729\). We need 9 consecutive odd numbers starting from \(9×8+1=73\)
So, \(9^3 = 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89 = 729\)
2. Find the value of \(15^3 - 14^3\) using the pattern.
Using the pattern: \(15^3 - 14^3 = 1 + 15 \times 14 \times 3 = 1 + 630 = 631\)
3. How many consecutive odd numbers are needed for \(12^3\)?
For \(12^3 = 1728\), we need 12 consecutive odd numbers.
They start from \(12×11+1=133\)
4. Without calculating, determine if these are even or odd: 18³, 25³, 32³
18 is even → 18³ is even
25 is odd → 25³ is odd
32 is even → 32³ is even
Closure - 4 Minutes
Summary:
- Cubes can be expressed as the sum of consecutive odd numbers
- The number of odd numbers needed equals the cube root
- The pattern for difference of consecutive cubes follows: \(n^3 - (n-1)^3 = 1 + n \times (n-1) \times 3\)
- New Pattern: The cube of an even number is always even, and the cube of an odd number is always odd
Homework:
- Find the sum of consecutive odd numbers for \(11^3\)
- Calculate \(20^3 - 19^3\) using the pattern
- Explore if a similar pattern exists for fourth powers
- Determine if 100³ is even or odd without calculating
