Enrol to start learning
Reading is open to everyone. Enrolling is free, and it is what unlocks the audio lessons, practice tests and progress tracking.
5.4.1. Other patterns in squares
Interactive Audio Lesson
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
Today, we will learn about a specific pattern with square numbers, especially for numbers that end in 5. Can anyone tell me what 25 squared equals?
Isn't it 625?
Exactly! Now, do you know why that is the case?
Maybe there's a rule for it?
Yes! We can use the formula (10a + 5)². When we expand it, we see it always ends with 25. The rest involves a simple calculation of a(a + 1) hundred. Can someone provide a value for 'a'?
What if a = 2?
Great choice! So we calculate 2 × 3, which gives us 6. Therefore, 25 squared can be visually understood as 625.
Can we try another example?
Of course! How about 35 squared?
That should be 1225!
Correct! Let’s summarize: Whenever we calculate the square of numbers ending in 5, remember it will follow the a(a + 1) hundred + 25 pattern!
Unlock the classroom podcast
The transcript is above and free to read. A free account plays the conversation back.
Create a free account
Let’s practice! Who is ready to find the square of 95?
I can try! a is 9, right?
Exactly! What’s a times a + 1?
That would be 9 × 10 = 90!
Good job! So according to our rule, what’s (10a + 5)²?
It equals 9025!
Perfect! Now you see how quickly we can find squares without multiplying directly. Can anyone give me the square of another number ending in 5?
How about 105?
Yes! Apply our formula! Remember the last digits will always be 25, and now calculate the rest.
It's 11025!
Excellent work, everyone! Always remember this pattern for quick calculations.
Overview
Short Summary
This section discusses various patterns found in square numbers and techniques for calculating them, particularly focusing on numbers ending in 5.
Medium Summary
The section explores specific patterns associated with square numbers, especially for numbers that end in the digits 5. It examines how to compute the squares of such numbers using formulas and highlights the relationship between unit digits and perfect squares.
Detailed Summary
Other Patterns in Squares
In this section, we delve into interesting patterns that emerge with square numbers, particularly concentrating on numbers that end in 5. The pattern we observe is that the square of any number that has 5 as its unit digit can be simplified using the expression:
Formula for Squares Ending in 5
For a number represented as a5:
-
To calculate (10a + 5)^2, we can expand this into:
(10a + 5)^2 = 100a^2 + 100a + 25 = 100a(a + 1) + 25
This reveals that the last two digits of any square ending in 5 will always be 25, while the preceding digits can be found by computing a(a + 1).
Pattern Illustration
- 25^2 = 625: Here, (2 * 3) hundreds + 25 gives 625.
- 35^2 = 1225: Similarly, (3 * 4) hundreds + 25 results in 1225.
- 75^2 = 5625: Following the same logic, we have (7 * 8) hundreds + 25.
- 125^2 = 15625: This continues with (12 * 13) hundreds + 25.
Conclusion
By utilizing this pattern, it's easy to compute the squares of numbers ending with 5 without extensive multiplication, showcasing an elegant relationship between square numbers and their properties.
Reference YouTube Videos
Audio Book
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountConsider the following pattern:
25^2 = 625 = (2 × 3) hundreds + 25
Consider a number with unit digit 5, i.e., a5
35^2 = 1225 = (3 × 4) hundreds + 25
(a5)^2 = (10a + 5)^2
75^2 = 5625 = (7 × 8) hundreds + 25
125^2 = 15625 = (12 × 13) hundreds + 25
Now can you find the square of 95? =100a(a + 1) + 25
= a(a + 1) hundred + 25
Detailed Explanation
This section discusses a pattern found in squares of numbers ending with the digit 5. When you square a number that ends in 5, the squared result always ends in 25. Moreover, the part before the 25 can be calculated using a specific pattern: for a number 'a5', where 'a' is the other digit in the number, the first part of the square can be determined by multiplying 'a' by 'a + 1'. For example, for 25, we have 2 * 3 = 6, and then we write 625. This logic extends to any number ending in 5, providing an easier way to find their squares without needing to perform full multiplication.
Examples & Analogies
Imagine you’re baking cupcakes and decide to bake them in batches of 5. If you notice that the number of batches you make follows a simple counting rule, like every number ending in a 5 resulting in a special pattern—625 for 25 batches or 1225 for 35 batches—this might help you remember how many cupcakes you’ve baked and make it easier to plan your baking day!
Unlock the audio lesson
The script is above and free to read. A free account plays it back, in the voice you pick.
Create a free accountTRY THESE
Find the squares of the following numbers containing 5 in unit’s place.
(i) 15 (ii) 95 (iii) 105 (iv) 205
Detailed Explanation
Here, students are prompted to apply the pattern discussed earlier to find the squares of various numbers ending in 5. By following the outlined method, they should calculate the square of each number knowing they can multiply the preceding digit and the next higher number. For instance, for 15, the square is 225, found by calculating 1 * 2 = 2, and then appending '25'. This reinforces the importance of recognizing number patterns and applies their understanding practically.
Examples & Analogies
Using patterns can be likened to a chef who always follows specific recipes. Just as they know that a pinch of salt elevates a dish, recognizing that any number ending in 5 will have a predictable square can simplify calculating large scaled recipes in mathematics and everyday calculations.
--
Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Square Numbers: Numbers formed by multiplying an integer by itself.
Patterns in Squares: Specific rules applicable to numbers ending in 5.
Expansion Formula: Expanding (10a + 5)² gives a clear methodology for finding squares.
Examples
Memory Aids
Interactive tools to help you remember key concepts