5.4.1 - Other patterns in squares
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Understanding Squares Ending in 5
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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!
Applying the Formula
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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.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
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
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.
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Identifying a Pattern in Squares
Chapter 1 of 2
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Chapter Content
Consider 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!
Finding Squares of Numbers with 5 in the Unit's Place
Chapter 2 of 2
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Chapter Content
TRY 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
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Square Numbers: Numbers formed by multiplying an integer by itself.
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Patterns in Squares: Specific rules applicable to numbers ending in 5.
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Expansion Formula: Expanding (10a + 5)² gives a clear methodology for finding squares.
Examples & Applications
Example of square of 25: 25² = 625 using the formula.
Calculating 35² = 1225 through the unit digit pattern.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
For numbers that end in five, the pattern will come alive, squares always end in twenty-five!
Stories
Once upon a time, a mathematician discovered that any time they squared a number ending in 5, they always ended with the beautiful 25, making number games much more fun!
Memory Tools
Remember '5+5=10' in the formula for squares with 5: (10a + 5)².
Acronyms
SQUAR
Squares' Quality Uniform Always Reveal (the last two digits
Flash Cards
Glossary
- Square Numbers
Numbers that can be expressed as the square of an integer.
- Unit Digit
The digit in the one's place of a number, which affects the properties of squares.
- Formula
A mathematical relationship or rule expressed in symbols.
Reference links
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