CLASS 8 WORKSHEET
CH-1 squares and square roots
Worksheet: Exploring Squares and Square Roots
Instructions: Read each question carefully and use the information from the provided text to answer. Show your work where applicable.
Part A: Understanding Squares
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Define a "square number" or "perfect square" in your own words.
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The area of a square is found by multiplying its side length by itself.
- If a square has a side length of 5 units, what is its area? Express this using square notation. Area = _______________________
- What notation do we use for any number 'n' multiplied by itself? n × n = __________
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List all the perfect squares between 1 and 100.
Part B: Properties of Perfect Squares
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What are the only possible digits that a perfect square can end with (its units place)?
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Which of the following numbers are not perfect squares? Explain your reasoning based on their unit digits.
- (i) 2032
- (ii) 1027
- (iii) 345
- (iv) 576
- (v) 2048
- (vi) 1089
Not Perfect Squares: ____________________________________________________ Reasoning:____________________________________________________________________
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Observe the pattern for the number of zeros at the end of a number and its square.
- If a number ends with three zeros (e.g., 1000), how many zeros will its square have at the end?
- What can you conclude about the number of zeros at the end of any perfect square?
- If a number ends with three zeros (e.g., 1000), how many zeros will its square have at the end?
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Complete the following pattern showing the sum of consecutive odd numbers resulting in square numbers:
- 1 = 1 = 1²
- 1 + 3 = 4 = 2²
- 1 + 3 + 5 = 9 = 3²
- 1 + 3 + 5 + 7 = __________ = __________
- 1 + 3 + 5 + 7 + 9 = __________ = __________
- 1 + 3 + 5 + 7 + 9 + 11 = __________ = __________
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Given that 35² = 1225, use the pattern of adding consecutive odd numbers to find 36².
- What is the 36th odd number? (Hint: The nth odd number is 2n-1)
- Calculate 36²: 36² = 35² + (36th odd number) = _______________________
- What is the 36th odd number? (Hint: The nth odd number is 2n-1)
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How many numbers lie between the squares of 16 and 17?
Part C: Finding Square Roots
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Define "square root" and state how it is denoted.
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Every perfect square has two integer square roots. What are they for the number 64?
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Determine if 1156 is a perfect square using prime factorization. If it is, find its square root. (Show your prime factorization steps)
Is 1156 a perfect square? __________ If yes, √1156 = __________
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Find the square root of 441 m² (area of a square) to determine the length of its side. Length of side = _______________________
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Estimate the square root of 250 without calculating it exactly. Explain your reasoning by identifying the closest perfect squares.
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Akhil has a square piece of cloth with an area of 125 cm². He wants to cut out the largest possible square handkerchief with an integer side length. What is the maximum side length he can cut?
Part D: Challenge Question
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Recall the puzzle given by Queen Ratnamanjuri involving 100 lockers.
- Person 1 opens every locker.
- Person 2 toggles every 2nd locker.
- Person 3 toggles every 3rd locker, and so on, until all 100 people have taken their turn.
Khoisnam immediately knew which lockers would remain open at the end. How did he know? (Hint: Think about how many times each locker is toggled and what type of numbers have an odd number of factors).
Which specific locker numbers (up to 100) would remain open?
Answer Key: Exploring Squares and Square Roots
Part A: Understanding Squares
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Define a "square number" or "perfect square" in your own words. A square number, or perfect square, is a number obtained by multiplying a number by itself. It can also be described as the area of a square figure. Example definition: A perfect square is a number that results from multiplying an integer by itself. For instance, 4 is a perfect square because 2 × 2 = 4.
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The area of a square is found by multiplying its side length by itself.
- If a square has a side length of 5 units, what is its area? Express this using square notation. Area = 5 × 5 = 25 square units, or 5² = 25.
- What notation do we use for any number 'n' multiplied by itself? n × n = n².
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List all the perfect squares between 1 and 100. The perfect squares between 1 and 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Part B: Properties of Perfect Squares
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What are the only possible digits that a perfect square can end with (its units place)? Perfect squares can only end with the digits 0, 1, 4, 5, 6, or 9.
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Which of the following numbers are not perfect squares? Explain your reasoning based on their unit digits.
- (i) 2032
- (ii) 1027
- (iii) 345
- (iv) 576
- (v) 2048
- (vi) 1089
Not Perfect Squares: (i) 2032, (ii) 1027, (v) 2048. Reasoning: If a number ends with 2, 3, 7, or 8, then it is definitely not a square.
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Observe the pattern for the number of zeros at the end of a number and its square.
- If a number ends with three zeros (e.g., 1000), how many zeros will its square have at the end? Six zeros (e.g., 1000² = 1,000,000).
- What can you conclude about the number of zeros at the end of any perfect square? Perfect squares can only have an even number of zeros at the end.
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Complete the following pattern showing the sum of consecutive odd numbers resulting in square numbers:
- 1 = 1 = 1²
- 1 + 3 = 4 = 2²
- 1 + 3 + 5 = 9 = 3²
- 1 + 3 + 5 + 7 = 16 = 4²
- 1 + 3 + 5 + 7 + 9 = 25 = 5²
- 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²
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Given that 35² = 1225, use the pattern of adding consecutive odd numbers to find 36².
- What is the 36th odd number? (Hint: The nth odd number is 2n-1) The nth odd number is 2n–1. So, the 36th odd number is (2 × 36) - 1 = 71.
- Calculate 36²: 36² = 35² + (36th odd number) = 1225 + 71 = 1296.
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How many numbers lie between the squares of 16 and 17? This information is not directly provided in the source material in an explicit statement. However, based on the pattern
n²and(n+1)², the numbers between them are(n+1)² - n² - 1 = (n² + 2n + 1) - n² - 1 = 2n. So, between 16² and 17² (where n=16), there are 2 × 16 = 32 numbers [16 (Implied by the question type)].
Part C: Finding Square Roots
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Define "square root" and state how it is denoted. If y = x², then x is the square root of y. It is the inverse operation of squaring a number. The square root of a number is denoted by the symbol √.
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Every perfect square has two integer square roots. What are they for the number 64? The square roots of 64 are +8 and –8.
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Determine if 1156 is a perfect square using prime factorization. If it is, find its square root. (Show your prime factorization steps) Prime factorization of 1156: 1156 = 2 × 578 578 = 2 × 289 289 = 17 × 17 So, 1156 = 2 × 2 × 17 × 17.
- Can factors be grouped into pairs? Yes, (2 × 2) × (17 × 17). This can be grouped as (2 × 17) × (2 × 17).
- Is 1156 a perfect square? Yes.
- If yes, √1156 = 2 × 17 = 34.
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Find the square root of 441 m² (area of a square) to determine the length of its side. Length of side = √441 = 21 m.
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Estimate the square root of 250 without calculating it exactly. Explain your reasoning by identifying the closest perfect squares. We know that 15² = 225 and 16² = 256. Therefore, the square root of 250 is between 15 and 16. Since 256 is much closer to 250 than 225, the square root of 250 is approximately 16 (but less than 16).
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Akhil has a square piece of cloth with an area of 125 cm². He wants to cut out the largest possible square handkerchief with an integer side length. What is the maximum side length he can cut? The nearest perfect square less than 125 is 121 (which is 11²). Therefore, the maximum side length he can cut with an integer side length is 11 cm.
Part D: Challenge Question
-
Recall the puzzle given by Queen Ratnamanjuri involving 100 lockers.
- Person 1 opens every locker.
- Person 2 toggles every 2nd locker.
- Person 3 toggles every 3rd locker, and so on, until all 100 people have taken their turn.
Khoisnam immediately knew which lockers would remain open at the end. How did he know? (Hint: Think about how many times each locker is toggled and what type of numbers have an odd number of factors). Khoisnam knew that a locker would remain open only if it was toggled an odd number of times. The number of times a locker is toggled is equal to the number of its factors. Most numbers have an even number of factors because factors usually come in pairs (e.g., for 6, factors are 1&6, 2&3). However, numbers that are perfect squares have an odd number of factors because one factor is paired with itself (e.g., for 36, 6 × 6). Thus, every locker whose number is a perfect square will remain open.
Which specific locker numbers (up to 100) would remain open? The locker numbers that remain open are the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
