Squares and Square Roots Study Notes

Squares and Square Roots

Lesson Starter: Speed Squaring Tests

• In pairs, test each other's knowledge of square numbers.
• Ask 10 quick questions, such as ‘3 squared', '5 squared' etc.
• Have two turns each.
• Time how long it takes each of you to answer the 10 questions.
• Aim to be quicker on your second attempt.
• Write down the first 10 square numbers.
• Begin to memorize these important numbers.
• Time how quickly you can recall the first 10 square numbers without looking at a list of numbers.
• Challenge: Can you go under 5 seconds?

Key Ideas

• Any whole number multiplied by itself produces a square number.
  • For example: $5^2 = 5 \times 5 = 25$. Therefore, 25 is a square number.
• Square numbers are also known as perfect squares.
• The first 12 square numbers (not including 0) are:
  • Index form: $1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2, 11^2, 12^2$
  • Basic numeral: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
• All non-zero square numbers have an odd number of factors.
• The symbol for squaring is $a^2$. The brackets are optional, but can be useful when simplifying more difficult expressions.

Building Understanding

• Consider a square of side length 6 cm. The area of this shape would be $6 \times 6 = 36 cm^2$. 36 is a square number.
• State the first 15 square numbers in index form and as basic numerals (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225).
• Confirming that 9 is a square number by drawing a 3x3 dot diagram.
  • Explain using dots, why 6 is not a square number: You can't arrange 6 dots into a perfect square.
  • Explain using dots, why 16 is a square number: You can arrange 16 dots into a perfect square with 4 dots on each side.
• Find:
  • a) $6^2 = 36$
  • b) 5 squared = $5^2 = 25$
  • c) $(11)^2 = 121$
  • d) 10 to the power of 2 = $10^2 = 100$
  • e) $7^2 = 49$
  • f) $12 \times 12 = 144$

Key Ideas (cont.)

• The square root of a given number is the 'non-negative' number that, when multiplied by itself, produces the given number.
• The symbol for square rooting is $\sqrt{a}$.
• Finding a square root of a number is the opposite of squaring a number.
• For example: $4^2 = 16$; hence, $\sqrt{16} = 4$.
• Read as: '4 squared equals 16, therefore, the square root of 16 equals 4.'
• Squaring and square rooting are 'opposite' operations.
  • For example: $(\sqrt{7})^2 = 7$ also $\sqrt{(7)^2} = 7$
• A list of common square roots is:
  • Square root form: $\sqrt{1}, \sqrt{4}, \sqrt{9}, \sqrt{16}, \sqrt{25}, \sqrt{36}, \sqrt{49}, \sqrt{64}, \sqrt{81}, \sqrt{100}, \sqrt{121}, \sqrt{144}$
  • Basic numeral: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Building Understanding (cont.)

• Find:
  • a) $\sqrt{25} = 5$
  • b) The square root of 16 = $\sqrt{16} = 4$
  • c) $\sqrt{100} = 10$
  • d) The side length of a square that has an area of 49 cm² = $\sqrt{49} = 7 cm$

Example 14 | Evaluating Squares and Square Roots

• Evaluate:
  • a) $6^2$
    • $6^2 = 36$
  • b) $\sqrt{64}$
    • $\sqrt{64} = 8$
  • c) $\sqrt{1600}$
    • $\sqrt{1600} = 40$

Example 15 | Locating square roots between positive integers

• State which consecutive whole numbers are either side of:
  • a) $\sqrt{43}$
    • Answer: 6 and 7 (since $6^2 = 36$ and $7^2 = 49$)
  • b) $\sqrt{130}$
    • Answer: 11 and 12 (since $11^2 = 121$ and $12^2 = 144$)

Example 16 | Evaluating expressions involving squares and square roots

• Evaluate:
  • a) $3^2 - \sqrt{9} + 1^2$
    • $3^2 - \sqrt{9} + 1^2 = 9 - 3 + 1 = 7$
  • b) $\sqrt{8^2 + 6^2}$
    • $\sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$

Exercises

• Exercise 2G
  • Q 2 acegik
  • Q 3 acegik
  • Q 5 aceg
  • Q 6, 7 - 9
  • Q 11, 12

For Mathematicians Only