Grade 8 - Unit 1: Square Roots & Pythagorean Theorem

Learning Objectives

By the end of this unit, students should be able to:

• Determine and identify the square of a number.
• List perfect squares between 1 and 144.
• Show that a number is a perfect square using symbols, diagrams, prime factorization, or by listing factors.
• Use terms: base, power, and exponent.
• Relate the area of a square to perfect squares and square roots.
• Determine and estimate the square root of a number both with and without a calculator.
• Identify whole numbers with square roots between two numbers.
• Explain and apply the Pythagorean Theorem.
• Solve word problems involving right triangles.

Organizational Tips

• Keep all hand-ins in your Doulas until the end of the unit.
• Handouts such as notes, examples, and reference pages should be placed in your binder and labeled with the date.

Area

• Definition: Amount of space a two-dimensional object occupies.
• Formula for area of a rectangle or square: (A = \text{base} \times \text{height}) or (\text{Area} = \text{length} \times \text{width}).

Squares

• Squares have equal side lengths, making the calculation of area straightforward.
• Perfect squares: Numbers like 1, 4, and 9 are perfect squares.
• Exercise: Determine whether given numbers are perfect squares.

Identifying Perfect Squares

Four methods to determine a perfect square:

1. Draw the square - Visual representation.
2. Division sentence - Quotient equals the divisor (e.g., (36 \div 6 = 6)).
3. List factors - A square number has an odd number of factors.
4. Prime Factorization - All factors are prime.

Example Exercises:

• Find factors and check if numbers like 49 and 14 are perfect squares.
• Prime factorization exercises: Complete for numbers such as 24, 81, 36, and 400.

Square vs. Square Root

• Square: Multiplying a number by itself.
• Square Root: A number which, when multiplied by itself, gives the original number.

Exercises:

1. Square numbers: 9, 3, 1, 23, 16.
2. Find square roots: 9, 64, 49, 1, 484.

Estimating Square Roots

• Steps to estimate square roots:
  1. List first few perfect squares.
  2. Determine which two perfect squares the number is between.
  3. Estimate a decimal value between these square roots.
• Exercise: Estimate square roots for numbers such as 14, 55, 100, 37, 62, 136.

Pythagorean Theorem

• Statement: In a right triangle, the sum of the squares on the legs equals the square of the hypotenuse.
• Formula: (a^2 + b^2 = c^2).
• Applications:
  • Find the hypotenuse when the legs are known.
  • Determine if three numbers form a right triangle.
  • Use to solve real-world problems like determining distances.

Practice Problems:

• Determine hypotenuse or legs of triangles when given partial information.
• Solve word problems using the theorem.

Practice Makes Perfect

• Complete designated exercises from textbook pages to reinforce understanding.
  • Pages 8-9: Questions #1, 2, 6, 7, 10, 15
  • Pages 13-14: Questions #2, 4, 7, 8, 9, 10, 11
  • Page 18: Questions #1, 2, 4, 5, 13, 14
  • Page 23: Questions #1-9
  • Pages 29-31: Questions #1, 2, 5, 7, 9, 10
  • Page 35: All questions
  • Chapter Self Test and Review

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These notes summarize the key points and exercises from the Unit 1 on Square Roots and the Pythagorean Theorem, providing a comprehensive overview and study aid.