Math Notes Unit 1
● Definition: A perfect square is a number that can be expressed as the product of two
equal factors. For example, a square with an area of 400 m² has a side length of 20 m
because 20 * 20 = 400.
● Square Roots: The side length of a square is the square root of its area. Squaring a
number and taking its square root are inverse operations.
● Fractions and Decimals:
○ A fraction in its simplest form is a perfect square if its numerator and denominator
can each be written as a product of two equal factors. For example, 8/18
simplifies to 4/9, which is a perfect square because 4 = 2x2 and 9 = 3x3.
○ A decimal is a perfect square if it can be written as a fraction that is a perfect
square. The square root of a perfect square decimal is a terminating or repeating
decimal. For example, 6.25 is a perfect square because it can be written as the
fraction 625/100, which simplifies to 25/4. The square root is 2.5.
Non-Perfect Squares
● Definition: A number that cannot be written as a product of two equal fractions is a
non-perfect square.
● Estimating Square Roots: To approximate the square root of a non-perfect square, you
can use benchmarks, or the perfect squares closest to the number. For example, to
estimate the square root of 7.5, you can use the perfect squares 4 and 9, as 7.5 is
between them. Since 7.5 is closer to 9, its square root will be closer to 3 than to 2. *
Pythagorean Theorem: This theorem can be used to find unknown lengths in a right
triangle. For example, if a right triangle has sides of 6.5 m and 1.5 m, the hypotenuse
(the length of a ramp) can be calculated using the formula a² + b² = c², which gives a
length of approximately 6.7 m
Square Roots
● Terminating decimal ends after a certain number of decimal places. example 0.5,
0.28, 0.265
● Repeating decimal has a repeating pattern of digits in the decimal example 0.333
or 0.454545 or 0.3, 0.45
● A non-terminating and non repeating decimal is like pi 3.1459 and so on
● The square root of a perfect square is either a turning decimal or a repeating
decimal for example is 1.96 a perfect square yes because the square root of 1.96
is 4.1 bracket terminating decimal bracket and 1.4 * 1.4 = 1.96 is 3.5 a perfect
square? no, because the square root of 3.5 equals 1.8708286
● Definition: A perfect square is a number that can be expressed as the product of two
equal factors. For example, a square with an area of 400 m² has a side length of 20 m
because 20 * 20 = 400.
● Square Roots: The side length of a square is the square root of its area. Squaring a
number and taking its square root are inverse operations.
● Fractions and Decimals:
○ A fraction in its simplest form is a perfect square if its numerator and denominator
can each be written as a product of two equal factors. For example, 8/18
simplifies to 4/9, which is a perfect square because 4 = 2x2 and 9 = 3x3.
○ A decimal is a perfect square if it can be written as a fraction that is a perfect
square. The square root of a perfect square decimal is a terminating or repeating
decimal. For example, 6.25 is a perfect square because it can be written as the
fraction 625/100, which simplifies to 25/4. The square root is 2.5.
Non-Perfect Squares
● Definition: A number that cannot be written as a product of two equal fractions is a
non-perfect square.
● Estimating Square Roots: To approximate the square root of a non-perfect square, you
can use benchmarks, or the perfect squares closest to the number. For example, to
estimate the square root of 7.5, you can use the perfect squares 4 and 9, as 7.5 is
between them. Since 7.5 is closer to 9, its square root will be closer to 3 than to 2. *
Pythagorean Theorem: This theorem can be used to find unknown lengths in a right
triangle. For example, if a right triangle has sides of 6.5 m and 1.5 m, the hypotenuse
(the length of a ramp) can be calculated using the formula a² + b² = c², which gives a
length of approximately 6.7 m
Square Roots
● Terminating decimal ends after a certain number of decimal places. example 0.5,
0.28, 0.265
● Repeating decimal has a repeating pattern of digits in the decimal example 0.333
or 0.454545 or 0.3, 0.45
● A non-terminating and non repeating decimal is like pi 3.1459 and so on
● The square root of a perfect square is either a turning decimal or a repeating
decimal for example is 1.96 a perfect square yes because the square root of 1.96
is 4.1 bracket terminating decimal bracket and 1.4 * 1.4 = 1.96 is 3.5 a perfect
square? no, because the square root of 3.5 equals 1.8708286
