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**Homework: Cumulative Review**

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Squares & Square Roots

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**Rational and Irrational Numbers Essential Question**

How do I distinguish between rational and irrational numbers?

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Vocabulary real number irrational number

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The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers. Irrational numbers Rational numbers Real Numbers Integers Whole numbers

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Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. 4 5 2 3 = 3.8 = 0.6 1.44 = 1.2

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Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational. A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution!

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**Reals Make a Venn Diagram that displays the following sets of numbers:**

Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Reals Rationals -2.65 Integers -3 -19 Wholes Irrationals Naturals 1, 2, 3...

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**Additional Example 1: Classifying Real Numbers**

Write all classifications that apply to each number. A. 5 5 is a whole number that is not a perfect square. irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 2 = = 2 4 2 16 C. whole, integer, rational, real

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**Write all classifications that apply to each number.**

Check It Out! Example 1 Write all classifications that apply to each number. A. 9 9 = 3 whole, integer, rational, real B. –35.9 –35.9 is a terminating decimal. rational, real 81 3 = = 3 9 3 81 C. whole, integer, rational, real

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A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

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**Additional Example 2: Determining the Classification of All Numbers**

State if each number is rational, irrational, or not a real number. A. 21 irrational 3 3 = 0 B. rational

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**Additional Example 2: Determining the Classification of All Numbers**

State if each number is rational, irrational, or not a real number. 4 0 C. not a real number

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**State if each number is rational, irrational, or not a real number.**

Check It Out! Example 2 State if each number is rational, irrational, or not a real number. A. 23 23 is a whole number that is not a perfect square. irrational 9 B. undefined, so not a real number

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**State if each number is rational, irrational, or not a real number.**

Check It Out! Example 2 State if each number is rational, irrational, or not a real number. 64 81 8 9 = 64 81 C. rational

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**Square Number Also called a “perfect square”**

A number that is the square of a whole number Can be represented by arranging objects in a square.

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Square Numbers

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Square Numbers 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16

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**Square Numbers 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 Activity:**

Calculate the perfect squares up to 152…

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**Square Numbers 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 5 x 5 = 25**

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**Activity: Identify the following numbers as perfect squares or not.**

16 15 146 300 324 729

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**Activity: Identify the following numbers as perfect squares or not.**

16 = 4 x 4 15 146 300 324 = 18 x 18 729 = 27 x 27

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Squares & Square Roots Square Root

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Square Numbers One property of a perfect square is that it can be represented by a square array. Each small square in the array shown has a side length of 1cm. The large square has a side length of 4 cm. 4cm 4cm 16 cm2

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**Square Numbers The large square has an area of 4cm x 4cm = 16 cm2.**

The number 4 is called the square root of 16. We write: 4 = 4cm 4cm 16 cm2

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Square Root A number which, when multiplied by itself, results in another number. Ex: 5 is the square root of 25. 5 = 25

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**Finding Square Roots 4 x 9 = 4 x 9 36 = 2 x 3 6 = 6**

We can use the following strategy to find a square root of a large number. 4 x 9 = x 9 = x 3 =

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**Finding Square Roots 4 x 9 = 4 9 36 = 2 x 3 6 = 6**

= We can factor large perfect squares into smaller perfect squares to simplify.

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**Finding Square Roots 256 = 4 x 64 = 2 x 8 = 16**

Activity: Find the square root of 256 256 = x 64 = 2 x 8 = 16

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**Estimating Square Root**

Squares & Square Roots Estimating Square Root

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**Estimating Square Roots**

25 = ?

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**Estimating Square Roots**

25 = 5

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**Estimating Square Roots**

49 = ?

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**Estimating Square Roots**

49 = 7

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**Estimating Square Roots**

27 = ?

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**Estimating Square Roots**

27 = ? Since 27 is not a perfect square, we have to use another method to calculate it’s square root.

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**Estimating Square Roots**

Not all numbers are perfect squares. Not every number has an Integer for a square root. We have to estimate square roots for numbers between perfect squares.

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**Estimating Square Roots**

To calculate the square root of a non-perfect square 1. Place the values of the adjacent perfect squares on a number line. 2. Interpolate between the points to estimate to the nearest tenth.

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**Estimating Square Roots**

Example: What are the perfect squares on each side of 27? 25 30 35 36

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**Estimating Square Roots**

Example: half 5 6 25 30 35 36 27 Estimate = 5.2

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**Estimating Square Roots**

Example: Estimate: = 5.2 Check: (5.2) (5.2) = 27.04

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CLASSWORK PAGE

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