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Student Learning Goal Chart

Lesson Reflections

3-10 LAST ONE!

- Learn to write rational numbers in equivalent forms (3.1)
- Learn to add and subtract decimals and rational numbers with like denominators (3.2)
- Learn to add and subtract fractions with unlike denominators (3.5)
- Learn to multiply fractions, decimals, and mixed numbers (3.3)
- Learn to divide fractions and decimals (3.4)
- Learn to solve equations with rational numbers (3.6)
- Learn to solve inequalities with rational numbers (3-7)
- Learn to find square roots (3-8)
- Learn to estimate square roots to a given number of decimal places and solve problems using square roots (3-9)
- Learn to determine if a number is rational or irrational (3-10)

Today’s Learning Goal Assignment

Learn to determine if a number is rational or irrational.

Page 169

#1-40 all

3-10

The Real Numbers

Warm Up

Problem of the Day

Lesson Presentation

Pre-Algebra

3-10

The Real Numbers

1. 119

2. – 15

3. 2

4. – 123

Pre-Algebra

Warm Up

Each square root is between two integers. Name the two integers.

Use a calculator to find each value. Round to the nearest tenth.

10 and 11

–4 and –3

1.4

–11.1

Problem of the Day

The circumference of a circle is approximately 3.14 times its diameter. A circular path 1 meter wide has an inner diameter of 100 meters. How much farther is it around the outer edge of the path than the inner edge?

6.28 m

Today’s Learning Goal Assignment

Learn to determine if a number is rational or irrational.

Vocabulary

irrational number

real number

Density Property

Biologists classify animals based on shared characteristics. The gray lesser mouse lemur is an animal, a mammal, a primate, and a lemur.

You already know that some numbers can be classified as whole numbers,integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number.

Animals

Mammals

Primates

Lemurs

Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat.

4 5

23

3 = 3.8

= 0.6

1.44 = 1.2

Helpful Hint

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

2 ≈1.4142135623730950488016…

Irrational numberscan only be written as decimals that do not terminate or repeat. If a whole number is not a perfect square, then its square root is an irrational number.

Real Numbers

Rational numbers

Irrational numbers

Integers

Whole

numbers

The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

16 2

4 2

= = 2

Additional Examples 1: Classifying Real Numbers

Write all names that apply to each number.

A.

5 is a whole number that is not a perfect square.

5

irrational, real

B.

–12.75

–12.75 is a terminating decimal.

rational, real

16 2

C.

whole, integer, rational, real

9 = 3

81 3

9 3

= = 3

Try This: Example 1

Write all names that apply to each number.

9

A.

whole, integer, rational, real

–35.9

–35.9 is a terminating decimal.

B.

rational, real

81 3

C.

whole, integer, rational, real

0 3

= 0

Additional Examples 2: Determining the Classification of All Numbers

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

A.

15 is a whole number that is not a perfect square.

15

irrational

0 3

B.

rational

2 3

2 3

4 9

=

Additional Examples 2: Determining the Classification of All Numbers

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

C.

–9

not a real number

4 9

D.

rational

Try This: Examples 2

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

A.

23 is a whole number that is not a perfect square.

23

irrational

9 0

B.

not a number, so not a real number

8 9

8 9

64 81

=

Try This: Examples 2

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

C.

–7

not a real number

64 81

D.

rational

The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3.

Find a real number between 3 and 3 .

2 5

3 5

2 5

1 2

3 5

2 5

3 5

5 5

1 2

3 + 3 ÷ 2

= 6 ÷ 2

= 7 ÷ 2 = 3

1 5

2 5

3

3

3

1 2

3

4 5

3 5

3

3

4

A real number between 3 and 3is 3 .

Additional Examples 3: Applying the Density Property of Real Numbers

There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.

Find a real number between 4 and 4 .

1 2

3 7

3 7

4 7

4 7

7 7

1 2

3 7

4 7

= 9 ÷ 2 = 4

4 + 4 ÷ 2

= 8 ÷ 2

5 7

1 7

6 7

2 7

3 7

4 7

4

4

4

4

4

4

1 2

4

A real number between 4 and 4 is 4 .

Try This: Example 3

There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.

3 4

3 8

Find a real number between –2 and –2 .

5 8

Possible answer –2 .

Lesson Quiz

Write all names that apply to each number.

16 2

2. –

1.

2

real, irrational

real, integer, rational

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

25 0

3.

4.

4 • 9

rational

not a real number

5.