Unit 1 Real Numbers - · PDF fileIrrational ... Strategies for Comparing and Ordering Rational Numbers In your groups, list at least 4 strategies for comparing and ordering rational

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8th Grade Math

Unit 1

Real Numbers

Name

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3

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Name Period

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Name Period

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Name Period

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Name Period

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Name Period

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Name Period

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In your group, write down as many different types of numbers as you can …. How many types did you come up with? How are these numbers related? When and why do we see different types of numbers?

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Unit 1

Main ideaMain idea

Real NumbersThe FRAME RoutineKey Topic

is about…

So What? (What’s important to understand about this?)

Essential details Essential details Essential details

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The Real Numbers: How are they related?

Type of Number

Definition Examples

Natural

Whole

Integers

Non-Integer Rational

Irrational

Real

“Zero is neither _____________________ nor ___________________”

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Another View of the Real Number System

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Concept Review

Real Numbers: Any number with a position on a number line is a real number.

Rational Numbers: Any number that can be written in fraction form as the ratio of

two _______________ is a _____________ number. This includes ____________,

______________________, ___________ and _______________ but we cannot

have ________________ as a denominator.

An integer can be written as a fraction simply by giving it a denominator of

________, so any integer is a rational number.

; ;

A terminating decimal can be written as a fraction simply by writing it the

way you say it: 3.75 = three and seventy-five hundredths = , then adding if

needed to produce a fraction: .

So, any terminating decimal is a __________________. Terminates also means

that if you convert the fraction to a decimal by dividing the number by the

denominator you will eventually get a remainder of ________________ when

doing the division.

A repeating decimal can be written as a fraction using algebraic methods, so

any repeating (in a pattern) decimal is a ___________________.

How do we convert a repeating decimal into a fraction?

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Classifying Real Numbers

For each number check all the categories in which the number belongs to complete the table. Two numbers have been included already has examples.

Number Natural Whole Integer Rational Irrational

1 X X X X

𝜋 X

0.46 1

3

64.1%

√3

-5.1

0

-3.2

√16

0.727272

17.2

142%

√17

-65%

-0.98

1%

-5

9 1

4

21.76%

1. Identify the least and greatest numbers in the list above.

2. “Zero is greater than all ______________ numbers”

3. “The number ___________ is a whole number but not a natural number”

4. “Rational numbers can be written as the ratio of two _________”

5. “The decimal form of rational numbers either ____________ or

______________”

6. “Zero is neither _________ nor ____________”

7. A real number is either _________________ or ____________________

8. ______________ cannot be the denominator of a rational number

9. The number √31 is an example of an _________________________

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Comparing and Ordering Rational Numbers

Everyone gets a card with a rational number on it. Create a ‘human number line’ by

lining up in correct ascending order. Ascending Order means from ________________

to _____________________.

Be prepared to:

1) explain what type of number in on your card (e.g. repeating decimal, terminating

decimal, fraction, etc.)

2) explain how you know your number is correctly placed on the number line.

Strategies for Comparing and Ordering Rational Numbers In your groups, list at least 4 strategies for comparing and ordering rational numbers 1. 2. 3. 4.

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Converting Between Different Types of Rational Numbers

Type of Conversion Process or Rule 2 examples

Fraction to Decimal

Fraction to Percent

Decimal to Fraction

Decimal to Percent

Percent to Fraction

Percent to Decimal

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Relating Rational Numbers

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Rational Number Practice

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Ordering Rational Numbers Group Work: Place the letter of each number in the table on the number line below (estimate).

Number Letter Number Letter

1

4

A −2

3

H

1.55 B √25 I

7.5 C 600% J

√81 D 3 K

21

10

E -8.25 L

-3.5 F −100

10

M

−√36 G -450% N

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Rolling Rational Numbers In pairs:

1) Roll two dice and write down the two numbers 2) Pick a mathematical operation and perform it on the two numbers 3) Complete the remaining columns of the table 4) The first 2 rows have been filled in as examples

Be creative with the numbers and operations, as well as the numbers that are greater than and less than the resulting number!

Number Number Operation Result Written

as a

Fraction

Written

as a

decimal

Written

as a

Percent

A

number

that is

greater

than

the

result

A

number

that is

less

than

the

result

3 4 Addition 7 (3+4) 7

1

7.00 700% 10 1

2 5 Division 2

divided

by 5

2

5

0.40 40% 1 0

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More Rational Number Practice

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#11 and #12

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Irrational Numbers What is an irrational number? In your groups, list 5 irrational numbers, then have someone from your group write These down on the board. As a group, we will put these in ‘ascending order’. How can we estimate the value of an irrational number?

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Estimating the value of imperfect square roots

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Irrational Number Practice

Complete the table below

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Irrational Number Practice

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Squares and Square Roots

Chart of Perfect Square Roots

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Rational and Irrational Number Practice

Skip #11

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Scientific Notation

What is scientific notation? What is a ‘base’ and an ‘exponent’ (powers)? In your groups, write down 5 examples of an ‘expression’ using different bases and exponents

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Unit 1

Main ideaMain ideaMain idea

Scientific NotationThe FRAME RoutineKey Topic

is about…

So What? (What’s important to understand about this?)

Essential details Essential details Essential details

.

The correct form of scientific notation is: 𝒂 ∙ 𝟏𝟎𝒏:

‘a’ must be greater than or equal to 1 but less than 10

‘a’ must be have exactly one digit to the left of the decimal point

‘a’ can have zero to many digits to the right of the decimal point

The base is ALWAYS ‘10’

‘n’ (the exponent or power) is positive for numbers greater than 1 and

negative for numbers smaller than 1 but more than zero.

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Scientific Notation Scientific Notation is directly related to place value and the ‘powers’ of 10.

Scientific Notation and Place Value

Number Place Value Scientific Notation

0.000001 Millionths 𝟏𝒙𝟏𝟎−𝟔

0.00001 Hundred Thousandths 𝟏𝒙𝟏𝟎−𝟓

0.0001 Ten Thousandths 𝟏𝒙𝟏𝟎−𝟒

0.001 Thousandths 𝟏𝒙𝟏𝟎−𝟑

0.01 Hundredths 𝟏𝒙𝟏𝟎−𝟐

0.1 Tenths 𝟏𝒙𝟏𝟎−𝟏

1 Ones 𝟏𝒙𝟏𝟎𝟎

10 Tens 𝟏𝒙𝟏𝟎𝟏

100 Hundreds 𝟏𝒙𝟏𝟎𝟐

1,000 Thousands 𝟏𝒙𝟏𝟎𝟑

10,000 Ten Thousands 𝟏𝒙𝟏𝟎𝟒

100,000 Hundred Thousands 𝟏𝒙𝟏𝟎𝟓

1,000,000 Millions 𝟏𝒙𝟏𝟎𝟔

10,000,000 Ten Millions 𝟏𝒙𝟏𝟎𝟕

100,000,000 Hundred Millions 𝟏𝒙𝟏𝟎𝟖

1,000,000,000 Billions 𝟏𝒙𝟏𝟎𝟗

In your groups discuss:

1) Any patterns you notice. 2) The relationship between number of zeros in the number and the exponent

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The Powers of 10

In your group complete the following exercise:

1) Pick a number that is greater than 150 and less than 950 that has 4 digits to the right of

the decimal (not zeros!).

2) Use that number to complete the table below

Number Multiplied by Product

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

10,000

In your groups, discuss what patterns you noticed in this table.

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Scientific Notation Examples

• 𝟑 ∙ 𝟏𝟎𝟓 = 300,000

• 𝟕. 𝟏𝟐 ∙ 𝟏𝟎−𝟓 = 0.0000712

• 𝟓. 𝟎𝟓 ∙ 𝟏𝟎𝟑 = 5,050

• 𝟓. 𝟎𝟎𝟓 ∙ 𝟏𝟎𝟓 = 500,500

• 𝟑. 𝟐𝟑 ∙ 𝟏𝟎𝟒 = 𝟑𝟐, 𝟑𝟎𝟎

• 𝟑. 𝟐𝟑 ∙ 𝟏𝟎−𝟓 = 𝟎. 𝟎𝟎𝟎𝟎𝟑𝟐𝟑

In your groups, come up with 4 examples of different numbers written in scientific

notation and have one person from the group write these on the white board and

1. Make sure to list them in ascending order.

2. Focus on getting the FORM of the number correct!

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Scientific Notation Practice

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Applications of Scientific Notation

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More Scientific Notation Practice

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Challenge: Using Scientific Notation to Solve Problems

1. Light travels 186,000 miles per second. How far does it travel in one year (called a light

year)?

2. The speed of light is 1.86 ∙ 105miles per second. The distance from the Earth to the Sun

is 9.3 ∙ 107miles. How long does it take the light from the Sun to reach the Earth?

3. The volume of the Sun is 1.4 𝑥 1018 cubic kilometers. The volume of the Earth is

1.1 𝑥 1012 cubic kilometers. How many Earths can fit inside the Sun?

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