8 MTXESE052888 U1M01L3

1.3 LESSON Ordering Real Numbers

? ESSENTIAL QUESTION How do you order a set of real numbers?

Number and operations-- 8.2.D Order a set of real numbers arising from mathematical and real-world contexts. Also 8.2.B

Comparing Irrational Numbers

Between any two real numbers is another real number. To compare and order real numbers, you can approximate irrational numbers as decimals.

EXAMPLE 1

_

_

Compare 3 + 5 3 + 5 . Write , or =.

STEP 1

_

First approximate 3.

_

_

3 is between 1 and 2, so 3 1.5.

_

Next approximate 5.

_

_

5 is between 2 and 3, so 5 2.5.

8.2.B

Use perfect squares to estimate square roots.

12 = 1 22 = 4 32 = 9

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My Notes

STEP 2

Then use your approximations to simplify the expressions.

_

3 + 5 is between 6 and 7

_

3 + 5 is between 5 and 6

_

_

So, 3 + 5 > 3 + 5

Reflect

_

_

1. If 7 + 5 is equal to 5 plus a number, what do you know about the

number? Why?

_

2. What are the closest two integers that 300 is between?

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YOUR TURN

Compare. Write , or =.

_

3. 2 + 4

_

2 + 4

_

4. 12 + 6

_

12 + 6

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Lesson 1.3 21

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My Notes

Ordering Real Numbers

You can compare and order real numbers and list them from least to greatest.

EXAMPLE 2

8.2.D

Order

_

22 ,

+

1,

and

4

_ 1

2

from least to greatest.

_

STEP 1 First approximate 22.

_

22 is between 4 and 5. Since you don't know where it fal_ ls

between 4 and 5, you need to find a better estimate for 22 so

you

can

compare

it

to

4

_ 1

2

.

_

To find a better estimate of 22, check the squares of numbers

close to 4.5.

4.42 = 19.36 4.52 = 20.25 4.62 = 21.16

_

_

22 is between 4.6 and 4.7, so 22 4.65.

4.72 = 22.09

An approximate value of is 3.14. So an approximate value of +1 is 4.14.

STEP 2

Plot

_

22 ,

+

1,

and

4

_ 1

2

on

a

number

line.

+ 1

4

1 2

22

4 4.2 4.4 4.6 4.8 5

Read the numbers from left to right to place them in order from

least to greatest.

From

least

to

greatest,

the

numbers

are

+

1,

4

_ 1

2

,

and

_

22 .

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22 Unit 1

YOUR TURN

Order the numbers from least to greatest. Then graph them on the

number line.

_

_

5. 5, 2.5, 3

Math Talk

Mathematical Processes

0 0.5 1 1.5 2 2.5 3 3.5 4

6.

2 ,

10,

_

75

If real numbers a, b, and c are in order from least to greatest, what is the order of their opposites from

least to greatest? Explain.

8 8.5 9 9.5 10 10.5 11 11.5 12

Ordering Real Numbers in a Real-World Context

Calculations and estimations in the real world may differ. It can be important to know not only which are the most accurate but which give the greatest or least values, depending upon the context.

EXAMPLE 3

8.2.D

Four people have found the distance in kilometers across a canyon using different methods. Their results are given in the table. Order the distances from greatest to least.

STEP 1 STEP 2

Distance Across Quarry Canyon (km)

Juana

_

28

Lee Ann

_2_3 4

Ryne

_

5.5

_

Approximate 28.

_

_

28 is between 5.2 and 5.3, so 28 5.25.

Jackson 5_12

_2_3 4

=

5.75

_

_

5.5 is 5.555..., so 5.5 to the nearest hundredth is 5.56.

5

_ 1 2

=

5.5

Plot

_

28 ,

_24_3,

_

5.5,

and

5

_ 1 2

on

a

number

line.

28

5

1 2

5.5

23 4

5 5.2 5.4 5.6 5.8 6

From greatest to least, the distances are:

_

_

_2_3

4

km,

5.5

km,

5

_ 1

2

km,

28

km.

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YOUR TURN

7. Four people have found the distance in miles across a crater using different methods. Their results are given below.

Jonathan:

_13_0,

Elaine:

_

3.45,

Jos?:

3_12,

Lashonda:

_

10

Order the distances from greatest to least.

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Lesson 1.3 23

Guided Practice

Compare. Write , or =. (Example 1)

_

1. 3 + 2

_

3 + 3

_

2. 11 + 15

_

8 + 15

_

3. 6 + 5

_

6 + 5

_

4. 9 + 3

_

9 + 3

_

5. 17 - 3

_

-2 + 5

_

6. 10 - 8

_

12 -2

_

7. 7 + 2

_

10 - 1

_

8. 17 + 3

_

3 + 11

_

9. Order 3, 2, and 1.5 from least to greatest. Then graph them on the

number line. (Example 2)

_

_

3 is between

and

, so 3

.

3.14, so 2

.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

From least to greatest, the numbers are

.

10. Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least. (Example 3)

,

,

Leon

_

17 - 2

Forest Perimeter (km)

Mika

Jason

1

+

__ 2

_1_2_ 5

Ashley 2.5

?? ESSENTIAL QUESTION CHECK-IN

11. Explain how to order a set of real numbers.

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24 Unit 1

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