thinking 1 rationally - kyrene school district › cms › lib › az01001083... · m1-74 • topic...

LESSON 1: Thinking Rationally • M1-71

LEARNING GOALS• Understand that counting numbers, fractions,

and decimals are rational numbers.• Identify properties of rational numbers.• Identify models for rational numbers.• Fluently compare and order rational numbers.

KEY TERMS• positive rational number• benchmark fraction

You have learned about whole numbers, fractions, and decimals. How can you compare these types of numbers?

WARM UPDetermine the fraction represented by the shaded part of each grid. If necessary, rewrite in lowest terms.

1. 2.

3. 4.

1ThinkingRationally Identifying and Ordering Rational Numbers

C01_SE_M01_T02_L01.indd 71C01_SE_M01_T02_L01.indd 71 4/7/17 4:09 PM4/7/17 4:09 PM

M1-72 • TOPIC 2: Positive Rational Numbers

You can group numbers in many different ways.

1. Cut out the cards at the end of this lesson. Sort the cards into different groups. You may sort the cards in any way you think is appropriate, but you must sort them into more than 1 group. Give each group of cards a title.

Explain how you sorted the numbers and diagrams on the cards, including why you gave each group its title.

2. Compare your groupings with your classmates’ groupings. Create a list of some of the different ways to group the numbers.

Identifying Positive Rational

Numbers

ACTIVIT Y

1.1

Getting Started

How Many Can You Name?

You have learned about many different types of numbers. List as many types of numbers as you can. Give an example of each number type.

Keep these cards when you're finished. You'll need them in a future lesson.


NOTES


a. Show why Danika’s reasoning is correct.

b. Identify other numbers or diagrams that belong in Danika’s group.

c. Explain why Josh’s reasoning is not correct.

d. Identify pairs of cards which show equal values. How many pairs can you find?

DanikaI grouped these numbers together because they all represent whole numbers.

8 _ 8 , 5 _ 5 , 10 __ 5 , 1, 0 0 _ 1

JoshI grouped these numbers together because they are all equal.

3 _ 5 , 3 _ 4 ,

3 _ 8

3. Vivianne grouped these cards together. What reason could she give for why she put these cards into the same group?

, 1 __ 5 , 0.2

4. Danika and Josh explained how they sorted the numbers.



Writing Positive

Rational Numbers

ACTIVIT Y

1.2

A positive rational number is a number that can be written in the form a __ b , where a and b are both whole numbers greater than 0.

1. Show that the decimals 0.6, 0.1, 0.2, and 0.325 are positive rational numbers.

2. Which numbers, if any, that you sorted are not positive rational numbers? Explain your answer.

WORKED EXAMPLE

Is 0.75 a rational number?

To write a decimal like 0.75 in the form a __ b , where a and b are both whole numbers and b is not equal to 0:

• Read the decimal using place value.

0.75 seventy-five hundredths

• Write the decimal as a fraction.

0.75 75 ____ 100

The fraction 75 ____ 100 is written in the form a __ b , where a is equal to 75 and b is equal to 100. The numbers 75 and 100 are both whole numbers greater than 0.

So, 0.75 is a rational number.

Any decimal greater

than 0 that has a

limited number of

nonzero digits after

the decimal point (like

0.5) or whose digits

repeat in a pattern

(like 0.3333 . . .) is

a positive rational

number.



Benchmark Fractions

Benchmark fractions are common fractions you can use to estimate the value of fractions.

Three common benchmark fractions are 0 __ 1 , 1 __ 2 , and 1 __ 1 .

1. Name the closest benchmark fraction for each fraction given.

a. 4 __ 9 b. 8 __ 9 c. 6 ____ 9

d. 5 ___ 67 e. 7 ___ 15 f. 7 ___ 12

g. 5 __ 6 h. 14 ___ 27 i. 12 ___ 13

j. 1 ___ 17 k. 5 ___ 11 l. 3 __ 7

2. Write the unknown numerator or denominator so that each fraction is close to but greater than 0.

a. ( )

_____ 12 b. ( )

_____ 27

c. 8 _____ ( ) d. 7 _____ ( )

0 11–2

ACTIVIT Y

1.3

A fraction is close to 0 when the numerator is very small compared to the denominator.

A fraction is close to 1 __ 2 when the numerator is about half the size of the denominator.

A fraction is close to 1 when the numerator is very close in size to the denominator.



3. Write the unknown numerator or denominator so that each fraction is close to but less than 1 __ 2 .

a. ( )

_____ 12 b. ( )

_____ 27

c. 8 _____ ( ) d. 7 _____ ( )

4. Write the unknown numerator or denominator so that each fraction is close to but less than 1.

a. ( )

_____ 12 b. ( )

_____ 27

c. 8 _____ ( ) d. 7 _____ ( )

5. Describe the relationship between a and b when the fraction a __ b is:

a. close to 0. b. close to 1 __ 2 . c. close to 1.

6. Compare each pair of fractions using benchmark fractions. Insert a . or , symbol to make the inequality true. Explain your reasoning.

a. 11 ___ 12 5 __ 9 b. 5 __ 9 5 __ 7

c. 7 ___ 13 5 ___ 11 d. 5 ___ 10 7 ___ 10

7. Compare the fractions in each pair. Think about how close the fractions are to 0, 1 __ 2 , or 1.

a. 5 __ 8 and 7 ___ 12 b. 14 ___ 15 and 7 __ 8 c. 1 __ 9 and 1 ___ 23

An inequality is a

statement that one

number is less than

or greater than

another number.



Ordering Rational NumbersACTIVIT Y

1.4

Felipe and Corinne ordered the rational numbers 0.8, 0.06, and 3 __ 5 from least to greatest using different strategies. Felipe used benchmark numbers, and Corinne used equivalent fractions.

1. Use Felipe’s strategy of benchmark numbers to order the rational numbers from least to greatest.

2. Use Corinne’s strategy of equivalent fractions to order the rational numbers from least to greatest.

3. Use any strategy to order the rational numbers 0.6, 3 __ 4 , and 5 __ 8 from least to greatest.

4. List the fractions in each set in ascending order.

a. 1 __ 8 , 1 ___ 11 , 1 __ 9 , 1 __ 4 , 1 __ 7 , 1 __ 5 b. 4 __ 5 , 4 ___ 10 , 4 ___ 12 , 4 __ 7

c. 3 __ 8 , 3 ___ 11 , 3 __ 9 , 3 __ 4 , 3 __ 7 , 3 __ 5

5. What do the fractions in each part of Question 4 have in common? Explain how you determined the order of the fractions in each.


NOTES


TALK the TALK

Close to Half

Consider the fractions shown.

5 __ 9 , 7 ___ 13 , 2 __ 7 , 10 ___ 11

1. Write the fractions in ascending order. Use what you know about benchmark fractions to determine the order. Explain your reasoning.



1 ___ 12

3 __ 5 5 __ 6 3 __ 4 1 __ 5

1 __ 3 3 __

8 1 __ 4 2 __

3 4 __ 5

5 __ 8 8 __

8 8 __ 4 5 __ 5 10 ___ 5

0.4 0.6 0.1 0.2 0.06

0.75 0.5 1 0 0 __ 1


✂


PracticeOrder the rational numbers in each group from least to greatest.

1. 0.09, 0.1, 2 ___ 25

2. 5 __ 6 , 5 __ 8 , 3 __ 2

3. 0.55, 3 __ 5 , 2 __ 3

4. 4.2, 3.10, 4 1 __ 8 , 3.01, 2.3, 2 4 __ 5 , 3.017

5. 6.84, 8 5 __ 7 , 6.34, 6 1 __ 4 , 8 3 ___ 10 , 8.15

6. 1.98, 0.23, 0, 1.89, 1 3 __ 5 , 1.02, 3 __ 2

7. 2.35, 2.54, 2.01

8. 9.3, 5 3 __ 5 , 9.90, 9 8 ___ 11 , 3.78, 3.9, 5 1 __ 6

9. 0.02, 0, 6.98, 2 1 ___ 16 , 2.2, 6.89, 2.01

Assignment

WriteDescribe a way to compare

two positive rational numbers

that are not written in the

same form.

RememberA positive rational number is a number that can be written in the

form a __ b , where a and b are both whole numbers, and b is not equal

to 0.

An inequality is a statement that one number is less than or

greater than another number.

StretchUse reasoning to compare the fractions. Do not use common denominators.

Explain your reasoning.

1. 13 ___ 3 17 ___ 4

2. 3 ___ 16 6 ___ 31

3. 7 ___ 11 9 ___ 13



Review1. In a video game, a character needs to shine a light through two spinning wheels that have holes in them.

The first wheel makes a complete rotation in 7 seconds. The second wheel makes a complete rotation in

9 seconds. The holes are lined up at 0 seconds. How many seconds will pass before they are lined

up again?

2. Your aunt’s club is planning to sell small bags of different types of beads to people who want to make

their own bead jewelry. The table below lists the different types of beads and how many they have.

The club wants to divide these beads into bags so that each bag has exactly the same number of oval

beads and metal beads. What is the greatest number of bags that they can make so that all of the beads

are used and there is the same number of each bead in each bag?

3. Determine each sum or difference.

a. 1 __ 8 1 2 __ 3

b. 7 __ 6 2 6 __ 7

Type of Bead Quantity

Oval bead 24

Metal bead 18



thinking 1 rationally - kyrene school district › cms › lib › az01001083... · m1-74 • topic...

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