Lesson 1 Fractions on a Number Line

1 Lesson

Fractions on a Number Line

Problem Solving:

Partitioning the Number Line

Fractions on a Number Line

Where are fractions on a number line?

All the numbers on the first two number lines shown are wholenumbers. When we count with consecutive whole numbers, we can count forward or backward in a predictable way. We can tell what number will come next by adding 1 to the current number. We can tell what number came before by subtracting 1 from the current number. Counting with whole numbers can continueforever.

Whole Numbers on a Number Line

Counting from 6 to 7:

0

1

2

3

4

5

6

7

Counting from 153 to 154:

147 148 149 150 151 152 153 154

The Numbers between Whole Numbers on a Number Line

Are there any numbers between the whole numbers on a number line? Yes! These numbers are fractions, or numbers that have a fractional part, and they can be found between every pair of consecutive whole numbers on a number line.

1

3

3

4

Vocabulary

whole numbers consecutive predictable fractions infinite denominator numerator

0

1

2

3

153

1 4

153

1 2

152

153

154

155

3 Unit 1 ? Lesson 1

Lesson 1

Here are some interesting concepts. First, there are an infinite number of fractions between any two consecutive whole numbers on a number line.

Second, look at the fractions below. Do they appear to have a pattern? Can the fraction that comes next be predicted?

There are an infinite number of fractions between any two consecutive whole numbers.

0

11 43

1

5

2

8

15 16

1

The simplest way to count with fractions in a predictable manner is

to count by using fractions with the same denominator. Let's count

byfifths.

Numerator

1 5 Denominator

0

1 5

2 5

3 5

4 5

1

?

We can easily predict what fraction comes next when the denominators are the same: 16, 26, 36, . . .

We can follows

continue to

45,

6 5

follows

count beyond 55, and so on.

4 5

by

adding

1

to

the

numerator.

So

5 5

0

1 5

2 5

3 5

4 5

5 5

6 5

Notice

that

5 5

is

in

the

same

location

as

1.

When

the

numerator

and

denominator of a fraction are the same number, the fraction is equal

to 1. To make it easier to remember this fact, we can write 1 beside

the fraction.

5 5

is

the

same

as

1.

0

1 5

2 5

3 5

4 5

5 5

=

1

6 5

Apply Skills

Turn to Interactive Text, page2.

4 Unit 1 ? Lesson 1

Reinforce Understanding

Use the Unit 1 Lesson 1 Teacher Talk Tutorial to review lesson concepts.

Lesson 1

Problem Solving: Partitioning the Number Line

How do we partition the number line?

When we divide the number line into equal parts, we are partitioning

the number line. One way to partition the number line is to

repeatedly find the number that is halfway between two numbers

already on the number line. When we partition from 0 to 1 into two

equal parts,

from

0

to

1 2

1 2 is

is halfway between 0 and 1. In other words,

the

same

as

the

distance

from

1 2

to

1.

the

distance

0

1

0

1 2

2 2

Now partition each halfway between 0

of the

and

1 2

two and

p34airstshainlftwoatwyobeetqwueael npa12rtasn. dSo1.14

is

0

1 2

1

0

1 4

2 4

3 4

4 4

Again, partition each part into two equal parts. Look at the fraction halfway between 0 and 14. It is 18. Notice that when we partition by repeatedly finding the number that is halfway between two numbers, the denominator doubles.

0

1 4

1 2

3 4

1

0

1 8

2 8

3 8

4 8

5 8

6 8

7 8

8 8

It is important to understand that when we divide distances on the

number line in half, the denominators double even though the fractions

are getting smaller.

than

1 2

.

This

is

easy

For

example,

1 8

is

smaller

than

14,

to see on a number line where we

which is smaller are comparing

fractions by comparing their distances from 0. We call this model a

length model.

Vocabulary

partition length model

Problem-Solving Activity

Turn to Interactive Text, page 4.

Reinforce Understanding

Use the Unit 1 Lesson 1 Problem Solving Teacher Talk Tutorial to review lesson concepts.

5 Unit 1 ? Lesson 1

Lesson 1

Homework

Activity 1

Find the fractions for the letters on the number line. Remember that we can

write the whole number 1 as a fraction.

(b)

1.

0

(a)

1

2. 0

(f)

(c)

(d)

(e)

1

3. 0

(l)

(g)

(h)

(j)

(k)

1

Activity 2 Write the correct multiple in each empty box in the list. Model

0 2 4 6 8 10 12 14 16 18

1. 0 10 20

40

70

90

2. 0 5

25

40

3. 0 4

16

28

4. 0 6 12

30

42

54

Activity 3 ? Distributed Practice

Solve.

1. 354 2. 203 3. 112

+489

?177

? 32

4. 1,045 + 992

5. 431 ? 27

6. 4q248

6 Unit 1 ? Lesson 1

2 Lesson

Connecting Fractions and Fair Shares to Geometry

Problem Solving:

Noncongruent Fair Shares

Connecting Fractions and Fair Shares to Geometry

How can fractions on a number line be related to fair shares?

Think about how we partitioned a number line in Lesson 1. Each segment between fractions on the number line was the same length. Look at this number line. It is divided into fifths. Each segment is a fair share because each segment is the same length.

0

1 5

2 5

3 5

4 5

5 5

Each

1 5

segment

is

the

same

length.

One-dimensional objects like number lines are divided into fair shares that look the same. The fair shares are congruent line segments. Fair shares are not always congruent when we use two-dimensional shapes like rectangles or squares. Look at the rectangles in Example 1. Each has been divided into fair shares called fourths.

Example 1 Partition each rectangle into fair shares called fourths.

Vocabulary

fair share congruent

In each rectangle, the fair shares are congruent. The area and shape of each fair share is the same. Here is another way to think about these fair shares. In each rectangle, the fair shares can be stacked on top of each other and they would look exactly alike.

7 Unit 1 ? Lesson 2

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