Lesson
2 Comparing Fractions with Different Denominators
Problem Solving:
Rules for Drawing Number Lines
Lesson 2 Skills Maintenance
Lesson Planner
Name
Skills Maintenance
Skills Maintenance
Ordering Fractions
Activity 1
Ordering Fractions, Common Multiples
Order the fractions from least to greatest.
9, 2, 4, 1, 5
9 9 9 9 9
Building Number Concepts:
1
9
omparing Fractions with
C
Different Denominators
Homework
Students fill in missing fractions on a number
line, order a set of fractions, and draw two
number lines to compare two fractions.
In Distributed Practice, students practice
multiplying unit fractions by whole numbers.
114 Unit 2 • Lesson 2
9
9
greatest
Activity 2
1.
Write two common multiples.
4
Sample answer: 12, 24
12
2.
Write two common multiples.
3
Sample answer: 12, 24
12
3.
Write three common multiples.
2
Sample answer: 20, 40, 60
20
4.
Problem Solving:
Students will accurately draw two number
lines for comparing fractions with different
numerators and denominators to each other.
5
9
Write common multiples as directed in each problem.
Students will compare fractions with different
numerators and denominators by looking at
the relative size of the numerator compared
to the denominator and comparing the
fractions using number lines.
Objective
4
9
Common Multiples
Objective
Students learn two important rules for
comparing fractions using number lines: the
number lines must be lined up at 0 and at 1
so that the number lines represent the “same
whole,” and the number lines must be divided
into fair-share parts.
2
9
least
In this lesson, students learn that we often
compare fractions that do not have the same
numerator or denominator. When students
compare fractions in this way, it is important
to show accurate representations of the
fractions on the number line. Students will
need this skill to correctly compare fractions.
ules for Drawing Number
R
Lines
Date
Write four common multiples.
4
Sample answer: 12, 24, 36, 48
6
50
Unit 2 • Lesson 2
Skills Maintenance
Ordering Fractions, Common Multiples
(Interactive Text, page 50)
Activity 1
Students order a list of fractions with the same
denominator.
Activity 2
Students write common multiples for two numbers.
Lesson
2
Comparing Fractions with Different
Denominators
Problem Solving:
Rules for Drawing Number Lines
Building Number Concepts:
Comparing Fractions with Different Denominators
omparing Fractions with Different
C
Denominators
How do we use number lines to compare
fractions?
When comparing fractions, consider this question, “Are the fractions
far from 0 or are they close to 0?” Here is what it looks like when we
compare two fractions with the same denominator.
How do we use number lines to
compare fractions?
5
Which fraction is greater, 2
6 or 6?
(Student Text, pages 68–69)
Connect to Prior Knowledge
Remind students about the patterns they saw
in fractions from the previous lesson. Draw four
number lines on the board, one directly below the
other, with the 0s and 1s lined up. Divide each
number line into fourths. On the top number line,
Link to Today’s Concept
Tell students that we will extend our knowledge
about the locations of fractions on a number line
to compare fractions that do not have simple
patterns.
Demonstrate
•Have students turn to page 68 in the
Student Text. Review the process of using a
number line to compare fractions with the
same denominator by comparing 2 and 5.
6
6
5
2
Point out that is greater than because
6
6
5
2
is to the right of on the number line.
6
6
Also point out that when the denominator
is fixed and the numerator increases, the
fractions increase.
1
6
2
6
3
6
4
6
5
6
6
6
=1
0
1
6
2
6
3
6
4
6
5
6
6
6
=1
3
3
=1
2
2
=1
It is important to notice:
2
• 5
6 is greater than 6.
• When the denominator stays the same and the numerator
increases, the fractions get farther from 0.
Now let’s look at a different pair of fractions.
1
Which fraction is greater, 1
3 or 2?
1
3
0
draw a bar from 0 to 1. On the next number
4
line, draw a bar from 0 to 2. On the third
4
number line, draw a bar from 0 to 3. And on
4
the last number line, draw a bar from 0 to 4 or
4
1. Ask students to describe the pattern. As the
numerators increase, the fractions increase.
0
2
3
1
2
0
It is important to notice:
1
•1
2 is greater than 3.
• When the numerator stays the same and the denominator increases,
the fractions get closer to 0.
6868
Unit 2 • Lesson 2
•Now have students look at the number lines for
1 and 1. Have them compare the positions of
3
2
0 and 1 on the number lines. Point out that the
distance between 0 and 1 is the same on the
number lines. So when we compare the fractions
like this, the fractions use the same whole, and
we can clearly see the fair share for 1 is smaller
than the fair share for 1.
2
3
•Finally ask students which fraction is farther
to the right on the number line. Because 1 is to
2
1
1
1
the right of , is closer to 1, and is closer to
3 2
3
0. These are all important things for students
to notice.
Unit 2 • Lesson 2 115
Lesson 2
Lesson 2
Not all fractions being compared have the same numerator or the same
denominator. In this case, comparing the fractions becomes a bit more
difficult.
How do we use number lines to
compare fractions? (continued)
5
Consider the fractions 6
8 and 6. In both fractions, the numerator is close
to the denominator. So we know both fractions are close to 1. Which
fraction is closer to 1? Let’s use a number line to find out.
Demonstrate
•Have students look at page 69 of the
Student Text. In Example 1 , students will
Example 1
or 5?
Which fraction is greater, 6
8 6
Begin by showing each fraction on a number line.
compare 6 and 5. Before comparing these
0
fractions, discuss the magnitude of each
fraction by discussing the relative size of
each numerator as compared to the
0
8
1
8
2
8
3
8
4
8
5
8
6
8
7
8
8
8
=1
6
6
=1
6
1
6
2
6
3
6
4
6
5
6
6
5
6
Because 5
6 is to the right of 8, it is closer to 1. So 6 is greater than 8.
denominator. Think about the fraction 6.
8
How does the numerator compare to the
denominator? Because 6 is close to 8,
we know that 6 is close to 1. Think about
8
5
the fraction . How does the numerator
6
compare to the denominator? Because 5 is
close to 6, we know that 5 is also close to 1.
6
Because these fractions are both close to 1,
a number line can be used to compare the
fractions.
•Discuss all the things we can observe from
these number line models about the
fractions 6 and 5. We can see that 5 is to
8
6
6
the right of 6. So we know 5 > 6. This is the
8
6
8
way we have been comparing numbers on
number lines for a long time.
Apply Skills
Reinforce Understanding
Turn to Interactive Text,
page 51.
Use the Unit 2 Lesson 2 Teacher Talk Tutorial
to review lesson concepts.
Unit 2 • Lesson 2
Ask:
We will compare the fractions 1 and 7. How does
5
8
1
the numerator of compare to its denominator?
5
(The numerator is much smaller than the
denominator.)
Is 1 closer to 0 or closer to 1? (closer to 0)
5
Check for Understanding
Engagement Strategy: Think, Think
Ask students the questions that follow about
comparing fractions. Tell them that one of them
will be called on to answer a question. Ask them
to listen for their name to be called. After each
question, allow time for students to think of the
answer. Then call on a student.
116 Unit 2 • Lesson 2
How does the numerator of 7 compare to its
8
denominator? (The numerator is very close to the
denominator.)
Is 7 closer to 0 or closer to 1? (closer to 1)
8
Which fraction is greater? ( 7 )
8
Reinforce Understanding
Remind students that they can review lesson
concepts by accessing the online Unit 2
Lesson 2 Teacher Talk Tutorial.
69
69
Lesson 2 Apply Skills
Name
Date
Apply Skills
Strategies for Comparing Fractions
Apply Skills
Activity 1
(Interactive Text, pages 51–52)
Unit 2
Follow the directions for completing the number lines.
1. Show 1
4 on the number line.
Have students turn to pages 51 and 52 in the
Interactive Text, which provide an opportunity to
compare fractions on a number line.
1
4
0
2.
Show 3
2 on the number line.
0
3.
Activity 1
4.
3
2
1
1
2
Show 1
5 on the number line.
1
5
0
Students are shown a fraction on a number line
and they find the location of a different fraction
with the same denominator.
1
3
4
1
3
5
Show 1
3 on the number line.
1
3
0
1
2
3
Activity 2
Students use two number lines to compare
fractions with different denominators.
Monitor students’ work as they complete these
activities.
Watch for:
Unit 2 • Lesson 2
51
•Can students use the unit fraction to find
the other fraction?
Lesson 2 Apply Skills
•Can students find the unit fraction given the
Name
location of another fraction with the same
denominator?
Date
Activity 2
Follow the directions for completing the number lines. Then write the
inequality.
•Can students compare two fractions with
different denominators by locating them on
number lines?
2
Use the number lines to compare 1
2 and 3.
1
2
0
•Can students complete the inequality to
Model
1
3
0
compare the fractions?
The inequality is:
Reinforce Understanding
Remind students that they can review
lesson concepts by accessing the online
Unit 2 Lesson 2 Teacher Talk Tutorial.
1.
1
2
2
3
2
3 .
<
1
3
Use the number lines to compare 1
4 and 8.
1
4
0
0
1
8
2
4
2
8
The inequality is:
2.
1
3
8
1
4
<
4
8
3
8
3
4
5
8
1
6
8
7
8
1
.
2
Use the number lines to compare 3
6 and 5.
0
0
1
6
2
6
1
5
The inequality is:
3
6
2
5
2
5
<
4
6
3
5
3
6
5
6
4
5
1
1
.
Reinforce Understanding
Use the Unit 2 Lesson 2 Teacher Talk Tutorial to review lesson concepts.
52
Unit 2 • Lesson 2
Unit 2 • Lesson 2 117
Lesson 2
Lesson 2
Problem Solving: Rules for Drawing Number Lines
What are some common mistakes we make when
we compare fractions using number lines?
Problem Solving:
We have to remember two important rules when we use number lines to
compare fractions.
Rules for Drawing Number Lines
Rule 1: The distance between 0 and 1 has to be the same on both
number lines. (the same whole)
Rule 2: The fractional parts on each number line are fair shares. (equalsized parts)
What are some common mistakes
we make when we compare fractions
using number lines?
1
We know 1
3 < 2. The next two examples show incorrect comparisons
of these fractions because they fail to follow one of the rules above.
Example 1 shows us what happens if the distance between 0 and 1 on
the number lines is not the same.
(Student Text, pages 70–71)
Example 1
What happens when the number lines are not the same length
between 0 and 1?
Demonstrate
•Have students turn to page 70 in the
Student Text. Tell them that there are some
very common mistakes that we can make
when drawing number lines. One error is to
use two different number lines that do not
line up. The other mistake is failing to divide
the number line into fair shares.
0
0
importance of these two rules for drawing
number lines:
Number lines must be lined up at 0
and 1.
Number lines must be divided into fair
shares.
■
•Have students look at Example 1 . They
are shown one of these common errors.
Point out where the 1s are located on each
number line. Explain that if the 0s and the
1s are not aligned, we are not using the
same whole for the fractions. We can only
compare fractions from a whole that has
the same length.
118 Unit 2 • Lesson 2
1
1
3
ERROR
2
3
1
These number lines show that 1
2 < 3 because the distance between 0
and 1 is not the same for each number line. The number line for thirds
is longer, so it makes the fair-share segments much longer than they
should be.
•Make sure students understand the
■
1
2
7070
Unit 2 • Lesson 2
1
Lesson 2
Demonstrate
•Next have students look at Example 2 .
Students are shown the second common
error. Here the number lines are not divided
into fair shares. Point out on the top number
line that the distance from 0 to 1 is less
2
than the distance between 1 and 1. This
2
number line does not show fair shares.
Example 2 shows us what can happen if we do not draw fair shares on
the number line.
Example 2
What happens when the fractional parts on each number line are not
fair shares?
0
1
2
0
1
1
3
2
3
ERROR
1
1
These number lines show that 1
2 < 3. Look at the way each number line
is partitioned. The segments are not fair shares. That is why 1
2 looks
closer to 0 than 1
is an easy mistake to make, especially with
3.1This 1
fractions such as 8 or 10.
•Likewise, point out on the bottom number
line that the distance between 0 and 1
3
is much greater when compared to the
distance between the next two parts. This
number line also does not show fair shares.
•Be sure students understand that we cannot
trust our comparison of two fractions if we
use number lines that do not follow these
important rules.
Problem-Solving Activity
Turn to Interactive Text,
page 53.
Reinforce Understanding
Remind students that they can review
lesson concepts by accessing the online
Unit 2 Lesson 2 Problem Solving Teacher
Talk Tutorial.
Reinforce Understanding
Use the Unit 2 Lesson 2 Problem Solving
Teacher Talk Tutorial to review lesson concepts.
Unit 2 • Lesson 2
Unit 2 • Lesson 2 71
71
119
Lesson 2 Problem-Solving Activity
Name
Problem-Solving Activity
(Interactive Text, page 53)
Have students turn to page 53 in the Interactive
Text, which provides students an opportunity
to draw number lines in order to compare the
two fractions 5 and 1. Remind them to line the
8
4
number lines up at 0 and 1 and make sure that
the parts are fair shares.
Monitor students’ work as they complete these
tasks.
Watch for:
•Can students line up the two number lines
Date
Problem-Solving Activity
Drawing Number Lines and Comparing Fractions
Draw number lines to use to compare the given fractions. Be sure to
follow the rules for drawing accurate number lines. Then write the
inequality.
1
Use the number lines to compare 5
8 and 4.
0
1
8
2
8
3
8
1
4
0
The inequality is:
fair shares?
•Can students see that 14 < 58 from the
number lines they drew?
5
8
6
8
2
4
1
4
5
8
<
3
4
7
8
1
1
.
Challenge Problem
Error Analysis
5
Marilyn compared 2
3 and 6 using the two number lines below. From
2
her drawing, she decided that 5
6 < 3.
1
3
0
correctly?
•Can students divide up the number lines into
4
8
Unit 2
Lesson 2
0
1
6
2
6
2
3
3
6
4
6
5
6
3 =1
3
6 =1
6
Do you agree with Marilyn? (circle one)
YES or NO
Explain your reasoning. Sample answer: Although the number
lines line up at 0 and 1, the shares in the number line for
sixths are not fair shares because the distances between
the tick marks are not equal.
•Can students make other observations
about the fractions in terms of magnitude
by comparing the numerator to the
denominator? (e.g., Five is relatively large
when compared to 8, and 1 is relatively
small when compared to 4.)
Challenge Problem
Error Analysis
Students are shown another student’s answer
to a fraction comparison problem and they are
to determine whether the answer is correct
or incorrect. Then they are to explain their
reasoning.
120 Unit 2 • Lesson 2
Unit 2 • Lesson 2
53
Monitor students’ work as they complete this task.
Watch for:
•Can students see that the first number line has
been drawn correctly?
•Can students see that the 0s and 1s are lined up
correctly?
•Can students see that the second number line
has not been divided into fair shares?
•Do students realize that 23 < 56?
Reinforce Understanding
Remind students that they can review lesson
concepts by accessing the online Unit 2 Lesson
2 Problem Solving Teacher Talk Tutorial.
Lesson 2
Homework
Activity 1
Write the fractions for the letters on the number lines.
Homework
(b)
1.
Go over the instructions on page 72 of the
Student Text for each part of the homework.
0
(a)
1
2
1
0
(f)
1
4
(g)
2
4
0
(l)
1
3
(m)
2
3
2.
Activity 1
3.
Students fill in missing fractions on number lines
labeled by letters.
2
2
(d)
(e)
5
2
(h)
1
(k)
5
4
1
(p)
(r)
5
3
(c)
3
2
4
2
4
(j)
4
3
4
(n) 3
3
4
3
Activity 2
Order the fractions from least to greatest.
1, 1, 1, 1, 1
4 3 7 5 2
1
7
Activity 2
1
5
1
4
1
3
1
2
least
Students order a list of fractions with different
denominators from least to greatest.
greatest
Activity 3
Compare the fractions by drawing two number lines. Be sure that the
distance between 0 and 1 is the same on both number lines and the number
lines are divided into fair shares. See number lines below.
3 2
1. Which fraction is less: 4 or 5?
Activity 3
2 1
2. Which fraction is less: 3 or 6?
Students compare two fractions by drawing two
number lines. Remind students to line up the
number lines and divide them carefully into
fair shares.
2 is less
5
1 is less
6
Activity 4 • Distributed Practice
Solve.
1. 3 × 1
4
3
4
2. 8 × 1
5
8
5
3. 4 × 1
9
4
9
4. 2 × 1
3
2
3
5. 6 × 1
2
6
2
6. 9 × 1
6
9
6
Activity 4 • Distributed Practice
Students practice basic multiplication of unit
fractions by whole numbers to find other
fractions.
7272
Unit 2 • Lesson 2
Additional Answers
Activity 3
1.
1
4
0
1
5
0
2.
2
5
1
6
2
6
3
4
3
5
1
3
0
0
2
4
4=1
4
4
5
5=1
5
2
3
3
6
4
6
3=1
3
5
6
6=1
6
Unit 2 • Lesson 2 121