Problem
2.1
Topic 2: Comparing Numbers and Absolute Value
for use before
Thinking With Mathematical Models
Investigation 1
Rational numbers are numbers that can be expressed as one integer
divided by another non-zero integer. Examples of rational numbers are ,
,, 0.75, and
.
Irrational numbers are numbers that cannot be expressed as one integer
divided by another integer. Examples of irrational numbers are
,
,
and 0.474774777....
The
absolute value of a number
a
, represented as | |, is the distance
between the number
a
and zero. Because distance is a measurement, the
absolute value of a number is always positive.
Opposites, like –3 and 3, have the same absolute value, 3, because they are
each three units from zero.
Alexis, Brandon, Jacob, and Madison are playing a game. Each player
gets a card with a number on it. The reverse side of the card contains a
hidden letter. The four players line up from least to greatest. If they are
correct, the hidden letters spell a word.
A. 1. For Round 1, Alexis has 0, Brandon has –2, Jacob has 2, and
Madison has –1. How should the students line up?
2. When they reveal their letters, they spell the word NICE. Assign
each letter to the proper student.
3. Which students have numbers that have the same absolute value?
B. 1. For Round 2, Alexis has , Brandon has
, Jacob has
, and
Madison has . How should the students line up?
2. When they reveal their letters, they spell AMTH. What mistake do
you think they made?
1
10
2
1
2
2
3
5
2
3
01 23
3 units
3 units
?
3
?
2
?
1
a
p
"2
2"
3
3
"9
2
21
7
8
3
4
C. For Round 3, Alexis has
, Brandon has
, Jacob has 4, and
Madison has 2. Alexis is not quite sure where to stand. Madison
tells her that
is 2 and
is 3. Where should Alexis stand?
D. The students decide to make another set of cards,
?0.3
,,
and
.
1. List the numbers from least to greatest
2. Could OOPS be the hidden word?
Exercises
Use the following list of numbers for Exercise 1–5.
?0.5
4
?2
0.25
02
?
0.3
_
3
?4
3
2.5
1. List all of the rational numbers.
2. List all of the irrational numbers.
3. Give an approximate location for each number on a number line.
4. Which numbers have the same value?
5. Which numbers have the same absolute value?
6. a. Order the numbers , , , and , from least to greatest.
b. As the denominator of a fraction increases, does the resulting
positive fraction get larger or smaller?
c. Does your rule apply for
,
,
, 2 ? Explain.
1
2
5
1
2
4
1
2
3
1
2
1
5
1
4
1
3
1
2
2
3
5
1
"3
2"4
2
1
"12
4
22
1
p
"4
2
2
1
2
2
1
3 2p 2"2
3
"4
"9
"7
"9
Topic 2: Comparing Numbers and
Absolute Value
Guided Instruction
Mathematical Goals
• Compare rational and irrational numbers by using the symbols ?, ?, ?, ?,
and ?.
• Apply the concept of absolute value.
Vocabulary
•
rational numbers
•
irrational numbers
•
absolute value
At a Glance
Remind students that the prefix
ir
means “not,” so
irrational
means “not
rational.”
Tell students that every rational number corresponds to a point on the
number line. Draw a number line marked in units between –5 and 5.
Remind students of the concept of absolute value.
•
Locate
?
3 on the number line.
(From zero, go 3 units to the left.)
•
Locate
on the number line
. (Between zero and ?1, closer to ?1; it is
units to the left of zero.)
•
What are the absolute values of –3 and
?
(3 and , respectively)
•
Compare
?
3 and
using
?.(?3 ?
)
•
Locate
on the number line.
(Because
? 3, go 3 units to the
right.)
•
Is a rational number?
(No)
Why not?
(It cannot be expressed as a
ratio of two integers.)
•
Can be located on a number line?
(Not exactly, only an
approximation of can be located as a point on the line.)
•
Give your best estimate as to where to locate on the number line.
( is
approximately 3.14, so just to the right of 3.)
•
Compare
and using
?.( ?
)
When solving Problem 2.1, have four students act out the problem.
Provide cards with additional numbers and letters to give students
additional practice ordering rational and irrational numbers.
You will find additional work on comparing rational numbers in the
grade 6 unit
Bits and Pieces I
.
!
9
p
p !9
pp
p
p
p
!9
!9
27
2
7
8
8
7
2
7
8
8
7
27
8
8
PACING
1 day
ACE Assignment Guide
for Topic 2
Core
1–6
Answers to Topic 2
Problem 2.1
A. 1.
Brandon, Madison, Alexis, Jacob
2.
Brandon, N; Madison, I; Alexis, C; Jacob, E
3.
Brandon and Jacob
B. 1.
Brandon, Jacob, Madison, Alexis
2.
Answers may vary. Sample: Brandon and
Jacob got confused with the negative
fractions. Since ? ,
?
since it is
further from zero.
C.
Between 2 and 3, but closer to 3.
D. 1.
,
, then
and ?0.33 are equal.
2.
No. The word would end in a double letter.
Figure 1
Exercises
1.
2.
3.
See Figure 1.
4.
5.
6. a.
,,,
b.
The resulting fraction gets smaller.
c.
Answers may vary. Sample: No. For
negative numbers, an increasing
denominator results in larger fractions.
1
2
1
3
1
4
1
5
Z1
4Z
5 Z0.25Z;Z24Z 5 Z4Z; Z221
2Z
5 Z2.5Z
Z2
1
2Z
5 Z20.5Z 5 Z
1
2
Z;
Z2!4Z 5 Z22Z 5 Z2Z 5 Z !4Z;
2!4 522; 21
2
52 0.5;
1
4
5 0.25; 2 5 !4
p, !12, !3
2!4,
1
2
, 2, 2 0.33
, 24, 23
5
, 3, 2.5
21
2
, 221
2
, 2 0.5, !4
, 4,
1
4
, 22, 0.25, 0,
2
1
2p 2!2
3
2
1
2
2
3
5
1
2
3
5
?
4
0.25
3
2
?
0.33
2.5
0
?
?4
?3
?4
?12
?
2
1
?
2
2
1
2
1
4
1
2
?
0.5
?
3
5
?
4
?