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Grade 7 Math LESSON 6: RATIONAL NUMBERS IN THE NUMBER LINE
TEACHING GUIDE
GRADE 7 MATH TEACHING GUIDE
Lesson 6: Rational Numbers in the Number Line
Prerequisite Concepts: Subsets of Real Numbers, Integers
Objective:
In this lesson, you, the students, are expected to
1. Define rational numbers;
2. Illustrate rational numbers on the number line;
3. Arrange rational numbers on the number line.
Time: 1 hour
NOTE TO THE TEACHER:
Ask students to recall the relationship of the set of rational numbers to the set of integers and the set of non-integers (Lesson 4).
This lesson gives students a challenge in their numerical estimation skills. How accurately can they locate rational numbers between
two integers, perhaps, or between any two numbers?
Lesson Proper
I. Activity
Determine whether the following numbers are rational numbers or not.
NOTE TO THE TEACHER:
Give as many rational
numbers as class time can
allow. Give them in different
forms: integers, fractions,
mixed numbers, decimals,
repeating decimals, etc.
1
- 2, π, , 3 4 , 16 , -1.89,
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Now, try to locate them on the real number line below by plotting:
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-3
-2
-1
0
1
2
3
4
II. Questions to Ponder
Consider the following examples and answer the questions that follow:
a. 7 ÷ 2 = 3 ½ ,
b. (-25) ÷ 4 = -6 ¼
AUTHORS: Gina Guerra and Catherine P. Vistro-Yu, Ed.D.
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Grade 7 Math LESSON 6: RATIONAL NUMBERS IN THE NUMBER LINE
TEACHING GUIDE
c. (-6) ÷ (-12) = ½
1. Are quotients integers? Not all the time. Consider 107 .
2. What kind of numbers are they? Quotients are rational numbers.
3. Can you represent them on a number line? Yes. Rational numbers are real numbers and therefore, they are found in the real
number line.
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Recall what rational numbers are...
3 ½, -6 ¼, ½, are rational numbers. The word rational is derived from the word “ratio” which means quotient. Rational numbers are
numbers which can be written as a quotient of two integers, where b ≠ 0.
The following are more examples of rational numbers:
6
5 = 15
0.06 = 100
1.3 =
From the example, we can see that an integer is also a rational number and therefore, integers are a subset of rational numbers. Why is that?
1 3
, 4 , 16 , -1.89, the numbers π and 3 4 are the only ones that are
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not rational numbers. Neither can be expressed as a quotient of two integers. However, we can express the remaining ones as a quotient of
two intergers:
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€
€
−189
−2 = −21 , 16 = 4 = 41 , −1.89 =
100
€ check on your work earlier.
€ Among the numbers given, - 2, π,
Let’s
Of course,
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1
is already a quotient by itself.
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We can locate rational numbers on the real number line.
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Example
1. Locate ½ on the number line.
a. Since 0 < ½ < 1, plot 0 and 1 on the number line.
0 AUTHORS: Gina Guerra and Catherine P. Vistro-Yu, Ed.D.
1 2
Grade 7 Math LESSON 6: RATIONAL NUMBERS IN THE NUMBER LINE
TEACHING GUIDE
b. Get the midpoint of the segment from 0 to 1. The midpoint now corresponds to ½
½ 0 1 Example 2. Locate 1.75 on the number line.
a. The number 1.75 can be written as 47 and, 1 < 47 < 2. Divide the segment from 0 to 2 into 8 equal parts.
0
1
2
€ point 1.75.
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b. The 7th mark from 0 is the
0
Example 3. Locate the point
Note that -2 <
1
1.75
2
on the number line.
< -1. Dividing the segment from -2 to 0 into 6 equal parts, it is easy to plot
. The number
is the 5th mark from
0 to the left.
-2
-1
0
Go back to the opening activity. You were asked to locate the rational numbers and plot them on the real number line. Before doing that, it is
useful to arrange them in order from least to greatest. To do this, express all numbers in the same form –- either as similar fractions or as
decimals. Because integers are easy to locate, they need not take any other form. It is easy to see that
- 2 < -1.89 <
1
< 16 11
AUTHORS: Gina Guerra and Catherine P. Vistro-Yu,
€ Ed.D.
€
3
Grade 7 Math LESSON 6: RATIONAL NUMBERS IN THE NUMBER LINE
TEACHING GUIDE
Can you explain why?
Therefore, plotting them by approximating their location gives
-3
-2
-1
0 1
1
2
3
4
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-1.89 III. Exercises
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1. Locate and plot the following on a number line (use only one number line).
a.
−10
3
e. -0.01
b. 2.07
f. 7
2
5
g. 0
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€
c.
1
9
NOTE TO THE TEACHER:
You are given a number line to work on. Plot the
numbers on this number line to serve as your answer
key.
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d. 12
h. −
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1
6
0
12
2
2. Name 10 rational numbers that are greater than -1 but less than 1 and arrange them from least to greatest on the real number line?
Examples are:
1
3
1
1
1
1 2 8 9
, − , − , − , −
, 0, , ,
,
10 10
2
5 100
8 11 37 10
AUTHORS: Gina Guerra
€ and Catherine P. Vistro-Yu, Ed.D.
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Grade 7 Math LESSON 6: RATIONAL NUMBERS IN THE NUMBER LINE
3. Name one rational number x that satisfies the descriptions below:
−10 ≤ x < −9 ,
a.
46
48
8
Possible answers: x = − , − , − 9.75, − 9 ,−9.99
5
5
9
1
1
<x<
€ b.
10
2
46
48
8
Possible€
answers: x = − , − , − 9.75, − 9 ,−9.99
5
5
9
3< x <π
c.
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Possible answers: x = 3.1, 3.01, 3.001, 3.12
1
1
d.
<x<
€
4
3
€
13
299 3
Possible€
answers: x = , 0.27, 0.28,
,
50
1000 10
1
1
− <x<−
€ e.
8
9
3
17
Possible€answers: x = − , − 0.124, −
, − 0.112
25
144
TEACHING GUIDE
NOTE TO THE TEACHER:
In this exercise, you may allow
students to use the calculator to check
that their choice of x is within the
range given. You may, as always also
encourage them to use mental
computation strategies if calculators
are not readily available. The
important thing is that students have a
way of checking their answers and will
not only rely on you to give the correct
answers. € TO THE TEACHER:
NOTE
End this lesson with a summary as well as a preview to what students will be expecting to learn about rational numbers, their
€
properties, operations and uses.
Summary
In this lesson, you learned more about what rational numbers are and where they can be found in the real number line. By changing all
rational numbers to equivalent forms, it is easy to arrange them in order, from least to greatest or vice versa.
AUTHORS: Gina Guerra and Catherine P. Vistro-Yu, Ed.D.
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