Download Math 11th grade LEARNING OBJECT Recognition of the order

Math 11th grade
LEARNING OBJECT
LEARNING UNIT
Recognition of the order relation between real
numbers
Operating in the real
numbers (R) set
S/K
SCO: Identify
numbers.
the
order
properties
in
real
SKILL 1. Interpret the definition of the order relation.
SKILL 2. Recognize the symbology associated to the
order relation.
SKILL 3.
numbers.
Establish the properties of order in real
SCO: Order real numbers
SKILL 4. Make comparisons between real numbers.
SKILL 5. Order real numbers from lowest to highest
and vice versa starting from their location in the
number line.
SKILL 6. Order real numbers according to their
symbolic representation.
SKILL 7. Order real numbers in the number line by
approximation or geometric methods. (Pythagoras
and Thales).
SKILL 8. Find real numbers between two given real
numbers.
SKILL 9. Recognize the property of density in real
numbers.
Language
English
Socio cultural context of
Classroom and Educational Institution.
the LO
Curricular axis
Numerical Thinking and Systems of Numeration
Standard competencies
Recognize the density and incompleteness of the
rational numbers through numerical, geometrical, and
algebraic methods.
Background Knowledge
To know the subsets of real numbers, the real
number line, and the symbols greater than (>) and
less than (<).
Basic Learning
Understand that between two real numbers there are
infinite real numbers
Rights
English Review topic
● Ordinal numbers
● Comparatives and superlatives
● Passive voice
Vocabulary box
● To allow: To give permission for someone to do
something, or to not prevent something from
happening.
● Aid: help.
● Take into account: to consider.
● Therefore: For that reason.
● Tool: an instrument for doing work, especially
by hand
● Thus: (referring to something mentioned before
or after) in this or that way.
● Whenever: at any time that, at every time that
NAME: _________________________________________________
GRADE: ________________________________________________
Introduction
The order relation existing between real numbers is also applicable to all
of the subsets contained in these, namely, the natural, whole, rational,
and irrational numbers.
You need to have in mind that the order relation between two numbers
is presented based on the use of the symbols greater than (>), less
than (<), and equal to (=), considering the properties that such
relationships are ruled by.
Based on the above, comparisons can be made between real numbers to
define which is greater, less, or equal, and also the location of the real
numbers in the number line, keeping its corresponding place; it should
be made using approximation methods or geometric methods such as
Pythagoras’ or Thales’ methods.
Finally, the use of the order relation in real numbers may be used to find
numbers between two quantities, involving the property of density of
the real numbers.
Objectives
● To acknowledge the order relation established in the real
numbers set.
● To identify the properties of the usual order established in real
numbers.
● To order real numbers based on the properties of order.
Activity 1
Skill 1. Interpret the definition of the order relation.
Skill 2. Recognize the symbology associated to the order
relation.
Skill 3. Establish the properties of order in real numbers.
ORDER RELATION IN REAL NUMBERS
The real numbers, being composed by numerical subsets, meet
properties of relation, besides meeting the properties that indicate the
existing bond between two elements in the real numbers set.
These properties are:
● Reflexive property: let 𝐴 be a set of real numbers, then every
element of 𝐴 is related to itself. That is, ∀𝑥 ∈ 𝐴, 𝑥𝑅𝑥
Example:
1. Starting from the fact that a brother is a son of the same mother and
father of any given person, if Juan is an only child, it can be said that
Juan is Juan’s brother; therefore, John is John’s brother.
● Antisymmetric property: being A a set of the real numbers, if
two elements of A are related to each other, then they are equal. So,
● ∀𝑥, 𝑦 ∈ 𝐴, 𝑥𝑅𝑦, 𝑦𝑅𝑥 → 𝑥 = 𝑦
Example:
Andrés and Santiago are twin brothers; Andrés is 13 years old, so,
Andrés’ age is equal to Santiago’s age. 13 = 13
● Transitive property: Being A a set of the real numbers, if an
element of A is related to another one, and this other element is at
the same time related to a third element, then the first element is
also related to the last one. Therefore,
∀𝑥, 𝑦, 𝑧 ∈ 𝐴, 𝑥𝑅𝑦, 𝑦𝑅𝑧 → 𝑥𝑅𝑧
Example:
Sofía, Ángela, and Daniel are siblings; Sofía is older than Daniel, and
Daniel is older than Ángela, therefore, it can be said that Sofía is
older than Ángela.
So, taking into account the definition of the order relation, we observe
that there are three symbols utilized to compare the elements of the
real set. These are:
1
2
● Greater than(>): 2 > − 3
● Less than (<): 𝜋 < 2
● Equal to (=):
9
2
= 4.5
They are also based on properties that are true for each case:
1. Being
𝑎, 𝑏 ∈ 𝑅. If 0 < 𝑎 and 0 < 𝑏 then 0 < 𝑎 ∙ 𝑏, in the case that
and 𝑏 < 0, then 0 < 𝑎 ∙
Example: If 𝑎 = 3 and 𝑏 = 1.5 then 3 ∙ 1.5 = 4.5
𝑎<0
2. Being 𝑎, 𝑏 ∈ 𝑅. If 0 < 𝑎 and 𝑏 < 0 then 𝑎 ∙ 𝑏 < 0. or, if 𝑎 < 0 and 0 <
𝑏 then 𝑎 ∙ 𝑏 < 0
2
2
4
Example: If 𝑎 = −2 and 𝑏 = 3 then −2 ∙ 3 = − 3
3. Being 𝑎, 𝑏 ∈ 𝑅. If 𝑎 < 𝑏 then −𝑏 < −𝑎
2
2
2
1. Example: 𝑎 = 5 and 𝑏 = 1.4 If 5 < 1.4 then −1.4 < − 5
𝑎
4. Being 𝑎, 𝑏 ∈ 𝑅, 𝑏 ≠ 0. If 0 < 𝑏 then 0 < 𝑎 ∙ 𝑏, or, If
4
𝑎
𝑏
< 0 then 𝑎 ∙ 𝑏 < 0
Example: 𝑎 = 4 and 𝑏 = 3 If 0 < 3 then 0 < 4 ∙ 3
5. Being 𝑎, 𝑏, 𝑐 ∈ 𝑅. If 𝑎 < 𝑏 then 𝑎 + 𝑐 < 𝑏 + 𝑐, in the same way for
multiplication.
1
5
1
5
1
5
Example: 𝑎 = 2 , 𝑏 = 4 and 𝑐 = 3, If 2 < 4 then 2 + 3 < 4 + 3
Did you know that...?
There are two kinds of order relation: the relation of partial order that
meets the reflexive, antisymmetric, and transitive properties; and the
relation of Total Order which meets the condition of being partial and all
of its elements are comparable.
Learning Activity
Mark true or false according to the case:
1
1. Being that −3 < 2 and that
1
2
< 2 then −3 < 2 due to the transitive
property. ____
2. The reflexive property is about the relationship between two
elements that belong to the set of real numbers. ____
3. The antisymmetric property refers to the relationship of two
elements in such a way that they are equal. ____
5
3
4. Being that − 2 < − 2 , then, by multiplying both figures by 4, we get
−10 < −6. ___
1. If
9
4
> 0 then 9 ∗ 4 < 0. ____
Activity 2
Skill 4. Make comparisons between real numbers.
Skill 5. Order real numbers from lowest to highest and vice versa
starting from their location in the number line.
Skill 6. Order
representation.
real
numbers
according
to
their
symbolic
Skill 7. Order real numbers in the number line by approximation
or geometrical methods. (Pythagoras and Thales).
REAL NUMBERS ORDER
Real numbers can be located in the number line, taking into account
that every negative number will always be less than a positive one.
To order natural numbers in the number line, they should be
enumerated from zero on, considering that the distance between the
numbers must be the same, and counting from left to right and from
lowest to highest.
Whole numbers, being composed by natural numbers and their
opposites, will be located in the line to the right and left of the origin
point meaning zero.
To locate rational numbers, an auxiliary construction has to be made by
using a ruler and a compass to be able to achieve the highest exactitude
possible; in the same way geometrical methods can be applied to locate
irrational numbers.
The geometric method of Thales to locate fractions in the
number line
If Camilo wants to locate the fraction
2
, in the number line, he should
5
use the geometric method of Thales following the next steps:
1. Draw a ray from O to A.
2. Starting from point O, an auxiliary r line is drawn, and with the
compass a measure is taken which will be one unit and this same
measure is moved on the line, so we have 5 units.
3. Point A of the ray is connected with number 5.
4. Lines parallel to segment A5 are drawn through points 1, 2, 3 and
4.
5. Finally, the desired fraction is noted.
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Geometric Pythagorean method to locate non exact fractions in
the number line
Now Camilo wants to locate √2 in the number line. To locate the
numbers he should follow the next steps:
1. Draw a segment and locate the unit of work there; it will also be
drawn perpendicular to the initial segment.
Created by the author
The reason why unit 1 should be taken is because the following has
to be made when the Pythagorean Theorem is applied:
ℎ = √12 + 12
ℎ = √12 + 12
ℎ = √2
2. Then the end points of the drawn segments are connected to
complete the triangle.
Created by the author
Finally, using the compass, the measure of the hypotenuse of the
triangle is moved on the line, which corresponds to √2.
Examples
Location and order of the rational numbers in the line
5
1
Order the following numbers in the number line: − 2 , 2, 2 , −3, √3, √2
● Principally, take into account that every negative number is
less than any positive number.
5
1
Therefore, − 2 𝑎𝑛𝑑 − 3 are less than 2 , 2 , √3, √2
−
3
1
● Now negative numbers are compared with each other. To do
this, the denominator is added to the integer to compare the
fractions.
5
−
2
The numerator of each fraction is multiplied by the denominator of the
other one:
−3 ∗ 2
−5∗1
−6
<
−5
Thus:
−3 < −
5
2
Then, the same procedure is followed with the positive numbers.
2
1
2∗2
4>1
1
2
1∗1
Therefore:
1
<2
2
When we know the values of √3 = 1,732 and√2 = 1,4142, they can be
compared like this:
1,4142 <
1
< 1,732 < 2
2
Finally, they are represented in the number line making use of the
geometric method of Thales to locate the fractions and, Pythagoras’
method, to locate the roots.
Created by the author
Remember that…
Constructing the square root of some numbers such as 3 and 7, among
others, depends on the location of other roots previously created in the
number line.
Learning Activity
In groups, solve the following exercises:
1. Order from lowest to highest and locate in the number line, use
ruler and compass to make auxiliary constructions if necessary.
1 1
● −3.5, −2, 4, − 2 , 2
9 4 2
1
● −5,3,5,−3
1
● √5,− √2, 2, − 3
2
● −2,5, 1.3, −√3, − 3 , 2.3
● Indicate whether the following sets are properly ordered:
1. From lowest to highest: −3.5,
1 2
3
2 3
2
, , √2 ,
_____________
4
2. From lowest to highest:−√2, −√3, −1.2, 1.2, 3 ______________
3. From highest to lowest: √7, 3, √5, 1.3, −2, −√7 ______________
4. From highest to lowest: 2.3, √5, 1.5,
1
3
, −1.4,
7
2
, −√5 _____
Activity 3
Skill 8. Find real numbers between two given real numbers.
Skill 9.Recognize the property of density in real numbers.
DENSITY OF THE REAL NUMBERS
Between two real numbers there are other numbers that can always be
found using the midpoint formula between two numbers.
It should be taken into account that if this procedure is repeated several
times, more numbers are going to be found each time between other
two in an infinite way.
To find a number between two given real numbers the following steps
are used:
1. Select two numbers 𝑥, 𝑦 to find a number between them.
2. Add both numbers 𝑥 + 𝑦.
3. The result of the addition is divided in 2,
𝑥+𝑦
2
.
The obtained number will be between the two numbers proposed
Examples
Two real numbers are meant to be found between
2
3
4
and5.
Created by the author
The numbers are added:
2 4 10 + 12
22
+ =
=
3 5
15
15
The result is divided in two:
22 2 22
÷ =
15 1 30
Simplifying
11
15
22 11
=
30 15
2
4
is the midpoint between 3 and 5
Created by the author
Then the procedure is repeated taking one of the initial numbers and the
one just found:
2 11
10 + 11 21
+
=
=
3 15
15
15
21 2 21
÷ =
15 1 30
Simplifying:
21
7
=
30 10
Created by the author
Consequently, it can be said that the numbers
between
2
3
4
7
10
and
11
15
are found
and 5.
Did you know that...?
The property of density in real numbers is applied to the rational
numbers set, thus, the result will always be a rational number.
Learning Activity
Answer True or False:
1. From these numbers
3
other numbers is 4.
1 3 5
, , the number located in between the two
2 4 9
__________
2. The correct way of finding a real number between two other
numbers is to add them and divide the result in two. __________
3. The property of density is the one that indicates that between any
two whole numbers there will always be another whole number.
__________
8
1
4. The midpoint between numbers 2, − 3 is − 3. __________
Abstract
ORDER RELATION OF REAL NUMBERS
Real numbers (R), being composed by the set of the natural numbers
(N), the rational whole numbers (Q) and the irrational numbers (I);
meet a relation of order; meaning, that they can be ordered from
highest to lowest or lowest to highest.
Rational numbers meet certain properties:
● Reflexive property: every element is related to itself.
● Antisymmetric property: if a number 𝑥 is related to another 𝑦,
and 𝑦 is related to 𝑥, then 𝑥 = 𝑦.
● Transitive property: if 𝑎 < 𝑏 and 𝑏 < 𝑐, then 𝑎 < 𝑐.
The symbols greater than (>), less than (<), and equal to (=) are used
to compare real numbers and organize them. Depending on the case,
these symbols can be used to locate numbers in a number line.
There are two geometric methods to locate rational and irrational
numbers in the number line. They are explained below:
1. The geometric method of Thales: is utilized to locate fractions
in the number line making use of a construction, in which a
segment is taken to be the unit and another segment is drawn
which is divided in the desired amount of fragments. Then, the
divisions are moved to the original segment by using parallel lines.
2. Pythagoras’ geometric method: this method is used for
constructing non exact square roots in the number line, starting
from the creation of a right triangle on the line and moving the
measure of the hypotenuse to that line using a compass.
Real numbers meet the property of density, that is to say, between two
real numbers another real number can always be found. To this end, a
midpoint should be used between two numbers. The midpoint is
obtained by adding the given numbers and dividing the result by two.
Infinite numbers can be obtained between two given number by
repeating this process.
Homework
In groups, solve the next exercises:
1. Indicate in each case what are the highest, the lowest, and the
equal elements, if there are any:
● −√5, −3,
2
5
1
, √2, − 4 ,
4
3
● −4.25, √3,
●
5
2
,−
12
5
8
7
9
, 1.35, − 2 , −
17
4
1
, −√7, 1.42, −2.4 , − 5
2. Locate the following numbers from lowest to highest in the
number line:
● −√3, 1, − 1, √2, √3
● 1.5,
4
5
3
, − 4 , − 2,
7
4
●
3
10
, 1.25, −√7, − 2.5,
6
5
3. Find three numbers between the two given numbers, and locate
them in the number line:
●
5
3
y 2
1
● −4 y
●
3
4
y
1
2
−3
4
4. Sara, Camila, Andrea, and Sonia went shopping; Sara spent less
than Camila, but more than Andrea. Sonia Spent more than Sara,
but less than Andrea. Organize the expenses of each girl in the
line from lowest to highest. Indicate who spent less and who spent
more.
5. Santiago and Carlos are heavier than Juan, while Andrés is less fat
than Santiago, but, more than Carlos. After representing them in the
number line, indicate who the least heavy boy is.
Evaluation
Answer true or false in questions 1 to 4, as appropriate:
1
1
1. If 0 < 2 and 0 < 3 then 0 < 2 ∙ 3 ____________________________
1
2. If− 4 < 0 then −1 ∙ 4 > 0
4
2
3. If − 5 < 5 then
1
4
5
______________________________
2
< − 5 _________________________________
1
4. If −3 < − 2 then −3 + 4 < − 2 + 4 ________________________
Select the correct answer in questions 5 to 9.
5. The geometric method of Thales allows us to construct one of the
following numbers in the number line:
a. 5
b. -7
c. 3/8
d.√10
6. The most adequate method to locate the number√7 in the number
line is:
a.
b.
c.
d.
Pythagoras’ Method.
Approximation Method.
Thales’ Method.
Location of exact numbers.
7. The simplest way to find a number in-between two real numbers is
to find their midpoint. Consequently, it can be affirmed that the
2
midpoint between 3and
𝑎.
1
4
𝑏.
3
4
𝑐.
5
6
is:
2
6
𝑑.
4
9
8. In the following sets of numbers, we can state that number √2 is
the highest in:
𝑎. √2,
2.5,
1 3
,−
2 5
𝑏. 1.4,
3
,
2
2
√2, − ,
7
2
𝑐. − , √2,
5
𝑑.
8
,
9
7
,
6
1
3
1
− ,
2
9
, −1.6,
4
2
3
1.33, √2
3
9. The correct ascending order for these numbers −2, 2 , 1.8, √2, −√3
is:
3
a. −√3, −2, 1.8, 2 , √2
3
b. −2, −√3, √2 , 2 , 1.8
c. 1.8,
3
2
, √2, −√3, −2
d. −√3, 1.8, −2,
3
2
, √2
Bibliography
● Rodríguez, J., & Astorga, A. Orden en el conjunto de los números
reales.
Retrieved
on
March
the
30th,
2016
from
https://tecdigital.tec.ac.cr/revistamatematica/cursoslinea/MATEGENERAL/t1-reales-expresionesalgebraicas/T1-1numeros-reales-julioetall/node12.html
● Ferrari, G., & Tenembaum, S. (2015). Resumen de Relaciones y
Funciones. Retrieved on March the 31st, 2016 from
http://www.x.edu.uy/
y
http://www.x.edu.uy/inet/RELACIONES_FUNCIONES.pdf
● OCWUPM, UPM. División de segmentos en n segmentos iguales.
Retrieved on March the 31st, 2016 from http://ocw.upm.es/ y
http://ocw.upm.es/apoyo-para-la-preparacion-de-los-estudios-deingenieria-y-arquitectura/dibujo-preparacion-para-launiversidad/contenidos/archivos/proporcionalidad/04-division-deun-segmento-en-n-partes-iguales.pdf