Download Teacher`s guide

SUBJECT
GRADE
LEARNING UNIT
MATHEMATICS
11th
OPERATING IN THE REAL NUMBERS
(R) SET.
TITLE OF LEARNING
OBJECT
RECOGNITION OF THE ORDER RELATION BETWEEN REAL
NUMBERS
CURRICULAR AXIS
STANDARD
COMPETENCIES
Numerical thinking, and Systems of Numeration.
Recognize the density and incompleteness of the rational
numbers through numerical, geometrical, and algebraic methods.
To determine the set of real numbers through its properties.
SCO: Identify the order properties in real numbers.
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.
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.
LEARNING OBJECTIVES
SKILLS/ KNOWLEDGE
LEARNING FLOW
ASSESSMENT
GUIDELINE
Introduction
Objectives. The teacher presents the objectives and may
establish others if desired.
Development
Activity 1. Relation of order of real numbers (Skills: 1, 2, 3).
Activity 2. Order of real numbers (Skills: 4, 5, 6, 7).
Activity 3. Density of real numbers (Skills: 8, 9).
Abstract
Homework
Evaluation
Glossary
Vocabulary box
Bibliography
English review topic
Students are expected to recognize the definition of the order
relation, and its properties, which he/she will be able to use to
order different numbers form the subsets contained in the real
numbers and to locate them in the number line, making use, if
necessary, of the geometric methods of Thales and Pythagoras.
Stage
Learning flow
Introduction
Introduction
Teaching/Learning Activities
Recommended
Resources
Using the animation, the teacher will Interactive resource.
present the relation of order in real Animation.
numbers, and the properties numbers
must meet to keep such relation; the
teacher will also introduce the symbols
utilized to establish numerical relations,
namely: greater than (>), less than (<),
and equal (=)
The location of numbers in the number
line is also applied, which implies the
construction of rational numbers and
non-exact square roots; to this end, the
geometric methods of Thales and
Pythagoras are used, accordingly.
Development
Activity 1
Finally, the property of density of real
numbers, and the way to find a number
in-between two given real numbers are
presented.
Activity 1
RELATION OF ORDER OF REAL
NUMBERS
Using the animation, the teacher
introduces the definition of the order
relation in real numbers, the symbology
utilized to organize numbers, and the
properties numbers should meet to keep
such relation.
The teacher uses the Did you know
that...? tool to introduce the existence of
two kinds of order relation.
There are two kinds of order relation:
partial order relation, which meets the
reflexive, antisymmetric, and transitive
properties; and the relation of Total Order
Interactive resource
animaction
which meets the condition of being partial
and all of its elements are comparable
Learning Activity
The student develops the learning activity Interactive resource
making use of the True or False tool in
the platform to answer 5 proposed
situations.
Recreational resource
Activity 2
Activity 2
ORDER OF REAL NUMBERS
The teacher used the infographic to
explain students how to compare real
numbers in order to know which number
is greater or less than other number
among a given set; besides he/she
presents the way of constructing and
locating rational numbers and roots in
which answers are not exact, with the
purpose of locating numbers in the
number line.
The teacher used the animation to
present the methods of Thales and
Pythagoras for the construction of
rational numbers, and non-exact roots.
The teacher gives practical examples of
the comparison and location of rational
and irrational numbers corresponding to
non-exact roots in the number line.,
The teacher introduces the information
making use of the fun fact 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 the Classroom
In
groups,
the
students
solve
the
proposed 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
●
−5,3,5,−3
●
√5,− √2, 2, − 3
●
−2,5, 1.3, −√3, − , 2.3
9 4 2
1
1
True or false game.
2
3
Learning Activity
The student develops the learning activity
making use of the True or False tool in
the platform to answer:
1. From lowest to highest:
−3.5,
1 2
, ,
2 3
√2 ,
3
2
_____________True
2. From lowest to highest:
−√2, −√3, −1.2, 1.2,
4
3
______________False
3. From highest to lowest:
√7, 3, √5, 1.3, −2, −√7 _____________False
4. From highest to lowest: 2.3, √5, 1.5,
Activity 3
−1.4,
7
, −√5
2
1
,
3
_____True
Activity 3
DENSITY OF REAL NUMBERS
Using a dialogue infographic, the teacher
presents the formula used to find a real
number in-between two given numbers,
as well as the steps that need to be
followed; which helps the student
recognize the density property of real
numbers.
Example
The teacher presents an example in
which two real numbers should be found
2
4
between fractions 3 and 5, resulting in
their location in the number line.
Recreational resource
The teacher uses the fun fact Did you
know that...? to present important
information.
The property of density of real numbers is
applied to the rational numbers set, and
consequently, the result will always be a True or false game.
rational number.
Learning Activity
The student develops the learning activity
making use of the True or False tool in
the platform to answer the proposed
situations.
1 3 5
1. From these numbers , , the number
2 4 9
located in between the two other
3
numbers is 4. FALSE
2. The correct way of finding a real
number between two other numbers is
to add them and divide the result in 2.
TRUE
3. The property of density is the one that
indicates that between any two whole
numbers there will always be another
whole number. FALSE
8
4. The midpoint between numbers 2, − 3
1
3
is − .TRUE
Abstract
Assignment/
homework
The
teacher
presents
through
an Conceptual map.
infographic the most relevant information
about the order relation of real numbers,
the properties that numbers should meet
to belong to such relation, how to
represent different numbers in the
number line taking into account the
geometric methods of Thales and
Pythagoras; as well as the property of
density of real numbers, and how to find
a number between two other given
numbers.
In group, answer the following exercises.
1. Indicate in each case what are the
highest, the lowest, and the equal
Student material
elements, if there are any, in each
case
2
1 4
● −√5, −3, 5 , √2, − 4 , 3
●
●
−4.25, √3,
5
12
,− 5 ,
2
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
●
3
,
10
1.25, −√7 , − 2.5,
− 4 , − 2,
7
4
6
5
3. Find three numbers between the two
given numbers, and locate them in
the number line.
5
● 3 and 2
●
−
●
3
4
1
4
and
and
−3
4
1
2
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
TRUE
Feedback:
Whenever
two
positive
numbers are multiplied, the result will be
positive.
1
2. If − 4 < 0 then −1 ∙ 4 > 0
FALSE
Feedback: Whenever a positive number is
multiplied by a negative number the
result will be negative
4
2
3. If − 5 < 5 then
4
5
2
< −5
FALSE
Feedback: If each number of
inequality is multiplied by −1,
inequality is reversed.
1
2
the
the
1
2
4. If −3 < − then −3 + 4 < − + 4
TRUE
Feedback: If the same number is added
to both sides of an inequality, the
inequality will not change.
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) Natural numbers.
b) Whole numbers.
c) Rational numbers.
d) Irrational numbers.
Answer key:/ c
Feedback: Thales’ method allows dividing
the unit in as many segments as the
denominator of a fraction indicates.
6. The most adequate method to
locate the number √7 in the number
line is:
a) Pythagoras’ Method.
b) Approximation Method.
c) Thales’ Method.
d) Location of exact numbers.
Answer key:/ a
Feedback: Pythagoras’ method is used to
construct non exact square roots.
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 midpoint between
2
5
and 6 is:
3
𝑎
1
4
𝑏.
3
4
𝑐.
2
6
d.
4
9
Answer key:/ b
Feedback: When comparing
the
given
numbers
it
with
becomes
evident that it is less than
greater than
3
4
5
6
and
2
3
8. In the following sets of numbers,
we can state that number √2 is the
highest in:
1
3
,−5
2
3
2 1
1.4, 2 , √2, − 7 , 3
2
7
1 2
− 5 , √2, 6 , − 2 , 3
8 9
, , −1.6, 1.33, √2
9 4
a) √2, 2.5,
b)
c)
d)
Answer key:/c
Feedback: After converting all the
numbers to decimals and comparing
them, √2 is the highest only in answer c.
9. The correct ascending order for
3
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, −√3, −2
2
d) −√3, 1.8, −2,
Glossary
Answer key:/b
Feedback: When comparing the numbers
in pairs, answer b is the only one
organized from lowest to highest.
 Relation:
correspondence
or Interactive resource
connection between two things.


Vocabulary
box
3
, √2
2
Reflection: Relation of an element
to itself.
Antisymmetry: Relationship of two
elements in which both of them are
equal.

Transitivity: Relationship between
three elements in such a way that
if 𝑎𝑅𝑏 and 𝑏𝑅𝑐 then 𝑎𝑅𝑐.

Thales’ Method: Based on the
Thales’
theorem
in
which
correspondences between triangles
can be found by creating congruent
segments.

Pythagoras’ Method: Based on the
Pythagorean Theorem, in which the
legs of a right triangle are related
with the measure of the
hypotenuse.
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
●
●
●
●
●
●
●
Bibliography
English
review topic

every time that.
Rodríguez, J., & Astorga, A. Orden
en el conjunto de los números
reales. Retrieved on March the
30th, 2016 from
https://tecdigital.tec.ac.cr/revista
matematica/cursoslinea/MATEGENERAL/t1-realesexpresionesalgebraicas/T1-1numeros-realesjulioetall/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/RELACIO
NES_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-lapreparacion-de-los-estudios-deingenieria-y-arquitectura/dibujopreparacion-para-launiversidad/contenidos/archivos/pr
oporcionalidad/04-division-de-unsegmento-en-n-partes-iguales.pdf
Ordinal numbers
Comparatives and superlatives
Passive voice
●
●
●