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Math 11th grade
LEARNING UNIT
Operating in the set
of real numbers
S/K
LEARNING OBJECT
Construction of some irrational numbers
SCO: Recognize the set of irrational numbers.
SKILL 1: Find out about the origin of irrational
numbers.
SKILL 2: Relate characteristics of repeating decimals in
a number to the set of irrational numbers.
SKILL 3: Identify irrational numbers as those related to
perfect squares and cubes.
SKILL 4: Infer from perfect squares and cubes the
definition of irrational number.
SCO: Identify the set of irrational numbers.
SKILL 5: Distinguish rational and irrational numbers.
SKILL 6: Classify irrational numbers into transcendental
and algebraic.
SKILL 7: Establish a hierarchy among the sets of
natural, integer, rational, and irrational numbers.
SKILL 8: Determine the properties of irrational
numbers.
SKILL 9: Approximate irrational to rational numbers for
calculation purposes.
SCO: Model the set of irrational numbers with constructions.
SKILL 10: Deduct that transcendental irrational
numbers are not constructible with ruler and compass.
SKILL 11: Support conclusions on non-constructible
irrational numbers.
SKILL 12: Relate segmentation with ruler and compass
to the construction of rational numbers.
SKILL 13: Construct irrational numbers with ruler and
compass.
SKILL 14: Construct with interactive programs to verify
procedure.
Language
Socio cultural
context of the LO
Curricular axis
SKILL 15: Identify application of some irrational
numbers (pi, e, golden ratio, etc.)
English
Colombia
Numerical thinking and number system.
Standard
competencies
Background
Knowledge
Basic Learning
Rights
English Review
topic
Vocabulary box
Compare and contrast the properties, relationships and
operations of numbers (natural, integer, rational and real) to
construct, manage, and use different number systems
appropriately.
Segment, circumference, intersection, number sets, real
number operations
Understand that between two real numbers there are infinite
real numbers.
May and might
Awning: a piece of material supported by a frame and used
to protect part of a building from the sun or rain.
Retrieved
from:
http://dictionary.cambridge.org/us/dictionary/english/awning
Establishment: a business or other organization, or the
place where an organization operates.
Retrieved
from:
http://dictionary.cambridge.org/us/dictionary/english/establis
hment
Forearm: the lower part of the arm, between the wrist and
the elbow.
Retrieved
from:
http://dictionary.cambridge.org/us/dictionary/english/forear
m
Navel: the small round part or hollow place in the middle of
the stomach which is left after the umbilical cord has been cut
at birth.
Retrieved
from:
http://dictionary.cambridge.org/us/dictionary/english/navel
Sole: the bottom part of a foot which touches the ground
when you stand or walk, or the front part of the bottom of a
shoe.
Retrieved
from:
http://dictionary.cambridge.org/us/dictionary/english/sole
NAME: _________________________________________________
GRADE: ________________________________________________
Objectives
 To construct some irrational numbers with ruler and compass.
 To recognize the set of irrational numbers based on historical
processes.
 To classify irrational numbers and their properties.
Introduction
After watching the image, determine a strategy to calculate the length
of the diagonal of a square.
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Activity 1. Origin of irrational numbers
Surface designs
In this section, we are going to analyze some concepts we must
consider in the construction and design of awnings for stores or business
establishments.
The company D&T will be taken as a reference as it designs its awnings
on bases having the following specifications: length and width of the
same size, as shown in the prototype (see the side of the prototype).
If the dimensions of the awning base are known, how should we
calculate its width?
Define irrational measures
A business engaged in fast food requests an awning to place it in front
of its establishment. The owners inform the company D&T that the base
̅̅̅̅
is already built and they have all the measures except for the 𝐴𝐵
diagonal.
Front of the awning:
Image 1. By the author.
Side of the awning:
Image 2. By the author.
The measurements of the awning width are defined by letter x. To find
this measurements, we will use the Pythagoras’s theorem:
ℎ2 = 𝑙 2 + 𝑙 2
ℎ2 = 12 + 12 (We substitute the values of legs)
ℎ2 = 1 + 1
ℎ2 = 2
√ℎ2 = √2
ℎ = √2
The measurement of the awning width is √2 m.
Characteristics of the awning width
In calculating the length of the diagonal, we have:
√2 = 1.414213562373095 …


It is a decimal number from
which infinite decimal digits
are derived.
There is not a repeating
decimal in the number.
The numbers with these characteristics are called irrational.
Definition of irrational number based on perfect squares and
cubes
Square numbers are those arising from 𝑎2 , such that 𝑎 ∈ 𝑁. See some
examples of square numbers:
Figure 3. Retrieved from
http://recursostic.educacion.es/gauss/web/materiales_didacticos/eso/ac
tividades/algebra/pautas/numeros_figurados/teoria.htm
Cube numbers are those arising from 𝑎3 , such that 𝑎 ∈ 𝑁. See some
examples of cube numbers:
Figure 4. Retrieved from
http://vemqueteexplico.blogspot.com.co/2012/01/numeros.htmlhttp://v
emqueteexplico.blogspot.com.co/2012/01/numeros.html
Irrational numbers from squares
If a natural number is a square, the result is another natural number.
√1 = 1
√4 = 2
√9 = 3
√16 = 4
√25 = 5
⋮
Based on the behavior of the square root of squares, we can state that
any square root of a natural number 𝑛, such that 𝑛 is not a square, is
an irrational number.
The same happens with the cube root of a cube. It will result in another
natural number; therefore, based on the behavior of the cubic root of
cubes, we can state that any cube root of a natural number 𝑛, such that
𝑛 is not a cube, is an irrational number.
Learning activity
1) Find out who Hippasus of Metapontum was and his relationship
with the origin of irrational numbers.
2) An irrational number is one that:
a. Has finite digital digits.
b. Is periodic.
c. Has infinite decimal digits but no repeating decimal.
d. Is on the same straight line.
3) Now imagine that the sides of the base are 3 m long each.
Calculate the measure of the awning width and determine which
number set the measure belongs to.
Figure 5. By the author.
4) Decide whether the following statements are true or false. Give
one or several examples supporting each:
a. The product between two irrational numbers is a rational
number. (t)__ or (f)__
b. The quotient between two irrational numbers is always an
irrational number. (t)__ or (f)__
c. The quotient between two irrational numbers is an irrational
number. (t)__ or (f)__
d. The difference between two irrational numbers is a rational
number. (t)__ or (f)__
5) Classify the following numbers into natural, integer, rational or
irrational and order them decreasingly.
34
3 √5
4
√14 − 8 − √3
√121 − √81
7
20 √3
Activity 2. Transcendental and algebraic irrational numbers.
Introduction
The divine number
One of the most outstanding
mysteries in the field of
mathematics is the golden
ratio, also known as the divine
proportion,
golden
mean,
golden section or phi (ɸ). This
Figure 6. Vitruvian Man by Leonardo number is found in nature, the
Universe, famous logo designs,
da Vinci. Retrieved from
important works of art and,
http://artivinilo.com/fr/home/231even, the human body.
hombre-de-vitruvio.html
Algebraic irrational numbers
The square framing the Vitruvian Man drawn by Leonardo da Vinci.
To solve for the value of the diagonal, we should find the solution to the
equation 𝑥 2 − 2.
𝑥 2 − 2 =0
𝑥2 = 2
√𝑥 2 = √2
𝑥 = √2
The diagonal is √2 and is an irrational number, which is referred to as
algebraic irrational number due to the following characteristics:



It is an irrational number.
It is the solution to a polynomial equation.
It is the square root of an integer.
Non-algebraic (transcendental) numbers
If the irrational number does not have the characteristics of an algebraic
irrational number, it is called transcendental number. For instance,
let’s define the ratio of the perimeter enclosing the Vitruvian Man to its
diameter.
Figure 7. Retrieved from http://artivinilo.com/fr/home/231-hombre-devitruvio.htmlhttp://artivinilo.com/fr/home/231-hombre-de-vitruvio.html

We will determine the ratio of the circumference (18.85 cm) to its
diameter (6 cm).
18.85
= 3.1416́
6

Let’s assume that now the measures are circumference equal to
6.28 cm and diameter equal to 2 cm.
6.28
= 3.14
2
If we continue with other circumferences of other sizes and get
the ratio of the measure of the circumference to the measure of
the diameter, we will obtain an approximate value to
3.141592653589793…, which is referred to as pi number (𝜋).
Learning activity
1) Calculate the following quotients and conclude on the results
obtained.
a. The quotient of the measure from the knees to the navel to
the measure from the sole of the foot to the navel.
b. The quotient of the measure from the elbow to the ring
fingertip to the measure from the elbow to the forearm.
c. The quotient of the measure from the knees to the neck to
the measure from the knees to the navel.
2) Construct
in
the
(https://www.geogebra.org/m/YhMm8vgX)
the
numbers:https://www.geogebra.org/m/YhMm8vgX
a.
25
8
b. √7
c.
d.
2
11
18
9
platform
following
e. √11
3) Classify the following irrational numbers into algebraic or
transcendental.
4)
a. √21
b. −√13
c. 𝑒
d. 2𝜋
e. sin (2)
f. √10
Abstract
Rational and irrational numbers
1. The differences between rational and irrational numbers are
defined.
2. Construction of rational and irrational numbers with ruler and
compass.
3. Classification of irrational numbers into algebraic irrational
numbers and transcendental irrational numbers.
4. Transcendental irrational numbers are not constructible with ruler
and compass; for example, the 𝜋 numbers, the 𝑒 number and the
golden ratio.
5. Origin of irrational numbers and discovery of classic
transcendental irrational numbers.
6. Pythagoras’s Theorem as a basis for the construction of irrational
numbers in the form √𝑎, such that 𝑎 ∈ 𝑧.
7. Approximation of irrational numbers to rational numbers for
calculation purposes.
Homework
a)
Find out about the Euler’s number and its origin.
b)
Construct with ruler and compass the numbers √2 and √3 on
the same real straight line and determine an approximate
position of the 𝑒 number.
2) Calculate the first 10 values starting from zero in the following
generality.
1 𝑛
3) (1 + 𝑛)
a. Calculate the sum of the first 10 terms.
b. What relationship do they have with the 𝑒 number?
Evaluation
Test your knowledge
For questions 1 and 2, complete the sentence.
1) The π number is obtained from the _______ of the perimeter of
the circle to its diameter.
a. products
b. quotient
c. product
d. quotients
2) One characteristic of an algebraic irrational number is the form
_______, such that 𝑎 ∈ 𝑧.
a. √𝑎
b. 𝑎𝑏
c.
𝑎
𝑏
d. 𝑎𝑏
3) Indicate if the operation results in an irrational number:
a. √2 × √8
b.
c.
√27
√3
3
2
+ √6
d. √12 − √12
4) Decide whether each of the following statements are true or false:
a. The ratio of the height of a person to the height from the
ground to his/her navel is the 𝜋 number. (f)___ or (t)___
b. Any irrational numbers are constructible with ruler and
compass. (f)___ or (t)___
5) Classify each number into rational, algebraic irrational
transcendental irrational. Mark with an X the correct answer.
Number
√3 × √5
cos (3)
√30
√2
√36
√9
or
Classification
Rational (_)
Algebraic irrational (_)
Transcendental irrational (_)
Rational (_)
Algebraic irrational (_)
Transcendental irrational (_)
Rational (_)
Algebraic irrational (_)
Transcendental irrational (_)
Rational (_)
Algebraic irrational (_)
Transcendental irrational (_)
Bibliography




González, P. (s.f) Historia de la matemática para la enseñanza
secundaria. Los elementos de Euclides. Retrieved from:
http://www.xtec.cat/sgfp/llicencies/200304/memories/elementseu
clides1.pdf
(s.a) (s.f) Números irracionales. Matemáticas 2 A y B. Retrieved
from:
https://investigacionmatematica.wikispaces.com/file/view/5+num
eros+irracionales.pdf
(s.a) (s.f) Sección Áurea en Arte, Arquitectura y Música. Retrieved
from:
http://matematicas.uclm.es/itacr/web_matematicas/trabajos/240/
La_seccion_aurea_en%20arte.pdf
Rivas, D. (s.f) Los números, operaciones y sus propiedades.
Apuntes de las clases de Cálculo 10. Retrieved from:
http://webdelprofesor.ula.ve/ingenieria/derivas/apuntes_calculo10
/%20losnumeros.pdf