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SUBJECT
MATH
TITLE OF LEARNING
OBJECT
CURRICULAR AXIS
STANDARD
COMPETENCIES
LEARNING
OBJECTIVES
SKILLS/ KNOWLEDGE
GRADE
MATH 11th GRADE
LEARNING UNIT
OPERATING IN THE SETS OF REAL NUMBERS
Characterization of real numbers
Numerical thinking and systems of numeration
 Analyze decimal representations of real numbers to differentiate between rational and irrational
numbers.
 Recognize the density and incompleteness of rational numbers through numeric, geometric and
algebraic methods.
 Compare and contrast the numeric properties (natural, whole, rational and real) and their relations
and operations to properly create, handle and use the different number systems.
Based on their properties, establish the set of Real numbers.
SCO: Recognizes the set of real numbers
SKILL 1: Research the history and importance of real numbers in mathematical processes.
SKILL 2: Research the origin of imaginary numbers and their difference with real numbers.
SKILL 3: Find characteristics in the Cauchy, Weierstrass and Cantor’s theorems regarding real
numbers.
SKILL 4: Establish features of the Dedekind cut to define real numbers.
SKILL 5: Build a definition of real numbers.
SCO: Represent ray intervals
SKILL 6: Understand the definition of real interval.
SKILL 7: Identify the types of intervals (finite, infinite, semi-infinite) and their corresponding
notation.
SKILL 8: Relate the continuity notion to real intervals.
SKILL 9: Represent real intervals on a line.
LEARNING FLOW
ASSESSMENT
GUIDELINE
SCO: Recognize the properties of real numbers
SKILL 10: Identify the number sets to which a certain real number belongs to.
SKILL 11: Identify real numbers as the intersection between the set of rational and irrational
numbers.
SKILL 12: Locate real numbers on the number line, and if possible represent them using a ruler
and a compass.
SKILL 13: Do usual operations between real numbers.
SKILL 14: Use technological tools to calculate using real numbers.
SKILL 15: Recognize the addition and multiplication properties among real numbers.
SKILL 16: Understand the distributive property as a property that relates sums and multiplication
between real numbers.
Introduction
Objectives. The teacher presents the objectives and may establish other ones if desired.
Development
Activity 1. History of real numbers (Skills: 1, 2, 5)
Activity 2. Real number interval (Skills: 6, 7, 8, 9)
Activity 3. Rational and irrational numbers (Skills: 10, 11, 12)
Activity 4. Addition and multiplication properties of real numbers (Skills: 13, 14, 15, 16)
Activity 5. Conceptions of real numbers (Skills: 3, 4) Summary
Homework
With this assignment we hope the student identifies the sets that make up the real numbers, apply
the addition and multiplication properties for problem solving, expresses real numbers on the number
line, has the ability to work using real number intervals in order to present and solve everyday life
problems, and finally that they may define real and imaginary numbers
Stage
Introduction
Learning
flow
Introduction
Teaching/Learning Activities
Recommended
Resources
Using the animation, the teacher will present the historical processes that Animation
contributed to creating real numbers (from the Egyptian time until the “Numbers”
theories that formed the concept of real numbers proposed during the
XVIII century).
On the other hand, we show the importance of representing real numbers
using intervals that describe the sets, which may be: open, closed, halfclosed or half-open and infinite, as well as the graphical representation of
the number line.
Some real numbers cannot only be placed on the number line, but can also
be represented geometrically using a ruler and a compass by forming
triangles and rectangles (this allows to see the real dimensions of certain
numbers which are hard to measure).
Content
The teacher
presents the
topic
Finally, the addition and multiplication properties of real numbers are
formalized, presenting different methods and tools to work with the set of
real numbers.
Activity 1
SKILL 1: Research the history and importance of real numbers in
mathematical processes.
SKILL 2: Research the origin of imaginary numbers and their difference
with real numbers.
SKILL 5: Build a definition of real numbers.
History of Real Numbers
Using the computer graphics, the teacher will present the history of real
numbers, keeping in mind the time period when each subset of real
numbers is acknowledged: natural, whole, rational and irrational numbers, Map Inphograps
as well as, sets that are not part of real numbers, such as imaginary ones. “History Of The
Numbers”
Learning activity
Activity
1.1.
In groups, and based on the information given, create the subsets that Exercise 1.
make up the real numbers. Give an example for each case.
The teacher uses the Did you know that? tool to teach a basic concept
of real numbers.
Interactive
resource Did you
Real numbers are made up of rational and irrational numbers. Meaning
know that?
that all natural and whole numbers may be written as a fraction.
The teacher
presents the
topic
Activity 2
SKILL 6: Understand the definition of real interval.
SKILL 7: Identify the types of intervals (finite, infinite, semi-infinite) and
their corresponding notation.
SKILL 8: Relate the continuity notion to real intervals.
SKILL 9: Represent real intervals on a line.
Real number intervals
Using an animation, the teacher presents the type of intervals of real
numbers (closed, open, half-open and infinite intervals), as well as their
representation on the number line and their symbolic expression.
Animation “ Real
Numbers
Intervals”
Furthermore, the Very important resource shows information about the Interactive
continuity of functions in a real number interval.
resource
Very
Important that?
Starting from a problematic situation, present the applicability of intervals
and how they may vary depending of the characteristics of each situation.
Furthermore, develop an example of continuity of intervals using real
number graphics.
Learning Activity
Students solve the activity through an interactive game.
Activity
2.3.
Match the data on the left to the type of interval on the right.
exercise 1
f(x) = {x/2 ≤ x ≤ 9}
f(x) = {x⁄−3 < 2𝑥
< 10}
f(x) = {x⁄4 < 𝑥 ≤ 15}
f(x) = {x/x ≤ 5}
Closed interval
Open interval
Hal-open interval
Infinite interval
Activity
Exercise
2Activity
Exercise 2.
2.3.
2.3.
Classroom learning activity
Students do the suggested exercises.
1. In groups:



Match each of the situations to the corresponding type of interval
Represent it algebraically
Represent it on the number line.
Situations:
Real numbers between 0 and 5.





Real
Real
Real
Real
Real
numbers
numbers
numbers
numbers
numbers
from 1 to 10
greater than -4
less than 3
from -3 and less than 12
greater than -5 and up to 6.
2. Find the intervals where the functions are continuous.
 f(x) = x 3 + 3x + 1
1
 f(x) = x
x2, 𝑥 < 2
 f(x) = {
4, 𝑥 ≥ 2
−x, 𝑥 < −3
 f(x) = {
x, 𝑥 > −3
 f(x) = √x 2 − 1
Activity 3
Skill
SKILL 10: Identify the number sets to which a certain real number belongs
to.
SKILL 11: Identify real numbers as the intersection between the set of
rational and irrational numbers
SKILL 12: Locate real numbers on the number line, and if possible
represent them using a ruler and a compass
Rational and Irrational numbers.
Activity 3.1. and
Using computer graphics of a dialogue, the teacher presents different activity 3.2. Map
subsets that are part of the rational numbers, meaning natural and whole Inphograps and
numbers, fractions, periodical decimals and irrational numbers.
Image, text.
The teacher presents the representation of number √2 on the number line,
by using a ruler and a compass.
Learning activity
Along with your classmates, develop the following exercises on your
notebook.
1. Indicate to which number set or sets (natural, whole, rational irrational)
the following events correspond to.
a. Counting how many students have shoes size 38 in the classroom.
b. Dividing 8 portions of pizza among 6 people.
c. The diagonal of a room with 2m * 3m sides.
d. The minimum temperature of New York is -3°.
Activity
Exercise 1.
3.1.
2. Complete the chart using the symbols “it belongs” ( ∈ ) or “it does not Activity
belong” ( ∉ ) depending on the case. If a number belongs to more than Exercise 2
one number set, mark it as such.
ℕ
ℤ
ℚ
I
3.1.
−8
3
4
20
√9
√7
π
−
7
12
3. Express the following irrational numbers.
a. √3
b. √5
c. √7
d. √10
Activity 4
Skill
SKILL 13: Do usual operations between real numbers.
SKILL 14: Use technological tools to calculate using real numbers.
SKILL 15: Recognize the addition and multiplication properties among real
numbers.
Activity
Exercise 1.
3.1.
SKILL 16: Understand the distributive property as a property that relates
sums and multiplication between real numbers
Properties of Addition and Multiplication of Real Numbers
Using animation, the teacher presents the addition and multiplication
properties for real numbers, as well as their symbolic expression and
usefulness.
Animation
“properties
of
Addition
and
Multiplication Of
Real Numbers”
Technological tools for performing operations
Activity
4.2.
Table
Different technological tools which can be used to do operations with real Inphograps and
numbers as well as graph them on the number line and Cartesian plane, audio.
are presented. Some examples like Geogegra, Cabri, Derive and the page
http://www.wolframalpha.com, among other, are presented.
Question Activity
4.1. Exercise 1
Learning Activity
The student matches each of the operations to the corresponding property.
1
−3 ∗ (4 + )
5
= −3 ∗ 4 + (−3)
1
∗
5
1
−12 ∗
=1
−12
√2 + (−√2) = 0
Distributive
Inverse for multiplication
Inverse for addition
1
−4 ∗ (3 ∗ )
2
1
2
1
−9
+ (−5) =
2
2
√3 ∗ 1 = √3
Associative for multiplication
= (−4 ∗ 3) ∗
Internal for addition
Neutral element for multiplication
Activity 5
SKILL 3: Find characteristics in the Cauchy, Weierstrass and Cantor’s
theorems regarding real numbers.
SKILL 4: Establish features of the Dedekind cut to define real numbers.
Conceptions of Real Numbers
The teacher presents information about the conceptions that different
mathematicians such as, Cauchy, Cantor and Weierstrass, have about real
numbers. Dedekind cut were also presented.
Activity 5.1. Map
Inphograps
The teacher uses the Did you know that? resource to present data about Activity 5.2.
the conception of infinity.
Image and Audio
Abstract
Abstract
Using computer graphics, the teacher presents the most relevant Map Inphograps
information regarding subsets that are part of the real numbers, the
operations that can be done and the addition and multiplication properties.
The way to represent real number subsets by using the different types of
intervals, or their location on the number line, is also taught.
Finally, the conceptions that have defined real numbers are taught.
Homework
Homework
1. On your own, represent the following algebraic expressions as an
interval in the number line.
5
a. {x⁄−2 < 𝑥 < }
Homework 1
Table
4
b. {x⁄0 ≤ x < √2}
3
6
c. {x⁄ ≤ x ≤ }
2
7
d. {x⁄−2 > 𝑥}
e. {x⁄x < √5}
2. The following problems can be solved using the different addition and
multiplication properties of real numbers, indicate which property Homework 2
Text and Image
applies for each problem and solve.
a. In a park there were 5 firs and 4 pines. The gardener planted
another 6 pines. How many trees are there in total now in the park?
b. At a bakery they made 5 boxes of French bread and 4 boxes of
churros, each box weighs 3kg. How many kilos of bread are there?
c. If Camila has 20 packages of 12 candies each and Samuel has 12
packages of 20 candies each, how many candies does Camila have
and how many does Samuel have?
d. At a soccer tournament the winning team gets 3 points, the loosing
team 0 and the team that ties 1. If team A ties 8 games and team
B loses 7, how many points does each team have?
Evaluation
Evaluation
1. The set of real numbers is made up of two number subsets that are:
a. Natural and Irrationals
b. Rational and Irrational
c. Rational and Imaginary
d. Irrational and Whole
Evaluation
Answer key: b Feedback: The set of real numbers is made up of rational
and irrational numbers
2. In a Company they are offering three positions. You must be between
20 and 30 years old to position A, for position B from 25 to 35 years
old, and for position C from 24 to 32. The age range for people can who
apply to the three positions is:
a. [25,32]
b. [25,30]
c. [20,35]
d. [24,30]
Answer key: b Feedback: It is the only range where the three intervals of
age meet
3. Match
Number
set
that goes from 2 to 5.
A number set
which elements
are between -2
and 5.
Number
sets
which elements
go from -2 and
are less than 5.
[−2,5]
(−2,5)
[−2,5)
4. Complete:
a. Examples of ________ numbers are: 2, 5, 12.
b. _______ numbers consist of fractions and finite and periodical
decimals.
c. Roots that have an infinite and non-periodical decimal result are
part of the _______numbers.
d. Numbers -3, -5, 2, 10, are part of the set of _________numbers.
e. _________ numbers are not part of the set of real numbers.
f.
Answer key: 1. Natural 2. Rational 3. Irrational 4. Whole 5. Imaginary
For the following cases, indicate the property that applies for solving each
case:
5. Andrea and her daughter take a certain amount of money to the mall
2
1
in order to do some shopping; they spend 5 on clothes and 4 on toys.
In order to know how much money they used, they have to add the
expenses using which property?
a. Associative for addition
b. Neutral element for multiplication
c. Inverse addition
d. Internal addition
e.
Answer key: Feedback: The right answer is d: since adding to real numbers
results in another real number.
6. After shopping where they spent
2
5
of their money on clothes and
1
4
on
toys, they decided to eat something, reason why they used 0.3 of their
money. At the end, to know how much they spent in total, you have to
use which property?
a. Distributive
b. Associative for addition
c. Commutative for multiplication
d. Inverse addition
Answer key: Feedback: The right answer is b: when adding three real
numbers they have to be associated by two terms, added and then the
remaining term must be added.
7. Ana builds trash cans using plastic bottles she recycles. To build small
1
trash cans she uses 5 out of 150 bottles, and to build big trash cans she
uses
11
2
Glossary
Glossary
●
●
●
●
●
3
5
out of the 150 bottles. If Ana builds
21
2
of a small trash can and
of a large one, and I wish to know how many bottles she used in
total, I must apply which property:
a. Commutative for multiplication.
b. Associative for multiplication
c. Distributive
d. Internal for addition
e.
Answer key: Feedback: The right answer is c: since the final result is
obtained by adding the amount of bottles used for a big trash can and
the amount of bottles used for a small trash can; and then the resulting
amount is multiplied by the amount of both trash cans built by Ana
Rational numbers: number set made up of all the numbers that can be Table
Inphograps
expressed as fractions or finite or periodical decimals.
Irrational numbers: number set made up of radicals and infinite and
non-periodical decimals.
Complex numbers: reference to adding a real and an imaginary
number.
Interval: Measurable set of real numbers.
Periodical decimal: a decimal point with one or more decimals that
repeat themselves.
Bibliography
Bibliography
●
●
●
●
Vocabulary
Box
Vocabulary
Box
Stewart. J, Redlin. L, & Watson. S. (2012) Precálculo, Matemáticas para
el cálculo (6a. ed.) México: Cengage Learning.
http://www.profesorenlinea.cl/ (s.f.) Operaciones aritméticas y
propiedades con números reales. Retrieved on March 16 from
http://www.profesorenlinea.cl/matematica/Numeros_reales_propieda
des.html
http://tecdigital.tec.ac.cr/ (s.f.) Los números reales según Cantor u
Dedeking. Una propuesta didáctica. Retrieved on March 17 from
http://tecdigital.tec.ac.cr/revistamatematica/propuestas-didacticasem/v6n1-may-2005/
http://www.pedagogica.edu.co/
(s.f.)
Concepciones
históricas
asociadas al concepto de límite e implicaciones didácticas. Retrieved on
March
17
from
http://www.pedagogica.edu.co/storage/ted/articulos/ted09_08arti.pdf
● Cathetus: In general, a cathetus is a line falling perpendicularly on
a surface or another line. Retrieved on April 20, 2016 from:
http://mathworld.wolfram.com/
● Field: A field is any set of elements that satisfies the field axioms for
both addition and multiplication and is a commutative division algebra…
Examples include the complex numbers ( ), rational numbers ( ), and
real numbers ( ). Retrieved on
April 20, 2016
from:
http://mathworld.wolfram.com/
● Fir: a tall evergreen tree (= one that never loses its leaves) that
grows in cold countries and has leaves that are like needles. Retrieved
on April 20, 2016 from http://dictionary.cambridge.org/dictionary/
● Pythagoras: c582–c500 b.c, Greek philosopher, mathematician, and
religious
reformer.
Retrieved
on
April
20,
2016
from
http://www.dictionary.com/
Table
Inphograps
Glossary
Inphograps
● Theorem: (especially in mathematics) a formal statement that can
be shown to be true by logic. Retrieved on April 20, 2016 from
http://dictionary.cambridge.org/dictionary/
●
English
Review topic
English
Review topic
4. Simple present, Imperatives, Compound sentences
Index