Lesson 2: Fractions and proportionality

Lesson 2: Fractions and proportionality
? Fractions
? Rational numbers
? Decimal numbers
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Fractions: an object, several interpretations
(1) A part of the whole
We have colored = of ...
(2) Share (division)
We want to share 3 candy bars
among 5 kids.
How much chocolate eats each kid?
(3) A point in the number line (a number)
=
¿
?
The denominator fixes the unit
2
=
The numerator is the number of units
you take
Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Some examples
∗ What fraction of each figure is colored?
(a)
(b)
=
=
=
=
(d)
3
(c)
Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Some examples
∗ I have eaten 1/3 of the chocolates in a box and there are 12
left. How many chocolates had the box when it was full?
∗ Paul read = of the pages of a book on Monday, on
Tuesday he was busy and he was able to read only = of
the pages that he read on Monday, and on Wednesday he
was free so he was able to read 140 pages and finish the
book. How many pages does the book have?
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Definition of fraction. Option 1
∗ A fraction is the quotient of two integer numbers, i.e., an
expression of the form a=b, with b 6
.
∗ Interpretations:
? part of a whole.
? solution to a share-out problem.
∗ Fractions such = , = , = ; : : : represent the same
quantity, i.e., they are the same rational number.
∗ Def: We say that a number is rational if it can be
expressed as the quotient of two integers, i.e., if it can be
expressed as a fraction.
The set of rational numbers is denoted by Q.
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Definition of fraction. Option 2
∗ Using the number line.
Given two integers a and b (with b 6
), the fraction a=b
represents the following point of the number line:
? take segment ; and divide it into b equal parts.
? take a of those equal parts.
∗ The main advantadge of this option is that it makes clear,
from the beginning, that fractions are an extension of the
set of numbers already known.
∗ Doing exercises as Represent = and = in the number
line helps in developping some intuition about fractions and
its meaning.
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Definition of fraction. Option 2
∗ If we have introduced this interpretation of fractions, we
can ask directly the question:
How much is
?
∗ For a kid working on this question on the number line (and
with graph paper) it is much easier to understand the
impossibility of adding up fractions that do not have the
same denominator.
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Equivalent fractions. Addition and substraction
∗ The concept of equivalent fractions is one of the most
important ones in this subject.
Given the fraction a=b, fractions that are obtained
multiplying (or dividing) numerator and denominator by the
same (non-zero) integer are called equivalent to fraction
a=b.
∗ It is very important to understand that equivalent fractions
represent the same part of a whole, and the same point on
the number line.
=
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Equivalent fractions. Addition and substraction
∗ Once the concepts of fraction and equivalent fractions are
properly understood, addition and substraction should be
easy to understand.
a) Fraction with different denominators cannot be added
(nor substracted).
b) Therefore, before adding or substracting we have to
find equivalent fractions with the same denominator.
∗ Example:
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Improper fractions, mixed numbers, integer division
∗ A fraction such as = (in general, fractions a=b where
a ≥ b) are usually called improper fractions and can also be
represented as mixed numbers:
∗ In general, if a q · b r, fraction a=b can be represented
r
a
r
also as q . From my view, it is better
q
.
b
b
b
∗ It is important to keep in mind that if option 1 has been
used (part of a whole), improper fractions are a nontrivial
generalization: what does eight sevenths of something
mean?
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Multiplication of fractions
∗ If we think only in the algorithm, multiplying fractions is
much easier than adding them.
Nevetheless, the concept is much more complicated.
∗ A good possibility is to generalize from integers, as follows:
?
·
is “the double of 18”
?
·
is “one third of 18”, i.e.,
∗ It is worth to pay attention here to a very important
relation between multiplication and divisionr:
multiplying by
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n
is the same as dividing by n
Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Multiplication of fractions
∗ Once that we understand the mutiplication of =n by an
integer (as a division by n) we can multiply =n by other
fraction:
a)
·
b)
·
∗ Now, it is easy to understand that
·
·
·
·
·
·
· X means
·
of X:
·
·
·
·
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Multiplication of fractions. Option 2
=
×
=
Here it can also be seen that
×
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means
of
.
=
=
Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Division
∗ First, something what I think it is not a good alternative.
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Division
∗ Option 1: Reduce to common denominator.
Quotative division of “quarters” in “quarters” is easier to
understand.
∗ The number line can be useful:
→ how many times does
“fit” in
?
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Division
∗ Option 2: Division as inverse of multiplication.
Divide is the same as multiplying by the inverse.
∗ We had already found this idea:
multiply by =n is the same as divide by n.
∗ The inverse of a rational number a is a number b such that
a·b
.
∗ Every rational number not equal to zero has an inverse.
a) The inverse of an integer n is =n.
b) The inverse of a rational number p=q is q=p
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Division
∗ Therefore,
·
∗ Of course, once the operation has been defined, the best
way of fully understanding it is to solve problems such as:
We have a barrel with 350 liters of water, and we want to
fill up bottles with capacity = of liter. How many bottles
will we fill up?
∗ If operations with fractions have been understood, we should be able
to solve (without algebra) problems like this one:
A person leaves = of his heritage to his only daugther, = of the
remaining part to an old uncle, he has to pay as taxes = of the
total heritage and he donates the rest,
euros, to charity. What
was the total value of his heritage?
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Exercise
∗ Compute and express as an irreducible fraction
× × − ×
−
−
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Order in Q
∗ Order in Q is defined in the same way as in natural
numbers: given two rational numbers a and b, we say that
a < b if b − a > .
∗ Properties of inequalities:
a) If a < b then a
c<b
c (for every rational number c).
b) If a < b and c > , then a · c < b · c.
c) If a < b, then −a > −b.
Therefore, if a < b and c < , then a · c > b · c
∗ Exercise: Find all rational numbers x for which
−x<
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Order in Q
∗ Rational numbers are “dense”: in Q the “next number”
does not exist.
Remark: between any two rational numbers there exist an
infinite number of rational numbers.
∗ But there are
√ numbers that are not rational:
Theorem:
is not a rational number.
∗ Last remark: The set of rational numbers is “countable”
(we can make a list with all of them).
(Of course, the list is infinite).
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Problem
∗ We have a bathtub with two faucets. It takes 1 hour for
the hot water faucet to fill in the bathtub, while it takes 30
minutes for the cold water faucet. If both faucets are
opened at the same time, and the flow in each of them is
the same as when they were opened alone, how long will it
take for the bathtub to be full of water?
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Decimal numbers
∗ Decimal numbers with finite expression: decimal fractions.
A decimal fraction is a fraction that is equivalent to
another one whose denominator is a power of .
∗ Example: = is a decimal fraction because it is equivalent
to = .
∗ What can be said about the denominator of a decimal
fraction?
∗ In 1585 Belgian mathematician Simon Stevin proposed to
represent quantities smaller than the unit considering
divisions in tenths, hundredths, .... For instance:
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Decimal numbers
∗ Decimal part can be seen as generalizing to negative
powers of 10 what we have make for the integer part.
:
where
·
:
·
·
·
−
·
−
·
−
∗ Review of basic arithmetic with decimal numbers.
∗ Mental calculation is also useful here. For instance:
a)
0
0
b)
0
c)
·
0
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Decimals in the number line
:
?
:
?
:
0
:
?
0
?
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Decimal ↔ fraction
∗ For decimal fractions (decimal numbers with finite
expression) conversion is easy.
Examples:
a)
b)
:
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Decimal ↔ fraction. Infinite expressions
∗ A lot of rational numbers do not have a finite decimal
expression.
Examples: = , = , = , ...
∗ A number like 0
· · · is called repeating decimal
number, and it is denoted by : .
∗ In a repeating decimal number there are two possibilities:
1. The full decimal part is repeated, as in
:
:
· · · pure repeating decimal
2. Only a part of the decimal part is repeated, as in
0
· · · mixed repeating decimal
:
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Decimal expression of rational numbers
∗ Theorem: The decimal expression of every rational number
is either finite or repeating (pure or mixed).
∗ Examples:
:
:
:
∗ How to express a repeating decimal as a fraction: “fracción
generatriz”.
∗ Example: express as an irreducible fraction
a)
:
b)
:
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015
Final remarks
∗ Decimal expression is not unique.
a) :
b)
c)
:
···
:
0
:
0
∗ Theorem: The set of decimal numbers is not countable.
This means that it is not possible to put in a list the set of
decimal numbers (even the set of decimal numbers in the
interval ; ).
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Pedro Ramos. Mathematics I. Grado de Educación Primaria (bilingual track). Alcalá University. Year 2014-2015