ILLUSTRATING AND ARRANGING RATIONAL NUMBERS || GRADE 7 MATHEMATICS Q1
Summary
TLDRThis educational video script offers a comprehensive guide to understanding and working with rational numbers. It begins by defining rational numbers and illustrating their classification using a Venn diagram, which includes integers and non-integers. The script then delves into examples of rational numbers, explaining how integers, fractions, percentages, and decimals can all be expressed in the form of 'a over b'. It further clarifies the concept by differentiating between terminating and repeating decimals, and proper and improper fractions. The video also teaches how to compare and order rational numbers using cross-multiplication and provides practical examples to solidify the concepts. The script concludes with a tutorial on comparing decimals and arranging fractions, making it an invaluable resource for anyone looking to enhance their mathematical skills.
Takeaways
- 📚 Rational numbers include integers (negative, zero, positive) and non-integers (fractions, percentages, decimals).
- 🔢 A rational number is any number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \).
- 📉 Negative integers are rational because they can be written as \( \frac{a}{1} \), such as \( -5 = \frac{-5}{1} \).
- 💯 Percentages are rational numbers expressed as parts per hundred, like 50% which is \( \frac{50}{100} \).
- 🔄 Terminating decimals are rational because they can be represented exactly as a fraction (e.g., 0.5 is \( \frac{1}{2} \)).
- 🔁 Repeating decimals are rational and can be denoted with a bar over the repeating digits (e.g., 0.333... is \( \frac{1}{3} \)).
- 📋 Proper fractions (where numerator is less than denominator) and improper fractions (where numerator is greater or equal to denominator) are both types of rational numbers.
- ⬇️ To compare rational numbers, use the property that if \( \frac{a}{b} < \frac{c}{d} \), then \( ad < bc \) provided \( b, d > 0 \).
- 📈 When ordering fractions, those with larger numerators are greater if the denominators are the same, and cross-multiplication helps compare when they're different.
- 📊 Decimals can be compared by their place value, with non-negative decimals being greater than negative ones.
Q & A
What are rational numbers?
-Rational numbers are numbers that can be expressed in the form a/b, where a and b are integers and b is not equal to zero.
How are rational numbers represented in a Venn diagram?
-In a Venn diagram, rational numbers are represented as a family that includes integers (negative numbers, zero, and positive numbers) and non-integers (fractions, percentages, and decimals).
What are the two types of decimals that are considered rational numbers?
-The two types of decimals that are considered rational numbers are terminating decimals and repeating decimals.
What is a terminating decimal?
-A terminating decimal is an exact representation of a fraction, obtained by dividing the numerator by the denominator with a remainder of zero.
Can you provide an example of a pure repeating decimal?
-An example of a pure repeating decimal is 0.555..., which can be expressed as 0.5 with a bar over the repeating digit 5.
How are percentages related to rational numbers?
-Percentages are related to rational numbers as they are expressed as a fraction of 100, denoted as 'per hundred', and can be written in the form a/b where a is the numerator and 100 is the denominator.
What is the difference between a proper fraction and an improper fraction?
-A proper fraction is a fraction where the numerator (a) is less than the denominator (b), while an improper fraction has a numerator that is greater than or equal to the denominator.
How can you compare two rational numbers when one is a fraction and the other is a decimal?
-To compare a fraction and a decimal, you can convert the fraction to a decimal by dividing the numerator by the denominator, and then compare the resulting decimals using standard comparison methods.
What is the comparison property of rational numbers?
-The comparison property states that for any rational numbers a/b and c/d (with b and d greater than zero), if a/b is less than c/d, then a*d is less than b*c, and vice versa.
How can you arrange a list of rational numbers in descending order?
-To arrange rational numbers in descending order, first convert all numbers to a common form (like improper fractions), then compare them using the comparison property, and finally order them from the highest to the lowest value.
Outlines
📐 Introduction to Rational Numbers
The paragraph introduces the concept of rational numbers and their representation using a Venn diagram. It explains that rational numbers include both integers and non-integers. Integers are further divided into negative numbers, whole numbers (including zero), and positive numbers. Non-integers encompass fractions, percentages, and decimals, which are further categorized into terminating and repeating decimals. The paragraph also provides examples of rational numbers, such as integers (negative, zero, positive), fractions (proper and improper), percentages, and decimals. It concludes with a definition of rational numbers as any number that can be expressed as a fraction a/b, where 'a' and 'b' are integers and 'b' is not zero.
🔢 Understanding Percent and Decimals as Rational Numbers
This paragraph delves into the representation of percentages and decimals as rational numbers. It explains that percentages are ratios expressed as a fraction of 100, using the percent sign and providing examples such as 1% (1/100) and 50% (50/100). The paragraph also discusses terminating decimals, which are exact representations of fractions, and repeating decimals, which are divided into pure repeating decimals (e.g., 0.555...) and mixed repeating decimals (e.g., 0.45555...). The concept of the vinculum, used to indicate repeating digits, is introduced. The paragraph emphasizes how these decimal forms can be expressed as fractions, thus qualifying as rational numbers.
📉 Comparing and Ordering Rational Numbers
The focus of this paragraph is on the comparison and ordering of rational numbers. It introduces the comparison property of rational numbers, stating that if a/b < c/d and both b and d are positive, then a*d < b*c. Practical examples are given to illustrate this property, such as comparing two-thirds with four-fifths and three-fourths with four-sixths. The paragraph also discusses how to order fractions, including mixed numbers, from highest to lowest. It explains that with the same denominator, the fraction with the larger numerator is greater, and with the same numerator, the fraction with the smaller denominator is greater. The paragraph concludes with a practical example of comparing and ordering fractions in the context of a shopping list for a recipe.
📋 Practical Application of Rational Numbers in Shopping
This paragraph applies the concept of rational numbers to a real-life scenario of shopping for ingredients for a tinola recipe. It describes how to compare and order the quantities of different items bought, such as chicken, onion, ginger, and pepper, using rational numbers. The paragraph demonstrates how to convert mixed numbers to improper fractions for easier comparison and ordering. It concludes with a summary of which item was bought in the largest and smallest quantities, using rational numbers for comparison.
📏 Arranging Fractions and Decimals on a Number Line
The paragraph discusses how to arrange fractions and decimals on a number line. It provides a step-by-step guide on comparing fractions with the same or different denominators and numerators, emphasizing that a larger numerator results in a larger value when the denominators are the same. The paragraph also explains how to compare decimals using place value, showing examples of comparing decimals with different numbers of digits after the decimal point. It concludes with a method for placing fractions on a number line in descending order, illustrating the process with an example.
🎓 Conclusion and Encouragement to Learn More
In the final paragraph, the video script wraps up with a summary and a call to action for viewers. It encourages viewers to like, subscribe, and enable notifications for more video tutorials on math lessons. The paragraph serves as a conclusion to the video, reinforcing the channel's role as a guide for learning math.
Mindmap
Keywords
💡Rational Numbers
💡Venn Diagram
💡Integers
💡Fractions
💡Percent
💡Decimals
💡Terminating Decimals
💡Repeating Decimals
💡Comparison Property
💡Number Line
Highlights
Rational numbers are defined as any number that can be expressed in the form a/b where a and b are integers and b ≠ 0.
Rational numbers include integers, fractions, percents, and decimals.
Integers are categorized into negative numbers, zero, and positive numbers.
Non-integers include fractions, percents, and decimals, which are further divided into terminating and repeating decimals.
Negative integers are rational numbers because they can be expressed as a negative integer over one.
Whole numbers are rational numbers as they can be expressed as the number over one.
Percents are rational numbers expressed as a fraction of 100.
Terminating decimals are rational numbers because they can be expressed as an exact fraction without remainder.
Repeating decimals are categorized into pure and mixed types, both of which are rational numbers.
Proper fractions, where the numerator is less than the denominator, are rational numbers.
Improper fractions, where the numerator is greater than or equal to the denominator, are also rational numbers.
Mixed numbers can be expressed as improper fractions, thus they are rational numbers.
Negative rational numbers can be expressed in three ways: with the negative sign on the numerator, on the denominator, or both.
The comparison property of rational numbers states that if a/b < c/d, then a*d < b*c.
When comparing fractions, the one with the larger numerator when the denominators are the same is greater.
For fractions with different numerators and denominators, cross-multiplication is used to compare their values.
Decimals are compared by their place value, with the larger digit in any place value making the decimal larger.
The video provides a method to compare and order rational numbers, including fractions and decimals.
The video concludes with practical examples of comparing rational numbers and their application in everyday scenarios.
Transcripts
[Music]
in this video we will illustrate
and arrange and compare rational numbers
so first what will be our objectives we
will define and illustrate rational
numbers and we will compare and arrange
rational numbers so first
let us illustrate rational numbers and
the family of rational numbers using
venn diagram
so this is the family of rational
numbers we have
integers and non-integers
under integers we have negative numbers
and whole numbers where it includes zero
and positive numbers now for
non-integers we have fractions
percent decimals under decimals we have
the terminating decimals
and the repeating decimals so this is
how we illustrate
rational numbers or the family of
rational numbers
using venn diagram let's have and if
let's have an example and define what is
a rational number so a rational number
is any number that can be expressed in
the form
a over b or a divided by b so
again a over b or a divided by
b where a and b are integers and b
should not be equal to zero let me
repeat
your a and your b or your numerator and
denominator should be integers
and your b should not be equal to 0 so
that your
rational number would not become
undefined
let's have an example of rational
numbers so we have
integers and under integers we have
negative four
zero one seven and all negative integers
zero and positive integers under
integers
we have here the whole number so d two
papaso
c zero and all the positive
integers another example proper fraction
where
we have one half two thirds ten over
fifteen
we also have improper fraction five over
three
fifteen over twelve fourteen over eight
percent we have fifty percent three
percent seventy five percent
and for decimals we have three point
five zero point
twenty five and one point three vin
gillum
so these are some of the examples of
rational
numbers which can be expressed in the
form of
a over b
so what are integers so this one this
is one of the examples of
rational numbers so let me explain we
have negative integers
and whole numbers so under negative
integers marunta young negative five
all negative numbers shampre now
why do we consider negative integers as
um rational number a rational number
should be expressed in the form of
a over b now i only have here negative 5
so where is your b
so the body this is your a so where is
your b
here so negative 5
over one so negative five can also be
expressed as negative five over one
negative four can be expressed as
negative four over one
ganondin so therefore we
negative five can be expressed in the
form of
a over b so negative numbers or negative
integers are considered
rational numbers now under whole numbers
again and then i have 0 and positive
numbers so
these numbers can be expressed in the
form of a over b
like this so zero over one one over one
two over one three over one
four over one and so on backhead because
one over one is the one
three over one is still three all right
so negative integers and whole numbers
are
uh under rational numbers or
examples of rational numbers now let's
proceed to
non-integers let's have first the
percent
so a percentage comes from the latin
percentum means by a hundred or per
hundred
is a number or ratio expressed as a
fraction
of 100 that's why we have 100 percent
when you got perfect you got 100 100
percent
right so a percent is uh
it's by a hundred and it's uh
expressed in the ratio of a fraction of
100
now it is often denoted using the
percent sign so i know you are all
familiar with this
so this is the symbol for the percent
okay
let's have an example so if i have here
one percent so this is how we write one
percent
this means one per 100 because a percent
means by a hundred so one percent one
per 100
if we're going to express this as a
ratio so that is 1 over 100
all right so we have here a over b so
this is a rational
number now 50
so 50 means 50 per 100 so that is 50
over 100
we have a nb a over b so this is a
rational number
now 100 is 100 over 100 or exactly one
because 100
divided by 100 is equal to
one now 100 percent
of any number is just the number
so meaning unchanged hindi magbabago
what is one hundred percent of five that
is still five one hundred percent of two
that is still two
so unchanged okay and then two hundred
percent
is two hundred over one hundred so if
we're going to divide this
that is exactly 2 so 200 percent
is actually twice the number
so if you were asked one what is 200
percent of
50 so you're going to multiply it by 2
that will become 100
all right so 200 percent of 2
that is 4 all right
next we have the decimals
so under decimals we have two kinds of
decimals under rational numbers
the terminating decimals and the
repeating decimals so let us
discuss first the terminating decimals
so a terminating decimal is the exact
representation of a
fraction so what do we mean by exactly
okay so ending all right
so when we say terminating decimal
exact it is obtained by dividing the
numerator by the denominator with
a remainder of zero so
exact because there is no remainder
so you want a remainder zero
so as um
remainder terminating decimals that was
that's why
it is called exact representation of a
fraction so if i have let's say
5 over 10 if we're going to divide this
that is 0.5 or 0.50
all right so exacto sha walang remainder
exact and adding decimal
all right now for repeating decimals so
terminating the exact representation
when we say repeating decimals we have
two kinds of it
so first we have the pure repeating
decimal so what do we mean by pure a big
sub
hand if i have 0.555
and atom and time three that's so
ellipse
0.555 now we can rewrite this
repeating sometimes if you're going to
divide a number using calculators
so ah this is an example of repeating
decimal
under pure repeating
we can express this since this is too
long machado
we can have 0.5 and then you put the bar
down above the answer repeating digit
not end
so since i'm repeating digit muay 5
puedi kanalang magla
another example if i have 0.181818 at
that that
so i'm gonna write this as zero point
eighteen angbor cos above
one and eight bucket one and eight
digits eight one
eight one
all right another kinds of decimal under
the rational numbers
are mix repeating decimals
0.555
all right so we're gonna what you're
going to do is
you're gonna write zero point four
five don't again versus five cases
repeating okay so the only repeating
digit is five so therefore we're gonna
place the bar
above five only all right
another example i have zero point three
one eight one eight one eight all right
so i'm gonna place the bar
above one and eight hindi kasama
kasih dinaman repeating so these two are
examples of
mix repeating decimal now
anong tawag
above the repeating digits so that is
what we called
the vinculum so the wind kilum is the
board that
indicates the repeating block you're
going to place
binky loom the ansati manga repeating
digits
only
next we have fractions so for fractions
we have proper fraction
proper fraction is a fraction where your
a is less than b so the bank a nation is
our numerator
and b is our denominators rational
numbers
so a is less than b so if your a
is less than the value of b then that
is a proper fraction that's why this is
also
considered as a rational number
another example under fraction is
improper fraction
improper fraction is also an example of
rational numbers because
even a is greater than b
greater than b
like 5 over 3 15 over 12 14 over 8
still we have a over b the bar so we can
still express this as a over b
so therefore improper fractions are also
rational numbers
and then for mixed numbers at the young
mere and chance whole number
and a fraction so this is a combination
of a whole number and a fraction so if i
have two
and two thirds one and three-fourths
three and four-fifths now
if you will be asking uh a rational
number
can be expressed in the form of a over b
yes we can express mixed numbers as a
over b
the bow we can express this as
an improper fraction how
you're gonna multiply the denominator by
the whole number
and then you're going to add the
numerator so three times two
that is six plus two that is eight and
then copy the denominator
so we have eight over three four times
one four
plus three that is seven and then copy
the denominator so that is seven over
four
and then five times three that is
eighteen a fifteen
plus four that is nineteen so nineteen
over five
all right so we already have a over b
so under uh none
integers is negative 1
over 2 equal to 1 over negative
two and negative one half
okay so try to analyze
negative signs or equal
yes they are all equal because
there are three ways to write a negative
rational number so even if we put the
negative sign
on the numerator or even in the
denominator or di to sagitta nila
that is still equal all right so
parepare
and they are still equal because we have
three ways
in writing a negative rational
number
all right so what is the relationship of
these three
so under none integers we have fraction
percent and decimals
how are you going to express a half
a half expression half is a fraction
percent and decimal all right so for a
fraction we can have it one half
for fifty first at four percent that is
fifty percent and for decimal that is
zero point five or zero point
fifty now let's have the comparison
property of rational numbers
so for any rational numbers a over b
and c over d with b greater than zero
and d is greater than zero
okay so remember that your a if
a over b is less than c over d
then a d is less than b c
so if a d is less than b c
then a b is less than c d so vice versa
if c d is less than a b
then um bc
is less than a d so vice versa
all right let's have an example so i
have here two thirds and four
fifths so how are we going to apply the
comparison property of rational numbers
so two times five that is ten
and then on the other side we have three
times four that is twelve
okay so ano which is less than
ten all right so since
therefore two-thirds is less than
four-fifths so
conditions
or four-fifths
is greater than two-thirds
another example three-fourths and four
over six
so this is eighteen this is sixteen
so which is less than we have this so
nandito so therefore four over six
is less than three fourths or
puederina three-fourths is greater than
four over six okay
now let's have the comparing and
ordering of fractions
mary bought four items in the market she
needed for
her tinola recipe which item did she buy
the most
and the list so first step is
arrange the fractions in descending
order when we say this ending
from the from highest to lowest
how are you gonna arrange the fractions
or rational numbers
if you have two or more fractions
so i have this now we all know
that this mixed number can be expressed
as
improper fraction so 2 times 1
2 plus 1 that is 3 over 2
all right so now let us arrange the
given rational numbers
from in descending order so from highest
to
lowest now it's obvious
that uh from the given items
chicken is
[Music]
all right now so what's next
anna young second highest not in so i
have here
three fourths nine over ten and four
over five so what are you gonna do
compare using the comparison property
so i compare nothing to this is 30
this is 36 so which one is bigger
36 so there therefore mas mata asi
9 over ten all right
and then four fifths
so compare nothing c three and five that
is fifteen
and then four and four that is sixteen
so which one is bigger
four
all right so let us go back to the
question
which item did she buy the most so
he picks a b hand three over two that
is chicken and which item
uh did she buy the least so and that
is ginger that is only three-fourths
kilogram okay how are you going to
illustrate this
on the number line so
i think number line i
lowest or in descending order
so three halves is the um largest
sodium so three halves is actually one
and one half so
d to the n and then so monodyto
and ayansha so this is how are we going
to place
the fractions on the number line
okay so how are we going to compare
naman
compare and arrange fractions
now remember uh there is a short
shortcut in arranging fractions if they
have the same denominators and
numerators
okay and like in previous leidner and
that they have
different numerators and denominators
okay
so if you were given a
fractions with the same denominators
remember that the bigger the numerator
the bigger the value
the bigger the numerator the higher the
value
so if you're gonna arrange this in
ascending when we say ascending from
smallest to largest
we have 2 over 5 followed by
four over five and then five five seven
over five and then eight over five
all right
the bigger the numerator the higher the
value
all right what if your numerator is not
in
okay numerators nothing are the same
so then
that is the greater value so we will
have
if we're going to arrange this in
ascending order ibx7 from this
simulate
five four over six four over eight and
the list
is four over nine
okay let us compare decimals how are you
gonna compare decimals so
let us use less than greater than or
equal sign
in comparing decimals so i have here
0.48
which is uh which
is bigger or lesser
than the other or are are they equal
compared to 0.84 so
one's value so tens hundred
this is four this is it so therefore
mas matasi 0.84
another so it's very obvious
that the other uh decimal is
a negative number or a negative um
there is a negative sign automatic
positive okay next i have 2.5 and 2.50
so which one is bigger or lesser
they are
that's why they are equal so 2.5
is equal to 2.50 another example
so 0 once at a 0 so
0 again tens quantity is a hundreds this
is zero this is already five
so e big sub hand this is bigger than
this so we have zero point zero five
zero
is greater than zero point zero zero
five and the last one
i have eighty point one two five and
eighty point one zero two five
so pannu yan so
that is
so this is uh greater than
this why i know some of you will wonder
mom this is one zero two five this is
bigger than this
no always compare that
a place value
this is zero all right so this is bigger
than this
thank you for watching this video i hope
you learned something
don't forget to like subscribe and hit
the notification bell for
updated ko for more video tutorials this
is your guide in learning your math
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