ILLUSTRATING AND ARRANGING RATIONAL NUMBERS || GRADE 7 MATHEMATICS Q1

Summary

This 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

00:00

📐 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.

05:02

🔢 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.

10:03

📉 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.

15:07

📋 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.

20:08

📏 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.

25:10

🎓 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.

Keywords

💡Rational Numbers

Rational numbers are a fundamental concept in mathematics, defined as any number that can be expressed as the quotient or fraction \( \frac{a}{b} \) where 'a' and 'b' are integers and 'b' is not zero. In the video, rational numbers are illustrated using a Venn diagram, which includes integers and non-integers. The script explains that integers encompass negative numbers, zero, and positive numbers, while non-integers include fractions, percentages, and decimals. Rational numbers are central to the video's theme as they are the focus of the tutorial.

💡Venn Diagram

A Venn diagram is a graphical tool used to illustrate the logical relationship between sets of numbers. In the context of the video, a Venn diagram is used to show the family of rational numbers, which includes integers and non-integers. This visual representation helps viewers understand the classification of rational numbers and how they are organized into different subsets.

💡Integers

Integers are whole numbers that can be positive, negative, or zero. The video script clarifies that integers are a subset of rational numbers and can be expressed in the form \( \frac{a}{b} \), such as negative five being \( \frac{-5}{1} \). This example illustrates how integers fit within the broader category of rational numbers.

💡Fractions

Fractions are part of the rational numbers family and are represented as the division of two integers, where the numerator is divided by the denominator. The video distinguishes between proper fractions (where the numerator is less than the denominator) and improper fractions (where the numerator is greater than or equal to the denominator). Fractions are used to demonstrate how different parts of a whole can be expressed as rational numbers.

💡Percent

Percent means 'per hundred' and is used to express a ratio or fraction of 100. The script explains that percentages can be converted into rational numbers by expressing them in the form \( \frac{a}{100} \). For instance, 50 percent is equivalent to \( \frac{50}{100} \) or 0.5, showing how percentages are a type of rational number.

💡Decimals

Decimals are another way to express rational numbers and are discussed in the video as having two types: terminating and repeating. Terminating decimals have a finite number of digits after the decimal point, while repeating decimals have one or more digits that repeat infinitely. The video uses examples such as 0.5 (terminating) and 0.333... (repeating) to illustrate this concept.

💡Terminating Decimals

Terminating decimals are a type of decimal number where the fractional part has a finite number of digits. The video provides an example of 0.5, which is the decimal representation of the fraction \( \frac{1}{2} \). This concept is important as it shows how decimals can be exact representations of fractions.

💡Repeating Decimals

Repeating decimals are decimals where one or more digits repeat infinitely. The video explains pure repeating decimals, such as 0.333..., and mixed repeating decimals, such as 0.515151..., using a vinculum (a bar above the repeating digits) to denote the repeating part. This concept helps viewers understand the infinite nature of some rational numbers.

💡Comparison Property

The comparison property of rational numbers is a rule that helps in comparing two fractions. If \( \frac{a}{b} < \frac{c}{d} \) where both 'b' and 'd' are positive, then \( ad < bc \). The video uses this property to compare fractions like \( \frac{2}{3} \) and \( \frac{4}{5} \), demonstrating how to determine which is larger by multiplying the numerators and denominators and comparing the results.

💡Number Line

A number line is a straight line that shows numbers in order, extending in both directions. The video script mentions placing fractions on a number line in descending order, which helps visualize the relative sizes of different rational numbers. This visual tool is crucial for understanding the arrangement and comparison of rational numbers.

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

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