lesson math 5

Component and Reading of Decimal numbers Component Each decimal number includes three parts: - Decimal point is the most important part of a decimal number.. Convert Decimal Fractions t

Whole numberConvert

Improper fraction

Improper fractionDenominatorWhole numberConvert

Combine

NumeratorProper fraction

Recalling knowledge

You can use either an improper fraction or a mixed fraction

to show the same amount

Do you remember?

• The numerator of a proper fraction is less than

its denominator

• The numerator of an improper fraction is greater

than (or equal to) its denominator

Example

Lan has two cakes She wants to bring some pieces for her mother.

We can see that there are one cake and a quarter of a cake.

Definition

What is a Mixed Fraction?

A Mixed Fraction is a whole number and a proper fraction

Rule and Remark

 Reading way: whole number -> fraction.

Example:

135135 : “one and three over five”, or “one and

three-fifths”

234234 : “two and three-quarters”

 In a mixed fraction, the fraction is less than one (the

numerator is less than the denominator)

Mixed Fractions or Improper Fractions

You can use either an improper fraction or a mixed fraction

to present the same amount

For example 114=54114=54, as shown below:

UNIT 2

Decimal part

/'desiməl pɑ:t/

phần thập phânDecimal part

Decimal point

/'desiməl pɔint/điểm thập phân

Integer partDecimal numberDecimal fraction

Decimal numberInteger partDecimal partDecimal point

Unit

Decimal fractionDecimal pointDecimal partDecimal numberInteger part

Unit

Recall knowledge

Do you remember?

A Decimal Fraction is a fraction

whose denominator (the bottom

number) is 10, 100, 1000, etc (in

other words, a power of ten)

Component and Reading of Decimal

numbers

Component

Each decimal number includes three parts:

- Decimal point is the most important part of a decimal

number

- On the left of the decimal point is a whole number

- On the right of the decimal point is a decimal part

Example: The number 17.591 has 3 parts with the whole

number (such as 17), the decimal point, and the decimal part(such as 591)

Place value:

When we write numbers, the position (or "place") of each digit is important

• Consider the whole number (integer part).

As we move to the left, each position is 10 times bigger from units, to tens, to hundreds, to thousands, to ten

thousand, to hundred thousand, to millions,…

• Consider the decimal part.

The first digit on the right means tenths (1/10) As we move

to the right, every place gets 10 times smaller (one tenth as big)

We can see the example:

Reading way

Whole number -> Decimal point ->

Decimal part.

Example: 35.168: thirty-five point one hundred sixty- eight

1.34: one point thirty- four

Convert Decimal Fractions to Decimals

(one space from right to left for every

zero in the bottom number)

Example:

1 Convert 210000210000 to a decimal number?

We write down 2 with the decimal point 4 spaces from the right (because 10000 has 4 zeros)

So, 210000=0.0002.210000=0.0002

2 Convert 12341001234100 to a decimal number?

We write down 1234 with the decimal point 2 spaces from the right (because 100 has 2 zeros)

So, 12341

Convert Fractions to Decimals

The simplest method is to use a calculator

Just divide the top of the fraction

by the bottom

To convert a Fraction to a Decimal manually, follow two

 Find a number you can multiply by the bottom of the fraction to make it 10, or 100, or 1000, or any 1 followed by0s.

 Multiply both top and bottom by that number

Step 2 Convert a decimal fraction to a decimal number.

00=12.34

Example: a) Convert 3535 to a decimal number?

Step 1: Convert a fraction to a decimal fraction:

- We can multiply 5 by 2 to become 10

- Multiply both the numerator and the denominator by 2:

Step 2: Convert 610610 to a decimal number?

We write down 6 with the decimal point one spaces from the right (because 10 has one zero)

So, 610=0.6.610=0.6

Answer = 0.6

b) Convert 720720 to a decimal number?

Step 1: Convert a fraction to a decimal fraction:

- We can multiply 20 by 5 to become 100

- Multiply both top and bottom by 5 :

Step 2: Convert 3510035100 to a decimal number?

We write down 35 with the decimal point 2 spaces from the

right (because 100 has 2 zeros)

Step 1: Write down the decimal divided by 1, like this: decimal/1.

Step 2: Multiply both top and bottom by 10 or 100 or 1000, such that the

number after the decimal point are equal to the number of zero in the bottom

Step 3: Simplify (or reduce) the fraction.

Example:

a) Convert 0.35 to a fraction ?

Step 1: Write down 0.35 divided by 1:

0.3510.351

Step 2: Multiply both top and bottom by 100 (there are 2

digits after the decimal point):

Answer: 720720

Note: 3510035100 is called a decimal fraction

and 720720 is called a common fraction

A special note

Convert 0.333… to a fraction?

If you really meant 0.333… (in other words 3 repeating

forever which is called 3 recurring) then we need to follow aspecial argument.

In that case we write down:

Step 1: Write down 0.333… divided by 1: 0.333…

10.333…1

Step 2: Then MULTIPLY both top and bottom by 3:

/'ekstrə/ thêmExtra

/ri'dju:s/

giảmReduce

ExtraValue

DigitReduceCompare

DigitExtraReduceValueCompare

Recall

Do you remember?

Each decimal number includes three parts:

• The decimal point is the most important part of a decimal number

• On the left of the decimal point is the whole number

• On the right of the decimal point is the decimal part

Now we will compare decimal numbers

Equivalent Decimal Numbers

Writing extra or reduce zeros to the

right of the last digit of a decimal

does not change its value

Comparing Decimal Numbers

Compare whole numbers if they are

We see that 532 < 682, therefore 532.48 < 682.26

Compare the decimal parts if the whole numbers are the same

Example

However, it is not correct !

So, we can write one extra zeros to the right of the last digit

of one decimal so that both decimals have the same number

number of decimal digits.

• We start at the left and compare digits

in the same place-value position

Unit 4

/səb´trækt/ trừ

Subtract

Multiply

/'mʌltiplai/ nhân

Multiply

CombineMultiplySubtract

DivisionCombineAdditionMultiplySubtract

DivisionAdditionMultiplyCombineSubtract

Adding Decimal Numbers

Do you still remember?

Adding the Natural

numbers:

We can do addition by

writing one number below

the other and then add one

To add the decimals, follow these steps:

 Write down the numbers, one under the other, with the decimal points lined up

 If there is no number in the column, add as many

zeroes as you need to the end of the number

 Then add using column addition, remembering to put the decimal point in the answer

Add zero at the end of the number 8 + 8.0 18.5

To subtract decimals, follow these steps:

• Write down the two numbers, one under the other, with the decimal points lined up

• If there is no number in the column, add

as many zeroes as you need to the end of the number

• Then subtract normally, remembering to put the decimal point in the answer

To multiply decimals, follow these steps:

• Multiply normally, ignore the decimal points

• Then put the decimal point in the answer

(it will have as many decimal places as the two original numbers combined).

Answer: 123 × 0.62 = 76.26

Dividing Decimal Numbers

Do you still remember?

Now, we will study decimal numbers division in two cases

Divide a Decimal Number by a Whole Number

(the number being divided).

Step 5: Bring down the next digit of dividend and the repeat the process.

18 ÷ 3 = 6

6 × 3 = 18

18 – 18 = 0

Answer: 16.8 ÷ 3 = 5.6.

Step 5: Bring down the next digit of dividend and the repeat the process

(the number being divided)

Step 5: Bring down the next digit of dividend and the repeat the process

11 ÷ 3 = 3

3 × 3 = 9

11 – 9 = 2

Step 6: If remainder 0, we put zero on the right of this remainder and repeat the process

Step 1: We move from dividing a Decimal Number

by a Whole number to dividing a Decimal Number

by a Decimal Number by shifting the decimal point

of both numbers to the right.

Step 2: We divide by a whole number and can

Unit 5

Percent

Percent means “per hundred” A percent can be written as a fraction or a decimal.

Percent Fraction Decimal

15% = 151001510

Percent can be written as a fraction

Percent can be written as a decimal

We need to move “Decimal point” two place to the left, erase “%” and put “0” on the left of “Decimal point”

What about 4%?

4.% > 04.% -> 0.04

Writing“digit 0” on the left of “digit 4”

Percentage with four operators (+, –, ×, ÷)

How to convert fractions to percentages?

First method

Divide the top of the fraction by the bottom, multiply by 100 and add a "%" sign

Example : What is 3434 as a percent?

• Divide the top of the fraction by the bottom

3 ÷ 4 = 0.75.

• Multiply the result by 100, put the “%” after the result

0.75 × 100 = 75%.

Example :

There are 20 students in

John’s class Nine of them

are male

a) What percentage of the

students in John’s class is

male?

b) What percentage of the

students in John’s class is

11 ÷ 20 × 100 = 55%

Second method

Percent means "per 100", so try to change the fraction to a100a100 form.

Example: The price of a pizza is 20

dollars The shop decreased 30%

from the initial price for each pizza

What is the price of each pizza after

the sale off?

Example: You have 70

dollars Can you buy the

So you can’t buy the shoes.

Example: A store sells a puppy

with the price of 80 dollars

John bought it with 35% the total

So John paid 28 dollars.

Problem of fresh nuts, dry nuts

Formula

Fresh nuts = original nuts +

waterDry nuts = original nuts + water

a × x = b × y

x: weight of fresh nutsy: weight of dry nutsa%: percentage of original nuts

in fresh nutsb%: percentage of original nuts

in dry nuts

Example : The amount of water in fresh nuts is 20% We

dry 200 gram of fresh nuts and get 180gram of dry nuts

What is the percentage of water in dry nuts?

Sulution:

The percentage of original nuts

in fresh nuts is:

a = 100% – 20% = 80%

The percent of original nuts in

dry nuts is:

The amount of water in fresh nuts is 20%

The amount of water in dry nuts is 10%

How many kilograms of dry nuts when

We make drying 180 kilograms of fresh nuts?

Analyze problems

Fresh nuts = original nuts + water

Unit 6

Definition

Time is the ongoing sequence of events taking place

We measure time by using seconds, minutes, hours, days, weeks, months and years

There are lots of other things we can measure, but those are the most common

A hours C minutes + B hours D minutes

Speed is measured as the distance

travelled per unit of time

• Speed = Distance ÷ Time

• Time = Distance ÷ Speed

• Distance = Speed × Time

Speed = Distance ÷ Time

Solve:

Distance: 50 metersTime: 2 secondsThe average speed of football:

Speed = Distance ÷ TimeSpeed = 50 ÷ 2 = 25 (m/ s)

Answer: 25 m/ s Example : Time = Distance ÷ Speed

Steve ran 5000 metres from a swarm of bees at an average speed of 5 metres/second before he drove into the pond How long did he run?

Solve:

Distance: 5000 meters

Speed: 5 metres/second

The time which Steve ran:

Time = Distance ÷ Speed

Time = 5000 ÷ 5

Time = 1000 (s)

Answer:1000 s

Example : Distance = Speed × Time

Mike rides his motorcycle at an average

speed of 20 metres/second for 500

The distance which Mike ride:

Distance = Speed × Time

Unit 8

Triangles

Definition of triangles

The triangle ABC has:

• Three sides: AB, AC, BC

• Three vertices: A, B, C

Acute triangle is a triangle with three acute angles.

Right triangle is a triangle with one right angle and two

acute angles

Obtuse triangle is a triangle with one obtuse angle and two

acute angles

Heights and bases

BC is the base, AH is the height corresponding to base BC The length of AH is the height

 Acute triangle ABC

 Obtuse triangle ABC

 Right triangle ABC

AB is the height corresponding base BC

Area of a triangle

The area is half of the base times

height.

"a" is the distance along the base

"h" is the height (measured at right

angles to the base)

Area = a × h ÷ 2

The formula works for all triangles

Note: a simpler way of writing the

formula is ah ÷ 2

Example

1 What is the area of this triangle?

Solution: We have the distance along the base “a” = 8 cm.

Solution: We have the distance along the base “a” = 5 cm.

Height = Area × 2 ÷ Base

Base = Area × 2 ÷ Height

2 Find the base of a triangle

1 A triangle ABC has the height AH = 6 cm corresponding

to the base BC = 12 cm and AC = 9 cm What is the length

of the height BK of the triangle ABC?

Solution: The area of the triangle ABC is AH × BC ÷ 2 = 6

× 12 ÷ 2 = 36 cm2

The height BK of triangle ABC is Area × 2 ÷ AC = 36 ×

2 ÷ 9 = 8 cm

2 A triangle ABC has the height AH = 4 cm corresponding the base BC = 4.5 cm and the height BK corresponding the base AC = 3cm What is the length of the base AC of the triangle ABC?

Solution: The area of the triangle ABC is AH × BC ÷ 2 = 4

What is the length of the base BC of the triangle ABC?

Solution:

We draw the height AH So, AH is the height of both

triangles ABC and ABD

The height AH of the triangle ABD is 37.5 × 2 ÷ 5 = 15 cm

So, the base BC of the triangle ABC is: area (triangle ABC)

Solution:

We see that AH is the height of both triangles ABC and ACD

The height AH of the triangle ABD is 50 × 2 ÷ 5 = 20 cm

So, the area of the triangle ABC is AH × BC ÷ 2 = 20 × 25

÷ 2 = 250 cm2

Unit 9

Definition of Trapezoid

A trapezoid is a four-sided shape that has two sides that are parallel and two sides that are not parallel

Component

The parallel sides are the "bases"

The other two sides are the "legs"

The distance (at right angles) from one base to the other is

called the "height"

We have AM=23ABAM=23AB

So, AM=27÷3×2=18AM=27÷3×2=18 cm

On the other hand, CD = AB and AD = BC

So, The area of a trapezoid AMCD

is: (AM+CD)×AD÷2=(18+27)×15÷2=337.5cm2.(AM+CD)×AD÷2=(18+27)×15÷2=337.5cm2

Unit 10

Definition

A circle is the set of all points on a plane that are a fixed distance from a center.

How to draw a circle?

A circle is easy to make:

Draw a curve that is "radius" away from a central point.All points are the same distance from the center

The tip of the compass draws a circle

The Radius is the distance from the center outwards.

The Diameter goes straight across the circle, through the

center

The length of a diameter of a circle is 2 times the length of its radius

The circumference is: 5 × 3.14 = 15.7 cm.

2) Find the circumference of a circle with a radius of 7 cm

The circumference is: 7 × 2 × 3.14 = 43.96 cm

Find the diameter and radius when we know the circumference of a circle

The diameter is: 15.7 ÷ 3.14 = 5 m

2) Find the radius of a circle with circumference C = 18.84 dm

The radius is: 18.84 ÷ 2 ÷ 3.14 = 3 m

How to find the semicircle?

The diameter of this circle is: 4 + 4 = 8 cm

The circumference of a semicircle is: 12.56 + 8 = 20.56 cm

Area of a circle

How to Calculate the Area?

To find the area of a circle, we multiply the radius by itself