Yearly Archives: 2022

Objects Around Us

When we look around, we find ourselves surrounded by a number of objects. Some of these different objects are made from a number of different materials, while others are made using the same material. For Example, both desk and chair are made from wood while pen and dustbins are made using plastic. The material from which an object is made depends on its properties.

Properties of Materials

1. Appearance

Materials can be classified on the basis of how they look or appear to be. Some materials have lustre, which is a very gentle sheen or soft glow to them while others are plain and dull looking. Materials that have such lustre can usually be classified as Metals. Examples include gold, copper, aluminium, iron etc. Usually, a metal loses its lustre after some time due to the action of moisture and air on it. Therefore only freshly-cut metals appear to have lustre on them.

2. Hardness

Materials can also be classified on the basis of hardness.

Materials that can be easily compressed or scratched are called Soft.

Materials that cannot be scratched and are difficult to compress are termed as Hard.

3. Soluble or Insoluble

Materials that can be dissolved in water upon stirring are said to be soluble materials. For Example, Sugar and Salt can be dissolved in water.

Materials that cannot be dissolved in water no matter how much we stir them are said to be insoluble materials. For Example, Stones and Clothes cannot be dissolved in water.

Not just solid materials, even liquids have the property of being soluble or insoluble. For Example, Lemon juice can easily dissolve in water while oil does not dissolve and deposits as a thin layer on the uppermost layer of water.

4. Objects may float or sink in water

There are some insoluble objects or materials which sink to the bottom of the surface when dissolved in water while some other float on the surface of the water. For Example, leaves and corks float in water while rocks and coins sink in water.

5. Transparency

Objects or materials which can be seen through are said to be transparent objects. For Example, Glass, clear water and some plastics can be seen through and are hence transparent materials.

Objects and materials through which things can be seen but only partially are called Translucent objects. Butter paper and frosted glass are some examples of translucent objects.

Objects which cannot be seen through are known as opaque objects. For Example, Metals, wood and cardboard are some examples of opaque materials as you cannot see through them.

Thus, we can group objects on the basis of their appearance, whether they are hard or soft, whether or not they can be compressed, if they dissolve in water or not and if they don’t do they float or sink and lastly if they can be seen through clearly, partially or at all. In this way, materials can be grouped on the basis of their similarities and differences.

Why do we need to group objects?

We need to group objects for a number of reasons:

• Convenience to store: We often group objects in order to store similar objects together in order to make locating them easier in the future. Even in our homes, we store spices together in the kitchen while storing washing products in our bathrooms.
•  Convenience to study: We also group objects so that it becomes easy for us to study their features as well as the patterns of these features.

Variety in Fibres

Yarn: Yarn is defined as a long, twisted and continuous strand composed of interlocked fibres or filaments which are used in knitting and weaving to form cloth.

Fibres: The thin threads or filaments which form a yarn are called Fibres.

Where do fibres come from?

Fibres can be broadly classified into two broad categories:

Natural Fibres: Fibres that come from plants and animals i.e. are found in nature are called Natural Fibres. Examples:

• We get jute and cotton from plants.
• Wool is acquired from the fleece of a goat or sheep. It can also be acquired from the hair of yak, rabbits and camels.
• Silk fibre can be procured from the cocoon of silkworms.

Synthetic Fibres: Fibres that are made of chemical substances i.e. substances not found directly in nature are classified as synthetic fibres. Examples include nylon, acrylic and polyester.

	Natural Fibre	Synthetic Fibre
1.	Natural fibrers are fibers that are found in nature. Ex: Woo, Silk and Cotton etc.	These fibres are man made or simply prepared in lab. Ex: Nylon, Teflon etc.
2.	They are good absorbents and so able to absorb heat, temperature, cold, sweat etc. depending on conditions and nature of fibres.	They do not have such pores as they are made up of chemical and so do not act as good asorbents.
3.	No spinning process is required for filament production.	Melting, wet or dry spinning processes are used for filament production.
4.	Comfortable in use.	Not as comfortable as natural fibres.
5.	Their length is naturally obtained and it is not possible to change the fibre structure.	Their lengths can be controlled by man and the fibres can easily be changed to different structures.

Some Plant Fibres

1. Cotton

A field of cotton

Where does cotton wool come from?

• Cotton plants are grown in fields usually at places having a warm climate and black soil.
• Some cotton producing Indian states are Punjab, Gujarat, Madhya Pradesh, Karnataka, Maharashtra etc.
• Cotton plants bear fruits the size of a lemon called Cotton Balls which burst open upon maturing and the seeds wrapped up in cotton fibre become visible. Cotton is generally picked by hand from these balls.

Ginning: Ginning of cotton can be defined as the process of separating cotton fibres from cotton seeds. Traditionally, ginning used to be done by hand but these days machines called double roller cotton ginning machines are widely in use.

In the above figure, we see a boy ginning by hand.

2. Jute

A jute plant

• Jute fibre is obtained from the stem of the plant.
• Unlike cotton, jute is cultivated in the rainy season.
• Some jute producing Indian states are Bihar, Assam and West Bengal.
• The plant is harvested during its flowering stage.
• The stems of these harvested plants are then soaked in water for four to five days
• The stems are left to rot and then the fibres are picked out by hand.

Yarn: Yarn is the spun thread that is made from fibres in order to produce a fabric.

Spinning Cotton Yarn

Spinning: Spinning is the process of constructing yarn from fibres in which fibres from a huge heap of cotton wool are taken out and twisted which brings them together to form a yarn.

There are two major devices called Takli which is a hand spindle and Charkha which is also a hand-operated device, are used for spinning.

The spinning of yarn on a bigger scale is done using spinning machines following which these yarns are used to weave fabric.

Khadi was the term used to denote clothes which were made from homespun yarn.

On the left we can see a charkha and on the right we can see a simple takli.

Yarn to Fabric

There are two major ways using which yarn is converted to fabric, namely, Weaving and Knitting.

• Weaving: The process of entwining two sets of yarn simultaneously to make fabric is called Weaving. The process is done using a loom (which can either be operated by hand or by a machine) which interlaces two sets of yarn at right angles to each other.

The above figure represents the process of weaving. 

• Knitting: Knitting is the process by which a single strand of yarn is used to make a piece of fabric. Socks, sweaters, mufflers and a lot of other winter clothes are made of knitted fabrics. Knitting can be done by hand as well as by machines.

History of Clothing Material

• In earlier times, when people did not have access or the knowledge to process fibre, big leaves and the bark of trees were used by people to cover themselves.
• After settlement began in agricultural communities, they learnt how to weave. They used grass and twigs to make mats and baskets. Animal hair or fleece and vines were warped together into stretched out strands which were then woven into fabrics.
• There was an abundant growth of cotton in areas near Ganga, which the early Indians readily used to make fabrics for themselves.
• There is another plant named flax which yields natural fibres.
• The early Egyptians cultivated both cotton and flax and used them for creating fabrics. These plants grew near the river Nile.
• But in those days, people were not aware of the process of stitching. They simply used to wrap around the fabric around different parts of their bodies. Even today unstitched clothes like sarees, dhotis, lungis or turbans are widely in use.
• It was with the advent of the sewing needle that people learnt how to stitch fibres to make fabric.

Nutrients: A nutrient can be defined as components that are needed by our body to grow, survive and carry on different daily activities.

Our food contains mainly five major kinds of nutrients namely vitamins, minerals, carbohydrates, proteins and fats. Additionally, food also contains water and dietary fibres/roughage and water which are also required by our bodies.

Fig. 1: Major nutrients of food

Carbohydrates: Carbohydrates main function is providing energy to the body. These are found in our food in the form of sugar and starch i.e. simple and complex carbohydrates. Example It is found in bread, potatoes etc.

Simple Carbohydrates: These are also referred to as simple sugars, containing single monosaccharide units and found in natural sources of food i.e. milk, fruits and vegetables. These carbohydrates add certain sweetness to the food. They raise the level of blood glucose quickly but are easier to break down.

Complex Carbohydrates: These are also referred to as polysaccharides, meaning they contain hundreds or thousands of such monosaccharide units. These are typically found in wheat grain, white bread, kernel and cakes. They are relatively less sweet than simple carbohydrates and also raise blood glucose level rather slowly. However, these are tougher to break down. Cellulose is present in plant cell wall. It is a complex carbohydrate. Humans cannot digest cellulose.

Test for carbohydrates: We can test whether a particular food item contains carbohydrates by pouring 2 to 3 drops of dilute iodine solution on it. If the iodine changes its colour to blue–black, then we can ascertain that the food item does, in fact, contain carbohydrates.

Fig.2: The Two Types of Carbohydrates

Proteins: Proteins performs the very essential function of helping our body grow and repair itself. These are found in food items such as milk, pulses, eggs, meat etc. Foods containing proteins are called ‘body-building’ foods.

Test for proteins: To test whether a food item contains proteins, first we need to grind it into a paste or powder form and add 10 drops of water. To this mixture when we add 2 drops of copper sulphate solution and 10 drops of caustic soda solution. After a few minutes, if the mixture turns violet, it is indicative of the presence of protein.

Fats: Fats are also responsible for providing energy to our body. In fact, they provide more energy than carbohydrates. The body uses fat as a fuel source. Fats are essential for the absorption of vitamins A, D, E and K in the body.  Butter, cheese, oil are all examples of fat-rich foods.

Test for fats: To test for fats, take a small quantity of food item and wrap a piece of paper around it and proceed to crush it. After removing the paper, allow it to dry. If you see a film of an oily patch on the paper when holding it against light, it is proof that the food contains fats.

Vitamins: Vitamins help in protecting our bodies from various kinds of diseases. They also help in keeping our eyes, gums, bones and teeth in good shape. Different types of vitamins and their uses:

Vitamin Type	Sources	Functions	Deficiency Diseases
Vitamin A	Leafy green vegetables, oranges, carrots, Pumpkin, Soy, Sweet potatoes	Forms and helps maintain bones, skin, tissue and teeth	Color blindness, night blindness- poor visibility at night.
Vitamin B1 (thiamine)	Dried herbs, sunflower seeds, whole grain cereals, sesame seeds, brown rice	Enables cells to turn carbohydrates into energy	Beriberi-  loss of appetite, loss of weight.
Vitamin B2 (riboflavin)	Almonds, Asparagus, bananas, green beans, wheat bran, dried spices	Maintains Body growth and RBCs i.e. Red Blood Cells	Skin disorders, Cheilosis-breaking of lips
Vitamin B12 (cyanocobalamin)	Mutton, fish, beef, lobster, clams, eggs, oysters, crab	Helps in maintenance of central nervous system and RBCs	Pale skin, lack of RBC, Less stamina and less appetite.
Vitamin C	Fresh herbs, cauliflower, papaya, oranges, strawberries, guava	Promotes healthy gums and teeth	Scurvy i.e. gum disease (gingivitis).
Vitamin D	Sunshine, Mushrooms, liver, fish and eggs	Necessary for the healthy development of bones and teeth	Rickets and Osteomalacia – weakening and softening of bones.
Vitamin E	Soyabean oil, red chilli powder, pine nuts, apricots, green olives and cooked spinach	Helps in processing vitamin K and formation of RBCs	Muscle weakness and transmission problems in nerve impulses
Vitamin K	Green leafy vegetables, Soyabean oil.	Essential for blood coagulation	Excessive bleeding from wound.

Minerals: Minerals are used by the body to perform various functions like building strong bones, maintaining the heartbeat, making hormones etc. The major five minerals are Calcium, Phosphorus, Magnesium, Sodium and Potassium. Examples of mineral-rich foods include leafy vegetables, fish, beans etc.

Mineral Type	Sources	Functions	Deficiency Disease
Calcium	Tofu, Dairy products, Salmon, Cabbage, Kale and Broccoli	Essential for efficient functioning of nervous system and healthy bones	Weak bones,  lower than normal bone density and stunted growth
Phosphorous	Lean meats, grain and milk	Essential for the maintenance of acid-base balance in body	Loss of appetite, bone fragility, muscle weakness, poor physique
Iodine	Green leafy vegetables, Seafood, iodised salt	Formation of thyroid hormone	Goitre- Enlargement of thyroid gland, mental disability.
Sodium	Table salt, celery	Helps keep control on blood pressure	Nausea, irritability
Iron	Whole grain, eggs, leafy vegetables and meats	Essential for haemoglobin formation in rbc.	Anaemia – weakness, fatigue, shortness of breath

Fig.3: The five major minerals and their source foods

Dietary Fibres/Roughage: While dietary fibres do not provide any such nutrition to our bodies but nevertheless are an important component of food. They help in easy absorption of food, helps in movement of bowel and prevents constipation. It helps our body get rid of undigested food. Cereals, fruits and vegetables are some of the roughage rich foods.

Water: Water performs the essential function of absorbing nutrients from our food. It also helps in releasing waste from our body in form of sweat and urine.

Balanced Diet

A balanced diet is one that contains a variety of food items providing different types of nutrients in adequate amounts necessary for maintaining good health. The diet should contain a good amount of dietary fibre and water as well.

A balanced diet includes a combination of protein-rich pulses, sprouted seeds etc. with combinations of various flours and cereals for carbohydrates and fats along with fruits and vegetables which provide the necessary vitamins and minerals.

• In addition to making sure that the right amount of food is eaten, it should also be ensured that food is properly cooked so that it does not end up losing its nutrients.
• Repeated washing of fruits, pulses, rice and vegetables can result in the loss of essential vitamins and minerals.
• Throwing away excess water which is used for cooking vegetables can result in the loss of considerable amounts of important proteins and minerals present in them.
• It’s a well-known fact that vitamin C gets destroyed in the heat while cooking.

Obesity: Obesity is a medical condition that results from excess intake of fat-rich foods. The excess fat gets accumulated to such an extent that it starts negatively affecting one’s health, well-being and the ability to carry out certain activities.

Deficiency Diseases

Deficiency: Sometimes simply getting adequate amounts of food might not be enough if the food does not contain the required nutrients in the right amounts. Prolonged usage of such nutrient-less food may result in a condition known as Deficiency.

Deficiency Diseases: Diseases that occur from the lack of an element in the diet, usually a particular vitamin or mineral are known as deficiency diseases.

• A diet lacking proteins may result in skin diseases, stunted growth, diarrhoea, swelling of face and discolouration of hair.
• A diet deficient in both carbohydrates and protein may hinder the growth completely and the person becomes so frail and lean that he or she might not even be able to move.
• Deficiency of certain vitamins and minerals can cause diseases like scurvy, goitre, anaemia etc. as mentioned in the tables above.

Hence one must always make sure to include all nutrients in their food and drink a good amount of water to maintain a healthy body that is free from diseases.

Malnutrition

Improper intake of nutrient-rich food may lead to another condition called malnutrition or undernutrition. Without adequate intake of food, our bones become brittle, our muscles weaken and our thinking becomes foggy. In such situations, our bodies are said to become malnourished.

When there is lack of protein in our bodies, repair of wounds and an injury becomes difficult. When this is combined with inadequate intake of calories, it leads to a condition known as Protein-energy Undernutrition or malnutrition.

There are two ways in which protein-energy malnutrition manifests itself:

Fig.4: Two types of protein-energy malnutrition diseases

Kwashiorkor: People with severe protein deficiency are often at high risk of developing this disease. People in rural areas are more likely to suffer from this disease, as there is lack of protein rich food. If one often indulges in diets that are high on carbohydrates and low on protein it can lead to them showing symptoms of kwashiorkor i.e. Edema (puffy appearance due to retention of fluid), inability to gain weight and bulging of abdomen.

Marasmus: Children and young adults are more likely to develop this disease. If there is inadequate intake of both energy and protein, it tends to manifest as marasmus. Chronic diarrhoea, weight loss, dehydration and stomach shrinkage are the major symptoms of marasmus.

Food Materials and Sources

Plants act as sources of food ingredients such as fruits, vegetables, grains, pulses etc.

Animals are sources of food ingredients such as milk, eggs, meat products etc.

These are shown in the figure above.

Some examples of edible plant parts are shown in the image above and their examples are given below:

• Roots: beets, carrots, radishes, turnips, ginger,
• Stems: broccoli stem, bamboo shoots, sugar cane, potato
• Leaves: spinach, lettuce etc.
• Fruits: apple, pear, tomatoes, grapes, cherries, oranges
• Edible Flowers: broccoli heads, cauliflower heads
• Seeds: sunflower seeds

What do Animals Eat?

Animals can be classified into three broad categories in terms of what they eat as can be seen in the image above,

First we have the herbivores i.e. plant eaters. They only consume plant parts. Examples: Cows, goats, deer, giraffe etc.

Next we have the carnivores i.e. meat eaters. They only consume meat of other animals. Examples: Lions, tigers, vultures etc.

Last we have omnivores i.e. animals who eat both plant parts and meat products. Examples: Humans, bears etc.

There are five major components of food namely vitamins, minerals, proteins, carbohydrates and fats. While the diet of carnivores is rich in fats and proteins, there are some necessary vitamins and minerals in plant-based foods that their diet lacks. Similarly an all plant-based diet lacks in a good amount of protein and certain minerals.

It is important to know that there is difference between vegetarians and herbivores. While vegetarians make a conscious decision to not eat meat, herbivores on the other hand are incapable of eating meat and dairy products. Similarly carnivores lack the necessary enzymes in their stomachs to digest cellulose which is a major component of green food like grass. 

Important Definitions

Nectar: The juicy sweet liquid secreted by within flowers which is sucked by bees and is made into honey by them

Sprouting: The process by which seeds shoot out small white structures as way of growth is called Sprouting.

Cellulose: It is a substance that is found in the cell walls of a number of plants. It is an indigestible fibre and is found in grass.

Enzymes: It is defined as a chemical substance that helps in bringing about changes to certain other substances without undergoing any changes in themselves.

A bowl of sprouts

Geometry tells us about the lines, angles, shapes and practical geometry is about the construction of these shapes.

Tools to be used for construction

1. The Ruler

A ruler is a straight edge which we sometimes call a scale. It is marked with the centimetres on one side and inches on the other side. It is used to draw line segments and also to measure them.

2. The Compass

The compass has two ends –one is a pointer and the other is for the pencil. It is used to draw arcs and circles and also to measure the line segment.

3. The Divider

It is a pair of pointers on both the ends. It is used to compare the length of line and arcs.

4. Set -squares

It is a set of two triangular pieces.

• One is having 45°, 45° and 90° angles at the corners. If we join two same triangular pieces, we will get a square. It is used to draw parallel and perpendicular lines.

• Other is having 30°, 60° and 90° angles at the corners. If we join two same triangular pieces, we will get a rectangle. It is also used to draw parallel and perpendicular lines.

5. The Protractor

It is a semi-circular device which is divided into 180-degree parts. It starts from the 0° from the right-hand side and ends with the 180° on the left-hand side and vice versa. It is used to measure and draw angles.

The Circle

It is a round shape in which all the points of its boundary are at equal distance from its centre.

Construction of a circle if the radius is known

Draw a circle of radius 5 cm.

To draw a circle, we need a compass and a ruler to measure the length.

Step 1: Open the compass and measure the length of 5 cm using a ruler.

Step 2: Mark a point O, which we will use as the centre of the circle.

Step 3: Put the pointer on the point O.

Step 4: Turn the compass to make a complete circle. Remember to do it in one instance. 

A Line Segment

A line segment is a part of a line which has two endpoints. It has a fixed length so it can be measured using a ruler.

1. Construction of line segment if the length is known

Draw a line segment of 5 cm.

To draw a line segment of a particular length, we need a compass and a ruler.

Step 1: First of all draw a line l of any length and mark a point named A on it.

Step 2: Measure the length of 5 cm by putting the compass on the ruler. Put the pointer on 0 and open the compass till 5 cm on the ruler.

Step 3: Put the pointer at point A on the line and draw an arc of 5 cm which intersects the line at B.

Step 4: Hence the line segment of 5 cm is

2. Construction of a copy of a given line segment

To make a copy of a line segment, there are three methods-

1. Measure the length of the given line segment using a ruler and draw the line segment of the same length with the ruler only.

2. Take a transparent sheet and trace the given line on another part of the paper.

3. These methods are not very accurate so we can use a ruler and compass to make an accurate line segment.

Draw a copy of the line segment

Step 1: is the given line segment whose length is known.

Step 2: Use the compass and put the pointer on A. open the compass to place the pencil side on point B. Now the open compass is of the same length as given

Step 3: Draw a new line l and mark a point C on it. Put the pointer on the point C.

Step 4: Using the same radius draw an arc on line l which cuts the line at the point D. 

Perpendiculars

If two lines intersect in such a way that they make a right angle at the point of intersection then they are perpendicular to each other.

1. Perpendicular to a line if a point is given on it (using a ruler and a set square)

Step 1: A line l is given and a point P on it.

Step 2: Put the ruler with one of its side next to l.

Step 3: Put the set square with one of its sides by the side of the already aligned edge of the ruler so that the right-angled corner comes in contact with the ruler.

Step 4:  Slide the set-square so that the right-angled corner coincides with P.

Step 5: Hold the set-square. Draw PQ along the edge of the set-square.

PQ is the required perpendicular to l from the given point P.

2. Perpendicular to a line if a point is given on it(using a ruler and compass)

Draw a perpendicular of the line at a given point A.

Step 1: Take A as the centre and draw a big arc with any radius so that it intersects the line at point B and C.

Step 2: Take B as the centre and draw an arc with the radius more than AC and then take C as center and draw an arc so that they intersect each other at point D.

Step 3: Join AD. AD is perpendicular to CB.

AD ⊥ CB.

3. Perpendicular to a line from a point which is not on it(using ruler and a set square)

Step 1: P is the point outside the given line l.

Step 2: Put the set-square on l so that one side of its right angle comes on l.

Step 3: Put a ruler on the hypotenuse of set square.

Step 4: Hold the ruler and slide the set-square until it touches the point P.

Step 5: Join PM.

Now PM ⊥ l.

4. Perpendicular to a line from a point not on it (using ruler and a compass)

Step 1: P is the point outside the given line l.

Step 2: Take P as the centre and draw a big arc of any radius which intersects line l at two points A and B.

Step 3: Take A and B as radius and draw arcs from both points with the same radius as taken before so that they intersect with each other.

Step 4: Join PQ.

PQ is the perpendicular to line l.

5. The perpendicular bisector of a line segment (using transparent tapes)

A line which divides the given line into two equal parts is known as a perpendicular bisector.

Step 1: Draw a line segment AB.

Step 2: Take a strip of transparent rectangular tape and put it diagonally on the line AB so that the endpoints of the line segment lie on the edges of the tape.

Step 3: Take another strip and put it over A and B diagonally across the previous tape so that the two tapes intersect at point M and N.

Step 4: Join MN. MN is the perpendicular of AB.

6. The perpendicular bisector of a line segment (using ruler and compass)

Step 1: Draw a line segment AB.

Step 2: Take A as the centre and draw two arcs – one upside and one downside with the radius of more than half of the length of AB. Or you can draw a circle taking A as the centre for the convenience.

Again make the arcs with taking B as the centre so that they intersect the previous arcs.

Step 3: Join the intersections of the arcs and name them as C and D.

Step 4: The required perpendicular bisector of AB is CD. Hence, AO = OB.

Angles

Angle is a stature formed by two rays. The two rays have a common endpoint which is called the Vertex of the Angle.

1. Construction of an angle with a given measure

Draw an angle of 60 ° using a protractor.

Step 1: Draw a line BA.

Step 2: Put the protractor on the line in such a way that the centre of the protractor lays on point B and the zero edge comes on the line segment BA.

Step 3: Start from 0° and mark the point C at 60°.

Step 4: Join BC.

∠CBA is the required angle.

2. Construction of a copy of an angle of unknown measure (using a ruler and a compass).

Draw a copy of ∠A.

Step 1: P is a point on the line l.

Step 2: Take A as the centre in the given angle and draw an arc of any radius which cuts the two rays at B and C.

Step 3: In the line l, take P as the centre and draw an arc with the same radius as above which cuts line l at Q.

Step 4: Open your compass to take the length of the arc BC.

Step 5: Take Q as the centre and draw an arc with the same radius, to cut the arc drawn earlier, at point R. 

Step 6: Join PR. It will make the angle of the same measure as given.

Hence, ∠P = ∠A

3. The bisector of an angle

An angle bisector is the line segment which divides a particular angle into two equal parts. It is also called the line of symmetry of the angle.

Construction of angle bisector (using a ruler and a compass)

Draw the angle bisector of ∠O.

Step 1: Put the pointer on O and draw an arc of any radius so that it cut the rays at point A and B.

Step 2: Put the pointer on point A and draw an arc of the radius of more than half of AB.

Step 3: While taking B as the centre we will draw an arc of the same radius so that it cut the previous arc at point C.

Step 4: Join OC.OC is the required angle bisector of ∠O.

Hence, ∠BOC = ∠COA.

4. Angles of special measures

There are some angles which we can construct accurately with the help of a compass without using a protractor.

a. Construction of 60° angle.

Step 1: Draw a line m and mark a point C on it.

Step 2: Take C as the centre and draw an arc of any radius to cut the line at point D.

Step 3: While taking D as the centre we need to draw an arc of the same radius to cut the previous arc.

Step 4: Join CE. ∠C = 60°.

b. Construction of 120° angle.

It is the twice of 60° angle.

Step 1: Draw a line m and mark a point C on it.

Step 2: Put the pointer on point C and draw an arc of any radius which cuts the line m at point D.

Step 3: While taking D as the centre we need to draw an arc with the same radius to cut the previous arc at E.

Again take E as the centre and draw an arc of the same radius to cut the first arc at point F.

Step 4: Join CF. ∠FCD = 120°.

c. Construction of 30° angle.

To construct an angle of 30°, we need to draw an angle of 60° as above then bisect it with the process of an angle bisector.

d. Construction of 90° angle

It can be made by two methods-

i. Draw a perpendicular bisector of 180° i.e. a straight line.

ii. Draw a bisector of 60° and 120°.

e. Construction of 45° angle.

Draw an angle of 90° then bisect it to make an angle of 45°.

When we fold a paper in such a way that the picture is divided into two equal halves then the line which divides the picture into two halves is called a Line of Symmetry.

Here the line divides the star into two halves so it is the line of symmetry. It is also called the Mirror Line because if we place the mirror on that line then one side of the picture will fall exactly on the other side of the picture.

Non-symmetrical Figure

This figure is not symmetrical as if we fold the image from the dotted line then it does not divide it into two equal halves.

Making Symmetric Figures: Ink-blot Devils

To make an ink-blot pattern-

• Take a piece of paper and fold it in half.
• Put some drops of ink on one side of the paper.
• Then press the halves together.
• It will make a symmetric pattern with the fold as the line of symmetry.

Inked-string pattern

To make an inked string pattern-

• Take a piece of paper and fold it in half.
• Dip a string in different colours and arrange it on the one side of the paper.
• Press the two halves together and pull the string.
• It will make a symmetric inked string pattern with the fold as the line of symmetry.

Two Lines of Symmetry

Some figures have two lines of symmetry.

1. A Rectangle

Take a rectangular sheet and fold it horizontally in two equal halves and then again fold it vertically in two equal halves. After opening it, we get two lines of symmetry of the rectangular sheet.

2. More Figures with two Line of Symmetry

If we take a rectangular piece of paper and double fold it to make two lines of symmetry and cut it in some new shape then after opening it we will get a new image that too with the two lines of symmetry.

Construction of figure with two Lines of Symmetry

1. To draw a figure with two lines of symmetry, take one figure.

2. Let L and M be the two lines of symmetry.

3. Draw the figure in such a way that L is the line of symmetry,

4. Now complete the figure by drawing the remaining part so that M will also become the line of symmetry.

Hence this is the final figure with two lines of symmetry.

Multiple Lines of Symmetry

Take a square sheet of paper and fold it in two halves vertically and again horizontally .open it and fold it in two equal halves diagonally then again open it and fold it along another diagonal.

When you will open the paper you will see four imaginary lines and these lines are the lines of symmetry.

Some more images with more than two lines of symmetry

• Equilateral triangle will have three lines of symmetry.
• Square will have four lines of symmetry.
• Regular pentagon will have five lines of symmetry.
• Regular hexagon will have six lines of symmetry.

Some Real-life Examples of Symmetry

In Taj Mahal and the butterfly there is one line of symmetry and there are so many other things also in our daily life which are having one or more line of symmetry.

Reflection and Symmetry

The line of symmetry is also called Mirror Line because the mirror image of an object is symmetrical to the image. When we see an object in the mirror then there is no change in the length and angles of the object except one thing i.e. the image is opposite to the original image.

Some Examples of Reflection Symmetry

1. Paper Decoration

We can use a rectangular sheet to fold and create some intricate patterns by cutting paper.

2. Kaleidoscope

In Kaleidoscope, mirrors are used to create pictures having various lines of symmetry. Two mirrors strips forming a V-shape are used. The angle between the mirrors determines the number of lines of symmetry.

Example

Which alphabet will remain same after reflection symmetry? Check for R, C, N, A and T.

Solution

In the alphabet reflection symmetry, the alphabets look opposite in the mirror i.e. the alphabet written from right to left will appear as written from left to right.

Hence C, N and R will not look the same after reflection.

Hence A and T will look same after reflection symmetry.

If we compare two quantities using division then it is called ratio. It compares quantities in terms of ‘How many times’. The symbol to represent ratio is “:”.

It reads as “4 is to 3”

It can also be written as 4/3.

Example

If there are 35 boys and 25 girls in a class, then what is the ratio of

• Number of boys to total students
• Number of girls to total students.

Solution

In the ratio, we want the total number of students.

Total number of students = Number of boys + Number of girls

35 + 25 = 60

• Ratio of number of boys to total number of students

• The ratio of the number of girls to the total number of students

The unit must be same to compare two quantities

If we have to compare two quantities with different units then we need to convert them in the same unit .then only they can be compared using ratio.

Example

What is the ratio of the height of Raman and Radha if the height of Raman is 175 cm and Radha is 1.35 m?

Solution

The unit of the height of Raman and Radha is not same so convert them in the same unit.

Height of Radha is 1.35 m = 1.35 × 100 cm = 135 cm

The ratio of the height of Raman and Radha 

Equivalent Ratios

If we multiply or divide both the numerator and denominator by the same number then we get the equivalent ratio. There could be so many equivalent ratios of the same ratio.

In the case of equivalent ratios only their value changes but they represent the same portion of the quantity.

Example

Find two equivalent ratios of 2/4.

Solution

To get the equivalent ratio we multiply both the numerator and denominator with 2.

To get another equivalent ratio we divide both the numerator and denominator with 2.

From the above figure, we can see that in all the equivalent ratios only the number of equal parts is changing but all the ratios are representing the half part of the circle only.

The Lowest form of the Ratio

If there is no common factor of numerator and denominator except one then it is the lowest form of the ratio.

Example

Find the lowest form of the ratio 25: 100.

Solution

The common factor of 25 and 100 is 25, so divide both the numerator and denominator with 25.

Hence the lowest ratio of 25: 100 is 1: 4.

Proportion

If we say that two ratios are equal then it is called Proportion.

We write it as a: b : : c: d or a: b = c: d

And reads as “a is to b as c is to d”.

Example

If a man runs at a speed of 20 km in 2 hours then with the same speed would he be able to cross 40 km in 4 hours?

Solution

Here the ratio of the distances given is 20/40 = 1/2 = 1: 2

And the ratio of the time taken by them is also 2/4 = 1/2 = 1: 2

Hence the four numbers are in proportion.

We can write them in proportion as 20: 40 : : 2: 4

And reads as “20 is to 40 as 2 is to 4”.
 

Extreme Terms and Middle Terms of Proportion

The first and the fourth term in the proportion are called extreme terms and the second and third terms are called the Middle or the Mean Terms.

In this statement of proportion, the four terms which we have written in order are called the Respective Terms.

If the two ratios are not equal then these are not in proportion.

Example 1

Check whether the terms 30,99,20,66 are in proportion or not.

Solution 1.1

To check the numbers are in proportion or not we have to equate the ratios.

As both the ratios are equal so the four terms are in proportion.

30: 99 :: 20: 66

Solution 1.2

We can check with the product of extremes and the product of means.

In the respective terms 30, 99, 20, 66

30 and 66 are the extremes.

99 and 20 are the means.

To be in proportion the product of extremes must be equal to the product of means.

30 × 66 = 1980

99 × 20 = 1980

The product of extremes = product of means

Hence, these terms are in proportion.
 

Example 2

Find the ratio 30 cm to 4 m is proportion to 25 cm to 5 m or not.

Solution 2

As the unit is different so we have to convert them into the same unit.

4 m = 4 × 100 cm = 400 cm

The ratio of 30 cm to 400 cm is

5 m = 5 × 100 cm = 500 cm

Ratio of 25 cm to 500 cm is

Here the two ratios are not equal so these ratios are not in proportion.

3: 40 ≠ 1: 20

Unitary Method

If we find the value of one unit then calculate the value of the required number of units then this method is called the Unitary method.

Example 1

If the cost of 3 books is 320 Rs. then what will be the cost of 6 books?

Solution 1

Cost of 3 books = Rs. 320

Cost of 1 book = 320/3 Rs.

Cost of 6 books = (320/3) × 6 = 640 Rs.

Hence, the cost of 6 books is Rs. 640.
 

Example 2

If the cost of 20 toys is Rs. 4000 then how many toys can be purchased for Rs. 6000?

Solution 2

In Rs. 4000, the number of toys can be purchased = 20

In Rs. 1, the number of toys can be purchased = Rs. 20/4000

Therefore, in Rs. 6000, the number of toys can be purchased = (20/4000) × 6000 = 30

Hence, 30 toys can be purchased by Rs. 6000.

Introduction to Algebra

There are so many branches of mathematics-

• The study of numbers is called Arithmetic.
• The study of shapes is called Geometry.
• The study to use the letters and symbols in mathematics is called Algebra.

Algebra

Algebra is a part of mathematics in which the letter and symbols are used to represent numbers in equations. It helps us to study about unknown quantities.

Matchstick Patterns

No. of matchsticks used to make 1^st square = 4

No. of matchsticks used to make 2^nd square = 7

No. of matchsticks used to make 3^rd square = 10

So, the pattern that we observe here is 3n + 1

With this pattern, we can easily find the number of matchsticks required in any number of squares.

Example

How many matchsticks will be used in the 50^th figure?

Solution

3n + 1

3 × 50 + 1

= 151 matchsticks

The Idea of a Variable

Variable refers to the unknown quantities that can change or vary and are represented using the lowercase letter of the English alphabets.

One such example of the same is the rule that we used in the matchstick pattern

3n + 1

Here the value of n is unknown and it can vary from time to time.

More Examples of Variables

• We can use any letter as a variable, but only lowercase English alphabets.
• Numbers cannot be used for the variable as they have a fixed value.
• They can also help in solving some other problems.

Example: 1

Karan wanted to buy story books from a bookstall. She wanted to buy 3 books for herself, 2 for her brother and 4 for 2 of her friends. Each book cost Rs.15.how much money she should pay to the shopkeeper?

Solution: 1

Cost of 1 book = Rs.15

We need to find the cost of 9 books.

No. of notebooks	1	2	3	4	…….	a	……..
Total cost	15	30	45	60	…….	15 a	…….

In the current situation, a (it’s a variable) stands for 9

Therefore,

Cost of 9 books = 15 × 9

= 135

Therefore Karan needs to pay Rs.135 to the shopkeeper of the bookstall.

The variable and constant not only multiply with each other but also can be added or subtracted, based on the situation.

Example: 2

Manu has 2 erasers more than Tanu. Form an expression for the statement.

Solution: 2.1

Erasers that Tanu have can be represented using a variable (x)

Erasers that Manu have = erasers that Tanu have + 2

Erasers with Manu = x + 2

Solution: 2.2

Erasers that Manu have can be represented using a variable (y)

Erasers that Tanu has = erasers that Manu have – 2

Erasers with Tanu = y – 2

Use of Variables in Common Rules (Geometry)

1. Perimeter of Square

The perimeter of a square = Sum of all sides

= 4 × side

= 4s

Thus, p = 4s

Here s is variable, so the perimeter changes as the value of side change.

2. Perimeter of Rectangle

Perimeter of rectangle = 2(length + breadth)

= 2 (l + b) or 2l + 2b

Thus, p + 2 × (l + b) or 2l + 2b

Where, l and b are variable and the value of perimeter changes with the change in l and b.

Use of Variables in Common Rules (Arithmetic)

1. Commutativity of Addition

5 + 4 = 9

4 + 5 = 9

Thus, 5 + 4 = 4 + 5

This is the commutative property of addition of the numbers, in which the result remains the same even if we interchanged the numbers.

a + b = b + a

Here, a and b are different variables.

Example

a = 16 and b = 20

According to commutative property

16 + 20 = 20 + 16

36 = 36

2. Commutativity of Multiplication

8 × 2 = 16

2 × 8 = 16

Thus, 8 × 2 = 2 × 8

This is the commutative property of multiplication, in which the result remains the same even if we interchange the numbers.

a × b = b × a

Here, a and b are different variables.

Example

18 ×12 = 216, 12 ×18 = 216

Thus, 18 × 12 = 12 × 18

3. Distributivity of Numbers

6 × 32

It is a complex sum but there is an easy way to solve it. It is known as the distributivity of multiplication over the addition of numbers.

6 × (30 + 2)

= 180 + 12

= 192

Thus, 6 × 32 = 192

A × (b + c) = a × b + a × c

Here, a, b and c are different variables.

4. Associativity of Addition

This property states that the result of the numbers added will remain same regardless of their grouping.

(a + b) + c = a + (b + c)

Example

(4 + 2) + 7 = 4 + (2 + 7)

6 + 7 = 4 + 9

13 = 13

Expressions

Arithmetic expressions may use numbers and all operations like addition, subtraction, multiplication and division

Example

2 + (9 – 3), (4 × 6) – 8 etc…

(4 × 6) – 8 = 24 – 8

= 16

Expressions with variable

We can make expressions using variables like

2m, 5 + t etc…..

An expression containing variable/s cannot be analyzed until its value is given.

Example

Find 3x – 12 if x = 6

Solution

(3 × 6) – 12

= 18 – 12

= 5

Thus,

3x – 12 = 5

Formation of Expressions

Statement	Expression
y subtracted from 12	12 – y
x multiplied by 6	6x
t Multiplied by 4, and then subtract 5 from the product.	4t – 5

Practical use of Expressions

Example

3 boys go to the theatre. The cost of the ticket and popcorn is $33 and $15 respectively. What is the cost per person?

Solution

Let’s say,

 x = cost of ticket per person

 y = cost of popcorn/person

Total cost of the movie (ticket + popcorn) per person = x + y

Total cost of ticket + popcorn for 3 boys = 3(x + y)

= 3 (33 + 15)

= 3 (48)

= 144

Hence the total cost of movie ticket and popcorn for 3 boys is $144.

Equation

If we use the equal sign between two expressions then they form an equation.

An equation satisfies only for a particular value of the variable.

The equal sign says that the LHS is equal to the RHS and the value of a variable which makes them equal is the only solution of that equation.

Example

3 + 2x = 13

5m – 7 = 3

p/6 = 18

If there is the greater then or less than sign instead of the equal sign then that statement is not an equation.

Some examples which are not an equation

23 + 6x  > 8

6f – 3 < 24

The Solution of an Equation

The value of the variable which satisfies the equation is the solution to that equation. To check whether the particular value is the solution or not, we have to check that the LHS must be equal to the RHS with that value of the variable.

Trial and Error Method

To find the solution of the equation, we use the trial and error method.

Example

Find the value of x in the equation 25 – x = 15.

Solution

Here we have to check for some values which we feel can be the solution by putting the value of the variable x and check for LHS = RHS.

Let’s take x = 5

25 – 5 = 15

20 ≠ 15

So x = 5 is not the solution of that equation.

Let’s take x = 10

25 – 10 = 15

15 = 15

LHS = RHS

Hence, x = 10 is the solution of that equation.

The closed 2-D shapes are referred to as plane figures.

Here “C” is the boundary of the above figure and the area inside the boundary is the region of this figure. Point D comes in the area of the given figure.

Perimeter

If we go around the figure along its boundary to form a closed figure then the distance covered is the perimeter of that figure. Hence the Perimeter refers to the length of the boundary of a closed figure.

If a figure is made up of line segments only then we can find its perimeter by adding the length of all the sides of the given figure.

Example

Find the Perimeter of the given figure.

Solution

Perimeter = Sum of all the sides

= (12 + 3 + 7 + 6 + 10 + 3 + 15 + 12) m

= 68 m

The Perimeter of a Rectangle

A rectangle is a closed figure with two pairs of equal opposite sides.

Perimeter of a rectangle = Sum of all sides

= length + breadth + length + breadth

Thus, Perimeter of a rectangle = 2 × (length + breadth)

Example: 1

The length and breadth of a rectangular swimming pool are 16 and 12 meters respectively .find the perimeter of the pool.

Solution: 

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the pool = 2 × (16 + 12)

= 2 × 28

= 56 meters

Example: 2

Find the cost of fencing a rectangular farm of length 24 meters and breadth 18 meters at 8/- per meter.

Solution:

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the farm = 2 × (24 + 18)

= 2 × 42

= 84 meter

Cost of fencing = 84 × 8

= Rs. 672

Thus the cost of fencing the farm is Rs. 672/-.

Regular Closed Figure

Figures with equal length of sides and an equal measure of angles are known as Regular Closed Figures or Regular Polygon.

Perimeter of Regular Polygon = Number of sides × Length of one side

Perimeter of Square

Square is a regular polygon with 4 equal sides.

Perimeter of square = side + side + side + side

Thus, Perimeter of a square = 4 × length of a side

Example

Find the perimeter of a square having side length 25 cm.

Solution

Perimeter of a square = 4 × length of a side

Perimeter of square = 4 × 25

 = 100 cm

Perimeter of an Equilateral Triangle

An equilateral triangle is a regular polygon with three equal sides and angles.

Perimeter of an equilateral triangle = 3 × length of a side

Example

Find the perimeter of a triangle having each side length 13 cm.

Solution

Perimeter of an equilateral triangle = 3 × length of a side

Perimeter of triangle = 3 × 13

= 39 cm

Perimeter of a Regular Pentagon

A regular pentagon is a polygon with 5 equal sides and angles.

Perimeter of a regular pentagon = 5 × length of one side

Example

Find the perimeter of a pentagon having side length 9 cm.

Solution

Perimeter of a regular pentagon = 5 × length of one side

Perimeter of a regular pentagon = 5 × 9

= 45 cm

Perimeter of a Regular Hexagon

A regular hexagon is a polygon with 6 equal sides and angles.

Perimeter of a regular hexagon = 6 × Length of one side

Example

Find the perimeter of a hexagon having side length 15cm.

Solution

Perimeter of a regular hexagon = 6 × Length of one side

Perimeter of a regular hexagon = 6 × 15

= 90 cm

Perimeter of a Regular Octagon

A regular octagon is a polygon with 8 equal sides and angles.

Perimeter of a regular octagon = 8 × length of one side

Example

Find the perimeter of an octagon having side length 7cm.

Solution

Perimeter of a regular octagon = 8 × length of one side

Perimeter of a regular octagon = 8 × 7

= 56 cm

Area

Area refers to the surface enclosed by a closed figure.

To find the area of any irregular closed figure, we can put them on a graph paper with the square of 1 cm × 1 cm .then estimate the area of that figure by counting the area of the squares covered by the figure.

Here one square is taken as 1 sq.unit.

Example

Find the area of the given figure. (1 square = 1 m²)

Solution

The given figure is made up of line segments and is covered with some full squares and some half squares.

Full squares in figure = 32

Half squares in figure = 21

Area covered by full squares = 32 × 1 sq. unit = 32 sq. unit.

Area covered by half squares = 21 × (1/2) sq. unit. = 10.5 sq. unit.

Total area covered by figure = 32 + 10.5 = 42.5 sq. unit.

Area of a Rectangle

Area of a rectangle = (length × breadth)

Example

Find the area of a rectangle whose length and breadth are 20 cm and 12 cm respectively.

Solution

Length of the rectangle = 20 cm

Breadth of the rectangle = 12 cm

Area of the rectangle = length × breadth

= 20 cm × 12 cm

= 240 sq cm.

To find the length of a rectangle if breadth and area are given:

Example

What will be the length of the rectangle if its breadth is 6 m and the area is 48sq.m?

Solution

Length = 48/6

 = 8 m

To find the breadth of the rectangle if length and area are given:

Example

What will be the breadth of the rectangle if its length is 8 m and the area is 81 sq.m?

Solution

Breadth = 81/8

= 9 m

Area of a Square

Area of a square is the region covered by the boundary of a square.

Area of a square = side × side

Example

Calculate the area of a square of side 13 cm.

Solution

Area of a square = side × side

= 13 × 13

= 169 cm².