जाति न पूछो साधु की, पूछ लीजिए ज्ञान। मेल करो तरवार का, पड़ा रहन दो म्यान।।1।। कबीर की साखियाँ अर्थ सहित: कबीर की साखी की इन पंक्तियों में कवि कहते हैं कि हमें कभी भी सज्जन इंसान की जाति पर ध्यान नहीं देना चाहिए, बल्कि हमें तो उसके गुणों के आधार पर उसका सम्मान करना चाहिए। जैसे, तलवार की कीमत म्यान नहीं, बल्कि तलवार की धार में छिपी होतो है।
आवत गारी एक है, उलटत होइ अनेक। कह कबीर नहिं उलटिए,वही एक की एक।।2।। कबीर की साखियाँ अर्थ सहित: प्रस्तुत साखी में कबीरदास जी कहते हैं कि किसी के अपशब्दों का जवाब कभी भी अपशब्दों से मत दो। इससे वो अपशब्द बढ़ने के बजाय घटते-घटते ख़त्म हो जाएंगे।
माला तो कर में फिरै, जीभि फिरै मुख माँहि। मनुवाँ तो दहुँ दिसि फिरै, यह तौ सुमिरन नाहिं।।3।। कबीर की साखियाँ अर्थ सहित: प्रस्तुत दोहे में कबीर जी कहते हैं कि अगर आपका मन प्रभु की भक्ति में नहीं लगता है, तो फिर हाथ में माला लेकर घूमना, मुख से प्रभु का नाम लेना बेकार है। अगर प्रभु को पाना है, तो हमें एकाग्र होकर उनकी भक्ति करनी होगी।
कबीर घास न नींदिए, जो पाऊँ तलि होइ। उड़ि पड़ै जब आँखि मैं, खरी दुहेली होइ।।4।। कबीर की साखियाँ अर्थ सहित: प्रस्तुत दोहे में कबीर जी कहते हैं कि हमें कभी भी किसी को छोटा समझकर उसका निरादर नहीं करना चाहिए। जैसे, घास को छोटा समझ कर हर वक़्त दबाना नहीं चाहिए क्योंकि अगर इसका एक तिनका भी आंख में चला जाए, तो हमें बहुत पीड़ा होती है।
जग में बैरी कोइ नहीं, जो मन सीतल होय। या आपा को डारि दे, दया करै सब कोय।।5।। कबीर की साखियाँ अर्थ सहित: प्रस्तुत साखी में कबीर जी कहते हैं कि जिस मनुष्य का मन शांत होता है, दुनिया में उसका कोई शत्रु नहीं हो सकता है। यदि दुनिया का हर मनुष्य स्वार्थ, क्रोध जैसी भावनाओं का त्याग कर दे, ओ वो दयालु और महान बन सकता है।
If an object keeps on changing its position with time, it is said to be moving or in motion. Motion can be of different types:
Linear or straight in which the object travels in a straight line.
Circular in which the object travels along a circular path.
Curvilinear in which the object moves along a curve.
Figure 1: Examples of Motion
Slow and Fast Motion
If one object covers a particular distance in less time and another object covers the same distance in more time then the first object is said to be moving slowly while the second object is said to be moving faster.
The Speed of an object
The distance travelled by an object in unit time is called its Speed.
Types of Speed:
Uniform Speed – When the object travels a fixed distance same time gaps, it is said to have a uniform speed.
Non-uniform speed – When an object covers different distances in different time gaps, it is said to have a non-uniform speed.
Average speed – The total distance travelled by an object divided by the total time taken by the object is called its average speed
Figure 2: Finding Speed, Time and Distance
Measuring Time
There are many events in nature that repeat after a time interval:
Morning – The rising of the sun
Day and Night – The time between the sunrise and sunset
Month – The time between two new moons
Year – The time the earth takes to complete its one revolution around the sun
Time measuring devices or clocks – Clocks use the concept of periodic motion to measure time. It means that it uses motion that repeats itself in equal amounts of time. There are different types of time measuring devices.
Sundial – It uses the position of the sun to depict time
Sand Clock (hourglass) – It uses sand to measure time
Water Clock – It uses water to measure time
Pendulum Clock – It uses a pendulum to measure time
Quartz Clocks – They have an electric circuit that works with the help of cells. They provide accurate time.
Periodic Motion of a Simple Pendulum
Figure 8: Simple Pendulum
A simple pendulum contains a Bob. It is a metallic ball or a stone which is suspended from a rigid stand with the help of a thread.
Oscillatory motion – The to and fro motion of the pendulum is called as Oscillatory Motion. The bob of the pendulum does move from the centre (mean position) of the pendulum to its extreme positions on the other side.
Oscillation – When the bob moves from its centre (mean position) to its extreme ends it is said to complete one oscillation.
Time Period of a pendulum – The time taken by the pendulum bob to complete one oscillation is called its Time Period.
Speedometer – It is a device which is used in vehicles such as cars and trucks which measures the speed in kilometer per hour.
Odometer – It is a device which measures the distance travelled by a vehicle in meters or kilometers.
Figure 10: Measure of Distance and Speed of a car
Distance-time Graph
A graph which represents the distance travelled by an object with respect to time is called a distance-time graph.
Making a distance-time graph:
1. Mark the x-axis and y-axis and divide them in equal quantities.
Figure 11: Take the first quadrant
2. Choose one scale to represent distance (for example, x-axis to represent distance where 1 km = 1 cm) and the other to represent time (for example, y-axis to represent the time where 1 min = 1 cm).
Figure 12: Choosing the scale
3. Mark the values of time and distance in the graph. 4. Mark the set of values of time taken and distance covered in that time by the object in the graph. For example, if 1 km is covered in 1 minute then mark 1 unit on both the x-axis and y-axis.
Figure 13: Marking the values for time and distance
5. Now draw lines parallel to x-axis and y-axis at the points that you have marked. 6. Mark the points where these lines intersect on the graph. These points show the position of the moving object. 7. Now join all the points of intersection and obtain a straight-line graph. 8. This is the distance-time graph of a moving object.
Figure 14: Obtaining a straight line graph
The shape of the distance-time graph can be the following:
Shape of Graph
Interpretation
Straight line
The object has a uniform or constant speed
Parallel to time-axis
It is a stationary object
Curve shape
The object has a non-uniform speed
Figure 15 Distance-time Graphs
To find the speed of the distance-time graph
Speed = distance/time = (final position of object – initial position of object)/time taken by object
Also, the speed of the distance-time graph can be calculated by the Slope of a graph. The steeper the slope of the graph, the more is the speed of the object. For example, in the graph given below object A has a steeper slope. This means that object A is moving at a higher speed than object B.
All living organisms require food. The food gives energy to the organisms for growth and maintenance of their body functions. Carbohydrates, proteins, fats, vitamins and minerals are the components of food. These components of food are necessary for our body and are called nutrients.
Nutrition is the process of taking food by an organism and its utilisation by the body. Green plants prepare their own food while humans and animals are directly or indirectly dependent on plants for their food.
Modes of Nutrition On the basis of a different mode of nutrition, organisms are categorised into two major types, i.e.
(i) Autotrophs (auto-self, trobpos-nourishment) Autotrophic nutrition is the mode of nutrition in which organisms make their own food from the simple substance (e.g. CO2 and H2O) by the process of photosynthesis. Therefore, plants are called autotrophs.
(ii) Heterotrophs (heteros-other) Humans and animals do not contain chlorophyll and are dependent on plants for their food in readymade form. Those organisms which cannot prepare their own food and take food from green plants or animals are called heterotrophs and the mode of nutrition is called heterotrophic nutrition.
Photosynthesis: Food Making Process in Plants The process by which autotrophic green plants make their own food from simple inorganic substances (carbon dioxide and water) in the presence of sunlight and green pigment or chlorophyll is known as photosynthesis.
Site of Photosynthesis The process of photosynthesis takes place in green leaves, therefore leaves are referred to as the food factories of plants. The. the photosynthetic process can occur in other green parts of the plant-like stem but is not enough for meeting all the needs of the plant.
Reactions Involved in Photosynthesis The whole process of photosynthesis can be given by the following equation:
Cells All living organisms are made from small building units of catted cells. Cells are the structural and functional units of the body of all living organisms. They can only be seen under a microscope. The cell has a thin outer boundary called cell membrane, a distinct, centrally located spherical structure called nucleus and jelly-the substance surrounding the nucleus called cytoplasm.
The inorganic raw material, i.e. CO2 is taken from the air through the tiny pores present on the surface of leaves called stomata and water is absorbed through the roots of plants (from the soil) and is transported to leaves by vessels which act like pipes. These vessels form the continuous path from roots to leaves for the movement of nutrients.
Green plants possess chlorophyll in their leaves which captures the energy of the sunlight. This light energy is used to prepare food (starch). During the process, oxygen is also released. Photosynthesis is the unique process in which solar energy is captured by the leaves and stored in the plants in the form of food. Thus, ‘Sun is the ultimate source of energy for all the living organisms.’
Products of Photosynthesis The food produced by the process of photosynthesis is mainly carbohydrate. It produces glucose as food material which later gets converted into starch. The presence of starch in leaves indicates the occurrence of photosynthesis.
Importance of Photosynthesis If the plants do not perform photosynthesis, there would be no food on earth. Photosynthesis is also necessary for the production of oxygen gas in the atmosphere which is necessary for the respiration of organisms. Therefore, it can be said that no life is possible in the absence of photosynthesis.
Photosynthesis in Leaves of Various Colours In green pants, chlorophyll absorbs light energy from the sun to perform photosynthesis. Besides some green colour plants like Croton, maple, Colocasia, etc., have leaves that are red, brown, violet colour (variegated). These colours are present in large amounts and masks the green colour of chlorophyll in leaves. Thus, these leaves also perform photosynthesis and synthesise starch in them.
Synthesis of Plant Food other than Carbohydrates The starch or glucose is the simplest form of carbohydrate synthesised by the plants which is composed of carbon, hydrogen and oxygen. Sometimes these simplest forms of carbohydrate are utilised to synthesise other food nutrients like fats (oils), proteins, etc. Starch or glucose is rich in seeds like wheat, rice and various parts of plants like potato tuber. Sometimes the starch or glucose is stored in the form of oil in their seeds (oilseed), e.g. sunflower seed.
When the plant nutrient contains, carbon, hydrogen and oxygen along with nitrogen elements, it is called protein. The element nitrogen comes from soil in the form of nitrate by the actions of some bacteria present in soil and forms amino acid which is then converted into proteins. Therefore, plants also make fats and proteins as their food.
Other Modes of Nutrition in Plants There are some plants which do not contain chlorophyll in them and thus, cannot prepare their own food. These plants obtain their food from other plants or animal, i.e. they are heterotrophic in nature.
Parasitic Plants A parasitic plant is one that lines inside or outside the other organism and derive their food from them. The plant (non-green) which obtains their food from other organism is called a parasite and the living organism from whose body, food is obtained is called host, e.g. amarbel or Cuscuta. It takes readymade food from host through special type of roots called sucking roots which penetrate into host plant and suck food material from the host.
Insectivorous Plants There are some plants which can trap insects and digest them for their nutrition. These plants are green in colour but lack nitrogen elements. To overcome this problem, these plants eat insects. Hence they are called insectivorous plant or carnivorous plants. These have specialised leaves, the apex of which forms a lid that can open and close the mouth of pitcher. There are hair inside the pitcher which are used to entangle the insects.
When an insect comes in contact of the lid, it gets closed and traps the insects. The insect inside the pitcher is digested by digestive juices secreted by the pitcher to obtain nitrogen compounds (amino acids) from them. e.g. pitcher plant, sundew, Venus flytrap and bladderwort.
Since these can synthesise their own food but fulfil their nitrogen deficiency by eating insects, therefore these are called as partial heterotrophs.
Saprotrophic Plants The mode of nutrition in which organisms take their nutrients from dead and decaying matter is called saprotrophic nutrition. Plants which use the saprotrophic mode of nutrition are called saprotrophs, e.g. fungi like mushrooms are non-green plants that grow on the dead and decaying matter for their food. Bread moulds (fungi) and yeast are saprophytic plants.
Symbiotic Plants Sometimes, two plants of different species live together and help each other in obtaining food and shelter. This association is called symbiosis and such plants are called symbiotic plants. The relationship in which two different organisms live together and share shelter and nutrients is called symbiotic relationship, e.g. lichens and Rhizobium.
Lichen is an association in which algae and a fungus live together. The fungus provides shelter, water and minerals to the algae and in return, the algae provide food which it prepares by photosynthesis.
Replenishment of Nutrients in Soil Crops require a lot of nitrogen to make proteins. After the harvest, the soil becomes deficient in nitrogen. Plants cannot use the nitrogen gas available in the atmosphere directly. The action of certain bacteria can convert this nitrogen into a form readily used by plants. Rhizobium bacteria live in the root nodules of leguminous plants. These bacteria take nitrogen gas from the atmosphere and convert it into water-soluble nitrogen compounds making it available to the leguminous plants for their growth.
In return, leguminous plants provide food and shelter to the bacteria as Rhizobium cannot prepare its food. They, thus have a symbiotic relationship. This association is very important for the farmers, as they do not need to add nitrogen fertilisers to the soil in which leguminous plants are grown.
We hope the given CBSE Class 7 Science Notes Chapter 1 Nutrition in Plants Pdf free download will help you. If you have any query regarding NCERT Class 7 Science Notes Chapter 1 Nutrition in Plants, drop a comment below and we will get back to you at the earliest.
The figures which we can draw on a flat surface are called Plane Figures. They have two dimensions i.e. length and breadth, hence these are called 2 Dimensional Shapes.
Solid Figures (3-Dimensional Shape)
The figures which have three dimensions i.e. length, breadth and height are called 3 –Dimensional Shape. They occupy some space. Like the ball, box, tube etc.
Faces, Edges and Vertices
Faces – All the flat surfaces of the 3-D figure are the faces of that shape. The faces of 3D shapes are made by the 2-D shapes.
Edges – The line segment where the faces of the 3D shape meet with each other are the edges of that shape.
Vertices – The corners or the points where the edges meet with each other are the vertices of the 3D shape. The singular form of vertices is the vertex.
No. of Faces, Edges and Vertices in some common 3- D Shapes
S.No.
3 – D shape
Figure
Faces
Edges
Vertices
1.
Cube
6
12
8
2.
Cuboid
6
12
8
3.
Cone
2
1
1
4.
Sphere
1
1
0
5.
Cylinder
3
2
0
6.
Square based Pyramid
6
8
5
Nets for building 3 – D Shapes
If we draw the structure on the 2D form and fold it to make a 3D shape then it is said to be the net of that figure. Different figures have a different type of nets.
We can open a 3D shape from its edges to get the net of that figure.
Drawing Solids on a Flat Surface (2D representation of 3D Shapes)
As you know that the 3D shapes are the shapes which occupy some space but we can draw the 3D shapes on the flat surface also by some techniques. This is called a visual illusion.
1. Oblique Sketches
When we draw a shape in such a way that we are not able to see some of the faces of the 3D shape and the size of the length is also not equal but we are able to recognize that this is a cube then this is called an Oblique Sketch.
Steps to draw an Oblique Sketch
Draw the sketch of a cube with a side of 4 cm each.
Step 1: First, we need to draw a square of 2 × 2 on a grid sheet.
Step 2: Draw the opposite face of the same size which is offset to the front face.
Step 3: Now, join the respective corners, to form a cube.
Points to remember for the oblique sketch-
The front and back faces are of the same size.
All the edges are appearing as of the same length but their measurements are different.
2. Isometric Sketches
The isometric sheet is the sheet made up of dots which makes the equilateral triangles. When we draw the shape on the isometric sheet with the measurements proportional to the original figure then it is said to be Isometric sketch.
Visualizing Solid Objects
When we see any 3D shape from one side then some of its parts are not visible to us. But then also we can assume that which shape it is, this is called Visualization.
We can see that the front, top and side view of the above image is completely different and we cannot see some of its faces then also we can say that this figure is made up of four cubes.
Viewing different sections of a Solid
There are so many ways to see the different sections of the 3D shape-
1. View an object by Cutting or Slicing
When we cut any 3D object horizontally or vertically, we get a 2D face of that figure. This is called the Cross-section. The shape of the cross section depends upon the type of cut like horizontal or vertical.
If we give a horizontal cut to a rectangular pyramid then the cross section will be a rectangle in shape and if we cut it vertically then we get the shape of a triangle.
2. Viewing an object by its Shadow
If we throw the torchlight on any 3D object then we will get the shadow of that object on the plane in the form of 2D shape. The resultant shape depends upon the side of the object where we throw the light.
If we throw the light on the cylinder from its circular side then we will get the image of a circle.
If we throw the light on the cylinder while it is on standing position then its shadow will be rectangular in shape.
3. Viewing from different angles to get different views
All the 3D objects have a different view if we see them from different sides. By seeing them from different angles we can observe them easily.
In the above figure, we will have a different top view, front view and side view but we get the information about the shape of the figure by observing them.
If two or more parts of a figure are identical after folding or flipping then it is said to be symmetry. To be symmetrical the two halves of a shape must be of same shape and size.
If the shape is not symmetrical then it is said to be asymmetrical.
Line of Symmetry
It is an imaginary line which divides the image into two equal halves. It could be horizontal, vertical or diagonal. There could be one or more than one line of symmetry in a figure.
Lines of Symmetry for Regular Polygons
If all the sides and angles of a polygon are equal then it is said to be a regular polygon. Like the equilateral triangle, square etc.
All the regular polygons are symmetrical shapes.
In the regular polygon, the number of lines of symmetry is the same as the number of its sides.
Regular Polygon
Number of Sides
Line of Symmetry
Image
Equilateral Triangle
3
3
Square
4
4
Regular Pentagon
5
5
Regular Hexagon
6
6
Types of Symmetry
There are two types of Symmetry
1. Reflection Symmetry
If we draw a dotted line which gives the mirror reflection of the other half of the image then it is reflection symmetry. It is the same as basic symmetry which tells us that if the dotted line divides the image into two equal halves then it is the reflective symmetry of the figure.
2. Rotational Symmetry
If we rotate the image at a centre point of the image at 360° then the number of times the image looks the same, shows the rotational symmetry of the image.
Rotational Symmetry
If a figure rotates at a fixed point then that point is the centre of Rotation.
It could rotate clockwise or anticlockwise.
While rotation the measurement of the angle which we take is the angle of rotation. And a complete rotation is of 360°.
If the angle of rotation is 180° then it is called Half Turn and if the angle of rotation is 90° then it is called a Quarter Turn.
This image looks symmetrical but there is no line of symmetry in it i.e. there is any such line which divides it into two equal halves. But if we rotate it at 90° about its centre then it will look exactly the same. This shows that it has Rotational Symmetry.
While rotating, there are four positions when the image looks exactly the same. So this windmill has a rotational symmetry of order 4 about its centre.
Example
What is the Rotational symmetry of the given figure?
Solution:
To find the rotational symmetry, we have to find
The angle of rotation = 90°
Direction = clockwise
Order of rotation = 4
This shows that if the given figure rotates anticlockwise at 90° around its centre then it has rotational symmetry of order 4.
Line Symmetry and Rotational Symmetry
Some shapes have only line symmetry and some shapes have only rotational symmetry but there are some shapes which have both types of symmetry.
Example
Find whether the given image has rotational symmetry or line symmetry or both.
Solution:
Rotational Symmetry
If we rotate the image clockwise at an angle of 360° around its centre then it will have rotational symmetry of order 1 or no symmetry as every image will look same if we rotate it at 360°.
This will not look the same at every 120° because of the colour of the balls at its edges.
Line Symmetry
This figure will have three line of symmetry. As there are three possible lines which can divide the image into two equal halves.
Example
Tell whether the figures below have line symmetry or rotational symmetry or both.
Solution:
The first figure have 2 line of symmetry and rotational symmetry of order 2.
The second figure has no line of symmetry but have rotational symmetry of order 3.
The third figure has 1 line of symmetry but no rotational symmetry.
If we have to multiply the powers which have the same base then we have to add the exponents.
am × an = am + n
Example
83 × 84 = 83 + 4 = 87
2. How to divide powers with the same base?
If we have to divide the powers which have the same base then we have to subtract the exponents.
Example
3. How to take the power of a power?
If we have to take the power of a power then we have to multiply the exponents.
(am)n = amn
Example
(83)4 = 83 × 4 = 812
4. How to multiply the powers with the same exponents?
If we have to multiply the powers where the base is different but exponents are same then we will multiply the base.
ambm = (ab)m
Example
8343 = (8× 4)3 = 323
5. How to divide the powers with the same exponents?
If we have to divide the powers where the base is different but exponents are same then we will divide the base.
Example
6. Numbers with Exponent Zero
Any number with zero exponents is equal to one irrespective of the base.
a° = 1
Example
8° = 1
7. Numbers with Exponent One
Any number with one as the exponent is equal to the number itself.
a1 = a
Example
81 = 8
8. Power with a Negative Exponent
Negative exponents can be converted into positive exponents.
Example
Miscellaneous Examples
Example: 1
Example: 2
Expressing Large Numbers in the Standard Form
If we have to write very large numbers then to make them easy to read and understand we can write them in the standard form using decimals and exponents from 1.0 to 10.0.
An Algebraic Expression is the combination of constant and variables. We use the operations like addition, subtraction etc to form an algebraic expression.
Variable
A variable does not have a fixed value .it can be varied. It is represented by letters like a, y, p m etc.
Constant
A constant has a fixed value. Any number without a variable is a constant.
Example
1. 2x + 7
Here we got this expression by multiplying 2 and x and then add 7 to it.
In the above expression, the variable is x and the constant is 7.
2. y2
We get it by multiplying the variable y to itself.
Terms of an Expression
Terms
To form an expression we use constant and variables and separate them using the operations like addition, subtraction etc. these parts of expressions which we separate using operations are called Terms.
In the above expression, there are three terms, 4x, – y and 7.
Factors of a Term
Every term is the product of its factors. As in the above expression, the term 4x is the product of 4 and x. So 4 and x are the factors of that term.
We can understand it by using a tree diagram.
Coefficients
As you can see above that some of the factors are numerical and some are algebraic i.e. contains variable.The numerical factor of the term is called the numerical coefficient of the term.
In the above expression,
-1 is the coefficient of ab
2 is the coefficient of b2
-3 is the coefficient of a2.
Parts of an Expression
Here in the above figure, you can identify the terms, variables, constants and coefficients.
Like and Unlike Terms
Like Terms are the terms which have same algebraic factors. They must have the same variable with the same exponent.
Unlike Terms are the terms which have different algebraic factors.
2x2 + 3x – 5 does not contain any term with same variable.
2a2 + 3a2 + 7a – 7 contains two terms with same variable i.e. 2a2 and 3a2.so these are like terms.
Monomials, Binomials, Trinomials and Polynomials
Expressions
Meaning
Example
Monomial
Any expression which has only one term.
5x2, 7y, 3ab
Binomial
Any expression which has two, unlike terms.
5x2 + 2y, 2ab – 3b
Trinomial
Any expression which has three, unlike terms.
5x2 + 2y + 9xy, x + y – 3
Polynomial
Any expression which has one or more terms with the variable having non-negative integers as an exponent is a polynomial.
5x2 + 2y + 9xy + 4 and all the above expressions are also polynomial.
Remark: All the expressions like monomial, binomial and trinomial are also a polynomial.
Addition and Subtraction of Algebraic Expression
1. Addition of Like Terms
If we have to add like terms then we can simply add their numerical coefficients and the result will also be a like term.
Example
Add 2x and 5x.
Solution
2x + 5x
= (2 × x) + (5 × x)
= (2 + 5) × x (using distributive law)
=7 × x = 7x
2. Subtraction of Like Terms
If we have to subtract like terms then we can simply subtract their numerical coefficients and the result will also be a like term.
Example
Subtract 3p from 11p.
Solution
11p – 3p
= (11-3) p
= 8p
3. Addition of unlike terms
If we have to add the unlike terms then we just have to put an addition sign between the terms.
Example
Add 9y, 2x and 3
Solution
We will simply write it like this-
9y + 2x + 3
4. Subtraction of Unlike Terms
If we have to subtract the unlike terms then we just have to put minus sign between the terms.
Example
Subtract 9y from 21.
Solution
We will simply write it like this-
21 – 9y
5. Addition of General Algebraic Expression
To add the general algebraic expressions, we have to arrange them so that the like terms come together, then simplify the terms and the unlike terms will remain the same in the resultant expression.
While subtracting the algebraic expression from another algebraic expression, we have to arrange them according to the like terms then subtract them.
Subtraction is same as adding the inverse of the term.
Example
Subtract 4ab– 5b2 – 3a2 from 5a2 + 3b2 – ab
Solution
Finding the Value of an Expression
1. Expressions with One Variable
If we know the value of the variable in the expression then we can easily find the numerical value of the given expression.
Example
Find the value of the expression 2x + 7 if x = 3.
Solution
We have to put the value of x = 3.
2x + 7
= 2(3) + 7
= 6 + 7
= 13
2. Expressions with two or more variables
To find the value of the expression with 2 variables, we must know the value of both the variables.
Example
Find the value of y2 + 2yz + z2 if y = 2 and z = 3.
Solution
Substitute the value y = 2 and z = 3.
y2 + 2yz + z2
= 22 + 2(2) (3) + 32
= 4 + 12 + 9
= 25
Formula and Rules using Algebraic Expression
There are so many formulas which are made using the algebraic expression.
Perimeter Formulas
1. The perimeter of an equilateral triangle = 3l where l is the length of the side of the equilateral triangle by l and l is variable which can be varied according to the size of the equilateral triangle.
2. The perimeter of a square = 4l where l = the length of the side of the square.
3. The perimeter of a regular pentagon = 5l where l = the length of the side of the Pentagon and so on.
Area formulas
1. The area of the square = a2 where a is the side of the square
2. The area of the rectangle = l × b = lb where the length of a rectangle is l and its breadth is b
3. The area of the triangle = 1/2 × b × h where b is the base and h is the height of the triangle. Here if we know the value of the variables given in the formulas then we can easily calculate the value of the quantity.
Example
What is the perimeter of a square if the side of the square is 4 cm?
Solution
The perimeter of a square = 4l
l = 4 cm
4 × 4 = 16 cm
Rules for the Number Pattern
1. If we denote a natural number by n then its successor will always be (n + 1). If n = 3 then n + 1 will be 3 + 1 = 4.
2. If we denote a natural number by n then 2n will always be an even number and (2n + 1) will always be an odd number. If n = 3 then 2n = 2(3) = 6(even number), n = 3 then 2n + 1 = 2(3) + 1 = 7 (odd number)
3. If we arrange the multiples of 5 in ascending order then we can denote it by 5n. If we have to check that what will be the 11th term in this series then we can check it by 5n. n = 11 so 5n = 5(11) = 55.
Pattern in geometry
The number of diagonals which we can draw from one vertex of any polygon is (n – 3) where n is the number of sides of the polygon.
How many diagonals can be drawn from the one vertex of a hexagon?
The number of diagonals will be (n -3).
The number of sides in a hexagon is 8 so (n – 3) = (8 – 3) = 5
It refers to the length of the outline of the enclosed figure.
Area
It refers to the surface of the enclosed figure.
Area and Perimeter of Square
Square is a quadrilateral, with four equal sides.
Area = Side × Side
Perimeter = 4 × Side
Example
Find the area and perimeter of a square-shaped cardboard whose length is 5 cm.
Solution
Area of square = (side)2
= (5)2
= 25 cm2
Perimeter of square = 4 × side
= 4 × 5
= 20 cm
Area and Perimeter of Rectangle
The rectangle is a quadrilateral, with equal opposite sides.
Area = Length × Breadth
Perimeter = 2(Length + Breadth)
Example
What is the length of a rectangular field if its width is 20 ft and Area is 500 ft2?
Solution
Area of rectangular field = length × width
500 = l × 20
l = 500/20
l = 25 ft
Note: Perimeter of a regular polygon = Number of sides × length of one side
Triangles as Parts of Rectangles
If we draw a diagonal of a rectangle then we get two equal sizes of triangles. So the area of these triangles will be half of the area of a rectangle.
The area of each triangle = 1/2 (Area of the rectangle)
Likewise, if we draw two diagonals of a square then we get four equal sizes of triangles .so the area of each triangle will be one-fourth of the area of the square.
The area of each triangle = 1/4 (Area of the square)
Example
What will be the area of each triangle if we draw two diagonals of a square with side 7 cm?
Solution
Area of square = 7 × 7
= 49 cm2
The area of each triangle = 1/4 (Area of the square)
= 1/4 × 49
= 12.25 cm2
Congruent Parts of Rectangles
Two parts of a rectangle are congruent to each other if the area of the first part is equal to the area of the second part.
Example
The area of each congruent part = 1/2 (Area of the rectangle)
= 1/2 (l × b) cm2
=1/2 (4 × 3) cm2
= 1/2 (12) cm2
= 6 cm2
Parallelogram
It is a simple quadrilateral with two pairs of parallel sides.
Also denoted as ∥ gm
Area of parallelogram = base × height
Or b × h (bh)
We can take any of the sides as the base of the parallelogram. And the perpendicular drawn on that side from the opposite vertex is the height of the parallelogram.
Example
Find the area of the figure given below:
Solution
Base of ∥ gm = 8 cm
Height of ∥ gm = 6 cm
Area of ∥ gm = b × h
= 8 × 6
= 48 cm
Area of Triangle
Triangle is a three-sided closed polygon.
If we join two congruent triangles together then we get a parallelogram. So the area of the triangle will be half of the area of the parallelogram.
Area of Triangle = 1/2(Area of ∥ gm)
= 1/2 (base × height)
Example
Find the area of the figure given below:
Solution
Area of triangle = 1/2 (base × height)
= 1/2 (12 × 5)
= 1/2 × 60
= 30 cm2
Note: All the congruent triangles are equal in area but the triangles equal in the area need not be congruent.
Circles
It is a round, closed shape.
The circumference of a Circle
The circumference of a circle refers to the distance around the circle.
Radius: A straight line from the Circumference till the centre of the circle.
Diameter: It refers to the line from one point of the Circumference to the other point of the Circumference.
π (pi): It refers to the ratio of a circle’s circumference to its diameter.
Circumference(c) = π × diameter
C = πd
= π × 2r
Note: diameter (d) = twice the radius (r)
d = 2r
Example
What is the circumference of a circle of diameter 12 cm (Take π = 3.14)?
Solution
C = πd
C = 3.14 × 12
= 37.68 cm
Area of Circle
Area of the circle = (Half of the circumference) × radius
= πr2
Example
Find the area of a circle of radius 23 cm (use π = 3.14).
Solution
R = 23 cm
π = 3.14
Area of circle = 3.14 × 232
= 1,661 cm2
Conversion of Units
Sometimes we need to convert the unit of the given measurements to make it similar to the other given units.
Unit
Conversion
1 cm
10 mm
1 m
100 cm
1 km
1000 m
1 hectare(ha)
100 × 100 m
Unit
Conversion
1 cm2
100 mm2
1 m2
10000 cm2
1 km2
1000000 m2 (1e + 6)
1 ha
10000 m2
Example: 1
Convert 70 cm2 in mm2
Solution:
1 cm = 10 mm
1 cm2 = 10 × 10
1 cm2 = 100 mm2
70 cm2 = 700 mm2
Example: 2
Convert 3.5 ha in m2
Solution:
1 ha = 10000 m2
3.5 ha = 10000 × 3.5
ha = 35000 m2
Applications
We can use these concepts of area and perimeter of plane figures in our day to day life.
If we have a rectangular field and want to calculate that how long will be the length of the fence required to cover that field, then we will use the perimeter.
If a child has to decorate a circular card with the lace then he can calculate the length of the lace required by calculating the circumference of the card, etc.
Example:
A rectangular park is 35 m long and 20 m wide. A path 1.5 m wide is constructed outside the park. Find the area of the path.
Solution
Area of rectangle ABCD – Area of rectangle STUV
AB = 35 + 2.5 + 2.5
= 40 m
AD = 20 + 2.5 + 2.5
= 25 m
Area of ABCD = 40 × 25
= 1000 m2
Area of STUV = 35 × 20
= 700 m2
Area of path = Area of rectangle ABCD – Area of rectangle STUV
A line segment is a part of a line with two endpoints.
A line perpendicular to a line segment
Any line which is perpendicular to a line segment makes an angle of 90°.
Construction of a line parallel to a given line, through a point not on the line
We need to construct it using ruler and compass only.
Step 1: Draw a line PQ and take a point R outside it.
Step 2: Take a point J on the line PQ and join it with R.
Step 3: Take J as a centre and draw an arc with any radius which cuts PQ at C and JR at B.
Step 4: Now with the same radius, draw an arc taking R as a centre.
Step 5: Take the measurement of BC with compass and mark an arc of the same measurement from R to cut the arc at S.
Step 6: Now join RS to make a line parallel to PR.
∠ARS = ∠BJC, hence RS ∥ PQ because of equal corresponding angles.
This concept is based on the fact that a transversal between two parallel lines creates a pair of equal corresponding angles.
Remark: This can be done by taking alternate interior angles instead of corresponding angles.
Construction of triangles
The construction of triangles is based on the rules of congruent triangles. A triangle can be drawn if-
Three sides are given (SSS criterion).
Two sides and an included angle are given (SAS criterion).
Two angles and an included side are given. (ASA criterion).
A hypotenuse and a side are given for right angle triangle (RHS criterion).
Construction of a triangle with three given sides (SSS criterion)
Example
Draw a triangle ABC with the sides AB = 6 cm, BC = 5 cm and AC = 9 cm.
Solution
Step 1: First of all draw a rough sketch of a triangle, so that we can understand how to go ahead.
Step 2: Draw a line segment AB = 6 cm.
Step 3: From point A, C is 9 cm away so take A as a centre and draw an arc of 9 cm.
Step 4: From point B, C is 5 cm away so take B as centre and draw an arc of 5 cm in such a way that both the arcs intersect with each other.
Step 5: This point of intersection of arcs is the required point C. Now join AC and BC.
ABC is the required triangle.
Construction of a triangle if two sides and one included angle is given (SAS criterion)
Example
Construct a triangle LMN with LM = 8 am, LN = 5 cm and ∠NLM = 60°.
Solution
Step 1: Draw a rough sketch of the triangle according to the given information.
Step 2: Draw a line segment LM = 8 cm.
Step 3: draw an angle of 60° at L and make a line LO.
Step 4: Take L as a centre and draw an arc of 5 cm on LO.
Step 5: Now join NM to make a required triangle LMN.
Construction of a triangle if two angles and one included side is given (ASA criterion)
Example
Draw a triangle ABC if BC = 8 cm, ∠B = 60°and ∠C = 70°.
Solution
Step 1: Draw a rough sketch of the triangle.
Step 2: Draw a line segment BC = 8 cm.
Step 3: Take B as a centre and make an angle of 60° with BC and join BP.
Step 4: Now take C as a centre and draw an angle of 70° using a protractor and join CQ. The point where BP and QC intersects is the required vertex A of the triangle ABC.
ABC is the required triangle ABC.
Construction of a right angle triangle if the length of the hypotenuse and one side is given (RHScriterion)
Example
Draw a triangle PQR which is right angled at P, with QR =7 cm and PQ = 4.5 cm.
Solution
Step 1: Draw a rough sketch of the triangle.
Step 2: Draw a line segment PQ = 4.5 cm.
Step 3: At P, draw PS ⊥ PQ. This shows that R must be somewhere on this perpendicular.
Step 4: Take Q as a centre and draw an arc of 7 cm which intersects PS at R.
Revision Notes on Practical Geometry
Line segment
A line segment is a part of a line with two endpoints.
A line perpendicular to a line segment
Any line which is perpendicular to a line segment makes an angle of 90°.
Construction of a line parallel to a given line, through a point not on the line
We need to construct it using ruler and compass only.
Step 1: Draw a line PQ and take a point R outside it.
Step 2: Take a point J on the line PQ and join it with R.
Step 3: Take J as a centre and draw an arc with any radius which cuts PQ at C and JR at B.
Step 4: Now with the same radius, draw an arc taking R as a centre.
Step 5: Take the measurement of BC with compass and mark an arc of the same measurement from R to cut the arc at S.
Step 6: Now join RS to make a line parallel to PR.
∠ARS = ∠BJC, hence RS ∥ PQ because of equal corresponding angles.
This concept is based on the fact that a transversal between two parallel lines creates a pair of equal corresponding angles.
Remark: This can be done by taking alternate interior angles instead of corresponding angles.
Construction of triangles
The construction of triangles is based on the rules of congruent triangles. A triangle can be drawn if-
Three sides are given (SSS criterion).
Two sides and an included angle are given (SAS criterion).
Two angles and an included side are given. (ASA criterion).
A hypotenuse and a side are given for right angle triangle (RHS criterion).
Construction of a triangle with three given sides (SSS criterion)
Example
Draw a triangle ABC with the sides AB = 6 cm, BC = 5 cm and AC = 9 cm.
Solution
Step 1: First of all draw a rough sketch of a triangle, so that we can understand how to go ahead.
Step 2: Draw a line segment AB = 6 cm.
Step 3: From point A, C is 9 cm away so take A as a centre and draw an arc of 9 cm.
Step 4: From point B, C is 5 cm away so take B as centre and draw an arc of 5 cm in such a way that both the arcs intersect with each other.
Step 5: This point of intersection of arcs is the required point C. Now join AC and BC.
ABC is the required triangle.
Construction of a triangle if two sides and one included angle is given (SAS criterion)
Example
Construct a triangle LMN with LM = 8 am, LN = 5 cm and ∠NLM = 60°.
Solution
Step 1: Draw a rough sketch of the triangle according to the given information.
Step 2: Draw a line segment LM = 8 cm.
Step 3: draw an angle of 60° at L and make a line LO.
Step 4: Take L as a centre and draw an arc of 5 cm on LO.
Step 5: Now join NM to make a required triangle LMN.
Construction of a triangle if two angles and one included side is given (ASA criterion)
Example
Draw a triangle ABC if BC = 8 cm, ∠B = 60°and ∠C = 70°.
Solution
Step 1: Draw a rough sketch of the triangle.
Step 2: Draw a line segment BC = 8 cm.
Step 3: Take B as a centre and make an angle of 60° with BC and join BP.
Step 4: Now take C as a centre and draw an angle of 70° using a protractor and join CQ. The point where BP and QC intersects is the required vertex A of the triangle ABC.
ABC is the required triangle ABC.
Construction of a right angle triangle if the length of the hypotenuse and one side is given (RHScriterion)
Example
Draw a triangle PQR which is right angled at P, with QR =7 cm and PQ = 4.5 cm.
Solution
Step 1: Draw a rough sketch of the triangle.
Step 2: Draw a line segment PQ = 4.5 cm.
Step 3: At P, draw PS ⊥ PQ. This shows that R must be somewhere on this perpendicular.
Step 4: Take Q as a centre and draw an arc of 7 cm which intersects PS at R.
Rational Numbers are the numbers that can be expressed in the form p/q where p and q are integers (q ≠ 0). It includes all natural, whole numbers, fractions and integers.
Equivalent Rational Numbers
By multiplying or dividing the numerator and denominator of a rational number by the same integer, we can obtain another rational number equivalent to the given rational number.
Numbers are said to be equivalent if they are proportionate to each other.
Example
Therefore 1/2, 2/4, 4/8 are equivalent to each other as they are equal to each other.
Positive and Negative Rational Numbers
1. Positive Rational Numbers are the numbers whose both the numerator and denominator are positive.
Example: 3/4, 12/24 etc.
2. Negative Rational Numbers are the numbers whose one of the numerator or denominator is negative.
Example: (-2)/6, 36/(-3) etc.
Remark: The number 0 is neither a positive nor a negative rational number.
Rational Numbers on the Number Line
Representation of whole numbers, natural numbers and integers on a number line is done as follows
Rational Numbers can also be represented on a number line like integers i.e. positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.
Representation of rational numbers can be done on a number line as follows
Rational Numbers in Standard Form
A rational number is in the standard form if its denominator is a positive integer and there is no common factor between the numerator and denominator other than 1.
If any given rational number is not in the standard form then we can reduce it to its standard form or the lowest form by dividing its numerator and denominator by their HCF ignoring its negative sign.
Example
Find the standard form of 12/18
Solution
2/3 is the standard or simplest form of 12/18
Comparison of Rational Numbers
1. To compare the two positive rational numbers we need to make their denominator same, then we can easily compare them.
Example
Compare 4/5 and 3/8 and tell which one is greater.
Solution
To make their denominator same, we need to take the LCM of the denominator of both the numbers.
LCM of 5 and 8 is 40.
2. To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.
Example
Compare – (2/5) and – (3/7) and tell which one is greater.
Solution
To compare, we need to compare them as normal numbers.
LCM of 5 and 7 is 35.
by reversing the order of the numbers.
3. If we have to compare one negative and one positive rational number then it is clear that the positive rational number will always be greater as the positive rational number is on the right to 0 and the negative rational numbers are on the left of 0.
Example
Compare 2/5 and – (2/5) and tell which one is greater.
Solution
It is simply that 2/5 > – (2/5)
Rational Numbers between Rational Numbers
To find the rational numbers between two rational numbers, we have to make their denominator same then we can find the rational numbers.
Example
Find the rational numbers between 3/5 and 3/7.
Solution
To find the rational numbers between 3/5 and 3/7, we have to make their denominator same.
LCM of 5 and 7 is 35.
Hence the rational numbers between 3/5 and 3/7 are
These are not the only rational numbers between 3/5 and 3/7.
If we find the equivalent rational numbers of both 3/5 and 3/7 then we can find more rational numbers between them.
Hence we can find more rational numbers between 3/5 and 3/7.
Remark: There are “n” numbers of rational numbers between any two rational numbers.
Operations on Rational Numbers
1. Addition
a. Addition of two rational numbers with the same denominator
i. We can add it using a number line.
Example:
Add 1/5 and 2/5
Solution:
On the number line we have to move right from 0 to 1/5 units and then move 2/5 units more to the right.
ii. If we have to add two rational numbers whose denominators are same then we simply add their numerators and the denominator remains the same.
Example
Add 3/11 and 7/11.
Solution
As the denominator is the same, we can simply add their numerator.
b. Addition of two Rational Numbers with different denominator
If we have to add two rational numbers with different denominators then we have to take the LCM of denominators and find their equivalent rational numbers with the LCM as the denominator, and then add them.
Example
Add 2/5 and 3/7.
Solution
To add the two rational numbers, first, we need to take the LCM of denominators the find the equivalent rational numbers.
LCM of 5 and 7 is 35.
c. Additive Inverse
Like integers, the additive inverse of rational numbers is also the same.
This shows that the additive inverse of 3/7 is – (3/7) This shows that
2. Subtraction
If we have to subtract two rational numbers then we have to add the additive inverse of the rational number that is being subtracted to the other rational number.
a – b = a + (-b)
Example
Subtract 4/21 from 8/21.
Solution
i. In the first method, we will simply subtract the numerator and the denominator remains the same.
ii. In the second method, we will add the additive inverse of the second number to the first number.
3. Multiplication
a. Multiplication of a Rational Number with a Positive Integer.
To multiply a rational number with a positive integer we simply multiply the integer with the numerator and the denominator remains the same.
Example
b. Multiply of a Rational Number with a Negative Integer
To multiply a rational number with a negative integer we simply multiply the integer with the numerator and the denominator remains the same and the resultant rational number will be a negative rational number.
Example
c. Multiply of a Rational Number with another Rational Number
To multiply a rational number with another rational number we have to multiply the numerator of two rational numbers and multiply the denominator of the two rational numbers.
Example
Multiply 3/7 and 5/11.
Solution
4. Division
a. Reciprocal
Reciprocal is the multiplier of the given rational number which gives the product of 1.
Reciprocal of a/b is b/a
Product of Reciprocal
If we multiply the reciprocal of the rational number with that rational number then the product will always be 1.
Example
b. Division of a Rational Number with another Rational Number
To divide a rational number with another rational number we have to multiply the reciprocal of the rational number with the other rational number.
Example
Divide
Solution
Rational Numbers
Rational Numbers are the numbers that can be expressed in the form p/q where p and q are integers (q ≠ 0). It includes all natural, whole numbers, fractions and integers.
Equivalent Rational Numbers
By multiplying or dividing the numerator and denominator of a rational number by the same integer, we can obtain another rational number equivalent to the given rational number.
Numbers are said to be equivalent if they are proportionate to each other.
Example
Therefore 1/2, 2/4, 4/8 are equivalent to each other as they are equal to each other.
Positive and Negative Rational Numbers
1. Positive Rational Numbers are the numbers whose both the numerator and denominator are positive.
Example: 3/4, 12/24 etc.
2. Negative Rational Numbers are the numbers whose one of the numerator or denominator is negative.
Example: (-2)/6, 36/(-3) etc.
Remark: The number 0 is neither a positive nor a negative rational number.
Rational Numbers on the Number Line
Representation of whole numbers, natural numbers and integers on a number line is done as follows
Rational Numbers can also be represented on a number line like integers i.e. positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.
Representation of rational numbers can be done on a number line as follows
Rational Numbers in Standard Form
A rational number is in the standard form if its denominator is a positive integer and there is no common factor between the numerator and denominator other than 1.
If any given rational number is not in the standard form then we can reduce it to its standard form or the lowest form by dividing its numerator and denominator by their HCF ignoring its negative sign.
Example
Find the standard form of 12/18
Solution
2/3 is the standard or simplest form of 12/18
Comparison of Rational Numbers
1. To compare the two positive rational numbers we need to make their denominator same, then we can easily compare them.
Example
Compare 4/5 and 3/8 and tell which one is greater.
Solution
To make their denominator same, we need to take the LCM of the denominator of both the numbers.
LCM of 5 and 8 is 40.
2. To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.
Example
Compare – (2/5) and – (3/7) and tell which one is greater.
Solution
To compare, we need to compare them as normal numbers.
LCM of 5 and 7 is 35.
by reversing the order of the numbers.
3. If we have to compare one negative and one positive rational number then it is clear that the positive rational number will always be greater as the positive rational number is on the right to 0 and the negative rational numbers are on the left of 0.
Example
Compare 2/5 and – (2/5) and tell which one is greater.
Solution
It is simply that 2/5 > – (2/5)
Rational Numbers between Rational Numbers
To find the rational numbers between two rational numbers, we have to make their denominator same then we can find the rational numbers.
Example
Find the rational numbers between 3/5 and 3/7.
Solution
To find the rational numbers between 3/5 and 3/7, we have to make their denominator same.
LCM of 5 and 7 is 35.
Hence the rational numbers between 3/5 and 3/7 are
These are not the only rational numbers between 3/5 and 3/7.
If we find the equivalent rational numbers of both 3/5 and 3/7 then we can find more rational numbers between them.
Hence we can find more rational numbers between 3/5 and 3/7.
Remark: There are “n” numbers of rational numbers between any two rational numbers.
Operations on Rational Numbers
1. Addition
a. Addition of two rational numbers with the same denominator
i. We can add it using a number line.
Example:
Add 1/5 and 2/5
Solution:
On the number line we have to move right from 0 to 1/5 units and then move 2/5 units more to the right.
ii. If we have to add two rational numbers whose denominators are same then we simply add their numerators and the denominator remains the same.
Example
Add 3/11 and 7/11.
Solution
As the denominator is the same, we can simply add their numerator.
b. Addition of two Rational Numbers with different denominator
If we have to add two rational numbers with different denominators then we have to take the LCM of denominators and find their equivalent rational numbers with the LCM as the denominator, and then add them.
Example
Add 2/5 and 3/7.
Solution
To add the two rational numbers, first, we need to take the LCM of denominators the find the equivalent rational numbers.
LCM of 5 and 7 is 35.
c. Additive Inverse
Like integers, the additive inverse of rational numbers is also the same.
This shows that the additive inverse of 3/7 is – (3/7) This shows that
2. Subtraction
If we have to subtract two rational numbers then we have to add the additive inverse of the rational number that is being subtracted to the other rational number.
a – b = a + (-b)
Example
Subtract 4/21 from 8/21.
Solution
i. In the first method, we will simply subtract the numerator and the denominator remains the same.
ii. In the second method, we will add the additive inverse of the second number to the first number.
3. Multiplication
a. Multiplication of a Rational Number with a Positive Integer.
To multiply a rational number with a positive integer we simply multiply the integer with the numerator and the denominator remains the same.
Example
b. Multiply of a Rational Number with a Negative Integer
To multiply a rational number with a negative integer we simply multiply the integer with the numerator and the denominator remains the same and the resultant rational number will be a negative rational number.
Example
c. Multiply of a Rational Number with another Rational Number
To multiply a rational number with another rational number we have to multiply the numerator of two rational numbers and multiply the denominator of the two rational numbers.
Example
Multiply 3/7 and 5/11.
Solution
4. Division
a. Reciprocal
Reciprocal is the multiplier of the given rational number which gives the product of 1.
Reciprocal of a/b is b/a
Product of Reciprocal
If we multiply the reciprocal of the rational number with that rational number then the product will always be 1.
Example
b. Division of a Rational Number with another Rational Number
To divide a rational number with another rational number we have to multiply the reciprocal of the rational number with the other rational number.
Figure 11: Take the first quadrant
Figure 12: Choosing the scale
Figure 13: Marking the values for time and distance
Figure 14: Obtaining a straight line graph
