Simplification Tricks- Mathematics Notes For CTET Exam: Free PDF

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt.

To let you make the most of Mathematics section, we are providing important facts related to the Simplification. At least 1-2 questions are asked from this topic in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

Mathematics Study Notes For All Teaching Exams

Simplification

In simplification an expression, we must remove the brackets strictly in the order ( ), { }, [ ] and then we must apply the operations:

Of, Division, Multiplication, Addition and Subtraction.

‘BODMAS’ where B stands for bracket, O for of (‘Of’ means multiplication); D for division; M for Multiplication, A for Addition and S for Subtraction strictly in the order.

Division Algorithm:  Dividend = (Divisor × Quotient) + Remainder

Modulus or Absolute value : The absolute value of a real number X is denoted by the symbol |x| and is defined as –

Ex. : |5| = 5, |-5| = -(-5) = 5

In multiplication and division, when both the numbers carry similar sign, we get positive sign in the result otherwise we get negative sign in the result i.e.

(+) × (+)          = +

(+) × (-)           = –

(-) × (+)           = –

(-) × (-)           = +

(+) × (+)          = +

(+) × (-)           = –

(-) × (+)           = –

(-) × (-)           = +

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Important terms:

Identity elements of Addition: ‘0’ (zero) is called identity element of addition of ‘O’ in any number does not affect that number.

e.g. x + 0 = x (x ∈ Q)

Identity element of Multiplication: ‘1’ is called identity element of multiplication as multiplication of ‘1’ in any number does not affect that number.

e.g. x × 1 = x

Inverse element of Addition / Negative element of Addition / Additive Inverse: The number is called “Additive inverse” of a certain number, when it is added to the certain number and result becomes ‘0’ (zero).

Ex.

• x + (-x) = 0

Here (-x) is Additive inverse of x.

• 9 + (-9) is Additive inverse of ‘9’

Inverse element of Multiplication / Reciprocal element / Multiplicative Inverse: The number is called “Multiplicative inverse” of a certain number, when the product of number and multiplication inverse is 1.

How to Overcome Exam Fever, Especially When You Fear Maths

CONTINUED FRACTION: A continued fraction consists of the fractional denominators

Componendo and dividendo (C & D): It is a theorem on proportions that allows for a quick way to perform calculations and Reduce the amount of expansions needed It is particularly useful when dealing with equations involving fractions or rational functions.

Some Points

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Recurring Number:

Pure recurring decimals: These are recurring decimals where the recurrence starts immediately after the decimal point.

Impure recurring decimals: Unlike pure recurring decimals, in these decimals, the recurrence occurs after a certain number of digits in the decimal.

Mathematics Study Notes For All Teaching Exams

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt.

Mathematics is the crucial subject for clearing the CTET exam. CTET exam is in two forms Paper-1 (For Primary Level) and Paper-2 (Secondary Level).

How to Overcome Exam Fever, Especially When You Fear Maths

• The mathematics section in CTET paper 1 and paper 2 exam has 30 questions for 30 marks in each paper.
• In Paper I, 15 Questions covers content part which is related to Class 1-5 math books and other 15 Questions covers Teaching pedagogy part which is related to teaching methodologies, remedial teaching, errors and evaluation based on Class 1-5 students.
• In Paper II, 15 Questions covers content part which is related to Class 6-8 math books and other 15 Questions covers Teaching pedagogy part which is related to teaching methodologies, remedial teaching, errors and evaluation based on Class 6-8 students.

Mathematics Weightage in Different Teaching Exams

 Exam Total Marks Total Question CTET PAPER 1 30 30 CTET PAPER 2 30 30 DSSSB PRT 20 20 DSSSB TGT Maths 120 120

To let you make the most of Mathematics section, we are providing important facts related to the Important Mathematics Topics.We wish you all the best of luck to come over the fear of the Mathematics section.

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Important Topic of Mathematics study Notes

 S.NO TOPIC 1. Number System 2. Facts and Formulae 3. Properties of Integers 4. Divisibility Rule 5. Powers, Indices and Surds 6. Fractions 7. Linear Equation 8. Algebra 9. Percentage 10. Average 11. Mixture and Allegation 12. Ratio and Proportion 13. Simple Interest 14. Compound Interest 15. Time and Work 16. Speed Distance and Time 17. Lines and Angles 18. Shapes and Spatial Understanding 19. Congruence And Similarity Of Triangles 20. Properties Of Circle 21. Properties Of Circle Part 2 22. Mensuration-2D 23. Mensuration-3D 24. Data Handling 25. Unit Digit 26. Simplification Tricks 27. Basics Of Probability 28. Fundamental Concepts Of Geometry

Previous Year CTET (Paper 2) Maths Pedagogy Questions

 S.NO TOPIC 1. CTET 2012 Maths Pedagogy Questions 2. CTET 2014 Maths Pedagogy Questions 3. CTET 2015 Maths Pedagogy Questions 4. CTET 2016 Maths Pedagogy Questions

Mathematics Practice Quiz

 S.No Click on the link given below to download Maths question PDF for Teaching Exams: 1 Mathematics Miscellaneous Quiz 2. Mathematics Miscellaneous Quiz 3. Mathematics Miscellaneous Quiz 4. Mathematics Miscellaneous Quiz

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Study Notes on Important Facts and Formulae- Math Study Notes for CTET Exam

Study Notes on Important Facts and Formulae

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt. Generally, there are questions asked related to basic concepts, Facts and Formulae of the Arithmetic.

To let you make the most of Mathematics section, we are providing important facts related to the Number System, HCF, LCM and Decimals. At least 5-6 questions are asked from these topics in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

How to Overcome Exam Fever, Especially When You Fear Maths

• FACTORS AND MULTIPLES: If a number ‘a’ divides another number ‘b’ exactly, then ‘a’ is a factor of ‘b’. In this case, ‘b’ is called a multiple of ‘a’.

Ex. Find the total number of factors of 240.

Sol. 240 = 2×2×2×2×3×5

= 24 × 31 × 51

total factors = 4+1 × 1+1 × 1+1=20

• HCF (Highest common Factor): The HCF of two or more than two numbers is the greatest number that divides each of them exactly.

Ex. HCF of 36, 72, 108

Sol. 36 = 22 × 33

72 = 23 × 32

108 = 22 × 33

HCF = 22 × 32 = 36

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• LCM (Lowest Common Multiple): The least number which is exactly divisible by each one of the given numbers is called their LCM

Ex. LCM of 87 and 145

Sol. 87= 3×29

145 = 5×29

LCM= 3×5×29= 435

• Product of two numbers = Products of their HCF and LCM and LCM is always divisible by HCF.

Ex. The sum and difference of the LCM and HCF of two numbers are 112 and 72 respectively. If the one of the numbers is 46, find the 2nd number.

Sol.

L+H = 112

L-H = 72

___________

2H = 40

___________

H= 20

L+20 = 112

L= 92

1st number = 46

Let the 2nd No. = x

46×x = 92×20

x= 40

• HCF and LCM of fractions:

• HCF of two numbers = HCF of sum of the numbers and their LCM.
• Decimal fraction: Fraction in which denominators are power of 10 are known as decimal fractions

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• Recurring Decimal: If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.

• Basic formulae:
• (a+b) (a-b) = (a²-b²)
• (a-b)² = a²+b²-2ab
• (a+b)²= a²+b²+2ab

Ex. (x+y)=21 and xy=108, then what is the value of x²+y²

Sol. Sol. x+y= 21

xy= 108

(x+y)² = x²+y²+2xy

(21)² = x²+y²+2×108

441 = x²+y²+216

x²+y² = 225

Mathematics Study Notes For All Teaching Exams

• (a+b+c)² = a²+b²+c²+2(ab+bc+ca)
• (a³+b³)= (a+b) (a²+b²-ab)

• (a³-b³)= (a-b) (a²+b²+ab)

• (a³+b³c³-3abc)= (a+b+c) (a²+b²+c²-ab-bc-ca)
• When a+b+c = 0, then a³+b³+c³= 3abc
• (a+b)²+(a-b)²= 2(a²+b²)
• (a-b)³ = a³-b³-3ab(a-b)
• (a+b)²-(a-b)² = 4ab

• BODMAS Rule: The rule depicts the correct sequence in which operations are to be executed, so as to find out the value of given expression.

Ex. ‘B’ for Bracket, ‘O’ for of, ‘D’ for Division, ‘M’ for Multiplication, ‘A’ or Addition and ‘S’ for Subtraction.

Ex.  The value of (6.5×7.25+8.5×19.5+4.5-8.5)

Sol. (6.5×7.25+8.5×19.5+4.5-8.5)

= 47.125+165.75+4.5-8.5

= 217.375-8.5

= 208.875

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Fundamental Concepts Of Geometry – Mathematics Notes For CTET Exam: Free PDF

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt.

To let you make the most of Mathematics section, we are providing important facts related to the Simplification. At least 1-2 questions are asked from this topic in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

Mathematics Study Notes For All Teaching Exams

FUNDAMENTAL CONCEPTS OF GEOMETRY

Point: It is an exact location. It is a fine dot which has neither length nor breadth nor thickness but has position i.e., it has no magnitude.

Line segment: The straight path joining two points A and B is called a line segment AB . It has and points and a definite length.

Ray: A line segment which can be extended in only one direction is called a ray.

Intersecting lines: Two lines having a common point are called intersecting lines. The common point is known as the point of intersection.

Concurrent lines: If two or more lines intersect at the same point, then they are known as concurrent lines.

Angles: When two straight lines meet at a point they form an angle.

Right angle: An angle whose measure is 90° is called a right angle.

Acute angle: An angle whose measure is less then one right angle (i.e., less than 90°), is called an acute angle.

Obtuse angle: An angle whose measure is more than one right angle and less than two right angles (i.e., less than 180° and more than 90°) is called an obtuse angle.

Reflex angle: An angle whose measure is more than 180° and less than 360° is called a reflex angle.

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Complementary angles: If the sum of the two angles is one right angle (i.e.,90°), they are called complementary angles. Therefore, the complement of an angle θ is equal to 90° – θ.

Supplementary angles: Two angles are said to be supplementary, if the sum of their measures is 180°. Example: Angles measuring 130° and 50° are supplementary angles. Two supplementary angles are the supplement of each other. Therefore, the supplement of an angle θ. is equal to 180° – θ..

Vertically opposite angles: When two straight lines intersect each other at a point, the pairs of opposite angles so formed are called vertically opposite angles.

Bisector of an angle: If a ray or a straight line passing through the vertex of that angle, divides the angle into two angles of equal measurement, then that line is known as the Bisector of that angle.

Parallel lines: Two lines are parallel if they are coplanar and they do not intersect each other even if they are extended on either side.

Transversal: A transversal is a line that intersects (or cuts) two or more coplanar lines at distinct points.

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Triangles: Part-1

1. A centroid divides a median in 2 : 1 ratio.

Centroid a point of intersection of three medians.

1. Ratio of two adjacent sidesof a triangle is equal to the two parts of third side which makes by the internal angle bisector.
2. In anequilateral triangle internal angle bisector and the median are same.
3. Any two of the four triangles formed by joiningthe mid-point of the sides of a given triangle are-congruent.
4. Two triangles have the same area if they have thesame base and lie between two parallel lines.
5. In a triangle side opposite to smaller angle is smallerin comparison to the side which is opposite to greater angle.
6. Orthocenter → Point of intersection of three Altitudes.

Incentre → Point of intersection of the angle bisectors of a triangle

Circumcenter → Point of intersection of the perpendicular bisectors of the sides.

Median → Line joining the mid-point of a side to the vertex opposite to the side.

1. When the corresponding sides of two triangles are in proportion then the corresponding angles are also in proportion.

If the two triangles are similar, then we have the following results –

Ratio of area of two triangles = Ratio of squares of corresponding sides

Ratio of sides of two triangles= Ratio of height (Altitudes)

= Ratio of medians

= Ratio of angle bisector

= Ratio of Perimeter

1. In a right angled triangle, the triangle on each side of the altitude drawn from the vertex of the right angleto the hypotenuse is similar to the original triangle and to each other too.

Basics Of Probability -Mathematics Notes For CTET Exam: Free PDF

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt.

To let you make the most of Mathematics section, we are providing important facts related to the Simplification. At least 1-2 questions are asked from this topic in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

Mathematics Study Notes For All Teaching Exams

PROBABILITY

Probability: It is the numerical measurement of the degree of certainty. There are two types of approaches to study probability

Experimental or Empirical Probability: The result of probability based on the actual experiment is called experimental probability. In this case, the results could be different if we do the same experiment again.

Probability — A Theoretical Approach: In the theoretical approach, we predict the results without performing the experiment actually. The other name of theoretical probability is classical probability.

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Probability of Occurrence of an Event

Theoretical probability associated with an event E is defined as “If there are ‘n’ elementary events associated with a random experiment and m of these are favourable to the event E then the probability of occurrence of an event is defined by P(E) as the ratio mn “.

If P(E) = 1, then it is called a ‘Certain Event’.

If P(E) = 0, then it is called an ‘Impossible Event’.

The probability of an event E is a number P(E) such that: 0 ≤ P(E) ≤ 1

An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

For any event E, P(E) + P(E¯) = 1, where E¯ stands for ‘not E’. E and E¯ are called complementary events.

Favourable outcomes are those outcomes in the sample space that are favourable to the occurrence of an event.

Sample Space:

A collection of all possible outcomes of an experiment is known as sample space. It is denoted by ‘S’ and represented in curly brackets.

Examples of Sample Spaces:

A coin is tossed = Event

E1 = Getting a head (H) on upper face

E2 = Getting a tail (T) on upper face

S = {H, T}

Total number of outcomes = 2

Two coins are tossed = Event = E

E1 = Getting a head on coin 1 and a tail on coin 2 = (H, T)

E2 = Getting a head on both coin 1 and coin 2 = (H, H)

E3 = Getting a tail on coin 1 and a head on coin 2 = (T, H)

E4 = Getting a tail on both, coin 1 and coin 2 = (T, T)

S = {(H, T), (H, H), (T, H), (T, T)}.

Total number of outcomes = 4

How to Overcome Exam Fever, Especially When You Fear Maths

Important Note:

Coin: A coin has two faces termed as Head and Tail.

Dice: A dice is a small cube which has between one to six spots or numbers on its sides, which is used in games.

Cards: A pack of playing cards consists of four suits called Hearts, Spades, Diamonds and Clubs. Each suite consists of 13 cards.

Example 1.  A coin is tossed 10 times and the outcomes are observed as:

H, T, H, T, T, H, H, T, H, H (H is Head; T is Tail)

What is the probability of getting Head?

Ans.(a)

Example 2. A bag contains 3 white, 2 blue and 5 red balls. One ball is drawn at random from the bag. What is the probability that the ball drawn is not red?

Ans.(a)

Sol.

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Example 3. Which of the following statement is incorrect?

(a) Probability of an event lies between 0 and 1.

(b) Probability of an impossible event is 1 and that of a sure event is 0.

(c) Probability is the measure of the chance of an event happening.

(d) None of these

Ans.(b)

Sol. correct statement ⟹ Probability of an impossible event is 0 and that of a sure event is 1.

Diagnostic And Remedial Teaching In Mathematics -Notes For CTET Exam: Free PDF

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt. Generally, there are questions asked related to basic concepts, Facts and Formulae of the Mathematics.

Mathematics Study Notes For All Teaching Exams

To let you make the most of Mathematics section, we are providing important facts related to the Mathematics Pedagogy. At least 10-15 questions are asked from Mathematics Pedagogy in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

How to Overcome Exam Fever, Especially When You Fear Maths

Diagnostic and Remedial Teaching in Mathematics

The main aim of diagnostic evaluation is to determine the causes of learning problems and to formulate a plan for remedial action.

“A test that is sharply focused on some specific aspect of a skill or some specific cause of difficulty in acquiring a skill, and that is useful in suggesting specific remedial actions that might help to improve mastery of that skill is a diagnostic test.“ – Thorndike.

“A diagnostic test undertakes to provide a picture of strengths and weaknesses.” – Payne.

A diagnostic test is a useful tool for analyzing difficulties but it is simply a starting point. Supplementary information concerning the physical, intellectual, social, and emotional development of the pupil is also needed before an effective remedial programme can be initiated.

In diagnostic testing the following points must be kept in mind:

1. Who are the pupils who need help?
2. Where are the errors located?
3. Why did the error occur?

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The essential steps in educational diagnosis are:

Identifying the students who are having trouble or need help: First, one must know the learners who require help. For this you can administer a general achievement test based on the topics already taught. After evaluation you will be in a position to make lists of students who are below average, average or above average. Next, one has to locate the area where the error occurs in order to have a deeper insight into the pupils’ difficulties.

Locating the errors or learning difficulties: After identifying the students who need help and visualising the necessity of additional instructional material to improve the quality of learning, your main role is to find out the area where the learner commits mistakes or which is the area where learning difficulties lie.

Discovering the causal factors of slow learning: In some cases of learning difficulties, the causal factors are relatively simple. A student may be inattentive during teaching-learning or may be committing errors due to insufficient practice or irregular attendance. Sometimes the cause is ill-health or faulty work habits etc. It has also been observed sometimes that the basic cause of low achievement is a feeling of helplessness or the complexity of the subject-matter which perhaps is much above the level of their comprehension.

Remedial Teaching

Remedial instruction or teaching helps in overcoming the difficulties due to instruction. It helps the students to be with the normal students in acquiring the common level of achievement. The term ‘remedial teaching’ is generally used instead of remedial instruction by various educationists.

“Remedial teaching tries to be specific and exact. It attempts to find a procedure which will cause the child to correct his errors of the past and thus in a sense prevents future error.“ –Yokam

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Salient Features of Remedial Teaching:

• Remedial teaching is a dynamic side of the diagnostic testing. Hence it depends on the educational diagnosis.
• To overcome the difficulties in learning and in acquisition of skills is the main purpose of remedial instruction.
• Remedial teaching is not only useful to cure the shortcomings but also in preventive measures.
• Remedial teaching is a short-term treatment.
• Remedial teaching helps the below average students to be with the normal students in acquiring the common level of achievement.

The ultimate aim of diagnosis is to remove the weaknesses and difficulties of students. If some emotional or physical factors are responsible for the weaknesses, then efforts should be made to eliminate these factors with the help of concerned peoples. After eliminating the factors, remedial teaching should be done. The mathematics teacher may also prepare corrective material for this purpose. Thus, by remedial teaching the success can be achieved in removing the weaknesses of the student.

Mathematics Pedagogy Notes PDF (Hindi)

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Unit Digit – Mathematics Notes For CTET Exam: Free PDF

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt.

To let you make the most of Mathematics section, we are providing important facts related to the Unit Digit. At least 1-2 questions are asked from this topic in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

Mathematics Study Notes For All Teaching Exams

Unit Digit

In the exams like CTET or State TET, the Mathematics section is the most scoring section among all. Students can score full marks in the Mathematics section if they manage their time properly in the exam. Questions may seem time-consuming but shortcut methods can save our crucial time. Students who are unaware of the shortcut methods can’t beat time.

Question: Find the unit place digit in (657)85 – (158)37

TIPS AND TRICKS TO FIND UNIT DIGIT

Numbers are classified into three categories to find unit digit.

1. Digits 0,1,5,6
2. Digits 4,9
3. Digits 2,3,7,8

Digits 0,1,5,6

When we have these numbers (0,1,5,6) in the unit place, we get the same digit itself at the unit place when raised to any power, i.e. 0n=0, 1n=1, 5n=5, 6n=6. Let us apply this concept to the following questions.

Example: Find the Unit place digit of the following numbers:

1. 360244

1. 2974281307

1. 4575400000666

1. 5687686265749375

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Digits 4 & 9

Both these numbers have a cyclicity of only two different digits as their unit’s digit.

In the case of 4 & 9

• If the Power of 4 is Even, the result will be 6
• If the Power of 4 is Odd, the result will be 4
• If the Power of 9 is Even, the result will be 1
• If the Power of 9 is Odd, the result will be 9

Example: Find the Unit place digit of the following numbers:

1. 456847426734258

1. 3456445767843

1. 548574657895768454

1. 4576348567895627369765787

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Digits 2,3,7,8

For Digit 2

When we have number 2 in the unit place then follow the given steps to find the unit digit.

Step 1- Divide the last two digits of the power of a given number with 4

Step 2- You get the remainder n

Step 3- Since you have got n as a remainder, put it as the power of 2, i.e (2)n

Step 4- Have a look at the table below and mark your answer.

Example: Find the Unit place digit of the following numbers:

1. 4657233

Here, the unit place is 2 and power is 33. To solve follow the given steps

Step 1- Divide 33 by 4.

Step 2- You get remainder 1.

Step 3- Since you have got remainder 1, put it as a power of 2, i.e 21

Step 4- Have a look at the table above, 21=2.

1. 7657845678235

Here, the unit place is 2 and power is 33. To solve follow the given steps

Step 1- Divide 35 by 4.

Step 2- You get remainder 3.

Step 3- Since you have got remainder 3, put it as a power of 2, i.e 23

Step 4- Have a look at the table above, 23=8.

How to Overcome Exam Fever, Especially When You Fear Maths

For the digits 3,7,8

Repeat the steps

When we have the numbers 3,7,8 in the unit place then follow the given steps to find the unit digit.

Step 1- Divide the last two digits of the power of a given number with 4

Step 2- You get the remainder n

Step 3- Since you have got n as a remainder, put it as the power of 3,7,8, i.e 3n, 7n, 8n

Step 4- Have a look at the table below and mark your answer.

For Digit 3

Example: Find the Unit place digit of the following numbers:

1. 4657333

Here, the unit place is 3 and power is 33. To solve follow the given steps

Step 1- Divide 33 by 4.

Step 2- You get remainder 1.

Step 3- Since you have got remainder 1, put it as a power of 3, i.e 31

Step 4- Have a look at the table above, 31=3.

For Digit 7

Example: Find the Unit place digit of the following numbers:

1. 4657718

Here, the unit place is 7 and power is 18. To solve follow the given steps

Step 1- Divide 18 by 4.

Step 2- You get remainder 2.

Step 3- Since you have got remainder 2, put it as a power of 7, i.e 72

Step 4- Have a look at the table above, 72=9.

For Digit 8

Example: Find the Unit place digit of the following numbers:

1. 4657859

Here unit place is 8 and power is 59. To solve follow the given steps

Step 1- Divide 59 by 4.

Step 2- You get remainder 3.

Step 3- Since you have got remainder 3, put it as a power of 8, i.e 83

Step 4- Have a look at the table above, 83=2.

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt. Generally, there are questions asked related to basic concepts, Facts and Formulae of the Mathematics.

Mathematics Study Notes For All Teaching Exams

To let you make the most of Mathematics section, we are providing important facts related to the Mathematics Pedagogy. At least 10-15 questions are asked from Mathematics Pedagogy in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

How to Overcome Exam Fever, Especially When You Fear Maths

Mensuration 3D

3D Shapes: All 3D shapes can be described in terms of their faces, vertices and edges.

Face: A flat or curved surface

Edge: Line where 2 faces meet

Vertex: A point where 3 or more edges meet

• Cuboid: It is a solid figure which has 6 regular faces, 12 edges, 8 vertices and 4 diagonals.

Let length = L, Breadth = b and height = h

Volume = (l×b×h)

Total surface Area= 2(lb+bh+hl)

Ex.  The length, breadth and height of a cuboid are in the ratio 5:6:8 and its volume is 1920 cm³. The total surface area of cuboid is:

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• Cube: A cube has 6 equal faces, 12 equal edges, 8 vertices and 4 equal diagonals. Let edge = a

Volume = a³

Total surface area = 6a²

Lateral surface area = 4a²

• Cylinder: A right circular cylinder is a cylinder whose base is a circle and whose elements are perpendicular to its base. Let radius = r

Height = h

Ex.  The radius of a cylinder is 28cm and its height is 54cm. Its volume is

• Cone: A right circular cone is one whose axis is perpendicular to the plane of the base. Let radius = r

Ex. Curved Surface area of a cone whose volume is 4,224 cm³ and radius is 24cm, is

• Frustum of a cone: When a cone is cut by a plane parallel to the base of the cone then the portion between the plane and the base is called the frustum of the cone.

Let, radius of base= R, radius of top =r and height =h

• Sphere: A sphere is a solid bounded by a closed surface every point of which is equidistant from a fixed point called the center. Let the radius = r

Ex. The radius of solid metallic sphere is 8cm, find the volume of the sphere.

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• Hemisphere: A plane through the center of the sphere cuts it into two equal parts. Each part is called Hemisphere. Let the radius =r

Ex.  The radius of hemispherical bowl is 27cm. find the volume of the bowl.

• Pyramid: A pyramid is made by connecting a base to an apex. There are many types of pyramid and they are named after the shape of their base.

Total surface area= Area of base + Area of each of the lateral faces

Ex. Area of base of a pyramid is 89 cm² and height is 9 cm then its volume (in cm³) is:

• Prism: A prism is a solid object with identical ends and flat faces.

Volume = area of base × height

Total surface Area = 2(area of base) + (perimeter of base × height)

Curved surface Area = perimeter of base × height

Mathematics Notes PDF (Hindi)

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Bloom’s Taxonomy Interpreted For Mathematics – Notes For CTET Exam: Free PDF

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt. Generally, there are questions asked related to basic concepts, Facts and Formulae of the Mathematics.

Mathematics Study Notes For All Teaching Exams

To let you make the most of Mathematics section, we are providing important facts related to the Mathematics Pedagogy. At least 10-15 questions are asked from Mathematics Pedagogy in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

How to Overcome Exam Fever, Especially When You Fear Maths

Bloom’s Taxonomy Interpreted for Mathematics

Bloom’s Taxonomy is an educational tool developed by Benjamin S. Bloom (1913-1999) that ranks the relative cognitive complexity of various educational objectives. This taxonomy is often used as an aid when creating test questions and assignments.

Bloom’s Taxonomy of Cognitive Skills:

Knowledge – retention of terminology, facts, conventions, methodologies, structures, principles, etc.

Comprehension – grasping of meaning, translation, extrapolation, interpretation of facts, making comparisons, etc.

Application – problem solving, usage of information in a new way

Analysis – making inferences and supporting them with evidence, identification of patterns

Synthesis – derivation of abstract relations, prediction, generalization, creation of new ideas

Evaluation – judgement of validity, usage of a set of criteria to make conclusions, discrimination

Questions that encourage each of these skills often begin with:

Knowledge: List, define, describe, show, name, what, when, etc.

Comprehension: Summarize, compare and contrast, estimate, discuss, etc.

Application: Apply, calculate, complete, show, solve, modify, etc.

Analysis: Separate, arrange, classify, explain, etc.

Synthesis: Integrate, modify, substitute, design, create, What if…, formulate, generalize, prepare, etc.

Evaluation: Assess, rank, test, explain, discriminate, support, etc.

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This taxonomy can be used to invent test or assignment questions. Here is an interpretation of each cognitive skill in a mathematical context. The example questions are aimed for introductory level, single-variable calculus students, but could be modified to apply to other courses.

Knowledge: Questions include “State the definition”, “State the theorem”, or “Use the specified method.”

E.g., Take the derivative of the following rational function using quotient rule.

Comprehension: Questions ask the student to use definitions or methods to calculate something.

E.g., Find the slope of the tangent line to the following function at a given point.

Application: Questions which require the usage of more than one definition, theorem, and/or algorithm.

E.g., Find the derivative of the following implicitly defined function. (This question could be used to test logarithmic differentiation as well, for instance)

Analysis: Questions require the student to identify the appropriate theorem and use it to arrive at the given conclusion or classification. Alternatively, these questions can provide a scenario and ask the student to generate a certain type of conclusion.

E.g., Let f(x) be a fourth-degree polynomial. How many roots can f(x) have? Explain.

Synthesis: Questions are similar to Analysis questions, but the conclusion to be reached by the student is an algorithm for solving the given question. This also includes questions which ask the student to develop their own classification system.

E.g., optimization word problems where student generates the function to be differentiated.

Evaluation: Questions are similar to Synthesis questions, except the student is required to make judgements about which information should be used.

E.g., related rate word problem where student decides which formulae are to be used and which of the given numbers are constants or instantaneous values.

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Mathematics Pedagogy Notes PDF (Hindi)

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The Van Hiele Model Of Geometric Thinking- Mathematics Notes For CTET Exam: Free PDF

Mathematics is an equally important section for CTET, MPTET, KVS & DSSSB Exams and has even more abundant importance in some other exams conducted by central or state govt. Generally, there are questions asked related to basic concepts, Facts and Formulae of the Mathematics.

Mathematics Study Notes For All Teaching Exams

To let you make the most of Mathematics section, we are providing important facts related to the Mathematics Pedagogy. At least 10-15 questions are asked from Mathematics Pedagogy in most of the teaching exams. We wish you all the best of luck to come over the fear of the Mathematics section.

How to Overcome Exam Fever, Especially When You Fear Maths

Van Hiele theory

The theory has three aspects: the existence of levels, the properties of the levels, and the progress from one level to the next level.

Van Hiele levels

According to the theory, there are five levels of thinking or understanding in geometry:

• Level 0 Visualization
• Level 1 Analysis
• Level 2 Abstraction
• Level 3 Deduction
• Level 4 Rigor

Level 0 Visualization (Basic visualization or Recognition)

At this level pupils use visual perception and nonverbal thinking. They recognize geometric figures by their shape as “a whole” and compare the figures with their prototypes or everyday things (“it looks like door”), categorize them (“it is / it is not a…”). They use simple language. They do not identify the properties of geometric figures.

Level 1 Analysis (Description)

At this level pupils (students) start analyzing and naming properties of geometric figures. They do not see relationships between properties, they think all properties are important (= there is no difference between necessary and sufficient properties). They do not see a need for proof of facts discovered empirically. They can measure, fold and cut paper, use geometric software etc.

Level 2 Abstraction (Informal deduction or Ordering or Relational)

At this level pupils or students perceive relationships between properties and figures. They create meaningful definitions. They are able to give simple arguments to justify their reasoning. They can draw logical maps and diagrams. They use sketches, grid paper, geometric SW.

Pierre van Hiele wrote: “My experience as a teacher of geometry convinces me that all too often, students have not yet achieved this level of informal deduction. Consequently, they are not successful in their study of the kind of geometry that Euclid created, which involves formal deduction.”

Level 3 Deduction (Formal deduction)

At this level students can give deductive geometric proofs. They are able to differentiate between necessary and sufficient conditions. They identify which properties are implied by others. They understand the role of definitions, theorems, axioms and proofs.

Level 4 Rigor

At this level students understand the way how mathematical systems are established. They are able to use all types of proofs. They comprehend Euclidean and non-Euclidean geometry. They are able to describe the effect of adding or removing an axiom on a given geometric system.

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Properties of levels

The levels have five important characteristics:

Fixed sequence (order): A student cannot be at level N without having gone through level (N−1). Therefore, the student must go through the levels in order.

Adjacency: At each level, what was intrinsic in the preceding level becomes extrinsic in the current level.

Distinction: Each level has its own linguistic symbols and its own network of relationships connecting those symbols. The meaning of a linguistic symbol is more than its explicit definition; it includes the experiences which the speaker associates with the given symbol. What may be “correct” at one level is not necessarily correct at another level.

Separation: Two persons at different levels cannot understand each other. The teacher speaks a different “language” to the student at a lower level. The van Hieles thought this property was one of the main reasons for failure in geometry.

Attainment: The learning process leading to complete understanding at the next level has five phases – information, guided orientation, explanation, free orientation, integration, which are approximately not strictly sequential.

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Five phases of the learning process

Van Hieles believed that cognitive progress in geometry can be accelerated by instruction. The progress from one level to the next one is more dependent upon instruction than on age or maturity. They gave clear explanations of how the teacher should proceed to guide students from one level to the next. However, this process takes tens of hours.

• Information or Inquiry: Students get the material and start discovering its structure. The teacher holds a conversation with the pupils, in well-known language symbols, in which the context he wants to use becomes clear. (A teacher might say: “This is a rhombus. Construct some more rhombi on your paper.”)
• Guided or directed orientation: Students deal with tasks which help them to explore implicit relationships. The teacher suggests activities that enable students to recognize the properties of the new concept. The relations belonging to the context are discovered and discussed. (A teacher might ask: “What happens when you cut out and fold the rhombus along a diagonal? Along the other diagonal?”)
• Explanation or Explication: Students formulate what they have discovered, and new terminology is introduced. They share their opinions on the relationships they have discovered in the activity. The teacher makes sure that the correct technical language is developed and used. The van Hieles thought it is more useful to learn terminology after students have had an opportunity to become familiar with the concept. (A teacher might say: “Here are the properties we have noticed and some associated terminology for the things you have discovered. Let us discuss what these mean: The diagonals lie on the lines of symmetry. There are two lines of symmetry. The opposite angles are congruent. The diagonals bisect the vertex angles.”)
• Free orientation: Students solve more complex tasks independently. It brings them to master the network of relationships in the material. They know the properties being studied, but they need to develop understanding of relationships in various situations. This type of activity is much more open-ended. (A teacher might say: “How could you construct a rhombus given only two of its sides?” and other problems for which students have not learned a fixed procedure.)
• Integration: Students summarize what they have learned and keep it in mind. The teacher should give to the students an overview of everything they have learned. It is important that the teacher does not present any new material during this phase, but only a summary of what has already been learned. (A teacher might say: “Here is a summary of what we have learned. Write this in your notebook and do these exercises for homework.”)

Pierre van Hiele wrote: “A definition of a concept is only possible if one knows, to some extent, the thing that is to be defined.”

Mathematics Pedagogy Notes PDF (Hindi)

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