Caring about the development of students’ mathematical proficiency

Caring about the development of students’ mathematical proficiency

Students want to learn in a harmonious environment. Teachers can help  create such an environment by respecting and valuing the mathematics  and the cultures that students bring to the classroom. By ensuring safety, teachers make it easier for all their students to get involved. It is important, however, that they avoid the kind of caring relationships that  encourage dependency. Rather, they need to promote classroom  relationships that allow students to think for themselves, ask questions,
and take intellectual risks.

Classroom routines play an important role in developing students’ mathematical thinking and reasoning. For example, the everyday practice of inviting students to contribute responses to a mathematical question or problem may do little more than promote cooperation. Teachers need to go further and clarify their expectations about how students can and should contribute, when and in what form, and how others might respond. Teachers who truly care about the development of their students’ mathematical proficiency show interest in the ideas they construct and express, no matter how unexpected or unorthodox.  By modelling the practice of evaluating ideas, they encourage their students to make thoughtful judgments about the mathematical soundness of the ideas voiced by their classmates. Ideas that are showno be sound contribute to the shaping of further instruction.

Quardatic equation test paper cbse maths for classs 10

Math Adda

1. The sum of the reciprocals of Rehman’s ages 3 years ago and 5 years from now is 1/3. Find his present age.

2. In a class test the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in two subjects.

3. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

4. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

5. A train travels 360 kms at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

6. Two water taps together can fill a tank in  9 ³/₈  hours. The tap of the larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank

7. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

8. Sum of areas of two squares is 468 sqm. If the difference of their perimeters is 24 metres, find the sides of two squares.

9. Find two consecutive positive integers, sum of whose squares is 365.

10. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs. 90, find the number of articles produced and the cost of each article.

11. The altitude of a right angled triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

12. A train travels a distance of 480 kms at a uniform speed. If the speed had been 8 kmph less, then it would have taken 3 hrs more to cover the same distance. We need to find the speed of the train.

13. Rohan’s mother is 26 years older than him. The product of their ages 3 years from now will be 360. We need to find Rohan’s present age.

14. Solve by factorization     a. 4x² - 4a²x + (a⁴ – b⁴) = 0  b.    (x – 3) (x – 4) = 34/33²

cbse math for 10 mcq trigonometry

cbse maths types of Quadrilateral

cbse maths types of Quadrilateral

Quadrilaterals 


A quadrilateral is a closed plane figure bounded by four line segments. E.g. The figure ABCD shown here is a quadrilateral.


A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral. For example, AC is a diagonal of quadrilateral ABCD.

Types of Quadrilaterals

There are six basic types of quadrilaterals:
1. Rectangle: Opposite sides of a rectangle are parallel and equal. All angles are 90º.


2. Square
Opposite sides of a square are parallel and all sides are equal. All angles are 90º.

3. Parallelogram 
Opposite sides of a parallelogram are parallel and equal. Opposite angles are equal.

4. Rhombus

All sides of a rhombus are equal and opposite sides are parallel. Opposite angles of a rhombus are equal. The diagonals of a rhombus bisect each other at right angles.


5. Trapezium
A trapezium has one pair of opposite sides parallel. A regular trapezium has non-parallel sides equal and its base angles are equal, as shown in the following diagram.

6. Kite

Two pairs of adjacent sides of a kite are equal, and one pair of opposite angles are equal. Diagonals intersect at right angles. One diagonal is bisected by the other.

Theorem 3 Prove that the angle sum of a quadrilateral is equal to 360º.
Proof: To prove < A + <B + <C + <D= 360

In tri ABC p + u + B = 180 (angle sum property of triangle)-----1
Similarly
In Tri. ACD , q + v + D = 180--------2
Adding (1 ) and  (2)

(p +  q) + (u+  v ) + B+ D = 180+ 180
 < A + <B + <C + <D= 360

Hence the angle sum of a quadrilateral is 360º.

Source  http://cbseadda.blogspot.com

Real Numbers examples and guess paper for class10

Sample Question 1: Using Euclid’s division algorithm, find which of the following
pairs of numbers are co-prime:
(i) 231, 396 (ii) 847, 2160

Solution : Let us find the HCF of each pair of numbers.
(i) 396 = 231 × 1 + 165
231 = 165 × 1 + 66
165 = 66 × 2 + 33
66 = 33 × 2 + 0
Therefore, HCF = 33. Hence, numbers are not co-prime.
(ii) 2160 = 847 × 2 + 466
847 = 466 × 1 + 381
466 = 381 × 1 + 85
381 = 85 × 4 + 41
85 = 41 × 2 + 3
41 = 3 × 13 + 2
3 = 2 × 1 + 1
2 = 1 × 2 + 0
Therefore, the HCF = 1. Hence, the numbers are co-prime.

Sample Question 2: Show that the square of an odd positive integer is of the form
8m + 1, for some whole number m.

Solution: Any positive odd integer is of the form 2q + 1,
where q is a whole number.
Therefore, (2q + 1)2 = 4q2 + 4q + 1
                                 = 4q (q + 1) + 1,----- (1)
q (q + 1) is either 0 or even.
So, it is 2m, where m is a whole number.
Therefore, (2q + 1)2 = 4.2 m + 1 = 8 m + 1. [From (1)]

Sample Question 3: Prove that √2 + √3 is irrational.


Solution : Let us suppose that √2 + √3 is rational. Let 2 + √3 = a , where a is
rational.

Therefore, √2 = a− √3
Squaring on both sides, we get
2 = a²  + 3 – 2a√3
2a√3 = a²  + 3 -2
2a√3 = a²  + 1
Therefore,
√3 = (a² + 1)/2a

which is a contradiction as the right hand side is a rational number while 3 is irrational. Hence,√2 + √3 is irrational.


Sample Question 4: 
Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.

Solution : 

We know that any positive integer can be of the form 6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4 or 6m + 5, for some integer m.

Thus, an odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5
Thus we have:

(6 m +1)2 = 36 m2 + 12 m + 1 = 6 (6 m2 + 2 m) + 1 = 6 q + 1, q is an integer

(6 m + 3)2 = 36 m2 + 36 m + 9 = 6 (6 m2 + 6 m + 1) + 3 = 6 q + 3, q is an integer

(6 m + 5)2 = 36 m2 + 60 m + 25 = 6 (6 m2 + 10 m + 4) + 1 = 6 q + 1, q is an integer.

Thus, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3.

                                             Now Solve these Questions

1. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

2. Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

3. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

4. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

5. Show that the square of any odd integer is of the form 4q + 1, for some integer q.

6. If n is an odd integer, then show that n2 – 1 is divisible by 8.

7. Prove that if x and y are both odd positive integers, then x2 + y2 is even but not divisible by 4.

8. Use Euclid’s division algorithm to find the HCF of 441, 567, 693.

9. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

10. Prove that √ 3+ √ 5 is irrational.

11. Show that 12n cannot end with the digit 0 or 5 for any natural number n.

12. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

13. Show that the cube of a positive integer of the form 6q + r, q is an integer and

r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.

14. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is

any positive integer.

15. Prove that one of any three consecutive positive integers must be divisible by 3.

16. For any positive integer n, prove that n3 – n is divisible by 6.

17. Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible

by 5, where n is any positive integer.

[Hint: Any positive integer can be written in the form 5q, 5q+1, 5q+2, 5q+3,5q+4].

Real number Test paper

Real number Test paper  

1. Write whether every positive integer can be of the form 4q + 2, where q is an

integer. Justify your answer.

2. “The product of two consecutive positive integers is divisible by 2”. Is this statement

true or false? Give reasons.

3. “The product of three consecutive positive integers is divisible by 6”. Is this statement

true or false”? Justify your answer.

4. Write whether the square of any positive integer can be of the form 3m + 2, where

m is a natural number. Justify your answer.

5. A positive integer is of the form 3q + 1, q being a natural number. Can you write its

square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify

your answer.

6. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is

HCF (525, 3000)? Justify your answer.

7. Explain why 3 × 5 × 7 + 7 is a composite number.

8. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

9. Without actually performing the long division, find if 987/10500

will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.

10. A rational number in its decimal expansion is 327.7081. What can you say about

the prime factors of q, when this number is expressed in the form p/q ? Give reasons.

CBSE Maths X Triangles – Criteria for Similarity of Triangles

Math Adda


10th math Converse of Thales theorem


10th math Triangles Thales theorem and angle bisector theorem


10th math SAS similarity of triangle


10th SSS similarity for two triangles are similar

10th AA- similarity for two triangles are similar

10th math SAS similarity of triangle

Math Adda

Statement: Two triangles are similar to each other when one angle of a triangle equal to an angle of other triangle and sides making these angles are proportional.

Proof: We are given DABC and DPQR such that  < A = <P

And AB / PQ = AC / PR


We have to prove. DABC ~ DPQR

To prove this: On the sides PQ and QR of DPQR, take points M and N such that 

AB = PM and AC = PN Join MN

Now 

AB / PQ = AC / PR
PM / PQ = PN / PR [Because AB = PM and AC = PN] 

Thus, MR || QR [Converse of Thales theorem]

So, <1 =<Q And <2 = <R [corresponding angles]

Therefore, DPMN ~ DPQR [A. A Similarity]

So, PM / PQ = MN / QR = PN / PR --------------- (1) [Sides of similar triangles are proportional]

Now, in DABC and DPMN

AB = PM [we have constructed]

<A = <P [Given]

AC = PN [we have constructed]

Thus, DABC@ DPMN

Thus, <A = <P, <B = <M and <C = <N

So DABC ~ DPQR [Because DABC @ DPMN and DPMN ~ DPQR]

This condition of similarity is known as SAS similarity.

10th SSS similarity for two triangles are similar

Two triangles are similar when their corresponding sides are proportional.

Let us prove this condition as a theorem, with the help of Thales Theorem.

Statement: Two triangles are similar to each other when their corresponding sides are proportional.

Proof: We are given DABC and DPQR such that

AB / PQ = BC / QR = AC / PR

We have to prove D ABC ~ DPQR

To prove this: On the sides PQ and QR of DPQR take points M and N such that

AB = PM and AC = PN. Join MN
Now,
 AB / PQ = AC / PR [Given]
i.e. PM / PQ = PN / PR [Because AB = PM and AC = PN]

Thus, MN || QR [Converse of Thales theorem]

So, <1 = <Q and <2 = <R [corresponding angles]

Therefore, DPMN ~ DPQR [AA similarity]

So, PM / PQ = MN / QR = PN / PR ----------------- (1) [Sides of similar triangles are proportional]

Since AB / PQ = PM / PQ [Because AB = PM]

And AB / PQ = BC / QR [Given]

Therefore,

PM / PQ = BC / QR ----------------- (2)

From (1) and (2)

MN / QR = BC / QR

MN =BC

Now, in D ABC and DPMN

AB =PM [We have constructed]

BC = MN [Proved above]

AC = PN [We have constructed]

Thus, DABC@D PMN [SSS congruency]

Thus, < A = <P, <B = <M and <C = vN

So DABC ~ DPQR [Because DABC @ D PMN and D PMN ~ D PQR]

We represent this condition of similarity as side - side similarity or SSS similarity

Two triangles are similar when one angle of a triangle is equal to an angle of other triangle and the sides making these angles are proportional.

Let us prove this as a theorem with the help of Thales theorem.

AA similarity Conditions under which two Triangles are Similar

Math Adda    Conditions under which two Triangles are Similar



Two triangles are similar when their corresponding angles are equal.

Let us prove these conditions as a theorem with the help of Thales Theorem.

Statement: Two triangles are similar if their corresponding angles are equal.

Proof: we are given D ABC and D PQR such that

< A = <P 
<B = <Q
<C = <R

We have to prove: DABC ~ DPQR

To prove this: On the sides PQ and PR of DPQR take points M and N such that 

AB = PM and AC = PN. Join MN.

Now, in DABC and DPMN

AB = PM [We have constructed] 
<A = <P [Given] 
AC = PN [We have constructed] 

Therefore DABC ~ DPMN [SAS congruency] 

This means
<PMN = <B [c. p. c. t]
But <B = <Q [Given] 

Therefore, 
<PMN = <Q
These are corresponding angles.

Thus, MN || QR

Now, 

PM / MQ = NR / PN {Thales theorom}

i.e. MQ / PM = NR / PN

i.e. MQ / PM + 1 = NR / PN +1

i.e. MQ + PM / PM = NR + PN/ PN

i.e. PQ / PM = PN / PR

i.e. PM / PQ = PR / PN

Since PM = AB and PN = AC

Therefore

AB / PQ = AC / PR

Similarly we can prove

AB / PQ = BC / QR

Hence AB / PQ = BC / QR = AC / PR, we already have been given that <A = <P, <B = <Q and <C = <F.

This way both conditions of similarity are fulfilled.

Thus, DABC ~ DPQR when their corresponding angles are equal.

We denote this condition as 

Angle - Angle - Angle similarity or AAA similarity.

Note: When two angles of a triangle are equal to two corresponding angles of another triangle then their remaining angles are also equal to each other. This AAA similarity is also written as AA similarity

10th math Converse of Thales theorem

Math Adda

If a line divides any two sides of a triangles proportionally then the line is parallel to the third side

Statement: If a line divides any two sides of a triangles proportionally (in same ratio), then the line is parallel to the third side.

Proof: We are given ABC 

AD / BD = AE / CE

We have to prove: DE is parallel to BC

Let DE's not parallel to BC then an another line DE' is parallel to BC.

Now

AD / BD = AE / CE [Given]

And AD / BD = AE' / CE' [Thales Theorem]

Therefore AE / CE = AE' / CE'

i.e. AE / CE + 1 = AE' / CE' + 1

i.e. AE + CE / CE = AE' + CE' / CE' 

i.e. CE = CE'

But this is not possible until E and E' is coincident.

Thus, our assumption is wrong and DE is parallel to BC.

10th math Triangles Thales theorem and angle bisector theorem

cbse maths papers 2012

CBSE MATHS

10_sample_paper_1

Download File

10__sample_pape_mathematics_2

Download File

10_math-sample_paper_3
Download File

10_math-sample_paper-4

Download File

10th_maths_sample_paper_5

Download File

10th_maths_sample_paper_6

Download File

10th_maths_sample_paper_7

Download File

cbse math sample paper

CBSE SUMMATIVE ASSESSMENT-II SAMPLE PAPER 2014

CBSE  SUMMATIVE  ASSESSMENT-II  SAMPLE   PAPER [Links]

Sample Paper Mathematics

Sample Paper Science

Sample Paper Hindi, English and Sanskrit

Sample Paper Social Science