Mathematics Lab Manual Class XII | 14 Activities

Mathematics Lab Manual Class XII 14 lab activities for class 12 with complete observation Tables strictly according to the CBSE syllabus also very useful & helpful for the students and teachers. General instructions All these activities are strictly according to the CBSE syllabus. Students need to complete atleast 12 activity from the list of 14 activities. Students can make their own selection.

Students in Secondary Mathematics

Errors that students often make in doing secondary mathematics during their practice and during the examinations and their remedial measures are well explained here stp by step.

Some Common Errors in Mathematics

Teachers may often observe that there are certain errors which are frequently committed by the students. In order to help students in correcting such errors, it is imperative that the teachers understand the reasons behind such mistakes. The common errors that students commit may be categorized as conceptual errors and procedural errors.

Conceptual Errors

A teacher has to keep in mind that knowledge is not transferred. It is constructed through one's experiences. The students construct knowledge though various experiences, both inside and outside the classroom. Sometimes, flawed reasoning or lack of proper understanding may lead to the development of some pre-conceived notions. These misconceptions or naive theories may lead to errors. Such misconceptions arise when the content is not properly transacted. The teachers may sometimes fail to explain the concept behind a particular procedure and this may lead to faulty understanding on the part of learners. Teachers need to understand and address any such misconceptions that students might have. It is essential to address these because they are difficult to remove and addressing them calls for a lot of effort.

Only if such conceptual errors are recognized in time and the reasons behind these are figured out, can they be clarified and the resulting mistakes corrected. Understanding the justification behind a procedure is a prerequisite for conceptual understanding. Conceptual knowledge includes the decision regarding which procedure is to be applied.

Following are some common conceptual errors that students often commit:

2(a+b) = 2a +2b

Due to this statement students get an impression that everything works like this. Here is whole list , in which this does not work.

 Incorrect Correct (a + b)2 = a2 + b2 (a + b)2 = a2 + b2  + 2ab $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ $\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}$ $\sqrt{a^{2}+b^{2}}=\sqrt{a^{2}}+\sqrt{b^{2}}=a+b$ $\sqrt{a^{2}+b^{2}}\neq\sqrt{a^{2}}+\sqrt{b^{2}}$ $\frac{1}{a+b}=\frac{1}{a}+\frac{1}{b}$ $\frac{1}{a+b}\neq\frac{1}{a}+\frac{1}{b}$ sin(x + y) = sinx + siny sin(x + y) = sinx cosy + cosx siny

Improper use of square root

Students seem to be under misconception that $\sqrt{4}=\pm 2$  which is incorrect.

The proper working is as under

x2 – 4 = 0

⇒  x2 = 4

⇒ $x=\pm\sqrt{4}=\pm 2$

⇒  $x=\pm\sqrt{2}$

* Improper use of power of trigonometric functions

 Incorrect Correct sinnx = sinxn sinnx = (sinx)n

* Incorrect use of trigonometric results:
 Incorrect Correct cos85o = sin(90o – 5o) = sin5o cos85o = cos(90o – 5o) = sin5o  or cos85o = sin(90o – 85o) = sin5o

Solve : 3x2 = x

 Incorrect Correct 3x2 = x ⇒ 3x = 1 ⇒ x = 1/3 Here students divide both side by x without realising that x could be 0 also 3x2 = x ⇒ 3x2 – x = 0 ⇒ x(3x - 1) = 0 Either x = 0 or 3x – 1 = 0  x = 0, 1/3 Both values of x satisfy the given equation; hence both are the solution of quadratic equation.

Procedural Errors

Procedural knowledge means the knowledge regarding how to do something. This can also be understood as mathematical skills. Rectification of procedural errors requires practice and feedback.

Following are some of the common procedural errors that the students commit:

A). Many a time, students decide that parentheses are not needed at certain steps. They fail to understand the importance of parentheses.

Example 1: Square of 3x.

 Incorrect Correct (3x)2 = 3x2 (3x)2 = (3)2 (x)2 = 9x2 Here students are failed to follow the procedure of BODMAS

Example 2: Square of -2

 Incorrect Correct (-2)2 -22 = - 4 (-2)2 = 4

Example 3: Solve for x:     $\frac{x-1}{2}-\frac{2x+3}{3}=\frac{1}{5}$

 Incorrect Correct $\frac{x-1}{2}-\frac{2x+3}{3}=\frac{1}{5}$$\Rightarrow\frac{3x-3-4x+6}{6}=\frac{1}{5}$ $\Rightarrow\frac{-x+3}{6}=\frac{1}{5}$ ⇒ -5x + 15 = 6⇒ -5x = -9⇒ x = 9/5 $\frac{x-1}{2}-\frac{2x+3}{3}=\frac{1}{5}$ $\Rightarrow\frac{3(x-1)-2(2x+3)}{6}=\frac{1}{5}$ $\Rightarrow\frac{3x-3-4x-6}{6}=\frac{1}{5}$ $\Rightarrow\frac{-x-9}{6}=\frac{1}{5}$ ⇒ -5x - 45 = 6⇒ x = - 51/5

B) Improper Distribution

Example 1: Simplify: 3(2x2 - 6)

 Incorrect Correct 3(2x2 - 6) = 6x2 - 6 3(2x2 - 6) = 6x2 - 18

Example 2: Simplify: 2(3x-5)2

 Incorrect Correct 2(3x-5)2 =(6x-10)2                = 36x2 - 120x + 100 2(3x-5)2 = 2(9x2 – 30x + 25)               = 18x2 – 60x + 50

C) Error in balancing the equation:
Example: Simplify:  $\frac{2x^{4}-x}{x}$
 Incorrect Correct $\frac{2x^{4}-x}{x}=2x^{3}-x$ $\frac{2x^{4}-x}{x}=\frac{x(2x^{3}-1)}{x}=2x^{3}-1$

D) The following mistake largely on the part of authors and the mistake is carried over the students.
 Incorrect Correct Area of the rectangle having length and breadth 6cm and 5cm respectively is 6 Ñ… 5 = 30 cm2 Area of the rectangle having length and breadth 6cm and 5cm respectively is 6cm x 5cm = 30 cm2

Area of a rectangle has to be expressed in some square units.
However, it is to be understood that though conceptual and procedural knowledge are often discussed as distincts entries, they do not develop independently in mathematics and, in fact, lie on a continuum, which often makes them hard to distinguish.
Common Errors in Finding Squares and Square Roots

1) In right angled triangle ABC right angled at C, if tanA = 1, find SinA

 Incorrect Correct $tanA=\frac{BC}{AC}=1$ BC = AC = K$\Rightarrow AB=\sqrt{BC^{2}+AC^{2}}$ $\Rightarrow AB=\sqrt{k^{2}+k^{2}}$ $\Rightarrow AB=\sqrt{2k^{2}}$ $\Rightarrow AB=2k$ $tanA=\frac{BC}{AC}=1$ BC = AC = K$\Rightarrow AB=\sqrt{BC^{2}+AC^{2}}$ $\Rightarrow AB=\sqrt{k^{2}+k^{2}}$ $\Rightarrow AB=\sqrt{2k^{2}}$ $\Rightarrow AB=\sqrt{2}k$

2) Find the square of (7x - 3).

 Incorrect Correct (7x – 3)2 = 7x2 – 32                 = 7x2 – 9 (7x – 3)2 = (7x)2 - 2× 7x × 3– 32                 = 49x2 – 42x + 9

3) For what value of k the quadratic equation: (k + 4)x2 + (k + 1)x + 1 = 0 has equal roots ?

 Incorrect Correct Since the given equation has equal roots.∴ Discriminant = 0⇒ b2 – 4ac = 0⇒ (k + 1)2 - 4× (k + 4) × 1 = 0⇒ k2 + 1 - 4k + 4 = 0⇒ k2  - 5k + k + 5 = 0⇒ k(k – 5) + 1(k – 5) = 0⇒ (k – 5) (k + 1) = 0 K = 5 or k = -1 Since the given equation has equal roots.∴ Discriminant = 0⇒ b2 – 4ac = 0⇒ (k + 1)2 - 4× (k + 4) × 1 = 0⇒ k2 + 2k + 1 - 4k - 16 = 0⇒ k2  - 2k - 15 = 0⇒ k2  - 5k + 3k - 15 = 0⇒ k(k – 5) + 3(k – 5) = 0⇒ (k – 5) (k + 3) = 0⇒ k = 5 or k = - 3

4) Common errors committed by the students in Similarity of Triangles

 S. No. Common Errors Reasons Remedial Measures 1 Sometime students used ASA rule Connot differentiate congruency and similarity rules Congruency rules are: SSS, SAS, ASA, AAS, RHS Similarity Rules are: SSS, SAS, AAA, AA 2 AB = DE BC = EF AC = DF Cannot understand the congruency and similarity rule properly. Two triangles are congruent then corresponding angles and sides are equal. Two triangles are similar then corresponding angles are equal and sides are proportional. 3 If DE || BC then$\frac{4}{3}=\frac{6}{12}=\frac{7}{8}$ Cannot differentiate between similarity and Basic Proportionality theorem DE || BC, then first prove that △ADE ~ △ABC $\frac{4}{7}=\frac{6}{12}=\frac{7}{15}$

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