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5. a) i) Explain the 5 levels of development in geometric understanding proposed by the Van
Hieles. Illustrate your explanation in the context of learning the concept of volume.
ii) Further, do you agree that children in Class 6 usually think at Level 2? Give reasons for
your answer. (12)
b) Can you think of a planar figure with exactly two axes of symmetry? Can this figure be a
triangle? Give reasons for your answers. (4)
c) Explain what inductive and deductive logic are, and illustrate them in the context of measuring
time. (4)
d) Give two reasons why children usually find mathematical notations confusing. Support your
answer with illustrations pertaining to representing and reading time. How would you help your
learners become comfortable with the notation?
Hieles. Illustrate your explanation in the context of learning the concept of volume.
ii) Further, do you agree that children in Class 6 usually think at Level 2? Give reasons for
your answer. (12)
b) Can you think of a planar figure with exactly two axes of symmetry? Can this figure be a
triangle? Give reasons for your answers. (4)
c) Explain what inductive and deductive logic are, and illustrate them in the context of measuring
time. (4)
d) Give two reasons why children usually find mathematical notations confusing. Support your
answer with illustrations pertaining to representing and reading time. How would you help your
learners become comfortable with the notation?
4. a) Devise a game to help children improve their understanding of addition and subtraction of
fractions. Also give two distinct activities you would use for assessing the efficacy of this
game. (5)
b) Explain the following statements, giving examples from the context of operations on decimal
fractions (i.e., numbers like x y z r, where x, y, z, r are digits between 0 and 9):
i) Mathematics permeates every aspect of your life.
ii) In mathematics, truth is a matter of consistency and logic.
iii) Articulating reasons and constructing arguments helps children learn mathematical
processes.
fractions. Also give two distinct activities you would use for assessing the efficacy of this
game. (5)
b) Explain the following statements, giving examples from the context of operations on decimal
fractions (i.e., numbers like x y z r, where x, y, z, r are digits between 0 and 9):
i) Mathematics permeates every aspect of your life.
ii) In mathematics, truth is a matter of consistency and logic.
iii) Articulating reasons and constructing arguments helps children learn mathematical
processes.
a) Children have several misconceptions regarding negative numbers. List four of them. Also, for
any one of these misconceptions, give a detailed strategy for helping the children correct it.
(6)
b) The diversity in any classroom has major implications for teaching mathematics. Explain this
statement, with examples from teaching algebra to support your explanation. (5)
4
c) Consider a classroom situation in which a teacher is introducing Class 6 children to operations
on negative numbers. In this context, explain the different levels at which mathematics and
language are related.
any one of these misconceptions, give a detailed strategy for helping the children correct it.
(6)
b) The diversity in any classroom has major implications for teaching mathematics. Explain this
statement, with examples from teaching algebra to support your explanation. (5)
4
c) Consider a classroom situation in which a teacher is introducing Class 6 children to operations
on negative numbers. In this context, explain the different levels at which mathematics and
language are related.
2. a) Explain why the three pre-number concepts need to be developed by a learner for him/her to be
able to count. Your explanation needs to include specific examples. (6)
b) i) Outline a series of three activities (each requiring a different level of learner’s ability) to
help a learner develop an understanding of ‘place value’. (Note that giving a ‘series’
means that the links between the different activities must also be brought out.)
ii) How would you modify these activities if you were doing them with a class of 30
learners? (9)
c) There are broadly 5 different real-life situations which require multiplication. Give a word
problem each for these situations, in the context of children playing in a field.
able to count. Your explanation needs to include specific examples. (6)
b) i) Outline a series of three activities (each requiring a different level of learner’s ability) to
help a learner develop an understanding of ‘place value’. (Note that giving a ‘series’
means that the links between the different activities must also be brought out.)
ii) How would you modify these activities if you were doing them with a class of 30
learners? (9)
c) There are broadly 5 different real-life situations which require multiplication. Give a word
problem each for these situations, in the context of children playing in a field.
Explain the differences in the following processes involved in the growth in mathematical
understanding. Also provide an example of each, pertaining to ‘data handling’.
i) known to unknown;
ii) particular to general. (8)
c) Illustrate how the E – L – P – S sequence can be applied to help children understand the concept
of ‘angle’. (4)
d) Is there any difference in the way you would plan a unit and a lesson? Explain your answer,
with examples in its support.
understanding. Also provide an example of each, pertaining to ‘data handling’.
i) known to unknown;
ii) particular to general. (8)
c) Illustrate how the E – L – P – S sequence can be applied to help children understand the concept
of ‘angle’. (4)
d) Is there any difference in the way you would plan a unit and a lesson? Explain your answer,
with examples in its support.
Partial Differential Equations:
Please assume that the improper integrals ∫_(-∞)^∞▒u_0 (x)dx,∫_(-∞)^∞▒〖u_1 (x)dx〗,and ∫_(-∞)^∞▒f(x,t)dx ∀t are convergent.
Prove that if u(x,t) is the solution of
u_tt-c^2 u_xx=f(x,t) x∈R,t≥0
u(x,0)=u_0 (x)
u_t (x,0)=u_1 (x)
then:
∫_(-∞)^∞▒u(x,t)dx=∫_(-∞)^∞▒u_0 (x)dx+t∫_(-∞)^∞▒〖u_1 (x)dx〗+∫_0^t▒〖(t-τ〗)∫_(-∞)^∞▒f(x,τ)dx dτ
Hint: Use the D’Alembert’s Formula for u(x,t) and change the order of integration dx and ds in the 2nd and 3rd terms.
Under what conditions on u_0,u_1,f is it true that ∫_(-∞)^∞▒〖u(x,t)dt=constant?〗
Please assume that the improper integrals ∫_(-∞)^∞▒u_0 (x)dx,∫_(-∞)^∞▒〖u_1 (x)dx〗,and ∫_(-∞)^∞▒f(x,t)dx ∀t are convergent.
Prove that if u(x,t) is the solution of
u_tt-c^2 u_xx=f(x,t) x∈R,t≥0
u(x,0)=u_0 (x)
u_t (x,0)=u_1 (x)
then:
∫_(-∞)^∞▒u(x,t)dx=∫_(-∞)^∞▒u_0 (x)dx+t∫_(-∞)^∞▒〖u_1 (x)dx〗+∫_0^t▒〖(t-τ〗)∫_(-∞)^∞▒f(x,τ)dx dτ
Hint: Use the D’Alembert’s Formula for u(x,t) and change the order of integration dx and ds in the 2nd and 3rd terms.
Under what conditions on u_0,u_1,f is it true that ∫_(-∞)^∞▒〖u(x,t)dt=constant?〗
3. BIOMEDICAL: Cholesterol. An experimental drug lowers a patients blood serum cholesterol at the rate of t square root of 25-t^2 units per day, where t is the number of days since the drug was administered (0 < t < 5). Find the total change during the first 3 days.
What is the area of a square metre in square centimetres?
Show that the curvilinear coordinate system defined by the following equations is
orthogonal:
x=uvcos(a)
y=uvsin(a)
z=1/2(u^2-v^2)
note here a stands for alpha
orthogonal:
x=uvcos(a)
y=uvsin(a)
z=1/2(u^2-v^2)
note here a stands for alpha
Let A and B be two great circles in a sphere S^2={x^2+y^2+z^2=1} such that A intersects B in two and only two points (recall that any two great circles intersect exactly twise). Let X be a finite set of points in A such that the arc lenght between each two of these points is equal and the cardinality of X is prime, i.e X contains prime number of elements.. For example if X contains 5 points, then the arc between each two of the 5 points subtending an angle 72 degree. Similarly, let Y be a finite set of points in B with equal arc length between each two points. Suppose also #B is prime and distinct from #A.
The question is: Can we define a homotopy between X and Y so that X is homotopic to Y. If so, how
Thank you in advance
The question is: Can we define a homotopy between X and Y so that X is homotopic to Y. If so, how
Thank you in advance
