Rewriting an Exam Question

When answering exam questions some weak students normally rewrite the question in the Answers booklet before providing the solution.why do they rewrite it?it does not add any value. Moreover it is a waste of precious time. No matter how many times you tell them they always repeat.Some explain this behaviour by saying the tutor may confuse the solution and its respective item.

Basic techniques of differentiation

Techniques of Differentiating

The following are basic techniques of differentiating. 

(a) Power rule

So far we have seen that given; y=f(x)=kx^{n}

Then the gradient function is; dy/dx=knx^{n-1}

Example 1: Differentiate the function y=3x^{5} with respect to x

solution: dy/dx=15x^{4}

Example 2: Find the derivative of the function y=2x^{3}-6x+3

solution : dy/dx=6x^{2}-6

Example 3: Differentiate the function y= [6x^{3}-7x]/[x^{2}]

solution: we need to first simplify the function by rewriting it as;

y=6x^{3}/x^{2}-7x/x^{2}=6x-7x^{-1}

Hence we have;

          dy/dx =6+7x^{-2}

(b) Product rule

Let y=f(x)=uv where u and v are both differentiable functions of x, then the gradient function is given as;

dy/dx= u dv/dx+v du/dx

Alternatively;

 (uv)^/=uv^/ +vu^/

This rule is appropriate when it comes to integrating product functions.

Example 1: Given y=(3x^{4}+7x)(5x^{7}-3x^{2}+9x) determine the derivative.

Solution: by definition dy/dx=u dv/dx+ v du/dx

we let,

u=3x^{4}+7x then du/dx=12x^{3}+7  and we let v=5x^{7}-3x^{2}+9x  then dv/dx =35x^{6}-6x+9

Therefore; 

dy/dx=u dv/dx+ v du/dx=(3x^{4}+7x)(35x^{6}-6x+9)+(5x^{7}-3x^{2}+9x)(12x^{3}+7)

Example 2: Given y=3x^{5}sinx , determine the gradient function.

Solution: by definition dy/dx =u dv/dx+v du/dx

We let, u=3x^{5} then du/dx =15x^{4}

Also let, v = sinx    then  dv/dx =cosx

Therefore; 

dy/dx=u dv/dx+v du/dx =3x^{5}cos x+15x^{4}sin x=3x^{4}(xcos x+5sin x)

Example 3: Let y=3x^{2}e^{x}, find dy/dx

Solution: We let u=3x^{2} then du/dx=6x

Also v=e^{x} then dv/dx=e^{x}

Hence;

dy/dx=(3x^{2})(e^{x})+(6x)(e^{x})=3x^{2}e^{x}+6xe^{x}

(c)  Quotient rule

Let y=f(x)= u/v  where u and  v are both differentiable functions of  x, then the gradient function is given as;

dy/dx=[v du/dv –u dv/dx]/v^{2}

Example 1:  Given y= 4x^{3}/[3x^{2}-4x]  find dy/dx;

Solution: Let u=4x^{3} then du/dx =12x^{2} and v =3x^{2}-4x then dv/dx = 6x-4

dy/dx=[12x^{2}(3x^{2}-4x)-4x^{3}(6x-4)}]/[(3x^{2}-4x)^{2}]

=[36x^{4}-48x^{3}-24x^{4}+16x^{3}]/[{(3x^{2}-4x)^{2}}]

=[12x^{4}-32x^{3}]/[(3x^{2}-4x)^{2}]

=[4x^{3}(3x-8)]/[(3x^{2}-4x)^{2}]

Example 2: Find the derivative of the function;

y= 2x/sin x

Solution: Let;

u = 2x then du/dx  =2

Let; v= sin x then dv/dx=cos x

Hence;

dy/dx=[2sin x -2xcos x]/sin^2x=2[sin x – x cos x]/ sin^{2}x

(d) Chain rule

Let y be a differentiable function of  u  i.e.  y=f(u)  and that u  is a differentiable function of x i.e. u=g(x), then y is a differentiable function of x i.e. y=f(g(x)), and;

dy/dx=(dy/du )(du/dx)

Example 1: Suppose y=(3x^{5}+7x^{4})^{4}, find dy/dx

Solution:

Let u=3x^{5}+7x^{4} then  du/dx=15x^{4}+28x^3

Also y=u^{4} then  dy/du=4u^{3}

By definition

dy/dx=(dy/du)(du/dx)=(4u^{3})(15x^{4}+28x^3)

=4(15x^{4}+28x^3)(3x^{5}+7x^{4})^{3}

=4x^{3}(15x+28)(3x^{5}+7x^{4})^{3}

Introduction to Differentiation

Definition 1:
To understand the concept of basic differentiation, we need to consider the gradient or slope or steepness of a straight line of the form y=mx+c where y and x are variables, m the gradient of the line and c is the y-intercept. Now the gradient of a straight line is the change in the vertical distance over the change in the horizontal distance. That is;
Gradient, m=(change in vertical distance)/(change in the horizontal distance)

Note that the gradient of a straight line is constant, but the gradient of a curve keeps on varying.

Definition 2:
To determine the gradient of a curve at point we need to use a gradient function. Given a function;
y=f(x)=kx^{n}
Then the gradient function;
dy/dx=knx^{n-1}

Example 1: Let, y=3x^{5},then dy/dx=15x^{4}

Example 2: Let; y=2 , then dy/dx =0

Remark 1: Note that, y=2 can also be written as y=2x^{0}. Recall that x^{0}=1 where x is not 0

Example 3: Differentiate the following function with respect to x; y=(2x-3)(4x+5)

Solution: Note that another term that is used to mean, 'finding the gradient function' is 'differentiate' with respect to a certain variable. One can also refer differentiation as 'determining the derived function'.
In this example, we need to expand the RHS to get;
y=8x^{2}-2x-15 then dy/dx =16x-2

Remark 1: The rule so far discussed is the most basic rule of differentiating. Some functions may not be differentiated at all over certain intervals or may not be differentiated using the above rule. Some differentiating techniques are discussed in the next section while others are beyond the scope of this blog.

P.S.
1) Next post will be on techniques of differentiating
2) This is an excerpt from the textbook, Basic Mathematics; by Kahenya NP

Incident 4 The Missing Girls

It was on Thursday mid morning at around 10.25am. This was immediately after a short break and learners were expected to attend science lesson. Due to the harshness of their class teacher who was also their science teacher twenty pupils did not attend science lesson and therefore only twenty three learners were in the classroom ready to be taught. The other twenty hid behind the classroom and others in the latrine.

This was the day I was expecting to be assessed for the second time because we had already communicated by the assessor that morning and even gave her direction to my school. I was expecting her to be in the classroom for assessment in the same class. When the teacher realized that some of the learners were not in the classroom she decided to report the matter to the deputy headteacher who also reported the same matter to the headteacher. The headteacher did not take his time to know what could be the root cause of the misconduct instead ordered boys from standard seven and eight to go and look for the standard three pupils who hid within the school compound. They were assumed to be within the school compound because the school is fenced and has lockable gate and a gate keeper, later on the learners were found in the latrines except two girls who were missing for the whole day.

On Friday very early in the morning the parents of the two girls came in tears (sic) asking for their beloved daughters, None of the teachers could explain the where-about of the girls. The Headteacher asked the teachers on duty to assemble learners in the assembly ground so that he could make an announcement concerning the two girls. During the announcement one of the boys raised up his hand and informed the headteacher that the previous evening he saw the two girls in one of the herdmen house and he expected that they slept there.

In addition the boy declared (sic) to the Headteacher infront of the two parents that the herdman had been luring school girls using money and therefore he had a chain of girl friends from our school. Above all, the herdman had been sickly and some weeks ago he was diagnosed H.I.V. positive. When the two parents heard that, one of them fainted and the other jumped on the headteacher in front of his office (sic). The situation was unbearable teachers tried to bring calm but it was in vain. The parent who fainted later recovered and went with the other parent and the deputy headteacher and two other teachers to where the girls were. Then girls were found still sleeping. To them it was not a big deal; one of the them declared to the deputy headteacher that even her parents knew she was befriending the guy (herdman). And many are the occasions she has ever taken shopping to the mother from the guy.

The long story ended in the police station whereby the herdsman was jailed because of child abuse. The girls were  later transferred to another school.
P.S.
(1) This is unedited version of the incident.
(2) Give reflective comments on this incident. Reflect on so what... and now what situation of the incident in respect to the teaching/learning process.

Incident 3: Std 3 CRE lesson

The following is unedited version of the incident as reported by Mary (not her real name):

It was a standard 3 CRE (christian religious education) lesson. The topic was 'Jesus Resurrection'. We learnt the purpose of Jesus being born, dying and resurrection. I explained that when we die our soul goes to heaven (sic) and our body remains here on earth (sic). Therefore in the case of Jesus, it seems God put the soul back into Jesus' body after three days.

After the lesson, it was questions time to check whether i had achieved my objectives. Question 3: which part of our body goes to heaven when we die? Answers: Child 1 gave: brain. Child 2 gave: mind. Child 3 gave: Heart. Child 4 gave: legs. I got curious and i further inquired why he said legs. The boy replied,'Teacher! it is because the other day my dad was struggling (sic) my mom while lying on her. Her legs were raised up and she was screaming in pain while saying, ooh my god, am coming'.I was extremely shocked, breathless...a moment of silent followed. I hoped other learners had not heard this. I didn't know what to say or do.

I recollected myself very fast and advised the learners that it was not the legs but the soul that goes to heaven. Afterwards i called the boy and inquired further. He disclosed that they live in single room. I was a bit mixed up because i felt i needed to talk to his mother but i did not know how she would respond or react. I prayed about the issue, gathered courage and called her. On hearing the story, she was honest enough and apologized. she also promised to check on the situation. Being a very private matter i left it at that.

How would you interpret the above incident?
Now what... next?

Incident 2: Class 2 pupil’s Traditional Brew Present

It was during the second week of my teaching practice when my assessor Mr Achi* came for assessment during teaching (sic). Earlier I had prepared my learner’s psychologically (sic) that the external visitor (sic) would come to class. The learners are always excited about visitors.

Prior to the day (sic), I informed them t0 clean their uniform and to come to class early. When the assessor came, learners were quite cooperative. Learning took place well as intended but during the lesson one of the learners had brought a present to the visitor. The boy stood and asks for permission to present the present to the visitor but I told him to wait.

At the end of the lesson, the boy stood and walked steadily to the visitor and presented a bottle of traditional brew (traditional brewed alcohol) to my assessor. The rest of the class laughed and tried to run out of the class but I controlled them.

Immediately, I allowed the class to go for the long break and called the boy and asked why he did so. The boy told me that it is a normal drink at their home. I advised him not be using the brew and later I called the parent and talked with her on the dangers of brew to young kids and the health effect in general.

NOTE:

·         No editing has been done.

·         Names have been changed to protect privacy.

What was most interesting or bad aspect about the incident?

What new lessons did the teacher learn?

What is the way forward as far as the teaching/learning is concerned, in regard to this incident?

Incident 1: Class Two Incident

On the morning of 14th March 2015, i woke up early than usual since i was the teacher on duty and i was also expecting external supervisor from the university. I was able to arrive at school at about 7:00 a.m. Pupils began tickling in one by one and soon learning started.

At about 10:00 a.m our school chairman came to school to inform us that the county governor would be visiting our school in the afternoon. We hurriedly held a brief staff-meeting and shared duties amongst ourselves. I was charged with responsibility of ensuring that the environment (sic) becomes clean and tidy. i asked pupils to collect litter and some to sweep their classrooms.

It all begun when it started raining in the afternoon. It was a heavy downpour and word came around that the governor was not going to make it. Kamau* (not his real name), one of our trusted class three prefect came running to the office and reported that something weird was happening in class two. Being the teacher on duty  i hurried to investigate. As i neared their class i could hear the whole class singing loudly ....'kila mtu na demu wake.. '(everybody hold your girl). I could clearly recognized the voice of the most notorious boy nicknamed 'Jangili' (Thug) leading the singing. When i entered the class, i was shocked to find some of the kids were half naked.

We were able to guide and counsel the kids who led others into this ugly incident. Jangili and his close associates (sic) confessed  that they have been doing thus on their way home, the reason being that, they have been seeing their parents doing 'jig jig' at night. The kids vowed not to repeat it again.

It came to our understanding that class two pupils had not being doing their assignments and homework since their class teacher has been reluctant to mark the classwork. We also learnt that Jangili and his close associates (sic) had joined a group (sic) that had been abusing drugs.

I learnt that each child has a unique characteristics/traits and should be handled differently. Learners have individual differences.
NOTE:
The name of the university, school, author, teacher, pupil, governor and county has been omitted to protect privacy.

Critical incidences in our classrooms

In the next posts I will share some of the incidences that my students experienced when they were undertaking their teaching practice. They analyzed the incidents from a reflective point of view. They were guided by reflective models. The goals of these posts is to hear your comments on theories/opinions on why such incidents happen in our classrooms. Theories to explain the why. I would also like to hear what feelings you could have experienced if you were in such a situation. And finally the lessons we have learnt or can learn from such incidences and the way forward.