Math Modelling Experience

I just want to share my math modelling experience that I had with contract school. Schools usually don't use the term 'modelling' but 'performance task'. Typically, schools integrate real world contexts into performance tasks. I rememeber doing one performance task with the secondary one students on floor tiling. Given a fixed floor area and 2 different types of tiles of different prices, the students were told to tile the floor based on a few criteria: optimum cost, least tiles wasted and least cutting of tiles involved. This kind of task really set the students thinking and at the end of the day, there is no fixed solution. But however, before the students even start to attempt the task, they will ask questions like 'is this counted in the examination?' and blah blah blah. I feel that in order to really engage the students' thinking in math modelling, the tone must first be set right; if not, you will have students giving superficial solutions, which you know they have not given much thought to it.

Another issue that arises is the evalution process, aka, rubrics. Does the rubric really provide the best solution? How do we assess the students? If the students fail in the performance task, does it mean that their ability is low? How do we train the students in mathematical modelling? These questions really pose some challenges in setting a performance task.

Well, I just feel that it may even take ages to come up with a good performance task with rubric that allows the teacher to assess the students' learning accordingly.

Group F

Something interesting to think about for educators out there..

Hello guys,

not sure if u have seen this video on FB recently. This guy was sharing about his opinion of the education system and the need to reform it (using animation to depict his ideas). Pretty thought provoking and the animation is amazing as well. Take a look.

http://www.youtube.com/watch?v=zDZFcDGpL4U

Modelling....

The modelling activity really brings back my university memories... I think it is good to let students experience such modelling techniques as I am sure it will be very helpful to them especially if they were to enter engineering or research areas in future where modelling becomes part and parcel of their lives...

As I reflect back regarding the part on using "square errors" rather than "absolute errors", I do not agree totally. I guess it really depends on situations. From what I learnt in process control, we term them as "Integral square errors (ISE)" and "Integral absolute errors (IAE)" respectively (it is actually the same as summation of the square errors etc since integral is basically summing up the area under the curve). In cases where large errors are particularly unacceptable, we will use ISE so as to amplify such errors otherwise, IAE is also often used. Nevertheless, in most journal papers on process control, engineers usually used both ISE and IAE to illustrate their results. 

I have also shared my old lecture notes - Classification of process models as follows (feel free to download it, it shows the various types of models as well as difference between them) :

http://www.mediafire.com/?akmuzukd1jrgd20 

Just some acronym used:

CSTR: Continuous stirred tank reactor

ODE: Ordinary differential equation

PDE: Partial differential equation 

Group A

Mathematical Modelling

Here is a website with a lot of resources on mathematical modelling. (http://www.math.montana.edu/frankw/ccp/modeling/topic.htm) I hope it is the same "modelling" that we are talking about in class.

Personally I find it is quite similar to steps in scientific methodologies. If you do have a graphical calculator or a data logger, you can try out some of the activities there.

Anyway enjoy reading!

Are areas made of parallel line segments?

I am not sure how many of you are convinced by the idea that the area is made of parallel line segments while the volume is made of parallel planes. Personally I am a little confused by this.

First of all, Cavalieri's principle does not implies this result. The statement of the principle is that "If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal" (http://mathworld.wolfram.com/CavalierisPrinciple.html). Personally I think it is only looking at the cross-sections, but it does not imply that the solids are made of the thin slices. Similar argument applies to the area as well.

Regarding the activity that the parallel line segments shade area, personally I find it is a bit flawed. The line segments we have drawn in this activity, are not really line segments but thin rectangle blocks. If we are looking at the line segment itself, by applying a similar argument as the density of the real numbers, we can show that between any pairs of parallel line segments we have drawn, we can always find a space between these two line segments. Thus we cannot completely fill the area of the rectangle block with parallel line segments. Similarly in the activity with A4 papers, we have already assumed that each A4 paper has a specific volume. Thus we can then sum up the volume.

I am not sure how helpful it would be to the students at their age. However, I just find it would be a bit dangerous to let them carry this idea with them especially when they are working on problems at a higher level. Till now, I finally understand why some students have some difficulties in differentiating between projections and cross-sections, between point-groups and space-groups, when they are working with 3D chemical models.

Nevertheless, Cavalieri's principle is quite useful in deriving area or volume formula of figures/solids with special shapes. I have tried to look for a formal proof for this principle but so far I cannot find any. Shall we accept it as an inductive observation or a hypothesis? :)

Group A

Mensuration is covered this week and we were introduce to the Cavalieri's principle. In school, teachers usually do not explain how the formulas are derived or related to students as it is perceived to be too difficult for them to understand. Many times, students were just told to memorise the formulas and apply straight. It is good that we are introduce to Cavalieri's principle and we will finally be able to answer students when they ask how did all the formulas comes about. Though I agree that the principle is hard to them to understand and appreciate at their level, it is still good to bring it in as an introduction to the topic. Perhaps more 3D animations may help in enhancing understanding.

I found 2 sites that provide some java animations on the Cavalieri's principle. With it, students no longer have to try hard to visualise the diagrams, which I had a hard time too.

http://www.jimloy.com/cindy/cavalier.htm

http://www.matematicasvisuales.com/english/html/history/cavalieri/cavalierisphere.html

KSEG Free Interactive Geometry Software

KSEG is a Free (GPL) interactive geometry program for exploring Euclidean geometry. It runs on Unix-based platforms (according to users, it also compiles and runs on Mac OS X and should run on anything that Qt supports). You create a construction, such as a triangle with a circumcenter, and then, as you drag verteces of the triangle, you can see the circumcenter moving in real time. Of course, you can do a lot more than that--see the feature list below.
KSEG can be used in the classroom, for personal exploration of geometry, or for making high-quality figures for LaTeX. It is very fast, stable, and the UI has been designed for efficiency and consistency. I can usually make a construction in KSEG in less than half the time it takes me to do it with similar programs. Despite the name, it is Qt based and does not require KDE to run.

KSEG was inspired by the Geometer's Sketchpad, but it goes beyond the functionality that Sketchpad provides.

http://www.mit.edu/~ibaran/kseg.html

Group D

http://www.youtube.com/watch?v=Wv65HBdACEs

Hi All,

Here is something interesting to see. It is related to (but not exactly the same as) what we did today in tutorial.

Thanks,
Group D

Alternative proof for gradient of perpendicular lines

I thought the proof m1.m2 = -1 for gradient between perpendicular lines we discussed last week may be too difficult for lower abililty students.

Did some googling for an alternative. We could use simple diagram and rotate the line 90 degrees to demostrate the gradient relationship.



After rotating 90 degrees, from the diagram, the gradient becomes - B/A (vertical rise / horizontal run).
Is this way of proving good enough?

Reference: http://mathforum.org/library/drmath/view/54496.html

Group F

Reflection on Geometry Tutorial by Group F

Last week, we spent lots of time both during tutorial sessions and off it working on geometrical proofs. We believe that this is certainly the most challenging topic in the A. Math syllabus. After working on so many questions, we find that to obtain the correct proof or solution, determination and some intuition is required. Even for O level questions on this topic, we find that very often, we’ll have to spend a considerable amount of time in order to arrive at the solution. Putting ourselves in students’ shoes, we feel that it will be very tough on them to require them to solve such questions under the constraint of time during exams.

This is not the typical A.Math topic which can be mastered easily by drill and practice. From feedback gathered from some students in my previous school, their teachers actually left this topic to the last and told them to concentrate on other topics during exams. It is thus easy to imagine the lack of confidence even teachers face with regard to this topic.

To help students, teachers have to be really proficient and not simply brush the topic aside. Once the confidence on our part is acquired, we can help students by working with them through numerous examples of various difficulties. Allowing for group discussion is also very important as students can question one another on the claims made. For tougher questions, hints should be provided so as not to scare students into “surrendering”.

Group F

Group A: Lab reflection

We had a 'compact' time through the geometry lab. Firstly some of us were new to the GSP software. While we learned to familarise ourselves with the controls and uses in GSP, we also had to focus on the geometry questions at hand so sometimes we managed to solve the maths problem but forgot how to use GSP or 'remembered' to draw in GSP but did not finished the question before it's time to move on to the next question. But i believe this will improve once we get the hang of it and we look forward to learning 'new things' that we can do with GSP.

It is good that GSP is introduced to us because it is another useful tool for teaching maths and the 'schools' uses it. So, is there some where we can practise or revise on GSP after the geometry lessons are over? Or are there similar free programs that is compatible to port GSP files over for use? I think it would be helpful if we can have the software installed into our laptop so that we can still practise in our own time.

Also it is interesting to know about the conventions for certain maths topics and how they practise it in schools. For example, for the 'Similarity' topic the convention is to state the reason in the final statement, 'triange ABC is similar to triange EFG (ASA)'. The practise in some schools state the reason ASA but some do not. And the way such questions are marked in prelim exams are different from the 'O' levels.

Geometry Problems

This week, we have explored the geometric properties related to triangles and circles.

In the last session, we have worked on four questions. The solution has been compiled and you can get a copy from http://sps.nus.edu.sg/~wujiangw/geometry.pdf



Group C

Assignment killer question (1) is on Heron's Formula

Hey guys, I found the following website while doing my maths reflection just now. The geometry question 1 that we are given in the assignment is actually a proof for Heron's Formula:

Preview of the webpage that constructed the proof: (diagram differs from assignment in labelling though)

http://jwilson.coe.uga.edu/EMT669/Student.Folders/Jones.June/heron/heronpaper.html

Cheers again,
Euclid

Trivial Proof of the formula used to find triangle in coordinate geometry (which can be extended to n-sided polygons)

Let say we want to find the area of the pink triangle (A+B) given the coordinates of the three vertices

Similarly, we can prove by the box method and construct a different diagram to obtain the area a:

Cheers, 

Euclid

P.S sorry for the messy print screens. The equations cant show up properly in the text box.

Teaching Site

A hyperlink to the Teaching site is added at the  corner :)

Courtesy of Group E :D

TLS: Angle at center is twice angle at circumference

We have learnt that the above approach can be used to show the angle property of angle at center is twice angle at circumference.

Thanks to Junyang and Sharon's question, how will we facilitate the discussion for the explanation/proof of this property if it is of the case below? 



Hint: think about angle in semicircle. Some of you may think that this is circular argument as angle in semicircle is derived from this angle property. However angle in semicircle can also be explained by two isosceles triangles.

Group D: Ivan Sutherland

Hi All,

Did you know that the sketchpad was created by Ivan Sutherland?

Watch the videos below to learn more about Sketchpad.

http://www.youtube.com/watch?v=USyoT_Ha_bA
http://www.youtube.com/watch?v=BKM3CmRqK2o

Hope you will use it more in your lessons next time.

Group D

Reflection by Group D:
One thing we learnt in today's tutorial was to use the properties of congruent triangles.
Triangles are said to congruent if they have the same shape and same size.

Congruent Triangles
There are four properties of congruency in triangles.

1. The SAS Property:
   
   If two sides and the included angle of a triangle are respectively equal to two sides and the included angle of another triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     If AB = DE, BC = EF and the included angle BAC = the included angle EDF.
     If the above conditions are satisfied, then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called Side, Angle, Side property.
   
   
2. The ASA Property:
   
   If two angles and a side of one triangle are respectively equal to two angles and the corresponding side of another triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     If BC = EF, ÐB = ÐE and ÐC = ÐF.
     If the above conditions are satisfies then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called Angle, Side, Angle property.
3. The SSS Property:
   
   If three sides of a triangle are equal to the three sides of another triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     If AB = DE, BC = EF AND AC = DF.
     If the above conditions are satisfies then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called Side, Side, Side property.
4. The RHS Property:
   
   If the hypotenuse and a side of a right-angled triangle are equal to the hypotenuse and a side of another right-angled triangle, then the two triangles are congruent.
   
   
   • Let us consider two triangles ABC and DEF.
     
     Conditions:
     ÐB = ÐE = 90°, side BC = Side EF and hypotenuse AC = hypotenuse DF. If the above conditions are satisfies then we say that the two triangles ABC and DEF are congruent and we can write it as ABC @ DEF.
     This is called the Right Angle, Hypotenuse, Side property.

Reference:
http://www.kwiznet.com/p/takeQuiz.php?ChapterID=2817&CurriculumID=24

Geometer's Sketchpad

I am surprised that none of my teachers have taught me using Sketchpad before. Also many of the classmates in QCM520 have not used Sketchpad or been taught Sketchpad before.

According to my experience, i have not seen any in service teacher use sketchpad to teach before even though the Sketchpad is very useful.

I wonder if all schools have access to this tool or if there are other reasons why schools are using sketchpad more.

sketchpad documents can be exported to websites for posting

GROUP D - STEPHEN, HAFIZAH, DANIEL, JUNYANG

Reflection on Expostional Teaching

Last week, we have done expositional teaching in the tutorial sessions. Here are some of things that we have learnt from this exercise.

First, it is on time management. It is a natural tendency for us to try our best to teach as much as we can in the class. However, one thing that we have overlooked is how much the students can learn within one period. We are fortunate that we do not have classroom management issues last week. Once we stepped into a real classroom, we might need to spend some time on it thus we need to think of how much time we have and how much we can cover. Our group tried to cover four factorisation methods within forty minutes using quite a number of different examples. However, the time devoted to each method for the students to learn is limited. It would be more effective if we had focused on one method and give more time for the students to learn and apply.

Second, it is on the choice of instructional method. Our group made use of both powerpoint presentation and whiteboard for teaching. During the planning stage, we are not aware of the size of the whiteboard. On the day itself, we found the space available is quite small thus many workings and steps are not presented in the way that we have planned to do so. Only when group E was presenting, we realized that we could make use of the visualizer instead, because it would be independent of the room that we use. As learnt from yesterday's lecture, we could have checked the room in advance and plan the lesson based on the room that we use.

Since we are learning through this exercise, I think it would be helpful if other groups can provide us with some feedback on our teaching. Please feel free to talk to us in private or just leave a post here.

Group C
(topic: factorisation)

Reflections on Expositional Teaching

Realised through the expositional teachining review that teaching of the right concepts is of paramount importance in the teaching of mathematics. While creativity is welcomed in the presenting of concepts to our students, it should be properly thought through to ensure that teachers not do it at the expense of contradicting basic concepts.

Even last week's lecture has emphasised the point that even if one is using direct instruction to teach, the key thing is to teach the 'right thing', the key concepts, the fundamentals must be communicated through to the class so that there would not be any misconcepts that students need to unlearn as they progress through the lesson.

Grp D

Some thoughts on expo-teaching

Ref to lesson on simultaneous equations

Just felt that it would help to point out the number of equations needed to solve for certain number of unknowns. For example:

2 unknowns x,y will require 2 or more different equations involving x,y so as to solve.
goes on for 3, 4...

Might help the students to pick out enough key points in word problems so as to form the required equations.


Hope it helps
Group B