MATHEMATICAL MODELLING 1 (INTRODUCTION)

SETTING THE CONTEXT


The following photos were taken along pavements and roads. 
They show 'cracks' or 'gaps'. 

  • Are these cracks common phenomena?  
  • Do they serve specific functions?

PICTURE 2
PICTURE 1
PICTURE 3


REVISIT THE CONTEXT







MATHEMATICAL MODELLING 2 (OBJECTIVES)


OBJECTIVES


Understand the facets of Applied Learning through a Mathematics Lesson

Authentic (Real World Application) 
The focus on a science experiment as a basis to gather data and information

Active & Relevant
Students will be engaged throughout the process as they are the co-constructors of knowledge (making Prediction, Observing patterns and Explaining the final outcome(s))

Integration (through inter-disciplinary approach)
Learning of cooling curve and using the Mathematical modeling to draw conclusion.





MATHEMATICAL MODELLING 3 (POE)


RECALL - REVIEW - REFLECT

ACTIVITY 

Measuring of body temperature against time

PREDICT
(before the experiment)
What do you think will be the characteristics of the temperature against time graph of this experiment?

OBSERVE
(during and after the experiment)
What do you notice about the graphical function(s) obtained?
Describe the characteristics?
Any unusual / distinct patterns observed?

EXPLAIN
(after experiment)
Are the experimental observations similar to the prediction? Why?
Can the outcome(s) be generalised for all cases?




RECALL - REVIEW - REFLECT

TYPES OF GRAPHS 

Graphs







MATHEMATICAL MODELLING 4 (ROUTINE)


CLASS LEARNING ROUTINE


Task in a Collaborative learning environment
You will be working in groups of 4 with the following roles:
  • Student 1: Leader (lead the task and present findings)
  • Student 2: Scribe (capture learning and post in Mathematics Blog)
  • Students 3 & 4: Researchers (set-up apparatus for activities)
  • All members will be involved in the activities through
    • Group discussion
    • Prediction Making
    • Conduct of Experiment
    • Data Gathering (recording observations)
    • Presentation of Findings (explain the Mathematics model - graphical functions)
Use of Apparatus / Equipment
Each group will have
  • Learning Devices (for recording, researching, presenting)
  • One beaker
  • One TI-Nspire Calculator 
  • One temperature probe
Set-up

  • Go to your Class Math Blog.
  • Set up and connect the TI-Nspire Calculator to the Local network. Refer to the instructions given.  (ensure that your group appears on the teacher's screen)
Class Math Blog

MATHEMATICAL MODELLING 5 (STUDENT CENTRED EXPERIMENT)

STUDENT-CENTRED ACTIVITY 


STUDENT-CENTRED ACTIVITY 

In this experiment, You have to use the POE thinking routine approach to learn about the temperature change in 2 scenarios.





TASK STRUCTURE
in brief
  • Predict by sketching the possible relationships between Temperature against time for BOTH scenarios. Indicate any intersections, turning points and any other possible characteristics based on your prediction.
  • You are given 40 minutes to do prediction, experiments, discussions and uploading of presentation on the S3-05 class Math blog (create a new page). You have to decide on the time and task management. 
  • Follow the format of your Presentation for BOTH scenarios as shown below.

in detail
  1. Post your PREDICTION in the Class mathematics blog. 
    1. Include a Sketch (you may use any sketching tools) and 
    2. a brief writeup of the Predicted mathematical model Example. Quadratic with a maximum turning point at ...(BEFORE THE EXPERIMENT)
  2. Conduct the scenario and OBSERVE patterns, unusual phenomena /patterns /observations (if any). You may repeat it a few times. 
    1. Do a screen capture and post your experimental findings and 
    2. identify the Mathematical model(s) that best suit the graphical function(s). example Linear from 0-4 s and then Quadratic etc.  
  3. EXPLAIN your Prediction and Observations by posting the following in your blog.
    1. Compare and contrast the predictions with the actual results, are there any significant differences? Why?
    2.  What are the characteristics of the graphs?
      • a.    the slope from start to finish (steeper, gentler)
      • b.    intersections with axes (if any)
      • c.     possible turning point (if any)
      • d.    any asymptote?
      • e.    And any other evidence
  4. What are the assumptions you have to make for this experiment? What are the sources for discrepancies?
Sample format:



SCENARIO 1 

Start with your prediction: Predict the relationship between temperature against time when you place the probe in hot water ONLY

SCENARIO 2 
Predict the relationship between temperature against time when you add few ice cubes into the hot water


MATHEMATICAL MODELLING 6 (POST ACTIVITY REFLECTION)

Finding the centre and radius of a circle - Sherman Kashyap and Sheares

Finding the centre
        -First of all, we drew two tangents that are not directly opposite each other. Since tangents and radii form 90 degree angles, we can find the diameter of the circle. From there we can either split the diameter into two or find the intersection of the lines perpendicular to the tangent.

Finding the radius
       -We measure the diameter and split it into half. (In this case the circle we constructed had a radius of 6 cm

Finding the centre of the circle — Aziel and Charlene

Finding the centre of the circle
We drew two long arcs with the same radius from opposite sides of the circle, making a line that passes through where the arcs intersected. This finds the perpendicular bisector of the undrawn line connecting the two points the arcs originated from.
We turned the paper a rough 90° and repeated the same process. The instance where the two perpendicular bisectors intersect was the centre of the circle.



Proof of how we found the centre of the circle
The perpendicular bisector of any chord in a circle would be its diameter if extended from one point of the circumference to the other. This is because a line connecting the centre of the circle to a chord is perpendicular to the chord it connects the centre to.
What we have done is find the perpendicular bisector of two chords and, since the diameters would intersect in the centre, hence have used this method to find it.

Finding the diameter and radius of the circle
Extending or limiting the line drawn by the arc intercepts to exactly at the circumference and measuring this line, we found that the diameter was approximately 10.25 cm and that the radius was approximately 5.125 cm.

Finding the centre of a circle - Esther, Dawn & Dylaine

To find the centre of the circle, we drew a chord of the circle, then drew an arc from each of the two end points of the chord of the same radius length. From the two intersecting points formed by the arcs, we were able to construct a diameter of the circle, from the Bisecting Chord Theorem (perpendicular bisector of chord is diameter).

This was repeated to find another diameter, in which the intersection with the first diameter is the centre of the circle.


Finding the center and radius of a circle (Guo Jin, Luke, Jerome)

1) How to find the center of a circle:

Assume we have a circle

We will use the property: angles in a semicircle = 90º.
This property shows that if a triangle is drawn inside a semicircle, the angle opposite the diameter will be 90º.
With this property in mind, let us draw a 90º angle at a point on the circumference of the circle.


The 2 lines from the 90º angle form 2 sides of a triangle inscribed inside a semicircle. This can be derived by working backwards from the property of angles in a semicircle. Since the triangle is inscribed inside the semicircle, the 3rd side of the triangle is the diameter of the semicircle, and thus the diameter of the circle itself.


We can repeat the process again, and draw another right angle. By extending the lines formed by the right angle, another triangle inscribed in the semicircle is formed. Again, the 3rd side of this triangle is the diameter.

The point where the two diameters intersect is the center of the circle. (All diameters pass through the center, and all diameters intersect only at the center.)


2) How to find the radius of a circle 

Method 1
Assuming we have another circle

Draw 2 lines of the same length joined at a 90º angle at the circumference. Join the lines with another line.
These lines form the 2 sides of an inscribed triangle in the semicircle (the semicircle is the circle divided in half). The third side of the triangle would be the diameter of the semicircle which the triangle is inscribed in. The semicircle's diameter is the diameter of the circle itself.

Draw another line down from the point where the lines meet. Since the lines are of equal length, the line drawn here will bisect the diameter, forming 2 radii. The line that is drawn down to meet the diameter is also another radius.

Therefore, the radius can be determined.

Other Methods:
If the center has already been found, using methods such as in the previous part,


Draw a line from the center to the circumference, on one side. This line is the radius.
This method can be used for any other circle which the center is already known.

Otherwise, calculation can be used. If the diameter, circumference or area of the circle is known, their respective formulas can be applied to obtain the radius.

Measurement can also be used.

Circle Circle Circle


ACTIVITY 3A
introduction to task



TASK:
this a group task.
Post your responses of the following:

  • Why is a circle a 'circle' and why it is NOT a triangle?   Describe its characteristics.
  • Give at least 2 examples of daily objects, figures, shapes that could be classified as a circle.

Find my Centre - Authentic Task


ACTIVITY 3

Objective:  
This is a collaborative activity to consolidate the learning of angles in a circle viz a viz APPLIED LEARNING MODE.

ICT tools:  
your choice (Geogebra, Geometer Sketch Pad etc)


TASK:
You are a mechanical engineer that is supervising a mould making process. You have just been tasked to replicate a circular object in your production line. The products must be precise (congruent of the highest degree) and must be mass produced efficiently at the shortest possible time with minuscule margin of error. 

The first task is to produce a mould by which all the objects will be replicated from. 

To facilitate the process, you are required to first determine how you would determine the following
  1. centre of the circle
  2. radius of the circle / sphere
For the above activity, you are required to use the angles properties discovered in the topic. 
Do provide the logical reasoning in your answer.
Post your solution in blog page assigned to you.
Remember time is a factor so you have to plan your time very well.

Assessment in Term 4 2013

Dear SSTudents,

1. Performance Task 2 (in lieu of Elementary Mathematics)

As mentioned earlier the deadline of the PT2 is 16 September 2013 @ 2359. To date many students have submitted their products with high quality questions and 'proof's.  Effective use of ICT tools (google, Blog, Geogebra, Keynote, Powerpoint, Pretzi etc) have further enhanced the final product.

2. Paper 3 (in lieu of Additional Mathematics)

The assessment information will be as follows:
Date:    23  September  2013 (Monday)
            (Please be punctual and ensure you have a heavier meal in the morning)
Time:   3.00 pm to 4.00 pm
Venue: Auditorium
Note that you are required to sit according to your classes and index numbers. The teachers will supervise you on this.

Logistic:
TI84 Graphic Calculator (or other approved GC model)
(no other calculator will be allowed)
Stationery - pen and ruler

3. Information on EOY
Please refer to your class math blog or google site earlier entries on this.


All the best - you can do it because we have faith in you but do you!

Angles in a Circle - MULTIMODAL LEARNING

ACTIVITY 1

Objective:  
This is a self directed activity to consolidate the learning of angles in a circle. 

ICT tools:  
Math Blog for activity specifications
TI-Nspire CAS
tns file will be given at the beginning of session







ACTIVITY 2

Objective:  
This is a collaborative activity to consolidate the learning of angles in a circle viz a viz formative assessment. 
Error analysis will follow the assessment.

ICT tools:  
TI-Nspire CAS
tns -based poll will be used to assess learning.



ACTIVITY 3

Objective:  
This is a collaborative activity to consolidate the learning of angles in a circle viz a viz APPLIED LEARNING MODE.

ICT tools:  
TI-Nspire CAS

TASK:
Given the object, determine how you would determine the following
  1. centre of the circle
  2. radius of the circle / sphere
For the above activity, you are required to use the angles properties discovered in the topic. Do provide the logical reasoning in your answer.