Team Microteaching Reflection:

As usual, time management is always an issue when it comes to teaching new lessons.  Although my group had an excellent lesson plan set up, there was simply too much to cover within the 15minute time span.  We were able to complete the main objectives in our lesson plan, but it was sort of a rushed job.  Majority of the group felt as if the introduction was way too fast for any student to follow and understand fully.  The group also said that they needed more time to understand and use the algebra tiles, which is understandable since it is a new concept to learn. 

Overall, everyone enjoyed our lesson and was able to learn multiple methods of factoring quadratic trinomials (algebraically, using algebra tiles, and the virtual manipulatives) and was able to see that factoring can be related to the concept of the area of a rectangle.  I find these microteaching lessons to be highly useful in showing everyone what teachers have to deal with when teaching a lesson, that not everything will go as planned and adjustments may be needed due to timing or the lesson being interrupted by a student's curious question.  I think that within a lesson plan, there should be spaces for teachers to mention when to create a 'check point(s)' to make sure everyone is paying attention and caught up with the lesson and just to confirm that the teacher isn't going too fast or slow for the student's understanding of the lesson.

Lesson Plan: Factoring Quadratic Trinomials Using Algebra Tiles

Lesson Plan: Factoring Quadratic Trinomials Using Algebra Tiles
EDCP 342:  Lesson Plan
Topic:  Factoring Quadratic Trinomials Using Algebra Tiles
Group:  Howard, Maria, Raman

Intended Students:    Grade 10  Fundamentals and Pre-calculus

	WHAT	HOW LONG	MATERIALS
BRIDGE	Give everyone a small  sheet of paper.   In 5 seconds, write as many factors of 60.	1 minute
LEARNING OBJECTIVES	Using the algebra tiles, students will be able to: 1. Factor quadratic trinomials, including perfect square trinomials 2. Relate the dimensions of a rectangular area with finding the factors of a trinomial 3. Experience three modes of factoring trinomials:  algebraic method, concrete algebra tiles, and virtual  manipulatives
TEACHING OBJECTIVES	1. Maximum engagement of all students 2. Individual hands-on-learning using math manipulatives  (algebra tiles) 3. Demonstration of using virtual manipulatives in factoring trinomials
PRETEST	Each student will be given a worksheet sheet      1. Factor the trinomial: x^2 + 5x + 6.  Write answer in worksheet. Ask for answer.  Show of hands who got the correct answer.  Ask a student to briefly explain his/her answer.	2 minutes
PARTICIPATORY LEARNING	1. State the learning objectives.  Tie-up bridge and pre-test to objectives. 2. What are the factors of 6? (3 and 2)  How can we illustrate this geometrically? (Draw a 3 by 2 rectangle, divided into 6 squares).  How are factors related to dimensions (of length and width), and product related to area?  (Finding the factors of a number is the same as finding the dimensions of a rectangle whose area is the number).  Will this geometric representation work for finding factors of a trinomial? 3. Distribute/introduce the algebra tiles, as a geometric method of finding factors of trinomials.  Each student will be given a complete set of tiles, with a transparent tile board.  Walk the students through the 3 different tile sizes representing x^2 (green), x (white) and 1 (red).  Explain that x is a variable that can represent any positive number. 4. Assemble 2-green x^2, 5-white x tiles and 2-red 1-tiles.  If all the 9 pieces represent the area of a rectangle, what algebraic expression represents this area?    (2x^2 + 5x + 2)  How can we get the dimensions of this rectangle?  * In your worksheet, complete equation #2:  2x^2 + 5x + 2 = (2x + 1)(x + 2) 5. Empty your tile board.  For our second rectangle, assemble 1- green x^2, 6-white x and 9-red 1-tiles into a rectangle.  What expression represents the area of this rectangle?  (x^2 + 6x + 9).  What are the factors?  (x + 3) and (x + 3).  What do you notice with our rectangle?  (It is a square).  Introduce the perfect square trinomial (PST).  * In your worksheet, complete equation #3:  x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 6. Virtual Manipulatives:    Reiterate that finding the linear factors of a quadratic trinomial is very much related to finding the dimensions of a rectangle that contain the trinomial.  The internet is full of virtual manipulatives that offer fun, creative, and interactive ways of factoring trinomial, which may appeal to today’s technology-savvy students.  Factor x^2 + 7x + 12.  (x + 4)(x+3).	9 minutes	* Algebra tiles * Virtual manipulatives
POST-TEST	Using your algebra tiles, find values of k, where x^2 + kx + 6 factors into 2 binomials. (k = 5, 7).  Write answer in #5 of your worksheet.		* Algebra tiles
SUMMARY & WRAP-UP	Ask students what they have learned today, which should touch the following points: 1. That to the concept of factoring is very much related to finding the dimensions of a rectangle of a given area. 2. That a quadratic trinomial factors only if one can arrange it into a rectangle. 3. That we know that a trinomial is a perfect square if the tiles neatly arranges into a square, with 2 equal dimensions. 4. Ask students to complete # 6 & 7 of their worksheet.  Collect worksheets.	3 minutes

Suggested student worksheet format:
Name:  __________________                                                                                    Topic:  Factoring Quadratic Trinomials
1.    Factor:    x^2  +  5x  +  6
2.    2x^2 + 5x + 2 = (                )(                  )
3.    x^2 + 6x + 9   = (                 )(                  )  =  (                )
4.    x^2 + 7x +  2  = (                  )(                  )
5.    x^2 + kx + 6   = (                   )(                    ) or (                   )(                   )
6.    One thing I learned today is _____________________________________.
7.    Algebra tiles do/do not help in understanding factoring trinomials because_______________________________________________________.

Thinking Mathematically: Chapters 2 and 3 Reflection

The three phases: Entry, Attack, and Review are a great way to break down any problem solving question.  Although these phases seem obvious to many of us, students often take these phases for granted.  Students often end up struggling to complete a question, forget everything about that question and move onto another question. 

The Entry phase emphasizes on the important information gathering questions to ask oneself: What do I KNOW? What do I WANT? What can I INTRODUCE? 
Careful reading is crucial when it comes to approaching any problem solving question, I found that the St Ives and the "hole" questions were very interesting and would probably ask my own future class similar questions that provoke their attention to catch the KEY words to solve the questions.

Important concepts of the Attack phase are: reaching the point of getting STUCK!, realizing what exactly you are STUCK! on, being calm and collected to see that being STUCK! is a good thing, thinking of how to resolve this STUCK! issue and reaching the AHA! result (or some KEY aspect that can lead to the AHA! result). 

I think the most valuable concept a student should learn is the Review phase.  Within the Review phase, the important concepts are to: CHECK the resolution, REFLECT on the key ideas and key moments, EXTEND to a wider context.  Students should be able to check and accept that their solution is correct and try to expand on what else they can apply the KEY points to, in order to solve other questions relating the one they have solved.

The Unattainable: Division By Zero

To divide by zero,
Is no easy task.

People have tried endlessly,
Yet, no one has passed.

It is said to be impossible,
Which is seemingly correct.

As time goes on,
This property still holds true!

Maybe one day,
Inspiration from another world or dimension...

Will bring an answer!
To divide by zero,

Still is... an imagination...

Timed writing exercise

word: divide
- to share evenly among a number of people
- a mathematical notation
- to split apart
- removing oneself from an event
- i got nothing else atm...
- "/" or the other divide symbol
- mind has gone blank...
- d i v i d e..............................................hm............................................
- 30 seconds left...........................
- used in word problems, a basic math tool used for calculations

word: zero
- of no value
- "how zero was created, it's beautiful" joke from russell peters (comedian)
- the smallest neutral number, possibly the only neutral number?
- nothing
- cannot divide by zero, cannot share when there is no one to share with
- anything multiplied by zero is zero
- anything to the power of zero is 1, except 0 itself, which is undefined
- zero factorial is 1
- anything added to zero is that anything
- anything subtracted from zero is the negative form of the anything, anything subtract zero is that anything
- done

Simmt article on math education and citizenship Response

I really like how this article brings the power of mathematics into situations that occur in society.  Ever since I learned the words: formula, algorithm, statistics, patterns, and many other math terms; I have noticed that everything we do is based on mathematical logic of some sort.  For me, statistics or probability play major role in my decision making skills because the numbers and percentages helps give me an idea on what would benefit me the most.  In mathematics, the results are simple, if you have one input, you should have one output.  However, this doesn't mean that there is only one method of getting to the solution.  As many instructors have noted, a teacher should have multiple and diverse solutions at hand, because mathematics is not only about getting the right answer.  The emphasis should be put on "how" did the student get to the right answer, because in life, everyone has their own way of doing things and what matters the most is what works best for them.  I enjoyed the concept of interaction between people in order to address and solve problems that arise because it isn't always possible to solve everything by yourself.  It doesn't hurt to gain other people's opinions and to work together to figure something out as a group.  It was interesting to read about how mathematical models were created and became applicable to predict reality for certain social, political and economic structures.  This really shows the true benefit of how mathematics could be applied to society and not just in a school setting.