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Get Your Free Worksheet Here! 1. Probability 2. Congruence 3. Data Analysis (Basic Statistics) 4. Ratio and Proportion (Direct and...

28/06/2016

Congruence



In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometrics, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be re-positioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.

In elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects.
Two line segments are congruent if they have the same length.
Two angles are congruent if they have the same measure.
Two circles are congruent if they have the same diameter.

In this sense, two plane figures are congruent implies that their corresponding characteristics are "congruent" or "equal" including not just their corresponding sides and angles, but also their corresponding diagonals, perimeters and areas.

Probability


Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis (Reliability theory of aging and longevity), and financial regulation.

A good example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such group-think on pricing, on policy, and on peace and conflict.



In addition to financial assessment, probability can be used to analyze trends in biology (e.g. disease spread) as well as ecology (e.g. biological Punnett squares). As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring and can assist with implementing protocols to avoid encountering such circumstances. Probability is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.

The discovery of rigorous methods to assess and combine probability assessments has changed society. It is important for most citizens to understand how probability assessments are made, and how they contribute to decisions.

Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's warranty.

The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.

25/06/2016

10 Ways To Make Math Fun

In most cases, Mathematics becomes a frightening lesson when in fact it is an interesting and very useful in our daily lives. So, here are several things you may try in teaching math.



1. Avoid to memorize.
An experience when I was a primary learner, my math Teacher asked me to memorize a multiplication table from 1x1 until 10x10 as the preparation to deal with the test (Math Examination). So, what is the different between math and other lessons like history, biology, and language? It will make primary learners stressed and frustrated even dislike math. Memorizing is less effective than understanding. Do not try to force the learners to memorize the math formulas or processes which is less productive and less effective. It will be much better to focus on understanding the involved-process and logic. This will help the learners understand how they should approach such problems in the future. Because we know that is not wrong to use SMS (Short Message Service) to send the message to our friends without knowing how it can be sent, but at least we should know that nowadays wireless signal along with development of technology can deliver our messages over the cloud, so we do not longer think that SMS does not make sense because we know why. As well as math, the learners do not just know 6x7=42 but they understand how it can be 42.


2. Concrete first.
We probably know the way to solve addition, subtraction, multiplication and division abstractly just using paper and pen. It is good, but it would be better to introduce with materials (props) first to allow the learners to construct their cognitive models for abstract mathematical ideas and processes. Materials also provide a common language to communicate these models with the teacher and other learners. Experts in education field believe the media or we call it as props can increase the learning motivation. It helps the students to build a better "mental model" (assumption) or the comprehension of learner. For instance, a book which contains text only requires us to have prior knowledge of what was discussed in the book so that we could make a "mental model". Without prior knowledge about the subjects covered, the "mental model" which is built could be inaccurate. But if in the book is also included a picture, then learners will be easier to make "mental model" more completely and precise. Thus, through the media, a learner has the ability to explore the places in a virtual world that probably was never seen in person, so that learners are better and able to understand the concept. It can be concluded that the media supports the human capacity especially for the learners' encouragement.


3.  Take advantage of hot news or what learners like.
Usually, something we do like can help our mood, moreover for the kids. Make problems link with hot news or what learners like might be more fun to stimulate the learners. I read that in order to be a good conversationalist, we should know what our interlocutors like. But unfortunately not everyone comes with their passions written on their foreheads. Listen and observe seems a little obvious, but it is also the nice first step in finding out what someone likes. Do not be afraid of the banalities. Sometimes those seemingly-boring questions can be gateways to finding out what the other person is passionate about, we may just have to dig a little. For instance, "What did you do this weekend?" is potentially going to reveal what they're passionate about, because that's what they do in their off time. Ask open-ended questions. Do not just ask "yes or no" questions. Most questions can pretty easily be rephrased to meet the aim, For instance, "Was your holiday fun?" turns into "What did you do during holiday?. Then after we know it, we can help to give math lesson linked with what learner's like.


4. Using more word problems.
Abstract problems encourage the learners to simply find the numbers and use the operation-studied in a linear, as if all problems were solved by using a "recipe". Rather than giving problem as “3x + y = 9 and 2x + 3y = 13, find x and y!” we can make the problem more interesting such as “Theo paid $9 for 3 apples and 1 kiwi. If Diana paid $13 for 2 apples and 3 kiwis, how much is each price?”. But it is not that simple, sometimes the existing word problems are solved by rules those learners found in the beginning of the particular lesson routinely. Teacher must be creative to avoid this situation. Mix up the word problems with the intention of all problems are not only solved by the operation just studied. Give learners the word problems to analyze so that instead of finding the answers, they find which operations are needed to get the answer.


5. Tell the math in our daily lives.
Recognizing is important. Allow learners to realize that mathematics always exists in our daily, such as transaction in shopping, estimating time or arranging schedule, cooking with recipes, playing games, over tuning in playing music, determining the selling price to make a profit, banking transaction, etc., we are using math principles. Then let them encourage themselves to learn math. If we are well versed in this language of numbers, it can help us make important decisions and perform everyday tasks. Math can help us to shop wisely, buy the right insurance, remodel a home within a budget, understand population growth, or even bet on the horse with the best opportunity of winning the race. Mathematics is a part of our lives, whether we clean the house, make supper or mow the lawn. Wherever you go, whatever you do, you are using math daily without even realizing it. It just comes naturally.


6. Introduce step by step.
Mathematics is very complicated if you jump too far. It is very impossible to solve 5x^2 + 9x +2 = 0 if you did not know the way to solve 5x2 or 10-1. Start from the basic then continue it step by step. Through automaticity students free up their working memory and can devote it to problem solving and learning new concepts and skills (Geary, 1994).  Quite simply, a lack of fluency in basic math fact recall significantly hinders a child’s subsequent progress with problem-solving, algebra and higher-order math concept. Math could be equated to languages in some ways. Just like you have to learn to combine letters into words and words into sentences. Math anxiety starts when learners fall behind and cannot keep it up.  To avoid these anxieties, learner’ early primary years should focus on learning the foundation math skills needed for later years. We should know how far the learners understand the concept.


7. Giving less is not wrong.
If the learner was not really good at math, it is very helpful to take him or her step backward because it will be useless to teach them in the same materials with other learners even if you took more time with him / her before understanding the previous basic. That learner cannot reach the level 5 before achieve level 4. As teachers, we often focus too much on where we want learners to be, then we forget that we should begin from where we met them at the first time. When a building is constructed, scaffolds are placed around the building to give it additional support. As the building becomes more stable on its own, the scaffolds are removed, a few at a time. Eventually, the building stands independently of the scaffolds (Daisy Martin, 2010). Similarly, "scaffolding" your lessons provides additional support to students as they work toward understanding the content on their own. You can scaffold your lessons in a wide variety of ways, depending on your students' specific needs.


8. Giving more if needed
Some learners may feel bored if the problems were too easy to be solved. They need challenges to learn more and further. Give them extra practices or more difficult problems. If they were not able to complete the assignment well, do not under estimate but just reduce the difficulty. Provide it in the right time and with positive purposes. Express the agreement, appreciation, and encouragement from us to that learners, do not only use what we think about. Occasionally praise their high quality works. Ask learners to give responses about problem-solving as opposed or appropriate with the teaching on the books or teaching from us. Keep in our mind that one learner’s responses or question might provide the opportunity for other learners even ourselves to apply and demonstrate their thinking skills. Instead of answering the questions, turn some of those questions into opportunities for students to process and respond together. In one side, the learner who is thirsty of challenge will be more enthusiastic to learn more and more


9. Playing is studying.
Of course, no one will say playing is not fun. Not every time we play we do not study. Make the math problems as game like “Monopoly”, “Snakes and Ladders”, sport, puzzle or quiz. Most people have like to play bingo, why not turn that into a math learning experience that they will really get a kick out of while learning math. Most people also love collecting things, and this is a fun way to combine art with math and add a social element like we invite the learners to make math fact trading cards, specifically with themes and colors then trade them with siblings or friends who also take part. Young learner are interested in colorful and interactive things. Teacher should make full use of such instincts to push math-related games to them. To our learners, math games are just a games, not math. A good math game will layer in incentives and different levels of difficulty to attract the learner. If traditional assessment books are not working for our learners, it is time to consider using math games. While games may seem less educational, those will be much more effective for our learners.

10. Being tolerant and helpful
Patience is a must for the teacher, but tolerance and help is more needed to encourage the learners to have self-confidence. Appreciate the different ways of learners in learning either in terms of understanding duration or learning environment. Do not desire to get or to show your learners who is the boss. It will encourage more disrespectful behavior from our student. Do not let who does not like math will more dislike math just because the teacher. Teacher is not the boss, we are the learners’ helper. Stop forcing our learners like how we want them to do, but show them how. The highest result of education is tolerance (Helen Keller). Everybody is unique, no matter about how old they are and who they are, we have to respect in diversity, we cannot force everyone to be the same like us or same as what we want. Love our learner unconditionally.



24/06/2016

Characteristics of Triangles

A. Classified by side length













 1. Isosceles triangle
- Two sides of the triangle have equal length.
- The two base angles are equal.

2. Equilateral triangle

- All three sides have equal length.
- Its each angle is 60°.

3. Scalene triangle

- All three sides have unequal lengths.




B. Classified by angle size









1. Acute-angled triangle
All three angles are less than 90°.

2. Obtuse-angled triangle

One of the angles is more than 90°.


3. Right-angled triangle

One of the angles is exactly 90°.


Source: Mathematics Matters (Normal Textbook of Secondary 2, published by EPB Pan Pacific Singapore).

List of Mathematics Worksheet (Get It For Free)

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1. Probability