Midterm Lesson 1-Functions and its Operations PDF

Objectives: At the end of the lesson, the students must be able to: Perform operations on mathematical expressions/functions correctly. Organize one’s methods and approaches for proving and solving problems. Solve problems involving recreational problems following Polya’s four steps ...

Objectives: At the end of the lesson, the students must be able to: Perform operations on mathematical expressions/functions correctly. Organize one’s methods and approaches for proving and solving problems. Solve problems involving recreational problems following Polya’s four steps  ENGAGE KWL Chart In a ½ crosswise, divide your paper into three column. On the first column, write a label of What I know about Function? On the second column, write a label of What am I expecting to know about Function? Lastly, on the third column, write a label of What I learned ?  ENGAGE KWL Chart On the first column, answer the question: “What do you know about functions?” in 1-minute. On the second column, answer the question: “What are you expecting to know about functions?” in 1-minute.  ENGAGE Before doing the third column, let’s watch the video about Functions. https://www.youtube.com/watch?v=XtIJXELn8Bs  EXPLORE After watching the video, answer the question on the third column: “What are the things that you learned about Functions?” Answer the question: Why is it necessary to know the concept of 4 fundamental operation of Mathematics?  EXPLAIN and ELABORATE ❖ A relation is a set of ordered pairs. ❖ A relation may have more than 1 output for any given input. ❖ The set whose elements are the first coordinates in the ordered pairs is the domain of the relation. ❖ The set whose elements are the second coordinates is the range. ❖ A = { (1, 1), (2, 3), (2,4)} ❖ Domain: {( 1, 2)} Range: {(1, 3, 4)} ❖ Money won after buying a lotto locket. ❖ The high temperature on July 1st in New York City. Depends on the year. ❖ How many words your friend uses when answering, “How are you?” ❖ The number of calories in a fast food hamburger. ❖ Places you can drive to with 1 gallon left in your gas tank. ❖It is a relation in which each element in the domain is paired with exactly one element in the range. ❖A function can have no more than 1 output for any given input. Examples: ❖The amount of sodas that come out of a vending machine. depending how much money you insert. ❖ The amount of carbon left in a fossil after so many years. ❖ The velocity of an object in freefall after being dropped so many seconds, excluding air resistance. ❖ The height of a person at a given time in their life. ❖ The intensity of a light as you slide its dimmer switch. ❖The notation f(x) defines a function named f. This is read as “y is a function of x.” The letter x represents the input value, or independent variable. The letter y is replaced by f(x) and represents the output value, or dependent variable. FUNCTION NOTATION It involves only one value or accepts one value or operand. ❖It can act on two operands “+” and “ – ” ❖ It takes two values and include the operations of addition, subtraction, multiplication, division and exponentiation. B. Commutativity of Binary Operations Addition and multiplication of any two real numbers is commutative as seen in their mathematical symbols: Example: x + y = y + x and x y = y x C. Associativity of Binary Operations Given any three real numbers you may take any two and perform addition or multiplication as the case maybe and you will end with the same answer. D. Distributivity of Binary Operations Distributivity applies when multiplication is performed on a group of two numbers added or subtracted together. Example: z(x ± y) = zx ± zy  Objectives: At the end of the lesson, the students must be able to: 1. perform operations on functions; and 2. solve composite functions ❖= f(x)F ❖f + g (x) Example 2 Example 2 Example 3 Example 4 Example 1 Example 2 Example 3 Example 4 Example 5 Polya’s Four-Step Problem Solving Strategy Step 1 : Understand the problem. Step 2 : Devise a plan. Step 3: Carry out the plan. Step 4: Review the solutio ❖What is the goal? ❖What is being asked? ❖What is the condition? ❖What sort of a problem is it? ❖What is known or unknown? ❖Is there enough information? ❖Can you draw a figure to illustrate the problem? ❖Is there a way to restate the problem in your own words? ❖Act it out. ❖Be systematic. ❖Work backwards. ❖Consider special cases. ❖Eliminate possibilities. ❖Perform an experiment. ❖Draw a picture/diagram. ❖Make a list or table/chart. ❖Use a variable, such as x. ❖Look for a formula/formulas. ❖Write an equation or model. ❖Look for a pattern/patterns. ❖Use direct or indirect reasoning. ❖Solve a simple version of the problem. ❖Guess and check your answer (trial and error). Example  In this strategy, data or information are organized by listing them or recording them systematically in tables. The data are then analyzed to discover relationships and patterns and to draw out generalizations or solutions to the problem. There are 12 possible ways that a patient can be classified.  To use the guess-check strategy, one follows these steps: Making a logical guess at the answer. The student learns more about the problem. Checking the guess. It is important that computation is accurate to avoid wastage of time and effort by making more guesses when in fact, the solution might have found some guesses before.  To use the guess-check strategy, one follows these steps: Making a logical guess at the answer. The student learns more about the problem. Checking the guess. It is important that computation is accurate to avoid wastage of time and effort by making more guesses when in fact, the solution might have found some guesses before.  Using the information obtained in checking to make another guess if necessary. The student is left to make his guess skip around so he can bracket the right answer. As to whether the next guess would be a smaller or a bigger number depends on how good the skill of the learner is in estimating and logical thinking.  To use the guess-check strategy, one follows these steps: Continuing the procedure until the correct answer is obtained Note: Use logical reasoning to minimize the number of trials.  In the World Math Competition held in Bulgaria, the contestants were given ten items to be solved in four hours. Five points were given for each correct answer and two points were deducted for each wrong answer. Albert did all questions and scored 29, how many correct answers did he have?  Acting out the Problem is a strategy in which people physically act out what is taking place in a word problem. One may use people or objects exactly as described in the problem, or you might use items that represent the people or objects. Using this strategy, people visualize and simulate the actions described in the problem.  The “Work Backward” method works well for problems where a series of operations is done on an unknown number and you’re only given the result. To use this method, start with the result and apply the operations in reverse order until you find the starting number.  In a dancing competition all the contestants started dancing together. After three minutes half the people were eliminated. During the next ten minutes half of the remaining were eliminated. At the 15 minute mark, half again were eliminated, and at the 20 minute mark, half of those still remaining were eliminated. In the last two minutes one more contestant was eliminated leaving a winner of the competition. How many dancers were there in the beginning? “Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers.” – Shakuntala Devi THANK YOU!  EXTEND Reflection Vlog- Group activity Each member in a group will record a 1 minute video about the things they have learned in this topic and create a drive and submit through Brightspace.