Shenandoah University (PowerPoint) LESSON PLAN * NAME DATE 10/28/04 TIME REQUIRED 90 minutes SUBJECT Algebra I GRADE 6-9 OBJECTIVES AND PURPOSE (for each objective, show connection to SOL for your subject area): The student will be able to recite the definition of polynomial, term and degree. The student will be able to determine the degree of each monomial in a polynomial and the degree of the polynomial, demonstrated by working 3 problems. The student will be able to add polynomials, demonstrated by working 3 problems. SOL A.11 The student will be able to subtract polynomials, demonstrated by working three problems. SOL A.11 The student will be able to multiply polynomials, demonstrated by working three problems each using the Distributive property and the FOIL method. SOL A.11 The student will be able to use PowerPoint to enter the solution of a FOIL method polynomial multiplication. This input requires modifying fonts to create superscripts (exponents). ANTICIPATORY SET: For students to do upon arrival to class: Write this problem on the board with the instructions: You want to build a square garden divided into 5 rectangular flower beds. Each flower bed is as long as one side of the square. The perimeter of each flower bed is 60 feet. If you want to build a fence around the entire garden, how much fencing do you need to buy? Draw a diagram of the garden with the flower beds in it. Write and solve an equation to represent the perimeter of each flower bed. Write and solve an equation to determine the perimeter of the entire garden.
Allow 10 minutes while you do roll and bring up the PowerPoint program and turn on the projector (to verify that it works for use later in the lesson). Collect the papers and turn off the projector display. REVIEW: In order to be ready to move onto today s lesson on polynomials, let s review monomials. Write the word monomial on the board. Who can tell me the definition of a monomial? (an expression that is a number, a variable, or the product of a number and one or more variables) Who can give me an example of a monomial? (x, 7x, 10x 2, 2ab, 2ab 2 ) Write the appropriate examples on the board. Get a selection of variables, variables with coefficients, terms with more than one variable, and terms with more than one variable with an exponent) Point to the x and ask what term this is. (variable) Point to the 10 and ask what term this is. (coefficient) Point to the superscript 2 and ask what term this is. (exponent) Point to the 2ab 2 and ask what its degree is. (3) Remind students that the degree is the sum of the exponents of the variables and that a has exponent = 1. Now that we know the definitions, let s move onto monomial addition or simplification. Write on the board: 4a 3 b + 11a 3 b a 3 b Ask who can give the answer. (14a 3 b) If the student doesn t give the method used, ask how they got to the answer. (by adding up the coefficients of the like terms, or using the distributive property to sum up the coefficients of the like terms) Now for monomial multiplication. Write on the board (r 3 s 2 )(r 4 s 4 ) Ask who can give the answer. (r 7 s 6 ) Ask how they got the answer. (by adding the exponents of the like variables, or by writing out the number of multiplies for each variable and counting to
create the exponents) If there is no understanding about how the new exponents are reached, write on the board r r r s s s r r r r s s s s and count up the number of r s and the number of s s to get r 7 s 6 INPUT AND MODELING: We want to know the area of a piece of cardboard that is used to make a box. The original piece of cardboard was y by y inches square. (Draw a square on the board and label length and width as y.) In order to make it a box, a square that measures x by x inches is cut out of each corner. (Draw a smaller square in each corner and label length and width x.) y x Ask what is the remaining area of the piece of cardboard. (y 2 4x 2 ) If students have trouble identifying this, ask for the area of the original square of cardboard (y y or y 2 ), then ask for the area of the small corner square (x x or x2), then for the 4 corner squares (4x 2 ) and finally prompt them for the area of the yellow area (y 2 4x 2 ). Write y 2 4x 2 on the board and the word Polynomial beside it. Today s lesson is about polynomials and how to add, subtract and multiply them. A polynomial is a monomial or a sum of monomials. Put the following up on the board as examples of polynomials a 2 + 3a + 6 x 2 + 2xy + y 2 s 3 t 5 + 2 a 2 b + ab + ab 2
Each monomial in a polynomial is called a term. Chose one polynomial and point out each term and identify it as a monomial Polynomials have degree. The degree of a polynomial is the degree of the monomial in it with the highest degree. Point to the polynomial s 3 t 5 + 2 and ask for its degree. (8). Identify the exponent of 3 for s and the exponent of 5 for t and that the sum is 8 the degree of that monomial. Ask if it is the highest degree term in the polynomial. (yes). What about the 2? The term 2 has no variables and no exponents, so doesn t contribute to the degree. Point to the polynomial a 2 b + ab + ab 2 and ask its degree (3). In the first term, identify the exponent of 2 for a and exponent of 1 for b, and the sum is 3 the degree for that term. In the second term, identify exponent of 1 for a and exponent 1 for b and the sum is 2 the degree for that term. In the third term, identify exponent 1 for a, exponent 2 for b and the sum is 3 the degree for that term. The highest degree is 3. Put the degree below each term and circle the term with the highest degree. Now we re going to do addition and subtraction (or simplification) of polynomials. This is similar to monomial addition and subtraction because we are going to be looking at similar terms to add together. Let s look at (put on board): (4x 2 y 2xy 3 + y) + (6xy 3 x 2 y + 9y) Do you see any like terms, monomials with the same variables and exponents? (4x 2 y and x 2 y, 2xy 3 and 6xy 3, y and 9y). Group together the like terms and reorder the polynomial to look like this (4x 2 y x 2 y) + ( 2xy 3 + 6xy 3 ) + (y + 9y) Note the parentheses added to show the like terms grouped together. Remember, too, that adding a negative is the same as subtracting. Apply the distributive principle to pull the coefficients out: (4 1)x 2 y + ( 2 + 6)xy 3 + (1 + 9)y Complete the process: 3x 2 y + 4xy 3 + 10y For another example, use the polynomials in a subtraction problem: (4s 2 + 7st 2t 2 ) (2s 2 + 8st 5t 2 ) This can be rewritten as: 4s 2 + 7st 2t 2 2s 2 8st + 5t 2
Now look for common terms and group them together: 4s 2 2s 2 + 7st 8st 2t 2 + 5t 2 Combine the like terms by adding or subtracting their coefficients: 2s 2 st + 3t 2 Go to Guided Practice for degrees of polynomials and adding/subtracting polynomials. Allow 6 minutes. Walk through the classroom to answer questions. Call the class back to attention. Call on various students for the answers to the problems on the worksheet. If there seems to be trouble with a particular problem, ask a student volunteer with a correct answer to come to the board and show their work on the problem. Now we re going to multiply polynomials. Consider this problem: You want to find the area of this square (draw outside edge) and it s divided this way (draw the 2 intersecting internal lines) with the measurements a and b as shown (write in a and b). b ab b 2 a a 2 ab a b Point to each section and ask students for the area of the individual sections. Start with blue, then red, then purple. Put in the term that describes the area of that section (ab in the purple sections, a 2 in the red section and b 2 in the blue section). Write the polynomial for the sum of the areas in the square reminding students that the total area is just the sum of the areas of the parts:
a 2 + ab + ab + b 2 and simplify it to a 2 + 2ab + b 2 Ask students to tell you another way to compute the area of this square (hint that each side is (a+b) if they have trouble.) (a + b)(a + b) and write it on the board next to the previous statement. We are going to learn that (a + b)(a + b) = a 2 + 2ab + b 2 which is polynomial multiplication. (put the = sign in between the 2 statements making an equation) So let s start with the (a + b)(a + b) and use the distributive property to solve it. First the a from the first term multiplies the (a + b) in the second term, Write a (a + b) on the board. Next, the b from the first term multiplies the (a + b) in the second term. Write + b (a + b) next to what was just written. Apply the distributive property again on each term and get a 2 + ab + ab + b 2 which simplifies into a 2 + 2ab + b 2. Use another example (x + 2)(x + 10) for further reinforcement. Going through the steps above to get x (x + 10) + 2 (x + 10) followed by x 2 + 10x + 2x + 20 followed by x 2 + 12x + 20 There is another method of performing multiplication called the FOIL method. FOIL stands for First, Outside, Inside and Last. This mnemonic is telling us what order to multiply the terms in the 2 factors of the multiplication. Write on the board and circle the first letter of each item. First Outside Inside Last Write (x + 2)(x + 10) on the board. Draw a curved line from the x in the first factor to the x in the second factor and identify as multiplying the first terms. Ask students for the result of that multiplication (x 2 ), and write it on the board below the original statement. Erase the line just drawn and draw a new line from the x in the first factor to the 10 in the second factor and identify as multiplying the outside terms. Ask students for the result of that multiplication (10x), and write + 10x on the board next to the x 2. Erase the line just drawn and draw a new line from the 2 in the first factor to
the x in the second factor and identify as multiplying inside terms. Ask students for the result of that multiplication (2x), and write + 2x next to the 10x. Erase the line just drawn and draw a new line from the 2 in the first factor to the 10 in the second factor and identify as multiplying the last terms. Ask students for the results of this multiplication (20), and write + 20 next to the 10x This yields the equation x 2 + 10x + 2x + 20 which simplifies to x 2 + 12x + 20. Repeat this process of FOIL with (9a 3)(a + 4) to yield 9a 2 + 36a 3a 12 and finally 9a 2 + 33a 12 Turn on the projector for the PowerPoint exercise. Explain that the Guided Practice will have 2 parts. They should first get into their PowerPoint groups. Then they are to do the 6 problems on the Multiplying Polynomials worksheet as a group and then they are to transfer their solution for the final problem to their PowerPoint presentation. Display the PowerPoint and show which page they are to work on. Remind them that they will be bringing up their own copy of the PowerPoint to make the modifications. Tell them to print out the PowerPoint slide they have created and turn it in. Allow 10 15 minutes. Walk around the room to answer questions. GUIDED PRACTICE: Use the Guided Practice worksheet for degrees of polynomials and adding/subtracting polynomials. Allow 6 minutes. This sheet contains 3 problems each in finding degrees of polynomials, adding/subtracting polynomials using the distributive property, and adding/subtracting polynomials by adding/subtracting coefficients. This is done individually. Use the Guided Practice worksheet for multiplying polynomials. This sheet contains 3 problems each in multiplying polynomials using the distributive property and the FOIL method. This is done in groups. Together with the PowerPoint update, allow 10 15 minutes. The PowerPoint exercise is a continuing project in the class to create a study guide/review for use in preparing for 9 week assessments. Groups have been defined earlier and students will continue to work in these groups during this unit. CLOSURE: Today s lesson introduced polynomials definitions, and addition, subtraction which is simplification, and multiplication using the distributive property and the FOIL method.
INDEPENDENT PRACTICE: Homework will be selected problems from the book to reinforce the skill of determining degree of polynomials, addition/subtraction/simplification of polynomials, and multiplication using the distributive property and the FOIL method. MATERIALS: Guided practice worksheet copies for all students and the answer key. Group PowerPoint presentations. * be sure to review Lesson plan design available on my web.