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Lesson Plan in Grade 8 Math July 15, 2014 I.

Objectives: By the end of the lesson the students should be able to: A. Identify if a binomial is a sum or difference of two cubes. B. Describe the resulting factor of the sum or difference of two cubes. C. Factor the sum or difference of two cubes.

II.

Subject Matter: Factoring Sub Topic: Factoring the sum or difference of two cubes Reference: Math Time: Activity SourceBook, pg. 25 Making connections in Mathematics, Oronce ET. Al, pg. 334 Materials: Chalk, visual aids, activity sheets and flash cards Strategies: Discovery, cooperative learning, discussion Procedure:

III.

A. Daily Routine Teacher’s Activity Good morning! Let us start our today’s meeting with a prayer. Who is the prayer leader for today? At the count of one, pick all the trashes under your chairs. At the count of two, put it in your pocket and at the count of three sit down. (The teacher will check the attendance and the assignments.) B. Recall Teacher’s Activity How do we factor the difference of two squares? (The teacher will call one student.) Very good!

Students’ Activity Good morning Sir! (The student prayer leader will lead the prayer)

Students’ Activity (Some students will raise their hand.) Get the square root of the first and last term.After getting the square roots express them as product of the sum and difference of two terms.

C. Priming The teacher will give series of drills to be answered by the students orally. (The flash cards contain the following) √ √ √ and√ . √ √ √ √ √ ;√

Teacher’s Activity How did you get the cube root of a variable raised in certain power? How can we know if a monomial raised in certain power is a perfect cube?

Students’ Activity Divide the exponent by three. The exponent should be multiples of three.

D. Lesson Proper a. Activity Directions: Rewrite the monomials inside the table horizontally on the space provided to create six term polynomial and simplify. Fill the blank boxes by the GCMF of the aligned column or row. Write the factors of the polynomial inside the parentheses.

x3

x2y

8y3

-4y2

-x2y

-xy2

4y2

-2y

2y

-1

2

xy

3

y

=(

=(

-5a2

a3

5a2b

5a2

-25a

-5a2b

-25ab2

25a

-125

25ab2

125b3

)(

)

=(

27x3

-36x2

a6b3

3a4b2

36x2

-48x

-3a4b2

-9a2b

48x

-64

9a2b

27

=(

)

a3

=(

)(

)(

)

=(

b. Analysis Teacher’s Activity When you rewrite the monomials horizontally to make six term polynomial on your activity, what kind of polynomial did you get? Why did you think so? Aside from being a binomial, what other things did you observe? Since we are on factoring. What do you think is our lesson for today? How can we know if a binomial is a sum or difference of two cubes? Going back to your activity, describe the resulting factors of the sum or difference of the cubes. a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Is this familiar with you? On what lesson did you encounter these formulas?

)(

)

)(

)

)(

)

Students’ Activity Binomial

Some terms were cancelled so they become zero and there were only two terms left. It is the sum or difference of two cubes.

Factoring the sum or difference of two cubes.

The first and the last termsmust be perfect cubes. One of the factors is a binomial and the other one is a trinomial.

Yes! Special products. Products of (x ± y)(x2 ± xy + y2)

How are the terms of the product related to the terms of the binomial factor?

How do you relate your binomial factor to your trinomial factor?

All in all, how do we factor sum or difference of cubes?

Example: Factor x6 + y6 Is this a sum of two cubes? How did you say so?

What is the cube root of x6? How about y6? What is next?

What is the binomial factor? How about the trinomial factor? What are the factors of x6 + y6? Example 2: Factor x3 – 125 Is this a difference of two cubes? Why? What is the binomial factor? How did you say so?

What is the trinomial factor?

How?

What are the factors of x3 – 125? Do you want more examples? (The teacher will give more examples if necessary.)

c. Abstraction Teacher’s Activity How can we identify if a binomial is a sum or difference of two cubes? What are the resulting factors of the sum or difference of two cubes? How do we factor the sum or difference of two cubes?

The first and second term of the binomial factor is the cube root of the first and second term of the product and the sign was copied. Square the first and second term of the binomial factor and get the additive inverse of the product of the terms. Get the cube root of the first and last term. Expressed them in a binomial and copy the sign. Use the expressed binomial factor to get the trinomial factor by squaring the first and second term and getting the additive inverse of their product as the middle term. Yes? The exponents of the first and second terms are multiples of three. We can extract cube roots from it. x2. y2. Expressed them in binomial and copy the sign. x2 + y2 x4 - x2y2 + y4 (x2 + y2)(x4 - x2y2 + y4) Yes! x3 and 125 are both perfect cubes. x–5 Get the cube root of the first term and second term. Express them in a binomial and copy the sign. x2 +5x + 25 Use the expressed binomial to get the trinomial. The square of x is x2. The square of -5 is 25, and the additive inverse of (x)(-5) is 5x. (x – 5)( x2 +5x + 25) (The students will respond.)

Students’ Activity The first and second terms are both perfect cubes. A binomial and a trinomial. Get the cube root of the first and second term and expressed them in a binomial and copy the sign. Use the expressed binomial to get the other trinomial factor. Get the square of the first and second term and get the additive inverse of the product of the two terms.

d. Application Teacher’s Activity x y + 8z 27 + 64x3 125 – x12 x3- 216 (The teacher will ask the four students to factor the sum or difference of two cubes on the board.) (The teacher will choose four students to answer.) (The teacher will verify the answers.) 6 9

IV.

12

Students’ Activity (Some students will raise their hand.)

Evaluation What did the bee say to the rose?

To answer the question stated, just factor the given sum of two cubes in column A. Find the answer in column B and write the corresponding letter on the box. COLUMN A x3 + 27 8x3 + 125y6 x6y9 + z12 64x9 - 729y3 27x3-1 x3 - y3 x6y9 – 125z12 x3 + 64

1. 2. 3. 4. 5. 6. 7. 8.

COLUMN B (x y -5z )(x4y6 + 5x2y3z4 + 25z8) (x - y)(x2 + xy + y2) (2x + 5y2)(4x2 – 10xy2 + 25y4) (x2y3 + z4)(x4y6 – x2y3z4 + z8) (x + 4)(x – 4x + 16 ) (3x -1)(9x2 + 3x + 1) (x+3)(x2 - 3x + 9) 3 (4x – 9y)(16x6 + 36x3y + 81y2) 2 3

B H D ! I , U “

4

DECODER

4

6

8

5

1

2

3

V.

Agreement Teacher’s Activity For your assignment, answer the following on your Math Time on page 26.

7

Practice Ex. 1.12 A – Even numbered items B – odd numbered items C – even numbered items

Prepared by: Acejan L. Jadie Practice Teacher,

Students’ Activity (The students will take note the assignment.)

4

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Objectives: By the end of the lesson the students should be able to: A. Identify if a binomial is a sum or difference of two cubes. B. Describe the resulting factor of the sum or difference of two cubes. C. Factor the sum or difference of two cubes.

II.

Subject Matter: Factoring Sub Topic: Factoring the sum or difference of two cubes Reference: Math Time: Activity SourceBook, pg. 25 Making connections in Mathematics, Oronce ET. Al, pg. 334 Materials: Chalk, visual aids, activity sheets and flash cards Strategies: Discovery, cooperative learning, discussion Procedure:

III.

A. Daily Routine Teacher’s Activity Good morning! Let us start our today’s meeting with a prayer. Who is the prayer leader for today? At the count of one, pick all the trashes under your chairs. At the count of two, put it in your pocket and at the count of three sit down. (The teacher will check the attendance and the assignments.) B. Recall Teacher’s Activity How do we factor the difference of two squares? (The teacher will call one student.) Very good!

Students’ Activity Good morning Sir! (The student prayer leader will lead the prayer)

Students’ Activity (Some students will raise their hand.) Get the square root of the first and last term.After getting the square roots express them as product of the sum and difference of two terms.

C. Priming The teacher will give series of drills to be answered by the students orally. (The flash cards contain the following) √ √ √ and√ . √ √ √ √ √ ;√

Teacher’s Activity How did you get the cube root of a variable raised in certain power? How can we know if a monomial raised in certain power is a perfect cube?

Students’ Activity Divide the exponent by three. The exponent should be multiples of three.

D. Lesson Proper a. Activity Directions: Rewrite the monomials inside the table horizontally on the space provided to create six term polynomial and simplify. Fill the blank boxes by the GCMF of the aligned column or row. Write the factors of the polynomial inside the parentheses.

x3

x2y

8y3

-4y2

-x2y

-xy2

4y2

-2y

2y

-1

2

xy

3

y

=(

=(

-5a2

a3

5a2b

5a2

-25a

-5a2b

-25ab2

25a

-125

25ab2

125b3

)(

)

=(

27x3

-36x2

a6b3

3a4b2

36x2

-48x

-3a4b2

-9a2b

48x

-64

9a2b

27

=(

)

a3

=(

)(

)(

)

=(

b. Analysis Teacher’s Activity When you rewrite the monomials horizontally to make six term polynomial on your activity, what kind of polynomial did you get? Why did you think so? Aside from being a binomial, what other things did you observe? Since we are on factoring. What do you think is our lesson for today? How can we know if a binomial is a sum or difference of two cubes? Going back to your activity, describe the resulting factors of the sum or difference of the cubes. a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Is this familiar with you? On what lesson did you encounter these formulas?

)(

)

)(

)

)(

)

Students’ Activity Binomial

Some terms were cancelled so they become zero and there were only two terms left. It is the sum or difference of two cubes.

Factoring the sum or difference of two cubes.

The first and the last termsmust be perfect cubes. One of the factors is a binomial and the other one is a trinomial.

Yes! Special products. Products of (x ± y)(x2 ± xy + y2)

How are the terms of the product related to the terms of the binomial factor?

How do you relate your binomial factor to your trinomial factor?

All in all, how do we factor sum or difference of cubes?

Example: Factor x6 + y6 Is this a sum of two cubes? How did you say so?

What is the cube root of x6? How about y6? What is next?

What is the binomial factor? How about the trinomial factor? What are the factors of x6 + y6? Example 2: Factor x3 – 125 Is this a difference of two cubes? Why? What is the binomial factor? How did you say so?

What is the trinomial factor?

How?

What are the factors of x3 – 125? Do you want more examples? (The teacher will give more examples if necessary.)

c. Abstraction Teacher’s Activity How can we identify if a binomial is a sum or difference of two cubes? What are the resulting factors of the sum or difference of two cubes? How do we factor the sum or difference of two cubes?

The first and second term of the binomial factor is the cube root of the first and second term of the product and the sign was copied. Square the first and second term of the binomial factor and get the additive inverse of the product of the terms. Get the cube root of the first and last term. Expressed them in a binomial and copy the sign. Use the expressed binomial factor to get the trinomial factor by squaring the first and second term and getting the additive inverse of their product as the middle term. Yes? The exponents of the first and second terms are multiples of three. We can extract cube roots from it. x2. y2. Expressed them in binomial and copy the sign. x2 + y2 x4 - x2y2 + y4 (x2 + y2)(x4 - x2y2 + y4) Yes! x3 and 125 are both perfect cubes. x–5 Get the cube root of the first term and second term. Express them in a binomial and copy the sign. x2 +5x + 25 Use the expressed binomial to get the trinomial. The square of x is x2. The square of -5 is 25, and the additive inverse of (x)(-5) is 5x. (x – 5)( x2 +5x + 25) (The students will respond.)

Students’ Activity The first and second terms are both perfect cubes. A binomial and a trinomial. Get the cube root of the first and second term and expressed them in a binomial and copy the sign. Use the expressed binomial to get the other trinomial factor. Get the square of the first and second term and get the additive inverse of the product of the two terms.

d. Application Teacher’s Activity x y + 8z 27 + 64x3 125 – x12 x3- 216 (The teacher will ask the four students to factor the sum or difference of two cubes on the board.) (The teacher will choose four students to answer.) (The teacher will verify the answers.) 6 9

IV.

12

Students’ Activity (Some students will raise their hand.)

Evaluation What did the bee say to the rose?

To answer the question stated, just factor the given sum of two cubes in column A. Find the answer in column B and write the corresponding letter on the box. COLUMN A x3 + 27 8x3 + 125y6 x6y9 + z12 64x9 - 729y3 27x3-1 x3 - y3 x6y9 – 125z12 x3 + 64

1. 2. 3. 4. 5. 6. 7. 8.

COLUMN B (x y -5z )(x4y6 + 5x2y3z4 + 25z8) (x - y)(x2 + xy + y2) (2x + 5y2)(4x2 – 10xy2 + 25y4) (x2y3 + z4)(x4y6 – x2y3z4 + z8) (x + 4)(x – 4x + 16 ) (3x -1)(9x2 + 3x + 1) (x+3)(x2 - 3x + 9) 3 (4x – 9y)(16x6 + 36x3y + 81y2) 2 3

B H D ! I , U “

4

DECODER

4

6

8

5

1

2

3

V.

Agreement Teacher’s Activity For your assignment, answer the following on your Math Time on page 26.

7

Practice Ex. 1.12 A – Even numbered items B – odd numbered items C – even numbered items

Prepared by: Acejan L. Jadie Practice Teacher,

Students’ Activity (The students will take note the assignment.)

4

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