Miss Edwards



Factoring Polynomials

Learning Through Discovery

Topic:

- Factoring Polynomials

NJCCS:

4.3.12.D2: Evaluate and simplify expressions.

4.5.C6 : Understand how mathematical ideas interconnect and build on one another to

produce a coherent whole.

4.5.E1: Create and use representations to organize, create, and communicate

mathematical ideas.

Materials:

- Guided Discovery Worksheets

- Markers and Whiteboards

- Timer

Objectives:

Level One: Students will be able to factor polynomials by working in groups and listening

to others.

Level Two: Students will be able to discover and master their aspect of factoring

polynomials within their groups. Students will be able to explain their method to

others.

Level Three: Students will be able to discover and master their method of factoring

polynomials while working effectively in groups. Students will be able to

explain their method to others in a constructive way. Students will be able to

recognize how all of the methods of factoring polynomials are connected.

Motivation:

Put the following on the board and give students a total of 10 minutes to complete.

Entrance Quiz

Multiply the following:

1. (x + 3)(x – 4)

2. (x – 6) (x + 3)

3. (x – 3)(x – 7)

Have students hand in their quizzes. This quiz will be used for assessment purposes of last week’s lesson of multiplying polynomials and the distributive property.

Procedure:

1. Explain the Jigsaw cooperative learning strategy to the students. Students will have the half the block to learn the concept, and complete their sections of the worksheet, and discuss a plan to teach other groups with necessary resources. Students will split up and tend to other groups to teach their aspect of the project. Split students into the predetermined groups. Explain to them that they are like separate countries- and there are NO international relations. They must look within their country for resources and for the roaming CIA (the teacher).

2. Allow students to work for 45 minutes. At the end of that time, each individual student from the group that factored polynomials of the form ax2 + bx + c will be assigned to teach a group their concept. Students should use the part of that they have completed to provide the groups with problems. After 10 minutes, students will return to their home groups. The next group who factored ax2 – bx + c will then split up and tend to their assigned group. This will continue until all groups have learned all the concepts.

Closure:

Have students put their desks back and clean up their stations.

Ask students to give you feedback on their methods.

- Did your group have trouble learning the method?

- If you did, in what way?

- What are some of the problems your group encountered?

- Did anyone have trouble teaching the method? Why do you think?

Homework:

- Worksheet

Assessment:

Assessment will be done mainly in the next class during the presentations. The groupwork worksheets will also provide excellent feedback on the progress of each individual group. Group progress will be assessed by observation rather than interrogation.

++

Name: Date:

Algebra I, Block 4

Factoring Polynomials

Your job as a group is to become masters at factoring polynomials that are of the form ax2 + bx + c so that you can teach other groups how to become masters also.

x2 + 7x + 12

First Step: Set up the parenthesis. ( )( )

Second Step: Check the signs.

In order to check the signs, we need to understand where the signs are coming from. Let us use an example of multiplying polynomials to discover this.

Use the FOIL method and the Box Method to multiply the following

(x + 3)(x + 4)

|FOIL |Box Method |

|(x + 4)(x + 3) | |

| |x |

|F O I L |+4 |

|x2 + 3x +4x +12 | |

| |x |

|F + (O+I) + L |x2 |

|X2 + 7x + 12 |+ 4x |

| | |

| |+3 |

| |+3x |

| | |

| |+12 |

| | |

Notice that the signs are determined by the constants in the binomial. In this case: we have a ( + ) ( + ) in our factors which gives us a (+) constant in our answer.

How is the sign of the middle term of the trinomial affected by the signs of the factors?

Let’s think backwards.

Write a rule for determining the signs in our factors when we are given a trinomial.

Third Step: Draw your ears.

The next step is to factor the leading coefficient and the constant of the trinomial.

Vocabulary Check!

What is a Leading Coefficient?

What is a constant?

x2 + 7x + 12

Fourth Step: Set up your frame.

( + ) ( + )

Do we have any options for the leading coefficients of the factors?

( + ) ( + )

Do we have any options for the constants of the factors?

Fifth Step: Work with the OI until you find the pieces that work for the middle term.

( + ) ( + )

Check your smile until you get the numbers that work!

|(x + 3)(x + 4) |(x + 2)(x+6) |(x + 1)(x + 12) |

– –

Name: Date:

Algebra I, Block 4

Factoring Polynomials

Your job as a group is to become masters at factoring polynomials that are of the form ax2 - bx + c so that you can teach other groups how to become masters also.

x2 – 7x + 12

First Step: Set up the parenthesis. ( )( )

Second Step: Check the signs.

In order to check the signs, we need to understand where the signs are coming from. Let us use an example of multiplying polynomials to discover this.

Use the FOIL method and the Box Method to multiply the following

(x – 3)(x – 4)

|FOIL |Box Method |

|(x– 4)(x – 3) | |

| |x |

|F O I L |– 4 |

|x2 – 3x – 4x +12 | |

| |x |

|F + (O+I) + L |x2 |

|x2 – 7x + 12 |– 4x |

| | |

| |– 3 |

| |– 3 x |

| | |

| |+12 |

| | |

Notice that the signs are determined by the constants in the binomial. In this case: we have a ( – ) ( – ) in our factors which gives us a (+) constant in our answer.

How is the sign of the middle term of the trinomial affected by the signs of the factors?

Let’s think backwards.

Write a rule for determining the signs in our factors when we are given a trinomial.

Third Step: Draw your ears.

The next step is to factor the leading coefficient and the constant of the trinomial.

Vocabulary Check!

What is a Leading Coefficient?

What is a constant?

x2 – 7x + 12

Fourth Step: Set up your frame.

( – ) ( – )

Do we have any options for the leading coefficients of the factors?

( – ) ( – )

Do we have any options for the constants of the factors?

Fifth Step: Work with the OI until you find the pieces that work for the middle term.

( – ) ( – )

Check your smile until you get the numbers that work!

|(x – 3)(x – 4) |(x – 2)(x – 6) |(x – 1)(x – 12) |

– +

Name: Date:

Algebra I, Block 4

Factoring Polynomials

Your job as a group is to become masters at factoring polynomials that are of the form ax2 + bx – c and ax2 – bx – c so that you can teach other groups how to become masters also.

x2 – x – 12

First Step: Set up the parenthesis. ( )( )

Second Step: Check the signs.

In order to check the signs, we need to understand where the signs are coming from. Let us use an example of multiplying polynomials to discover this.

Use the FOIL method and the Box Method to multiply the following

(x + 3)(x – 4)

|FOIL |Box Method |

|(x + 3)(x - 4) | |

| |x |

|F O I L |– 4 |

|x2 – 4x + 3x –12 | |

| |x |

|F + (O+I) + L |x2 |

|x2 – 1x – 12 |– 4x |

| | |

| |+ 3 |

| |+ 3 x |

| | |

| |–12 |

| | |

Notice that the signs are determined by the constants in the binomial. In this case: we have a ( + ) ( – ) in our factors which gives us a (-) constant in our answer.

How is the sign of the middle term of the trinomial affected by the signs of the factors?

Let’s think backwards.

Write a rule for determining the signs in our factors when we are given a trinomial.

Third Step: Draw your ears.

The next step is to factor the leading coefficient and the constant of the trinomial.

Vocabulary Check!

What is a Leading Coefficient?

What is a constant?

x2 – 1x – 12

Fourth Step: Set up your frame.

( + ) ( – )

Do we have any options for the leading coefficients of the factors?

( + ) ( – )

Do we have any options for the constants of the factors?

Fifth Step: Work with the OI until you find the pieces that work for the middle term.

( + ) ( – )

Check your smile until you get the numbers that work!

|(x + 3)(x – 4) |(x + 2)(x – 6) |(x + 1)(x – 12) |

[pic]Remember: If you get the correct number but the wrong sign when you are checking your smile- switch the signs in the factors!

Name: Date:

Factoring Quadratic Equations

Jigsaw Style

Factoring Quadratics of the form: ax2 + bx + c

|x2 + 5x + 6 |x2 + 7x + 10 |x2 + 9x + 20 |

| | | |

| | | |

| | | |

|x2 + 18x + 32 |x2 + 11x + 24 |2x2+ 11x + 15 |

| | | |

| | | |

| | | |

|4x2 + 8x + 3 |3x2 + 4x + 1 |6x2 + 7x + 2 |

Factoring Quadratics of the form: ax2 – bx + c

|x2 – 8x + 12 |x2 – 28x + 75 |x2 – 8x + 7 |

| | | |

|x2 – 14x + 45 |x2 – 9x + 14 |3x2 – 16x + 5 |

| | | |

|8x2 – 9x + 1 |2x2 – 5x + 2 |8x2 – 9x + 1 |

| | | |

Factoring Quadratics of the form: ax2 + bx – c

|x2 + 6x – 7 |x2 + 3x – 10 |x2 + 9x – 22 |

| | | |

| | | |

| | | |

|x2 + 5x – 24 |x2 + 34x – 35 |5x2+ 4x – 1 |

| | | |

| | | |

| | | |

|8x2 + 18x – 5 |3x2 + 16x – 12 |6x2 + 7x – 5 |

| | | |

| | | |

| | | |

| | | |

Factoring Quadratics of the form: ax2 – bx – c

|x2 – 5x – 6 |x2 – 2x – 15 |x2 – 23x – 50 |

| | | |

| | | |

| | | |

|x2 – 3x – 28 |x2 – 7x – 18 |7x2 – 13x – 2 |

| | | |

| | | |

| | | |

|2x2 – 5x – 3 |8x2 – 14x – 15 |6x2 – 7x – 3 |

| | | |

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