Julia Kemp

Factoring Polynomials Louisiana Curriculum Framework Content Strand:   Algebra                                                          Grade Level : 9  

Objectives: To factor quadratic trinomials using algebra tiles.

  Teacher Information  

Benchmarks A1,A2,A4	Time Frame 1-2 days
Curriculum Integration Geometry(Area)	Materials  Algebra Tiles, Overhead Algebra Tiles
Applications Geometry (Area)	Student Groupings 2 -3 students to a group

 

Possible Obstacles to Student Learning: Some students will not want to take the time to manipulate the algebra tiles, because they can factor the polynomials more easily with pen and paper.

   

Opportunities for Assessment 1. Students can go to the overhead and factor a given polynomial. 2. Students can be given a short quiz of  3-5 polynomials to factor, have them draw the tiles on their paper. I would make this a group grade , especially the first day.

 

Lesson Procedure 1. Ask the students to model a particular polynomial with their algebra tiles such as x^2 +5x -3.Make sure everyone is comfortable with modeling a polynomial expression with the tiles. 2. Explain  and demonstrate on the board that factoring a polynomial  is the same thing as creating a rectangle of certain dimensions. The given polynomial will be the parts of  the rectangle and the goal is to rearrange those parts into a rectangle. The dimensions of the rectangle will be the factors of the polynomial. If the pieces cannot be rearranged into a rectangle, then the polynomial  is not factorable. 3.  Demonstrate factoring  x^2 + 5x + 4 using the algebra tiles. Stress that you will work first with the first and last terms of the polynomial.  This will parallel what you will do later  with paper and pen. Show that  4 can be arranged in either a 2x2 array or a 1x4 array. Try the 2x2 array and show how this will not form a rectangle with 5 x tiles. This means that, if this polynomial is factorable, that we must use a 4x1 array. Show how to arrange the 5 x tiles to fill in the space to create a rectangle. This means that the factors of x^2 + 5x + 4 are (x + 4) and ( x+1). Have the students draw  the diagram of the tiles in their notebook along with the polynomial and the dimensions of the rectangle formed. 4. Have students use their algebra tiles to show the factors of  several more polynomials. Students may come up to the overhead and demonstrate the solution after the majority of the class has completed the task. Make sure to include both positive and negative terms in your polynomials. 5. After several have been worked, write x^2 -x -6 on the board. This polynomial cannot be factored without introducing more x tiles in the form of zero-pairs.  Ask  what are  the various arrays that can be formed from 6 tiles. None of these will create a rectangle with only one x tile. Now we will have to find the correct array that will form a rectangle and have pairs of  positive and negative (zero sum ) tiles to add to the polynomial. The correct array is 3x2 . This will require you to add 2 positive and 2 negative x tiles ( zero sum) .Therefore the factors of x^2 - x - 6 is (x - 3)(x + 2). Have the students draw the tiles and copy the problem in their notebooks. 6. Have students work several more where the zero pair must be  added  to the polynomial  before they can be factored. They are to write these in their notebooks . Drawings should accompany each  example. 7. Have students try to factor x^2 + 3x - 1. Show how there is no possible way to create a rectangle even by adding zero pairs. This means that this is not factorable. Have students determine if several polynomials are factorable using the algebra tiles. If they are factorable, have students draw the rectangle formed.  8. Have students write a paragraph that explains how to determine if a trinomial is factorable or not. Include in the paragraph an example of one polynomial that is factorable and one that is not factorable

 

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