Insights Into Algebra 1: Teaching for Learning
Properties Lesson Plan 1: The X Factor – Trinomials and Algebra Tiles
This lesson will teach students to factor trinomial expressions of the form x2 + bx + c. Students will use algebra tiles to identify the binomial factors and the graphing calculator to verify the result. In addition, students will identify the x-intercepts and y-intercepts of each trinomial function and explore relationships between the trinomial x2 + bx + c and its factored form
(x + m)(x + n).
One 50-minute class period
Students will be able to:
- Factor trinomials of the form x2 + bx + c into two binomial factors.
- Identify the relationships that exist between b, c, m, and n when x2 + bx + c is factored as (x + m)(x + n).
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000.
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Teachers Activities and Assignment
1. Ask a student to define the word “factor.” Elicit that “factor,” when used as a verb, means to “to rewrite a number or expression as a product of two or more numbers or expressions.”
2. Have students, in pairs or small groups, talk about ways to factor 30. Students should come up with four ways: 1 x 30, 2 x 15, 3 x 10, and 5 x 6. Note that students might also include methods that break 30 into more than 2 factors, such as 5 x 3 x 2 or 10 x 3 x 1.
3. Explain that the factors of 30 are “numerical factors” and that during today’s lesson, students will look at algebraic factors.
1. Give some examples of algebraic expressions that can be written in factored form:
- 3x2 – 6x = 3x (x – 2)
- a2 + 7a + 12 = (a + 4)(a + 3)
2.Use the distributive property to multiply the factors and obtain a product, and then have students verify that both of these equations are true.
3.Reinforce the definition of factoring by asking, “Which side is in factored form?” Students should conclude that the right side is in factored form, because factoring means to rewrite an expression as a product or as a multiplication problem.
4.In groups, have students use algebra tiles to write the first trinomial on Factoring Gift in factored form. That is, explain that you would like them to find two binomials that, when multiplied, give this trinomial: x2 + 6x + 8. Reinforce that the algebra tiles must be arranged to form a rectangle with no gaps. It’s important to note that students may arrange the tiles in any formation that results in a rectangle (for example: below, left). However, students must be able to identify the side length of the rectangle, which may be easier if the rectangle is arranged in a systematic manner (below, right). Students should understand that they can find the area of the rectangle by using the area formula, namely A = lw = (x + 4)(x + 2), or by adding the individual pieces inside. These pieces are
x2 + x + x + x + x + x + x + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = x2+ 6x + 8.
Because the two expressions measure the same area, they must be equivalent.
5.Once students have used the tiles to find the factored form (x + 2)(x + 4),have them verify the product using the distributive property.
6.Have students graph the trinomial x2 + 6x + 8 and its factored form, (x + 2)(x + 4), on the graphing calculator. Students should see that the graphs are identical.
7.Ask students to identify the values of x at which the graph crosses the x-intercept. Using the TRACE feature, students should identify (-2, 0) and (-4, 0)as the x-intercepts.
8.Give students 30 seconds to determine where the y-intercept occurs. Using the TRACE feature, students should identify (0, 8) as the y-intercept.
9.Have students use the same process for the second and third trinomials on the Factoring Gift handout: x2 + 7x + 6 and x2 + 8x + 12.
- Factor the trinomial using algebra tiles, and write it in factored form.
- Verify the product with the distributive property.
- Identify the x-intercepts.
- Identify the y-intercept.
On the chalkboard or overhead projector, create a table that lists all of the information that has so far been obtained:
10.Give students one minute to discuss, in pairs or small groups, any patterns or relationships that exist in the table. Encourage them to look for several different patterns, and explain that there are many to find. Students should identify the following relationships:
- The y coordinate of the y-intercept is equal to the constant term in the trinomial; that is, if the polynomial is x2 + bx + c, the y-intercept occurs at (0, c).
- The x coordinates of the x-intercepts are equal to the opposite of the constant terms when the trinomial is written in factored form; that is, if the polynomial can be expressed as (x + m)(x + n), the x-interceptsoccur at (-m, 0) and (-n, 0) because x + m = 0 or x + n = 0.
- If a trinomial x2 + bx + c can be written as (x + m)(x + n), then b = m + n and c = m x n.
Note: The final pattern listed above, that b = m + n and c = m x n, is one of the keystones of this lesson. Students must realize that this relationship always holds, and that it is the key to factoring trinomials. If students have not identified this relationship by this point of the lesson, have them continue to factor trinomials from the Factoring Gift handout until they identify this pattern.
11.Using the patterns identified in the table, students should factor the fourth trinomial from the Factoring Gift handout, x2 + 7x + 10, without algebra tiles or the graphing calculator. At this point, students should have discovered that they must find two numbers for which the product is 10 and the sum is 7, resulting in (x + 2)(x + 5).
12.Ask students to factor x2 + 4x + 6. To factor this trinomial, students must identify two numbers that have a product of 6 and a sum of 4. Because no real numbers exist for which this is true, students should conclude that this trinomial cannot be factored. Define such a polynomial as a “prime trinomial.”
13.Have students graph x2 + 4x + 6 on graphing calculator. To make sure students understand this visual representation, which shows why this expression can’t be factored, point out or elicit that the graph does not cross the x-axis, so it has no x-intercepts and consequently no real factors.
In their math journals, have students identify all positive values of C such that x2 + 6x + C is a factorable trinomial. Students should identify at least three values for C, namely 5, 8, and 9. Students may also notice that C = 0 is a solution, but it produces a slightly different form than this lesson addresses.
- 9 x2 + 6x + 9 = (x + 3)(x + 3)
- 8 x2 + 6x + 8 = (x + 2) (x + 4)
- 5 x2 + 6x + 5 = (x + 1)(x + 5)
- 0 x2 + 6x = x (x + 6)
Related Standardized Test Questions
The questions below dealing with factoring polynomials have been selected from various state and national assessments. Although the lesson above may not fully equip students to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.
- Taken from the New York Regents High School Examination, Mathematics (August 2002):
A rectangular park is three blocks longer than it is wide. The area of the park is 40 square blocks. If w represents the width, write an equation in terms of w for the area of the park. Find the length and the width of the park.
Solution: w(w + 3) = 40, w2 + 3w – 40 = 0, (w + 8)(w – 5) = 0, w + 8 = 0 or w – 5 = 0, w = -8 or w = 5. The width must be positive, so w = 5 and l = 8. The width of the park is five blocks and the length of the park is 8 blocks.
- Taken from the New York Regents High School Examination (January 2003):
What are the factors of x2 – 10x – 24?
A. (x – 4)(x + 6)
B. (x – 4)(x – 6)
C. (x – 12)(x + 2) (correct answer)
D. (x + 12)(x – 2)
- Taken from the Virginia Standards of Learning Assessment (Spring 2002):
Which property justifies the following statement?
If 3a + 3b = 12, then 3(a + b) = 12
A. Commutative property of multiplication
B. Distributive property for multiplication over addition (correct answer)
C. Multiplicative identity property
D. Associative property of addition
- Taken from the Virginia Standards of Learning Assessment, Algebra I (Spring 2002):
Which is the complete factorization of 2x2 + 5x + 3?
A. (2x + 1)(x + 2)
B. (2x + 1)(x + 3)
C. (2x + 2)(x + 1)
D. (2x + 3)(x + 1) (correct answer)
- Taken from the Virginia Standards of Learning Assessment, Algebra I (Spring 2002):
- Taken from the Virginia Standards of Learning Assessment, Algebra II (Spring 2001):
Which is a zero of the function f(x) = x2 + 6x + 8?
B. -4 (correct answer)
Student Work: Factoring Assignment
First of all, I am impressed with the student’s neatness and organization. The notes are easy to follow. There is evidence that the student understands what it means to factor a number and to factor an expression. He/she appears to understand how to apply the distributive property. The student might not fully understand what is meant by an expression being prime. It is unclear that this student understood the last problem where he/she had to decide on the values for C to make the trinomial factorable. I would have preferred some written explanation such as, “we are looking for two numbers whose product is 8 and whose sum is 6.” Perhaps next time I will ask the students to summarize their findings in words and write them down on paper. I would make one change to this handout (remember I call it a “gift”): on #16, I would stipulate that C can be greater than or equal to zero.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.