## Insights Into Algebra 1: Teaching for Learning

# 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.

## Part 1: Factoring Quadratic Expressions

**Factoring** is the process of rewriting a number or expression as a product of two or more numbers or expressions. It can be used to break a polynomial into smaller parts. By writing the factors of a polynomial, it is often easier to solve equations. The distributive property plays a big role when multiplying factors to get a product.

## Part 2: Understanding Basic Recursion

**Recursion** is the process of describing the next term in a sequence in relation to preceding terms. Recursive formulas can model population growth patterns, the distance traveled on a trip, the interest earned on a bank account, and other real-life events.

### Part 1: Explanation

A factor (of a polynomial) is a polynomial that, when multiplied by another polynomial, yields the original polynomial. For instance, x and (x – 3) are factors of x^{2} – 3x because x(x – 3) = x^{2} – 3x. Similarly, (x – 5) and (x + 2) are factors of x^{2} – 3x – 10, because (x – 5)(x + 2) = x^{2} – 3x – 10.

When used as a verb, factor means to divide a number or expression into a product of other numbers or expressions. As an example, the number 30 can be factored as 1 x 30, 2 x 15, 3 x 10, 5 x 6, or 2 x 3 x 5. The last set of factors is called the prime factorization of 30 because the factors are prime numbers. On the other hand, there is typically only one way to factor a polynomial, especially when the leading coefficient of x^{2} is 1. For instance, the expression x^{2} + 2x – 24can only be factored as (x + 6)(x – 4).

Many polynomials, such as x^{2} + 2x – 7, cannot be factored at all, and are therefore known as **prime polynomials**. Some polynomials, however, contain coefficients with common factors, and they may be factored in more than one way. For instance, 2x^{2} + 4x – 48 can be factored in several different ways: 2(x + 6)(x – 4) or (2x + 12)(x – 4) or (x + 6)(2x – 8).

Factoring is one method of finding the solutions of a polynomial equation. Using factoring, the quadratic equation x^{2} + 2x – 15 = 0 can be rewritten as (x + 5)(x – 3) = 0. This shows that either x + 5 = 0 or x – 3 = 0, because one or both of the factors must equal zero if their product is to equal zero. Therefore, either x = -5 or x = 3.

Other examples:

- The height (h) of a ball thrown into the air with an initial vertical velocity of 24 feet per second from a height of 6 feet above the ground is represented by the equation h = 16t
^{2}+ 24t + 6 where t is the time, in seconds, that the ball has been in the air. After how many seconds is the ball at a height of 14 feet? - The parabola y = x
^{2}+ 18x – 40 crosses the x-axis at (2, 0) and (-20, 0);the function can be factored as y = (x + 20)(x – 2). - A garden that is 20 feet by 30 feet is bordered by a cement walkway. If the width of the walkway is w feet, the area of the garden and walkway combined is (20 + 2w)(30+2w) = 600 + 100w+ 4w
^{2}or (2w+20)(2w+30) = 4w^{2}+ 100w + 600 - A rectangular box has a volume of 280 in
^{3}. Its dimensions are 4 in * (x + 2) in * (x + 5) in. The formula V = lwh can be used to find the value of x, as follows:

4(x + 2)(x + 5) = 280

4x^{2}+ 28x + 40 = 280

4x^{2}+ 28x – 240 = 0

x^{2}+ 7x – 60 = 0

(x + 12)(x – 5) = 0

x = -12 or x = 5

Because x = -12 does not make sense in the context of the problem, it cannot be an answer. Consequently, x = 5, and the dimensions are

4 in, 7 in, and 10 in. - The equation -0.00239d
^{2}+ 1.199d = 0 can be used to model a home run that Mickey Mantle hit on May 22, 1963, where d represents the distance of the home run in feet. By factoring the expression as 0.00239d (d – 501.6736) = 0, students can see that Mantle’s hit measured about 502 feet.

### Part 1: Mathematical Definition

A **factor** (of a number) is an integer that, when multiplied by another integer, results in the original number. For instance, 3 is a factor of 18, because 3 x 6 = 18, and a and b are factors of ab.

A **factor** (of a polynomial) is a polynomial that, when multiplied by another polynomial, results in the original polynomial. For instance, (x + 2) and (x + 4)are factors of x^{2} + 6x + 8, because (x + 2)(x + 4) = x^{2} + 6x + 8. Similarly, 4 and (x + 5) are factors of 4x + 20 because 4(x + 5) = 4x + 20.

To factor a number (or a polynomial) is to name the number by its factors; that is, to write the number (or polynomial) as a product of two or more integers (or polynomials). When used as a verb, factoring refers to the process opposite of multiplying. In a sense, it is similar to dividing – to factor a number or polynomial is to divide it into other numbers or polynomials.

Following is a definition of factor from a mathematics dictionary:

**Factor:** As a verb, to resolve into factors. One factors 6 when he writes it in the form 2 x 3. [As a noun,] a factor of an object (perhaps of some specified type) divides the given object.

factor of an integer:An integer whose product with some integer is the given integer. For example, 3 and 4 are factors of 12, since 3 x 4 = 12; the positive factors of 12 are 1, 2, 3, 4, 6, 12, and the negative factors are -1, -2, -3, -4, -6, -12.

factor of a polynomial:One of two or more polynomials whose product is the given polynomial. Sometimes one of the polynomials is not allowed to be the constant 1, but usually in elementary algebra a polynomial with rational coefficients is considered factorable if and only if it has two or more nonconstant polynomial factors whose coefficients are rational (sometimes it is required that the coefficients be integers). For example, (x^{2}– y^{2}) has the factors (x – y) and (x + y) in the ordinary (elementary) sense; (x^{2}– 2y^{2}) has the factors and in the field of real numbers; (x^{2}+ y^{2}) has the factors (x – iy) and (x + iy) in the complex field.

(Source: James, Robert C. and Glenn James. *Mathematics Dictionary* (5th edition). New York: Chapman & Hall, 1992)

### Part 1: Role in the Curriculum

Exposure to factoring is important for Algebra 1 students because it illuminates many aspects of the nature of mathematics and serves as a bridge to the study of advanced mathematical topics. The key word here is exposure. In the past, students would be entrenched in the study of factoring for an entire unit and spend a month learning myriad methods; today, the topic receives less emphasis. Students should understand the various representations for polynomials, one of which is its factored form, rather than mastering every factoring technique.

Read what teacher educator Diane Briars has to say about the role of factoring in the curriculum:

**Transcript from Diane Briars**

In terms of solving quadratics, we need to recognize that factoring is limited as a general method. The majority of quadratics are solved by the quadratic formula; the majority of quadratics are not factorable. But in looking at factoring, there’re some nice connections that you can make. I think Tom’s lesson illustrates how you can use factoring, or a factoring lesson, to really connect ideas of roots of quadratics to x-intercepts.

The NCTM *Principles and Standards for School Mathematics (PSSM) *does not explicitly mention factoring polynomials. However, the Algebra Standard lists the following expectation: “Students should understand relations and functions and select, convert flexibly among, and use various representations for them; … [and] understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.” (*PSSM* 2000, p. 395)

Recognizing equivalent forms of expressions and being able to convert flexibly among them means that a student should be able to write a polynomial in factored form. That is, a student should understand that x^{2} + 7x + 10 = (x + 2)(x + 5). Further, students should recognize that both expressions represent a quadratic function that crosses the x-intercept at (2, 0) and (-5, 0).

According to Workshop 5 video teacher Tom Reardon, factoring trinomials gives his algebra students their first exposure to quadratic functions and lays the foundation for in-depth study of quadratics later in the year. Students should be able to factor expressions by graphing the equivalent function and understanding the relationship between the x-intercepts and the factors. The strength of the graphical technique is that it is generalized to all polynomials and not just quadratic expressions. Computer Algebra Systems (CAS) can also assist in factoring complicated expressions. (CAS manipulate a formula symbolically using the computer. Factoring, finding roots, and simplifying polynomials are some of the typical uses of CAS.) Students should be able to factor some expressions mentally, such as x^{2} + 2x + 1, while more complex expressions, such as 3.2x^{2} + 7x – ½, should be solved graphically or with CAS. They should recognize when one technique is more efficient or beneficial than the others.

Read what teacher Tom Reardon has to say about the role of factoring in the algebra curriculum:

**Transcript from Tom Reardon**

It’s a technique that’s going to be used to solve quadratic equations later, or maybe help work with rational expressions … I like to bring in the algebra tiles, so that [the students] can see there’s at least a relationship to another part of mathematics … I think it’s still a very important topic, and not because I’m a traditional teacher, but because when I want to solve x

^{2}– x = 6, I’d like my students to be able to do that fairly easily, and factoring is the best way to do it. If they bring in the quadratic formula to factor that, I think I’m doing a great disservice.

### Part 2: Explanation

A recursive function contains two important components:

- A
**recursion formula**that tells how any term of a sequence relates to the preceding terms (e.g., x_{n}= x_{n – 1}+ 2, or Next = Now + 2); and - An initial condition that gives the starting point

(e.g., x_{1}= 1 or Start = 1).

The initial condition is necessary to ensure a uniquely defined sequence. The example above gives the sequence of odd numbers 1, 3, 5, 7, … . However, if the initial condition was modified to x_{1} = 2 or Start = 2, the recursive function would give the sequence of even numbers 2, 4, 6, 8, … .

Unlike a recursive formula, an explicit formula stands alone; that is, it has no additional qualifiers. The explicit formula y = x^{2} can be used to determine the x^{th} square number when you know the value of y. In contrast, the recursive formula for the square numbers is a_{n} = a_{n – 1} + 2n – 1 where a_{1} = 1. To find the n^{th} square number, you first need to find the previous n – 1 square numbers.

Other examples:

- The linear function y = 60x + 230, which describes the line with y-intercept 230 and slope 60, can be used to represent the distance from home (y) of a traveler who began the day 230 miles from home and drives at 60 miles per hour for x hours. Defined recursively, this relationship would be Next = Now + 60 where Start = 230.
- The balance of a bank account that earns 3% a year can be defined recursively as Next = Now + 0.03 * Now or as Next = 1.03 * Now. If the beginning balance in the account was $248, then Start= 248.
- In the game of Monopoly®, players may need to roll doubles to get out of jail. When rolling two dice, the probability of getting doubles is 1/6. The recursive formula Next = Now * 5/6, Start = 1/6 describes the probability of getting doubles for the first time on exactly the nth roll (meaning that doubles were not obtained on any previous roll.)
- The Fibonacci sequence is most often defined by the recursive formula a
_{n + 2}= a_{n + 1}+ a_{n}where a_{1}= 1 and a_{2}= 1; that is, each term is equal to the sum of the previous two terms, and the sequence begins 1, 1, 2, 3, 5, 8, 13, 21, … However, it would not be possible to write this equation using the Now-Next form because you have to reference the previous two terms instead of only the previous term.

### Part 2: Mathematical Definition

A **recursion** (or a **recursive function**) is an expression, such as a polynomial, each term of which is determined by application of a formula to preceding terms. The formula must include a start or seed value.

### Part 2: Role in the Curriculum

Informally, recursive functions are sometimes called Now-Next equations because they describe the next term in relation to the current, or now, term. In addition to laying the foundation for a more formal study of recursion, Now-Next equations provide an alternate form of representation and help students learn to develop an explicit equation.

Read what teacher educator Carol Malloy has to say about the role of recursion in developing mathematical understanding:

**Transcript from Carol Malloy**

This lesson is a precursor to students working with sequences and series. If the student did forget the formula for finding a term in a series, they would be able to come up with it themselves if they truly understand what was going on in this lesson. And for that reason, it’s an outstanding addition to their knowledge about sequences of numbers.

The National Council of Teachers of Mathematics (NCTM) suggests in *Principles and Standards of School Mathematics (PSSM) *, “Another type of representation that teachers might wish to introduce their students to is a NOW-NEXT equation, which can be used to define relationships among variables iteratively. The equation NEXT=NOW+10 would mean that each term in a pattern is found by adding 10 to the previous term.” (NCTM, *PSSM*, p. 284).

With many sequences, it may be easy to identify how terms change from one to the next. Consequently, students will sometimes have less difficulty writing a Now-Next equation than writing an explicit formula. As an example, consider the table of values below. It may be easier for students to write a Now-Next equation than to write an explicit equation for the table below.

Linear Equation: y = 8x + 5

Writing a Now-Next equation may also help students to develop explicit formulas. Both equations contain an 8, and students should see in both cases that 8 represents the amount of change between terms. Both equations also contain a 5, which is the y-intercept and the initial condition of the recursive equation.

Educator Sarah Wallick realizes that students may have a preference for either recursive or explicit equations, which is why she believes in teaching both. Sarah explains, “When kids are just getting started, they are coming at it from lots of different places. Some kids, when they look at a table of values, immediately go into a recursive mode. They will look at the relationship from one y value to the next y value in the table.”

For these kids, learning about recursive formulas and Now-Next equations is not only necessary to ensure they understand various representations, it may be the only way they can understand and interpret the mathematics of the situation.

Sarah continues, “Other kids look at the relationship and immediately connect whatever the x is to the y. They are working in the explicit mode.”

For these students, seeing Now-Next equations introduces them to an alternate representation, and it pushes them to think about the relationship in a slightly different way.

NCTM advocates that students understand the relationship between recursive and explicit formulas. In discussing a population growth problem, *PSSM* states, “Some students might generate an iterative or recursive definition for the function, using the population of a given year (NOW) to determine the population of the next year (NEXT):

Moreover, students should be able to recognize that this situation can be represented explicitly by the exponential function f(n)=6(1.02)^{n}, where f(n) is the population in billions and n is the number of years since 1999.” (NCTM, *PSSM*, 2000, p. 298).

Working with recursion formulas in algebra lays the foundation for the later study of sequences and series. Conceptually, understanding sequences and sequence notation is sometimes difficult for students. An adequate introduction to recursive formulas will help students make the transition seamlessly. The study of sequences and series in later math courses involves determining the initial condition and transitioning from recursive to explicit formulas and vice versa. It also involves replacing Now-Next equations such as

Now = Next + 8, Start = 5, with the formal notation a_{n} = a_{n – 1} + 8, a_{0} = 5.

Students may have difficulty identifying the start value. In the Workshop 5 video, some students chose the first term as the start value; others chose the y-intercept (or zero term). As Carol Malloy says, either is acceptable at the Algebra 1 level as long as the students can justify their choice.

When selecting situations for students to model, NCTM suggests that teachers “include examples in which models can be expressed in iterative, or recursive, form.” (NCTM, *PSSM*, p. 302). One of the examples that NCTM recommends involves the elimination of medicine from the body, based on a problem originally developed by the National Research Council. (For a discussion of this problem, see page 302 of *PSSM*, available online at

http://standards.nctm.org/document/chapter7/alg.htm.

This problem is available as an E-Example at

http://standards.nctm.org/document/eexamples/chap7/7.2/index.htm,

and the entire collection of NCTM E-Examples is available at

http://standards.nctm.org/document/eexamples/.)

Sarah Wallick believes in teaching recursion because of its relevance to real life.

**Transcript from Sarah Wallick**

It’s a big topic in technology today. It’s used extensively in computer applications [particularly spreadsheets]. It’s used in accounting. It’s used in biology – you keep looking at what’s happening from moment to moment, and that’s recursion. And the nice thing is, we have the technology to support recursion now. Before the advent of computers and calculators, a recursive algorithm wasn’t very useful, because it was tedious. You have to go through every iteration to get where you want to go. An explicit equation will take you there immediately. Well, sometimes you need all those iterations. Sometimes you might want those intermediate values, so a recursive algorithm is the right choice.

### Resources

The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.

**Patterns:**

Devlin, Keith. *Mathematics: The Science of Patterns*. Scientific American Library, 1994.

Julius, Edward H. *Rapid Math Tricks and Tips: 30 Days to Number Power*. John Wiley and Sons, 1992.

Seymour, Dale and Margaret Shedd. *Finite Differences: A Pattern-Discovery Approach to Problem-Solving.* Dale Seymour Publications, 1997.

**Rule of Four:**

Hughes-Hallett, Deborah, Andrew M. Gleason, et al. *Calculus, Single and Multivariable* (3rd edition). Wiley Publishers, 2001.

Gardner, Howard. *Frames of Mind: The Theory of Multiple Intelligences* (10th edition). Basic Books, 1993.

Phillips, Tony. “Multiple Mathematical Intelligences.” *What’s New in Mathematics*, June 1, 1999. *What’s New in Mathematics* is available online at http://www.math.sunysb.edu/~tony/whatsnew/column/mi-0699/mi1.html.

*Focusing on Learners.* http://www.epsilonlearning.com/focus.htm. Epsilon Learning Systems, 2003.

**Algebra Tiles:**

There are many resources available online regarding the use of algebra tiles in the classroom. Use a search engine to find them. In addition, there are a number of Web sites with applets involving algebra tiles. Some of the best applets can be found at:

National Library of Virtual Manipulatives:

http://matti.usu.edu/nlvm/nav/

Tom Reardon’s Algebra Tiles for SMART Board:

Algebra tiles (created by Tom Reardon) can be downloaded as AlgebraTiles.nbk. You’ll need a SMART Board to use this file. The SMART Board is a commercial product offered by SMART Technologies, Inc.