Insights Into Algebra 1: Teaching for Learning
Properties Teaching Strategies: Patterns
Educator Marilyn Burns describes algebra as “the generalization of arithmetic.” The same definition is given in The Mathematics Dictionary (4th edition) by James & James. The University of Cambridge Maths Thesaurus defines algebra as “The noticing of patterns and describing them in words and pictures … It is the branch of elementary mathematics that generalizes arithmetic by using variables…”
Hear what Diane Briars has to say about the use of patterns when factoring trinomials:
|Listen to audio clip of teacher educator
Transcript from Diane Briars
Looking for patterns is really the heart of mathematics. In fact, some people have defined mathematics as the study of patterns. By having students look for patterns in this lesson, Tom was really giving them ownership of discovering the procedures. It’s really significant when students can discover for themselves that the constant terms in the binomials are factors of the constant term in the quadratic. When they discover that for themselves, then they own it; it’s not just a rule they are being told. They own it, they’ll remember it, and they are more likely to be able to use it effectively.
With the themes of patterns and generalization appearing in most definitions of algebra, it is not surprising that NCTM lists an understanding of patterns in the Algebra Standard. The Principles and Standards for School Mathematics (PSSM) states:
(NCTM, PSSM, p. 222 and p. 296.)
Patterns have at least three roles in the mathematics classroom. First, generating and using patterns typically requires students to practice basic skills such as addition, subtraction, multiplication, and division. Second, students can identify and generalize patterns to make predictions and solve problems. Third, and most importantly, patterns stimulate learning: By noticing a pattern regarding the y-intercepts of linear equations or regarding the change from term to term in a sequence, students discover mathematical truths, and they glimpse the interconnectedness of mathematics.
Basic Skills Practice
The Activities Integrating Math & Science Educational Foundation (AIMS) states: “Pattern-seeking activities provide more basic skills practice than we would ever dare to impose in a pure drill approach and does so without numbing student interest.”
A pattern which has come to be known as the “hailstone problem” produces a sequence using the following rules:
- Choose any positive integer to begin the sequence. Then,
- If the integer is odd, multiply by 3 and add 1 to find the next number. If the integer is even, divide by 2.
- Return to Step 1 and plug in the result. Continue until at least a portion of the sequence begins to repeat.
Stated as a recursive function, the hailstone problem would be described as follows:
- Begin a sequence with a1, where a1 is any positive integer.
- If an is odd, then an + 1 = 3an + 1.
- If an is even, then an + 1 = an / 2.
For instance, if the sequence begins with a1 = 17, then the sequence formed by these rules is 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, … .
Students often enjoy investigating this pattern, especially when they learn that no one has ever found a number that won’t eventually repeat the cycle 4, 2, 1, 4, 2, 1, … . In the course of exploring numbers for this pattern, students have numerous opportunities to practice multiplication and division. Engaging students’ interest yields more substantive benefits than having them complete traditional worksheets.
Students can investigate other patterns, such as the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, …, where each term is the sum of the previous two terms. By generating terms of the sequence, students practice their mental addition skills. By investigating the ratio of consecutive terms, students practice their division skills.
A decreasing sequence in which students could practice subtraction skills might involve a runner’s time during training runs; if she starts at a pace of 9 minutes, 12 seconds per mile and her time decreases by 14 seconds per mile during each week of training, how fast will she be running in 10 weeks? Will she ever get to the point where she breaks the three-minute mile barrier? Or, to include subtraction with decimals, the scenario might involve an overweight man who is put on a diet by his doctor. On the diet, he will lose an average of 1.7 pounds per week. If he starts at 211 pounds, track his progress for 6 months. How long will he have to stay on this diet to get his weight below 180 pounds? Would you expect him to get down to 80 pounds?
Students often enjoy working with and discovering patterns. Think of a pattern or series of patterns that contains an embedded review of a skill that you would like your students to master, and describe ways they could practice that skill by investigating the pattern.
Problem Solving Strategy
In his book Mathematics: The Science of Patterns, mathematician Keith Devlin argues that mathematics is the study of patterns – whether the patterns occur in nature or in our minds, whether they are numerical or geometric. It’s no surprise, then, that NCTM suggests all students should be able to “represent, analyze, and generalize a variety of patterns.” (NCTM, PSSM, 2000)
In an informal way, students use patterns to solve problems all the time. A young student may know that one piece of candy costs 5¢, two pieces cost 10¢, and three pieces cost 15¢. From that information, a student would be able to find the cost for any number of pieces of candy, since the pattern is that “each piece of candy adds 5¢ to the total cost.”
In a review of literature on mathematical problem solving, looking for a pattern was recognized as a strategy in every article and publication. This is no coincidence. Identifying a pattern may shed more light on a solution than any other tactic.
Consider the following problem:
GlobalComm charges 65¢ for a one minute phone call, 85¢ for a two minute call, and $1.05 for a three minute call. How much do you think GlobalComm charges for a 15 minute phone call?
Without an understanding of patterns, a student may incorrectly use a proportion to determine that the price of a 15 minute call is $5.25, found by taking 5 × $1.05, because a 15 minute call lasts five times as long as a three minute call. On the other hand, a student who has a well developed pattern-recognition ability may notice that the price increased by 20¢ each minute and use that pattern to correctly conclude that the price of a 15 minute call is $3.45.
As Joshua Zucker of the Castilleja School in Palo Alto says, “Sometimes, you have to get messy.” Looking for patterns by experimenting with numbers may be a fruitful search that leads to a solution. Without looking for a pattern, however, a student may never uncover any of the underlying mathematics. But finding a pattern is only the first step to mathematical enlightenment. A deeper understanding develops when the pattern is generalized algebraically. While applying rules and formulas may be useful, students attain a much deeper level of conceptual understanding when they develop rules and formulas on their own.
Choose three problems from an algebra textbook that could be solved by identifying a pattern, but are typically solved with a different method. If students used patterns to solve these same problems, would there be increased conceptual understanding? Why or why not?
An Instructional Tool for Discovering Mathematics
Patterns are pervasive in mathematics and, therefore, fundamental to the effective learning of mathematics. According to PSSM, using patterns in the classroom results in the following:
- Students make and test conjectures.
- Students apply both inductive and deductive reasoning.
- In groups or individually, students explore, conjecture, analyze, and apply mathematics to a variety of mathematical and real world problems.
Patterns often serve a mathematical end. Pointing out a pattern to students is one method of explaining a mathematical phenomenon. However, having students discover patterns on their own is often a more effective method of learning. As the psychologist Jean Piaget said, “To understand is to discover.” When you ask students to find a pattern, you arouse their curiosity. Their heightened level of interest promotes learning and retention. (See the “Teaching Strategies” section of Workshop 3 for more discussion of discovery learning.)
One of the first topics covered in algebra is integers – positive and negative numbers. Students often have difficulty with this concept, especially the multiplication and division of signed numbers. However, patterns can be illuminating when students first encounter this concept. Students should consider the pattern of products below.
3 x 2 = 6
3 x 1 = 3
3 x 0 = 0
3 x (-1) = __
3 x (-2) = __
3 x (-3) = __
The pattern of products is 9, 6, 3, 0, … . Students will likely realize that the numbers decrease by 3; consequently, they will continue the pattern by subtracting 3 each time and show that the pattern proceeds with -3, -6, and -9. A student who recognizes this pattern is more likely to understand why “the product of a positive and a negative is negative” than the student who simply hears the rule from a teacher.
NCTM suggests that students explore linear patterns, like the example above, in algebra: “The study of patterns and relationships … should focus on patterns that relate to linear functions, which arise when there is a constant rate of change. Students should solve problems in which they use tables, graphs, words, and symbolic expressions to represent and examine functions and patterns of change.”
Tom Reardon used the discovery of patterns to exemplify the connections between trinomials, factored form, and x and y-intercepts. By organizing data into the table below, he asked students to identify relationships between the various pieces.
Based on this table, there are two important observations that students ought to make about equations in the form of x2 + bx + c:
- The product of the constant terms in factored form is equal to the constant term of the trinomial. In the first row, for example, 2 x 4 = 8, and in the second row 1 x 6 = 6.
- The sum of the constant terms in factored form is equal to the coefficient of x in the middle term of the trinomial. In the first row, for example, 2 + 4 = 6, and in the second row 1 + 6 = 7.
These two discoveries constitute the rule that we would hope students learn about factoring a trinomial. However, in order to discover this relationship, students must spend a significant amount of time gathering data for the table, and then they must spend time searching for patterns within the table. Can the use of that much class time be justified, given that students only learn one piece of information, especially when the same information could be told to them in fewer than 30 seconds?
Hear what Diane Briars has to say about presenting patterns to students for discovery in the video for Workshop 5 Part I:
|Listen to audio clip of teacher educator
Transcript from Diane Briars
One of the most striking features of this lesson was Tom’s use of color on the SMART Board to help focus students’ attention. So when students were looking at the table and looking for patterns, they weren’t just seeing the sea of numbers. You really clearly could focus on the different columns and really isolate and look for the pattern and then look across and see … that the two numbers I have as my x-intercepts multiply together to give me the constant in the quadratic term. I think it really did help students make those connections, to focus their attention so they knew where to look.
In the video for Workshop 5 Part II, Sarah Wallick’s students considered finite differences. The method of finite differences is easier to explain with an example than with a definition, so take a look at the terms of a linear sequence:
Coincidentally, the difference between terms is 3, which corresponds to the coefficient of x in the function y = 3x – 2. More importantly, the degree of a linear function is 1, and the first differences are constant (3).
Consider the terms of a sequence defined by the quadratic function y = 2x2:
For this sequence, the second differences are constant, and the sequence of terms is described by a polynomial of degree 2 – that is, a quadratic equation.
As a last example, consider the terms of a cubic sequence defined by the function y = x3 + 2x:
In this case, the third differences of the terms of a polynomial of degree 3 are constant.
These three observations – that the first differences of a polynomial of degree 1 are constant, that the second differences of a polynomial of degree 2 are constant, and that the third differences of a polynomial of degree 3 are constant – lead to the following generalization:
If the nth differences of a sequence are constant, the original pattern can be described by a polynomial equation of degree n.
This important discovery is useful for generating explicit formulas when the terms of a sequence are known. Take a look at the following sequence:
Because the second differences are constant, this sequence of terms can be described by a polynomial of degree 2 – that is, by a quadratic function of the form y = ax2 + bx + c. Knowing that, we will be able to generate a general formula for the sequence.
The sequence of terms generated by a quadratic function can be expressed generally as:
a(1)2 + b(1) + c, a(2)2 + b(2) + c, a(3)2 + b(3) + c, a(4)2 + b(4) + c
The first term is equal to the function when x = 1, the second term occurs for x = 2, and so on. When simplified, the general sequence becomes:
a + b + c, 4a + 2b + c, 9a + 3b + c, 16a + 4b + c
This may seem like pointless rambling, but now consider what happens when the differences of this sequence are found:
Although expressed with variables, the second differences are still constant! But more importantly, notice that the constant value of the second difference is 2a when the general form is used here; but above, for the sequence 5, 8, 13, 20, 29, …, the constant value of the second differences was 2. Setting these terms equal yields:
a = 1
Further, the first term of the first differences in the general sequence is 3a + b,and above, the value of the first term of the first difference was 3; because a = 1, this yields:
3(1) + b = 3
b = 0
Finally, the first term of the general sequence is a + b + c, and the first term of the sequence above was 5; because a = 1 and b = 0, this yields:
1 + 0 + c = 5
c = 4
Putting this all together, a = 1, b = 0, and c = 4. When these values replace a, b, and c in the general function y = ax2 + bx + c, the resulting function describes this sequence:
y = 1x2 + 0x + 4
y = x2 + 4
The power of the method of finite differences is that it can be used to find the explicit formula for any sequence that can be defined by a polynomial function.
Some sequences, however, will never have constant differences. For instance, the powers of 2 continually yield the same pattern of differences:
The same is true of the Fibonacci sequence, which has differences that are similar but not quite the same:
There is a corollary to the above theorem that is also important:
If none of the differences of a sequence are constant, the original pattern cannot be described by a polynomial equation.
The powers of 2 can be described by the exponential function y = 2x but cannot be described by a polynomial function. Likewise, the Fibonacci sequence can be described by the recursive function an = an – 1 + an – 2, a1 = 1, but it cannot be described by a polynomial function.
It is important to realize that many functions can be described by recursive formulas, but they cannot be described by explicit formulas.
The sequence that arose in Sarah Wallick’s class when students were investigating the get out of jail simulation cannot be described by a polynomial function. Recall that the sequence was 1/6, 5/36, 25/216, … .This sequence can be described by the exponential function 5n-1/6n, and it can also be described by the recursive function = 1/6.
Because many sequences cannot be described by a polynomial function but can be described by exponential, recursive, and other types of functions, the ability to recognize patterns is extremely important.
Hear what Carol Malloy has to say about the use of finite differences:
Transcript from Carol Malloy
If you are trying to find the general rule for a series of numbers, you can use finite differences to find a polynomial that will represent that particular relationship in the series of numbers. Sarah [Wallick] first asked the students to look at the finite differences for a function or a relationship that ends up being recursive, and they see that they can’t find a constant difference … She wants them to understand that in some cases there are finite differences that are constant, and in other cases there are not. And she goes through the linear function or the linear polynomial, the polynomial of degree 1, and then she does degree 2, degree 3, and that’s when she comes back to the recursive relationship with the probability and they realize that they have a different kind of a general term. They realize it’s no longer a simple polynomial that they are using to represent this particular series, but it is something that is different – in this case, an exponential function.
In their chapter “Strategy Discovery and Strategy Generalization” in How Children Discover New Strategies, authors Robert Siegler and Eric Jenkins state, “Strategy discovery involves a sudden burst of understanding and is accompanied by a conscious ‘aha’ (or ‘Eureka’) experience; the discoverer not only uses the strategy for the first time but immediately understands why it works and what types of problems it can solve.” (Source: How Children Discover New Strategies. Hillsdale, NJ: Lawrence Erlbaum Associates, 1989; p. 15.)
Students understand patterns and relationships that they discover themselves at a much deeper cognitive level than they do facts given to them. Allowing students to discover patterns may require more time than simply explaining the pattern, but the result is deeper understanding and greater retention.
In addition, the identification of patterns makes it easier to represent a situation algebraically. Students who notice a pattern regarding the probability of obtaining doubles on the nth roll of two dice, for instance, will have an easier time expressing the relationship recursively or explicitly. The table below shows the probability of obtaining doubles on the first, second, or subsequent rolls of two dice.
From this table, students will likely recognize that the probability changes by a factor of 5/6 from roll to roll, leading to the recursive relationship:
(1/6 is the value of the first toss)
They will also recognize the explicit relationship:
Choose a pattern – either one that you saw in this session, or one that you know from another source – that could be used to enhance the instruction of a particular topic. Explain how the discovery of the pattern would lead to increased student learning, and describe how you might include this pattern in a lesson.
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.