Learning Math: Patterns, Functions, and Algebra
Patterns in Context Part A: What Do You See? (40 minutes)
Session 2, Part A
In This Part
- Finding a Pattern
- Descriptions of a Pattern
Finding a Pattern
Patterns are everywhere around us. We use patterns to organize what we see and hear and to make sense of data whether we are driving in a car, listening to music, or solving mathematical problems.
Finding, describing, explaining, and using patterns to make predictions are among the most important skills in mathematics. These skills allow users of mathematics to impose order, meaning, and understanding on situations that at first seem like collections of random facts.
Finding patterns is a subjective activity. Different people notice different things, so what one person sees is often different from what another perceives. That’s why it’s so important to describe patterns in language that everyone understands — so others can see what you see. Algebra is a tool for describing patterns, and there are many others.
It’s important to keep in mind, however, that algebra is much more than a language. As you discovered in Session 1, algebra is also a way to reason about things. In fact, “making sense” is what doing mathematics is all about.
Describe several different patterns that you see in this table:
Tip: A description can be an equation, a sentence, or a rule for continuing the pattern.
In this segment, participants describe and compare different patterns they found in the table for Problem A1. Watch the segment after you have completed Problem A1 and compare the patterns you identified with those of the onscreen participants. If you get stuck finding patterns in the table, you can watch the video segment to help you.
If you wanted to extend the table of Problem A1, which of these descriptions would be most effective? Would they all produce the same table?
You can find this segment on the session video, approximately 4 minutes and 46 seconds after the Annenberg Media logo.
What is the 100th entry in the table? How do you know?
Tip: Try to answer this question by prediction rather than by making a long table of values.
Would 102 ever be in the “output” column of this table? What about 1004? Why or why not?
Tip: Try to answer this question by prediction rather than by making a long table of values.
Descriptions of a Pattern
Here is a recipe to perform on a number:
What does this recipe do to the numbers from 1 to 10? Record your answers in a table. Can you explain how this is related to the patterns you saw in the table from Problem A1 (below)?
Comment on each of the following descriptions of the table given in Problem A1. Do all of these descriptions produce the same list of outputs? Are all of these descriptions valid for the table?
- As the input increases by 1, the output increases by 4.
- If you add 2 to 1 and double it, you get 6. If you add 3 to 2 and double it, you get 10. If you add 4 to 3 and double it, you get 14. Or, if you add the input to the next input, double that, you get the output.
- The units digits are in the sequence 6, 0, 4, 8, 2, so the next number would be 26, then 30, 34, 38, and 42, and then 46, 50, 54, 58, 62, etc.
- To get the output, multiply the input by 4 and add 2.
- To get the output, triple the input, then add 2 more than the input.
- After 6 as an input, the output numbers repeat over again: 6, 10, 14, 18, 22, 26, etc.
- After 6, the output numbers remain constant: 26, 26, 26, etc.
Which number comes next in this sequence: 1, 2, 3, … ? Find as many different answers and explanations as you can.
Let’s take a closer look at the four skills for thinking about patterns.
- Finding patterns involves looking for regular features of a situation that repeats.
- Describing patterns involves communicating this regularity in words or in a mathematically concise way that other people can understand.
- Explaining patterns involves thinking about why the pattern continues forever, even down the line in cases you haven’t looked at.
- Predicting with patterns involves using your description to predict pieces of the situation that aren’t given.
Look back at Problems A1-A6. Describe how you use these skills in each problem. In other words:
- Finding is observing the pattern you see
- Describing is putting what you see into words or symbols
- Explaining is figuring out why the pattern continues
- Predicting is using your description or rule for a new value
The content of this session centers on patterns. Go over the processes of finding, describing, explaining, and predicting with patterns. These skills help us to understand and make sense of situations, something to remember while working on problems throughout the session.
Groups: Work in pairs on Problems A1-A4. When you’re finished, share your answers to Problem A1, coming up with as many different ways of describing the patterns as possible. At this point, we’re not necessarily interested in descriptions given in symbolic form. Descriptions such as “each output is 4 more than the previous” are valid because there are many ways to describe patterns, each with its own merit. At some point we will want to make a case for using a variable as a way of being concise, but that will come later.
Read Problem A5.
Groups: Talk about each part of the question as a full group. Some people may be troubled by parts (c), (f), and (g), rejecting these scenarios as being far-fetched. An important point we are making in this session, though, is that many different rules can describe a table of data. When the data is attached to a context or situation, we can be more certain of an associated unique rule. Therefore it is important to see parts (c), (f), and (g) as possible descriptions.
Some possible answers: each output number is 4 more than the last; the output numbers that appear are all the even numbers that aren’t multiples of 4 (starting with 6); the output number is 2 more than 4 times the input number. Also, adding one input to the following input yields half the first output.
You can’t be sure, because the pattern is not completely specified, but it would be likely that the 100th number is 402. This follows the third rule listed above — that the output number is 2 more than 4 times the input number.
Again, you can’t be completely sure, but it would be likely that the 25th number is 102, because 2 more than 4 times 25 is 102. Following the same pattern, 1004 would not appear in the output column, since 1004 is not 2 more than 4 times any whole number.
Example: Pick 7, and then follow the algorithm. 7 >> 21 >> 19 >> 38 >> 44 >> 30 is the output. The numbers at the end are the same as the pattern described in the table. Here’s why: Pick n instead, which stands for a variable number. Follow the algorithm. n >> 3n >> 3n – 2 >> 6n – 4 >> 6n + 2 >> 4n + 2 is the output, which is the rule described in Problem A1.
- This describes the way the table is built, but doesn’t say what values the table begins with.
- If the first input is n, the sum of the first input and the next is n + (n + 1) = 2n + 1. Doubling this gives 4n + 2, which is the formula for the table.
- This is a good digit-based description of the table.
- It’s 4n + 2 again. An efficient, closed-form description.
- Triple the input is 3n, then 2 more than the input is n + 2, so the sum is 4n + 2.
- There’s no way of being sure that 4n + 2 is the correct pattern. It is definitely not the only pattern that starts 6, 10, 14, 18, 22, 26, …
- This is another example of a different continuation to the pattern. Because the given table ends with 26, this is a valid continuation.
The next number could be any number with a justified pattern. 4 is the clear choice, but so is 5, the next Fibonacci number, or 10, the next number you would get when counting in base 4, or 1, the next beat in a waltz. A formula could be found for any 4th number in that sequence.
Session 1 Algebraic Thinking
In this initial session, we will explore algebraic thinking first by developing a definition of what it means to think algebraically, then by using algebraic thinking skills to make sense of different situations.
Session 2 Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of what algebra is.
Session 3 Functions and Algorithms
In Session 1, we looked at patterns in pictures, charts, and graphs to determine how different quantities are related. In Session 2, we used patterns and variables to describe relationships in tables and in situations like toothpicks and triangles. This session extends the exploration of relationships to include the concepts of algorithm and function. Note1
Session 4 Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help you to differentiate between relative and absolute meanings of "more" and to compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality, and interpreting graphs that represent proportional relationships or direct variation.
Session 5 Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.
Session 6 Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.
Session 7 Nonlinear Functions
Continue exploring functions and relationships with two types of non-linear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions and explore exponential and quadratic functions in real-life situations.
Session 8 More Nonlinear Functions
Investigate more non-linear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts where inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.
Session 9 Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.
Session 10 Classroom Case Studies, Grades K-2
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the K-2 grade band.
Session 11 Classroom Case Studies, Grades 3-5
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 3-5 grade band.
Session 12 Classroom Case Studies, Grades 6-8
Explore how the concepts developed in Patterns, Functions, and Algebra can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This video is for the 6-8 grade band.