Private: Learning Math: Patterns, Functions, and Algebra
Patterns in Context Part B: Patterns in Situations (45 minutes)
Session 2, Part B
In This Part
- Toothpick Triangles
- More Toothpick Triangles
Problem A5 describes several different ways to extend the pattern given in Problem A1. When a sequence of numbers is not tied to a context, there is no way to determine which extension of a pattern is “correct.” Note 5
Describing patterns is most useful when the regularity is embedded in some well-understood situation. For example, examine the toothpick pattern given here:
Describe the pattern in as many ways as you can.
Imagine you keep adding triangles. Complete the following table:
If someone tells you the number of triangles, explain how to tell them the number of toothpicks they would need. How do you know your description will always give the right number of toothpicks?
Use your explanation to complete this table:
In this segment, participants describe and explain different patterns they found in the toothpick triangles problem, and Professor Cossey discusses the importance of understanding where a pattern comes from. Watch the segment after you have completed the toothpick triangles problems, Problems B1-B4, and consider the different patterns that the onscreen participants identified.
Do Sue-Anne and Frederick’s rules extend the table in the same way? Is there only one correct way to extend the table in Problem B2?
You can find this segment on the session video, approximately 14 minutes and 10 seconds after the Annenberg Media logo.
More Toothpick Triangles
How does the table in Problem B4 compare to the table from Problem A1? Is there more than one way to extend the triangle and toothpick table? Explain.
Table from Problem B4
Table from Problem A1
Which of the descriptions below from Problem A5 are valid for the toothpick triangles? Why are the others invalid?
- 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 n
Explain your rule for calculating the number of triangles if you are given the number of toothpicks. Why does it work?
Will every number of toothpicks correspond to a number of triangles? What does this tell you about your rule?
Look at the toothpick pattern and complete Problems B1-B8.
Groups: Work in pairs. There are two common ways of thinking about the toothpick pattern, and one of the reasons it’s useful to work in pairs or small groups is to observe this variety.
Some people see the pattern as groups of 4 toothpicks, with 2 toothpicks added on to the right-most triangle, and they come up with the following rule: Take the number of triangles, multiply by 4, and add 2. Others see the pattern as groups of 6 toothpicks, with 2 toothpicks removed from all of the triangles except the right-most triangle. The rule they come up with is: Take the number of triangles, multiply by 6, and then subtract 2 times 1 less than the number of triangles.
When using a variable, the first rule turns out to be 4n + 2; the second becomes 6n – 2(n-1). The second rule may be harder to represent,”subtract 2 times 1 less than the number of triangles” can be tricky to express. Consider going through a derivation of this rule, starting with: If the number of triangles is 5, what’s 1 less? So we subtract 2 times that. If the number of triangles is 10, what’s 1 less? Then subtract 2 times that number. Finally, if the number of triangles is n, what’s 1 less? It is n – 1.
The second difficulty may be seeing the need for the parentheses when multiplying this quantity by 2. Explore the expressions 2(n – 1) and 2n – 1, testing some particular values for n to decide which accurately describes “2 times 1 less than n.”
Those who come up with a symbolic rule right away, without even looking at the drawing, should think about where the “4-ness” or the “6-ness” comes from in the picture. Make sure to make the link between what’s going on in the equation and what’s happening in the picture.
Groups: No matter how people come up with them, explore the two different rules as a full group. Do they continue the table in the same way? Are the rules equivalent, even though they look different? Or are they genuinely different?
Groups: Go over Problem B5 as a full group.
In this situation there is only one way to fill in the table because the data is linked to a situation. Whatever the number of triangles, we can visualize how many toothpicks are needed.
Be sure to go over Problem B7. Here, we are given the number of toothpicks and are asked to work backwards to find the number of triangles. Working backwards is a hallmark of algebraic thinking; we will revisit this idea in a later session.
Some possible descriptions:
- As the number of triangles increases by 1, the number of toothpicks increases by 4.
- To get the number of toothpicks, multiply the number of triangles by 4 and add 2.
- To get the number of toothpicks, triple the number of triangles, then add 2 more than the number of triangles.
- To get the number of toothpicks, take the number of triangles, multiply by 6, and then subtract 2 times 1 less than the number of triangles
Here’s the completed table:
Look familiar? It should — it’s the table from Problem A1.
The number of toothpicks is 4 times the number of triangles, plus 2. Look at what happens when each new shape is built. Specifically, imagine that you have the shape for 1 triangle already built, and you need to build the shape for 2 triangles. You would only need to add 4 toothpicks to the left of the existing shape. Sure enough, these 4 are what you’d need to build the 3rd shape on to the 2nd, and so on. So each new triangle means 4 new toothpicks. The “+ 2” comes from the lower right corner of the original triangle, which is not part of the shape added on each time.
You can be sure that the description always gives the correct number of toothpicks because there is a specific context to the problem. The method of creating new triangles never changes, so you can be sure that each triangle will have 4 more toothpicks than the last. Compare this to Problem A5; without a context, there is no guaranteed pattern.
One subtle change between this table and the table in Part A is that we can be sure the pattern in this table will continue as long as necessary.
Patterns like the ones listed in parts (f) and (g) in Problem A5 will not happen in this context. The others (parts (a) through (e)) are valid.
One way to calculate the number of triangles is to “undo” the way we turn triangles into toothpicks. To get toothpicks from triangles, we multiply by 4, then add 2. So to “undo” these operations, we have to subtract 2, then divide by 4. So the number of triangles can be determined by this process, or it can be determined by the rule n = (T – 2) / 4.
No. For example, there would never be a number of triangles for 1004 toothpicks. This means that the rule developed in Problem B7 only works for numbers that produce a positive whole number as the output.
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.