Learning Math: Patterns, Functions, and Algebra
Patterns in Context Part D: Counting Stairs (60 minutes)
Session 2, Part D
Here is a problem in which you can use patterns to make predictions.
Count the number of blocks in each of the following staircases. You can use cubes or blocks to construct the stairs. Then devise as many general methods as you can for predicting the number of blocks in any staircase. If you come up with a rule for predicting the number of blocks, explain why the rule works. If you used a variable in your rule, explain the meaning of the variable.
If someone tells you how many blocks there are in staircase n, describe how you could use that to find the number of blocks in staircase n + 1.
Tip:What is the difference between staircase n and staircase n + 1?
Suppose there are 37,401 blocks in the 273rd staircase. How many blocks are there in the 275th?
Tip: Use what you learned in Problem D2.
How many blocks are there in the 100th staircase?
Tip: As in Part A, try to do this problem by using prediction rather than just by extending the table.
Look at the following geometric solution for the third staircase. Imagine the rectangle made for the nth staircase. Write a rule to determine the number of blocks in the nth staircase.
Think about the different conceptions of algebra and the use of variables when working on these problems.
For those not using the Counting Stairs interactive activity, each individual working alone or group should have about 30 cubes to use.
Whether doing the Counting Stairs interactive activity or using blocks or cubes, come up with as many ways as possible for solving the problem. (There are at least 12 different solutions.) If you get stuck, think about building squares or rectangles from the stairs. In fact, the geometric solution to this problem is the most elegant: If you take any staircase with bottom row length n, form a second identical one, rotate it, and put it together with the first, you form a rectangle of dimensions n by n + 1. Therefore, the area of the original staircase is 1/2(n)(n+1).
Groups: Share answers to Problem D1. If anyone used variables, they should describe what conception of variable they were using.
Groups: It is important to talk about Problem D2 as a whole group, as this question underscores the importance and convenience of the recursive formula. If you know the number of stairs in the nth staircase, the number of stairs in the next staircase can be found by adding n + 1 onto the previous total.
Notice that the recursive formula can be applied only in situations where you know the previous term. Part of our job is to determine in which situations different kinds of representations are most useful. What makes a rule useful is how easy it is to apply — closed-form rules often win the battle here — and how easy it is to come up with. In the staircase problem, the recursive rule is much easier to come up with than the closed form.
Another example where the recursive rule is easier to find than the closed-form rule is the famous Fibonacci number sequence: 1, 1, 2, 3, 5, 8, 13, … .
The recursive rule for the sequence is elegant: Start with 1, 1. Then, to get any term in the sequence, add the previous two terms.
The closed-form rule is complicated:
It’s not clear that this function produces integer outputs, much less that the outputs are the terms of the sequence above. Imagine trying to come up with such a formula!
One method is that the number of blocks for the nth staircase is n more than the number of blocks for the (n – 1)st staircase, because you need to add on a row of n blocks to build the new staircase. This allows you to construct a table. One clever trick is to piece together 2 consecutive staircases; the 5th staircase fits nicely into the 6th staircase to form a 6-by-6 square!
There will be 1 new row in staircase n + 1, a row with n + 1 blocks in it. That means that to go from staircase n to staircase n + 1, you will need to add n + 1 blocks.
In the 274th staircase there will be 37,401 + 274 = 37,675 blocks. In the 275th staircase there will be 37,675 + 275 = 37,950 blocks. This answer can also be reached by using (275) * (276) / 2 = 37,950.
There are 5,050 blocks in the 100th staircase. The fastest way is to use n(n + 1) / 2.
A way to build a general formula is to notice that 2 copies of the 6th staircase will form a 6-by-7 rectangle. For the nth staircase this will be n(n + 1) blocks for 2 staircases, which means n(n + 1) / 2 for 1.
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.