Learning Math: Patterns, Functions, and Algebra
Classroom Case Studies, Grades K-2 Part D: Critiquing Student Lessons (20 minutes)
Session 10: K-2, Part D
In the K-2 curriculum, students are frequently asked to think about patterns, but often their “pattern sniffing” skills end with simply finding the next object. Note 7
Here is a “pattern” problem.
Subtract, then look at the differences in each row.
Continue the pattern.
What algebraic ideas are in this lesson?
How are patterns used in this lesson?
What mathematics do you think students would learn from this lesson?
Are there any misconceptions that students might develop from this lesson?
How would you modify the problem, or what additional questions might you ask, to incorporate the framework for analyzing patterns?
Problem taken from Addison Wesley Mathematics, Grade 2 (Menlo Park, Calif.: Scott Foresman-Addison Wesley, 1993).
You may be puzzled by this series of subtraction problems, as well you should! After working on Problems D1-D5, you may notice that in certain problems — where students “see” the pattern and then mindlessly fill in numbers that satisfy the pattern — the mathematical thinking and reasoning gets lost.
Answers will vary.
Answers will vary. The patterns shown only relate to the answers. For the first row, the answers increase by 10, but the problems themselves are not directly related. The second row’s answers are consecutive multiples of 11, but the problems are not related. The third row’s answers decrease by one with unrelated problems, and the fourth row’s answers decrease by five with unrelated problems.
Answers will vary. Students should strengthen their subtraction with regrouping skills.
Answers will vary. Students could learn to look for patterns in answers without actually looking at the problems. The answers may not have any pattern.
Answers will vary. One way to modify the problem would be to make the problems relate to the pattern of the answers. For example, the first row could show 63 – 55, 63 – 45, 63 – 35, 63 – 25, 63 – 15, then ask what comes next.
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.