Session 10, Grades 6-8, Part B:
An Example for Developing Algebraic Thinking (20 minutes)

The National Council of Teachers of Mathematics' Principles and Standards for School Mathematics (2000) identifies algebra as a strand for grades Pre-K-12. The Standards identify the following concepts that all students should cover and comprehend:
Note 4

 • Understand patterns, relationships, and functions • Represent and analyze mathematical situations and structures using algebraic symbols • Use mathematical models to represent and understand quantitative relationships • Analyze change in various contexts

For the classroom in grades 6-8, understanding patterns includes the following expectations:

 • Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules • Relate and compare different forms of representation for relationships • Model and solve contextualized problems using various representations, such as graphs, tables, and equations

In this part, we'll look at problems that foster algebraic thinking as it relates to these standards, and explore ways of asking questions that elicit algebraic thinking. The situations we'll be exploring are representative of the kinds of problems you would find in some existing texts; in fact, you may recognize some of them! The goal is for you to examine these problems with the critical eye of someone who has taken this course and is beginning to view algebraic thinking with a different perspective.

Consider the situation below, appropriate for exploration in a grade 6-8 classroom:

Tat Ming is designing square swimming pools. Each pool has a square center that is the area of the water. Tat Ming uses blue tiles to represent the water. Around each pool there is a border of white tiles. Here are pictures of the three smallest square pools that he can design, with blue tiles for the interior and white tiles for the border. Note 4

Problem B1

 Problem B2 How do your questions reflect the algebra content in the situation?

 Now focus on the questions you want the students in your classroom to consider. You may want to consider new ways to represent the relationships between the number of tiles of each color and the number of the square pools, and then use those representations to predict what will happen when the pools are very large.

 Problem B3 What patterns, conjectures, and questions will your students find as they work with this situation?

 Problem B4 What questions could you as the teacher pose to elicit and extend student thinking at your grade level?

 Problem B5 Recall the framework you explored in Session 2 in looking at patterns: finding, describing, explaining, and using patterns to predict. Which of these skills will your students use in approaching this problem?

 Problem B6 Read the article "Experiences with Patterning" from Teaching Children Mathematics. What ideas mentioned seem appropriate for your classroom? Note 5

 Problem B7 In this sequence there are 4 toothpicks in Term 1, 7 toothpicks in Term 2, and 10 toothpicks in Term 3. How many toothpicks are in Term 4? If you continued the pattern, how many toothpicks would you need to make Term 5? Term 6? Term 10? What questions could you ask to develop students' skills in describing this pattern?

 Problem B8 What questions could you ask to develop students' skills in predicting?

 The swimming pool problem adapted from Algebra in the K-12 Curriculum: Dilemmas and Possibilities, Final Report to the Board of Directors, by the NCTM Algebra Working Group (East Lansing, Mich.: Michigan State University, 1995). Pool problem and analysis of algebraic thinking discussed in "Experiences with Patterning," by Joan Ferrini-Mundy, Glenda Lappan, and Elizabeth Phillips, in Teaching Children Mathematics (February 1997), p. 282-288. Pattern problem taken from IMPACT Mathematics Course 1, developed by Education Development Center, Inc. (New York: Glencoe/McGraw-Hill, 2000), p.32. www.glencoe.com/sec/math
 Session 10, Grades 6-8: Index | Notes | Solutions | Video