Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 10, Grades 3-5, 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 Grade 3-5, understanding patterns includes the following expectations:

 • Describe, extend, and make generalizations about geometric and numeric patterns • Represent and analyze patterns and functions, using words, tables, and graphs • Represent the idea of a variable as an unknown quantity using a letter or a symbol

In this part, we'll look at problems that foster algebraic thinking as it relates to these standards, and we'll explore ways of asking questions that elicit algebraic thinking. The situations we will 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 3-5 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 begin by having them draw a table that shows the pool "number" and the relative number of blue and white tiles. Students may also want to begin by looking at the fraction of blue or white tiles out of the whole number of tiles.

 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

Here is a set of problems on patterns:

 37, 40, 43, _____, _____, _____ 27, 25, _____, 21, _____, _____ _____, 11, 15, _____, 23, _____ _____, _____, 36, 33, _____, 27

 Problem B7 What questions could you ask to develop students' skills in describing these patterns?

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

 The swimming pool problem is 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). 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 problems taken from Everyday Mathematics, Student Journal, Volume 1, Grade 3, developed by the University of Chicago Math Project (New York: SRA/McGraw-Hill, 2001). McGraw-Hill makes no representations or warranties as to the accuracy of any information contained in the McGraw-Hill Material, including any warranties of merchantability or fitness for a particular purpose. In no event shall McGraw-Hill have any liability to any party for special, incidental, tort, or consequential damages arising out of or in connection with the McGraw-Hill Material, even if McGraw-Hill has been advised of the possibility of such damages.
 Session 10, Grades 3-5: Index | Notes | Solutions | Video