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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
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