Home Mathematics Modeling mathematical ideas: developing strategic competence in elementary and middle school

# TEACHING STRATEGIES: PROMOTING THE ALGEBRAIC HABITS Of MIND

One way to develop algebraic reasoning and problem solving is to become pattern seekers for mathematical patterns and relationships, describe them using multiple representations, and analyze change. Literature connections are great ways to introduce the idea of becoming a pattern seeker. An investigation with the book, Two of Everything by Lily Toy Hong builds on opportunities from previous grades in which students learned about repeating and growing patterns through simple numeric and geometric patterns, repetitive songs, chants, and predictive poems.

Two of Everything by Lily Toy Hong (1993) is a retelling of a Chinese folktale. This picture book introduces the mathematical concepts of doubling and functions. In the story, old farmer Haktak and his wife dig up a large brass pot in their field and discover it has magic doubling powers. However, one day when Mrs. Haktak accidentally falls into the pot, they learn that not everything in life should be doubled. This is a great story to set the stage for exploring number patterns, relationships, and functions. The concept of doubling is critical to teaching addition strategies and is also a prerequisite to understanding multiplication facts for two. A teacher from one of our professional development workshops used this book in her classroom and reflects on her observations.

After reading the book, students explored doubling using a “magic pot” as a conceptual support. The class summarized the story by reenacting the doubling events with a coin purse, a hairpin, and Mr. and Mrs. Haktak. As they retold the story, the teacher had students keep a record of the items that went into the magic pot. By creating a table showing both input and output, the students found a pattern and generated a rule. At the end of the lesson, the class brainstormed what they might put in the magic pot besides the items from the story. The second day, the teacher told the students that the magic pot was doing something different and they had to figure out what was going on. The teacher asked them to write a number on a card and drop the card into the magic pot. For example, when a student dropped in a card with the number 5 written on it, the teacher took the card from the back opening of the magic pot and wrote the number 11 on it.

The class recorded the numbers on another input-output table. With this list of generated numbers, the students looked for the pattern and the relationship between input and output to determine the function. As the list grew, students made and tested their conjectures. If a few students thought they knew the rule before the rest of the class did, they thought of an input number and silently predicted the output number; they could determine whether they knew the function rule without denying other students an opportunity to discover the numeric pattern. The pot was doubling but adding one more. Depending on the students’ level of understanding, you can differentiate the challenge by creating a function rule appropriate for the class. Simple addition and subtraction rules are a good place to start, but more complex rules can be created. Starting the school year with this activity encourages students to constantly search for patterns and relationships as they investigate other concepts throughout the year.

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