Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of mathematical notation wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Ethnomathematics wikipedia , lookup
Mathematical model wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Moiré pattern wikipedia , lookup
Elementary mathematics wikipedia , lookup
Patterning and Algebra Suggested Reading (journal articles) posted on PJ Math Blog (Teaching Resources): o “Developing Algebra Eyes and Ears” o “Understanding the Equals sign” o “Integrating Math and Music” Before formal schooling, children develop beginning concepts related to patterns, functions, and algebra. They learn repetitive songs, rhythmic chants, and predictive poems / stories that are based on repeating and growing patterns. Their observations and discussions of how quantities relate to one another, lead to initial experiences with functional relationships (early number sense). Their representations of mathematical situations using concrete objects, pictures, and symbols are the beginnings of mathematical modeling. (NCTM Principles and Standards for School Mathematics, 2000, pg 91). Developmentally Appropriate Teaching Sequence Recognize → Copy → Extend - Finish - Middle - Begin → Create Patterning expectations should be explored and developed through integration within other strands (e.g.) patterns in numbers, geometry, measurement and data management, etc. Explore & Model Patterns in Multiple Ways: Numbers Colours Shapes Sizes Words/stories Concrete Materials Movement Body positions/dance Sample Prompting Questions • What comes next in the pattern? • What do you notice about the pattern? • How could you describe the pattern? • How can you repeat or extend the pattern? • How are these patterns alike? Different? – 2, 4, 6, 8,… AND 3, 5, 7, 9,… – 2, 4, 6, 8, 10,… AND 2, 4, 8, 16, 32,… • Model the patterning in another way. – RED, BLUE, YELLOW, RED, BLUE, YELLOW … • What is the “pattern unit”? Describe the pattern unit. – ABBABBABB… – ABCCABCCA… – NOTE: A “pattern core/unit” must repeat TWICE!! Sounds / Music What does “Algebra” look like? Understanding Variable Solving and Creating Equations Exploring Growing & Shrinking Patterns The main concepts include: 1. Equality (the meaning of the equals sign) … Use a “Balance” beginning in Grade 1! 2. Variable versus a Constant For example, for the “table problem” the number of tables is a “variable”, yet the “+2 people” at the end of each group of tables is a “constant”. Both are represented in the following equation: # of People = 2 (# of tables) + 2 3. Functional versus successor / recursive relationships 4. Representing relationships (or “rules”) in tables, graphs, words, pictorially, in real life contexts, and/or using concrete materials. x 1 2 3 4 y 9 19 29 39 Common responses include: • “x goes up by ones and y goes up by tens” (successor/recursive relationship identified) • “the x tells you what the next digit is in the next y” • “to work out y, put a zero onto x and take away 1” (functional relationship identified, yet clarification required) • “take 1 off the x, and write a nine after it” (place value relationship, yet clarification required) These descriptions can be used to extend the table to the 50th or 100th term … BUT, they cannot all be used to write an algebraic equation (generalization)! In the Junior grades “Successor / Recursive” relationships are important, but they also begin to explore “Functional” relationships.