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Hands on Algebra for K-2 Learners

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This workshop focuses on algebraic concepts and activities for K-2 learners

This workshop focuses on algebraic concepts and activities for K-2 learners

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• Transcript

• 1. Hands-On Algebra Presented by Michelle Flaming ESSDACK February 8, 2007
• 2. Agenda
• Icebreaker and Housekeeping
• Defining Algebra
• Pattern Investigations
• Variables and Equations
• Functions
• Assessing Algebraic Concepts
• Web Resources
• 3. Defining Algebra
• Brainstorm
• If a parent asked you, “What is algebra at your grade level, what types of things do you do?” How would you respond?
• Sort your list into categories.
• Record on chart paper with a partner.
• Principles and Standards K-2
• Algebra in the Early Years? Yes!
• Summarize your findings on chart paper.
• 4. Why Teach Algebra So Early?
• If a parent asked you, “Why do you teach algebra at such an early age? I didn’t learn it until high school.” How would you respond?
• 5. Our Response …
• Good for problem solving
• Gatekeeper to higher education.
• They encounter it at an earlier age.
• They may have encountered algebra earlier, just didn’t call it “algebra”.
• 6. The Why
• Early experience will lead to greater success in later grades. Creates a bridge.
• Algebraic concepts have a direct connection to other areas of math: numbers, measurement, and money.
• Describes important relationships in the world.
• 7. What is Algebra?
• Patterns
• Repeating
• Growing
• Variables and Equations
• Variables as Unknowns
• Variables that Vary
• Equality (Thomas Carpenter and Linda Levi’s work)
• Functions
• Regular
• Proportions
• Models
• 8. Patterns
• Activity: Card Patterns
• Repeating and Growing Patterns
• Color, shape, number, movement, and nature
• As early as 3 1/2 years of age.
• Songs, Rhythmic Chants, Nursery Rhymes
• Pattern recognition and generalization are central components not only in algebraic concepts but in our base-ten system.
• 9. Patterns
• Activity: Snakes and Rocks
• What skills would students work on?
• What type of pattern? Growing or Repeating?
• Replicate, complete, continue, describe, and create repeating patterns.
• 10. Patterns
• Activity: Changing Shapes
• What skills would students work on?
• What type of pattern? Growing or Repeating?
• Replicate, complete, continue, describe, create, and predict growing patterns.
• Describe the change between successive elements in a pattern that grows or decreases in a constant rate.
• Describe the change in words, and move towards symbols.
• 11. Growing Patterns
• Arithmetic change between pairs of items in the pattern.
• To help children recognize a growing pattern, geometric or visual clues are helpful.
• Builds toward functions: Determine the number of squares in position 30. This generalization is called a function.
• I + 1 = S
• 12. Pattern Terminology
• Item (element) - Appear in the same order in each repetition.
• A is an item or element
• B is an item or element
• Sequence - The pattern itself (used with repeating patterns)
• A, B, B, A, B, B, A, B, B, A, B, B
• The A,B, B sequence repeated 3 x
• 13. Patterns
• Activity: Pattern Round-Robin
• What skills would students work on?
• What type of pattern? Growing or Repeating?
• Replicate, complete, continue, describe, create, and predict growing patterns.
• Represent patterns in various ways and to move between those different representations. - How are these patterns alike?
• Generalize the relationship between elements and their locations in a sequence and use that information to continue and predict, what will the 30th item (element) be?
• 1, 2, 3, 4, 5, 6
• A, B, A, B, A, B (B’s are always even. The 20th position will be a B)
• 15. Variables and Equations
• Understanding the concept of variable is fundamental to algebra and, as noted by Schoenfeld and Arcavi (1988, pg. 420), “is necessary for the meaningful use of all advanced mathematics.”
• Student learning is strongly linked to personal concrete experience. Students can only move toward an understanding of symbols, only after a strong foundation has been built on the concrete level.
• 16. Variables and Equations
• Activity: Apples in a Basket
• What skills would students work on?
• Explore various use of symbols, including letters and geometric shapes to represent unknown quantities that both do and do not vary.
• Teachers Role:
• Help students see the relationship between the hands-on work to mathematical symbols and equations.
• 17. Variables as Unknowns
• “ A symbol for an element of a specified replacement set.” (Usiskin, 1988, pg 9)
• Students in early grades need to be taught the concept of variables.
• Quantities that are NOT given and DO NOT vary are referred to as “unknowns” or missing elements.
• D + 3 = 9; D can only be 6
• 18. Variables that Vary
• Activity: Gumballs in the Sack
• What skills would students work on?
• Explore equations where they may be more than one variable.
• Able to write missing addend equations.
• Learn to replace variables with values in order to compute or verify solutions.
• _____ + _____ = 10
• (4) + (6) 4 is independent .
• 19. Equality
• How would the students in your class respond to the following question: 8 + 4 = ____ + 5
• What % of students would get the correct answer in the following grades:
• 1st and 2nd ____________%
• 3rd and 4th ____________%
• 5th and 6th ____________%
• Goal: Think about how we might engage children in examining and revising their conceptions of the meaning of the equal sign.
• 20. Equality
• Video from Thomas Carpenter and Linda Levi’s research.
• As students explore ways balanced can be maintained, they learn that adding the same to both or removing the same amount from both sides, will not cause the balance to be disturbed.
• Students should see relationships between numbers.
• The understanding of equality and the appropriate use of the equal sign is critical for developing algebraic reasoning.
• 21. Functions
• Definition: A relationship between two sets A and B, expressed as an unambiguous rule which tells how to associate each member of A with one member of B (Karush, 1989).
• The function is considered by many to be the single most important idea in mathematics at all levels.
• Functions are the central idea that ties algebra and calculus together.
• 22. Functions
• A special type of function - Proportional
• Elements are related by multiplication. (Rule is In x 4 = Out)
• Examples:
• 1 table has 4 legs, 2 tables have 8 legs
• 1 foot = 12 inches
• 1 quart = 4 cups
• 1 ten = 10 ones
• 1 hundred = 10 tens
• 1 dime = 2 nickels
• 23. Functions
• Activity: Animal Functions
• What skills would students work on?
• Identify the relationship between pairs of data in a t-chart.
• Use the relationship “rule” to find the missing inputs and outputs, continue the table.
• Describe the rule that relates the pairs of data using words.
• Represent the function using symbols.
• 24.
• Representation systems:
• Equations
• Tables
• Function Tables (Input/Output Machines, T-tables)
• Graphs
• 25. Functions
• Activity: Stain Glass Designs
• What skills would students work on?
• Identify the relationship between pairs of data in a t-chart.
• Use the relationship “rule” to find the missing inputs and outputs, continue the table.
• Describe the rule that relates the pairs of data using words.
• Represent the function using symbols.
• 26. Assessing Algebraic Concepts
• Review the concepts on the Diagnostic Assessment Tool.
• Discuss with a partner any concept you are not clear about.
• 27. Web Resources
• http://www.lulu.com
• Search “Hands on Algebra”
• http://nlvm.usu.edu/en/nav/category_g_1_t_2.html
• http://www.bbc.co.uk/schools/laac/numbers/chi.shtml
• http://pbskids.org/cyberchase/games/algebra/algebra.html
• http://pbskids.org/cyberchase/games/patterns/patterns.html
• http://pbskids.org/cyberchase/games/functions/functions.html
• http://www.hbschool.com/activity/busy_bees/index.html
• http://www.tvokids.com/framesets/nook_new.html?game=141&
• http://www.kidsplaypark.com/games/jack/
• http://www.sesameworkshop.org/sesamestreet/?scrollerId=games