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Where Does that Algebraic Equation Come From? Moving From Concrete Experience to Symbolic Form "

Where Does that Algebraic Equation Come From? Moving From Concrete Experience to Symbolic Form "

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Where Does that Algebraic Equation Come From? Moving From Concrete Experience to Symbolic Form "

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  1. Where Does that Algebraic Equation Come From?Moving From Concrete Experience to Symbolic Form" Jim Rubillo JRubillo@verizon.net

  2. What is Algebra? The intensive study of the last three letters of the alphabet.

  3. The Policy Dilemma Algebra in Grade 7/8 or Algebra When Ready?

  4. Algebra When Ready Only when students exhibit demonstrable success with prerequisite skills—not at a prescribed grade level—should they focus explicitly and extensively on algebra, whether in a course titled Algebra 1 or within an integrated mathematics curriculum. Exposing students to such coursework before they are ready often leads to frustration, failure, and negative attitudes toward mathematics and learning. NCTM Position : Algebra: What, When, and for Whom (September 2011)

  5. Major Themes that Start in PreK and Go all the Way through Grade 12 • Exploring and extending patterns • Representing mathematical ideas with symbols and objects • Using mathematical models to represent quantitative relationships • Analyzing change in various contexts

  6. What We Know About Student Difficulties in Algebra • Lack of proficiency in proportionality (fractions, decimals, ratios, percent) • Understanding the equal sign (do something vs. equality, balance) • Using Variables (placeholder vs. changing value) • Making the transition from words (verbal or written) to symbols. • Understanding of function concept (rule or formula) • Lack of exposure to multiple representations (numbers, graphs, tables, symbols, etc.)

  7. What We Know About Student Difficulties in Algebra Understanding the equal sign (do something vs. equality, balance) 3 + 5 = or (3x + 5) – (x -3) = versus 4 + 6 = 6 + 4 or y + (-y) = 0 Using Variables (placeholder vs. changing value) 2x + 3 = 17 versus y = 3x2 – 19x - 14

  8. The Far Too Typical Experience! • Here is an equation: y = 3x + 4 • Make a table of x and y values using whole number values of x and then find the y values, • Plot the points on a Cartesian coordinate system. • Connect the points with a line. Opinion: In a student’s first experience, the equation should come last, not first.

  9. SHOW ME!

  10. Equations Arise From Physical Situations How many tiles are needed for Pile 5? ? Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

  11. Piles of Tiles A table can help communicate the number of tiles that must be added to form each successive pile? (the recursive rule) ? Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

  12. Piles of Tiles How many tiles in pile 457? ? Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

  13. Piles of Tiles A table can help communicate the number of tiles that must be added to form each successive pile? (the recursive rule) ? Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

  14. Piles of Tiles Physical objects can help find the explicit rule to determine the number of tiles in Pile N? Pile 1 Pile 2 Pile 3 Pile 4 3+13+3+1 3+3+3+13+3+3+3+1

  15. Piles of Tiles Tiles = 3n +1 For pile N = 457 Tiles = 3x457 + 1 Tiles = 1372

  16. Piles of Tiles Graphing the Information. Tiles = 3n +1 n = pile number

  17. Piles of Tiles The information can be visually analyzed.

  18. Piles of Tiles How is the change, add 3 tiles, from one pile to the next (recursive form) reflected in the graph? Explain. How is the term 3n and the value 1 (explicit form) reflected in the graph? Explain. Y = 3n + 1

  19. Piles of Tiles The recursive rule “Add 3 tiles” reflects the constant rate of change of the linear function. The 3n term of the explicit formula is the “repeated addition of ‘add 3’” Y = 3n + 1

  20. Piles of Tiles Tiles = 3n +1 What rule will tell the number of tiles needed for Pile N?

  21. Playing with the Four Basic Operations 1 + 1 = 1 – 1 = 1 × 1 = 1 ÷ 1 =_______ Total = 2 0 1 1 4

  22. Playing with the Four Basic Operations 2 + 2 = 2 – 2 = 2 × 2 = 2 ÷ 2 =_______ Total = 4 0 4 1 9

  23. Playing with the Four Basic Operations 3 + 3 = 3 – 3 = 3 × 3 = 3 ÷ 3 =_______ Total = 6 0 9 1 16

  24. Playing with the Four Basic Operations 4 + 4 = 4 – 4 = 4 × 4 = 4 ÷ 4 =_______ Total = 8 0 16 1 25

  25. Playing with the Four Basic Operations Willing to Predict?

  26. Playing with the Four Basic Operations Willing to Predict?

  27. Playing with the Four Basic Operations Willing to Generalize? n + n = n – n = n × n = n ÷ n =_______ Total = 2n 0 n2 1 _______________ n2 + 2n + 1 = (n + 1)2

  28. Equations Arise From Physical Situations What is the perimeter of shape 6?

  29. Find the Perimeter A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)

  30. Find the Perimeter A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)

  31. Find the Perimeter Can we find the perimeter of shape N without using the recursive rule? (the explicit rule) 2N + 2

  32. Equations Arise From Physical Situations What is the perimeter of shape 6?

  33. Find the Perimeter A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)

  34. Find the Perimeter A table can help communicate the length of the sides that must be added to a shape to find the perimeter of the next shape? (the recursive rule)

  35. Find the Perimeter Can we find the perimeter of shape N without using the recursive rule? (the explicit rule) 5N + 1

  36. How many beams are needed to build a bridge of length n? Bridge of length 6

  37. How many beams are needed to build a bridge of length n? Bridge of length n B = n + 2n + (n - 1) B = 3n + (n - 1) B = 4 + 3(n – 1) + (n – 2) B = 4n – 1 where n is the length of the bridge and B is the number of beams needed

  38. Follow the Fold(s)

  39. Follow the Fold(s)

  40. What’s the Graph?

  41. Total Cost Table: Example 1 P E N C I L S ERASERS

  42. Total Cost Table Example 2 P E N C I L S ERASERS

  43. Multiple Representations The understanding of mathematics is advanced when concepts are explored in a variety of forms including symbols, graphs, tables, physical models, as well as spoken and written words.