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

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

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

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Algebra When Ready Only when students exhibit demonstrable success with prerequisite skillsnot at a prescribed grade levelshould 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)

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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

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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.)

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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 = 3x 2 – 19x - 14

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The Far Too Typical Experience! 1.Here is an equation: y = 3x + 4 2.Make a table of x and y values using whole number values of x and then find the y values, 3.Plot the points on a Cartesian coordinate system. 4.Connect the points with a line. Opinion: In a students first experience, the equation should come last, not first.

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SHOW ME!

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

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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 12345678.. Tiles Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

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Piles of Tiles How many tiles in pile 457? ? Pile 12345678.. Tiles 47101316192225.. Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

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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 12345678.. Tiles 47101316192225.. Pile 1 Pile 2 Pile 3 Pile 4 Pile 5

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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+1 3+3+1 3+3+3+1 3+3+3+3+1

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Piles of Tiles Tiles = 3n + 1 For pile N = 457 Tiles = 3x457 + 1 Tiles = 1372 Pile 1234.. Tiles 3+13+13+3+13+3+13+3+3+13+3+3+1..

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Piles of Tiles Graphing the Information. Pile 12345678 Tiles 47101316192225 Tiles = 3n + 1 n = pile number

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Piles of Tiles The information can be visually analyzed. PileTiles 01 14 27 310 413 516 619 722 825 928 1031

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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

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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

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Piles of Tiles Pile 0123456 Tiles 14710131619 What rule will tell the number of tiles needed for Pile N? Tiles = 3n + 1

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Playing with the Four Basic Operations 1 + 1 = 1 – 1 = 1 × 1 = 1 ÷ 1 =_______ Total = 2011420114

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2 + 2 = 2 – 2 = 2 × 2 = 2 ÷ 2 =_______ Total = 4041940419 Playing with the Four Basic Operations

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

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

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n123456789 Total491625 Willing to Predict? Playing with the Four Basic Operations

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n123456789 Total49162536496481100 Willing to Predict? Playing with the Four Basic Operations

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Playing with the Four Basic Operations n1234567..n Total491625364964 Willing to Generalize? n + n = n – n = n × n = n ÷ n =_______ Total = 2n 0 n 2 1 _______________ n 2 + 2n + 1 = (n + 1) 2

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

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Find the Perimeter Shape12345678.. 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)

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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) Shape12345678.. Perimeter4681012141618

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Find the Perimeter Can we find the perimeter of shape N without using the recursive rule? (the explicit rule) Shape123456..N Perimeter468101214.. 2N + 2

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

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Find the Perimeter Shape12345678.. 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)

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Shape12345678.. Perimeter6111621263136 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) Find the Perimeter

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Can we find the perimeter of shape N without using the recursive rule? (the explicit rule) Shape123456..N Perimeter51116212631.. 5N + 1 Find the Perimeter

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

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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 How many beams are needed to build a bridge of length n?

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Follow the Fold(s) Folds012345678910 Sides

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Folds44--49--67--82--876-- Sides Follow the Fold(s)

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Whats the Graph? Folds012345678910 Sides45654565456

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7 6 5 4 3 2 17 03 012345678 PENCILSPENCILS ERASERS Total Cost Table: Example 1

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7 6 5 4 28 3 2 24 1 0 012345678 PENCILSPENCILS ERASERS Total Cost Table Example 2

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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.

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