logging in or signing up visual aSGuest800 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 62 Category: Science & Tech.. License: All Rights Reserved Like it (0) Dislike it (0) Added: October 13, 2008 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... Premium member Presentation Transcript Northwest Two Year CollegeMathematics Conference 2006Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations : Northwest Two Year CollegeMathematics Conference 2006Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie BurtonMathematics DepartmentWestern Oregon Universitywww.wou.edu/~burtonl Slide 2: These ideas useALGEBRA PIECES and the MATH IN THE MIND’S EYE curriculum developed at Portland State University (see handout for access) What are ALGEBRA PIECES? : What are ALGEBRA PIECES? The first pieces are BLACK AND RED TILES which model integers: Black Square = 1 Red Square = -1 INTEGER OPERATIONS Addition : INTEGER OPERATIONS Addition 2 + 3 5 black total = 5 INTEGER OPERATIONS Addition : INTEGER OPERATIONS Addition -2 + -3 5 red total = -5 INTEGER OPERATIONS Addition : INTEGER OPERATIONS Addition -2 + 3 Black/Red pair: Net Value (NV) = 0Total NV = 1 INTEGER OPERATIONSSubtraction : INTEGER OPERATIONSSubtraction 2 - 3 Take Away?? Still Net Value: 2 INTEGER OPERATIONSSubtraction : INTEGER OPERATIONSSubtraction 2 - 3 Net Value: 2 2 - 3 = -1 You can see that all integer subtraction models may be solved by simply added B/R--Net Value 0 pairs until you have the correct amount of black or red tiles to subtract. : You can see that all integer subtraction models may be solved by simply added B/R--Net Value 0 pairs until you have the correct amount of black or red tiles to subtract. This is excellent for understanding “subtracting a negative is equivalent to adding a positive.” : This is excellent for understanding “subtracting a negative is equivalent to adding a positive.” INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication 2 x 3 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication 2 x 3 Net Value = 62 x 3 = 6 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication -2 x 3 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication -2 x 3 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication -2 x 3 Net Value = -6-2 x 3 = -6 Slide 16: -2 x -3 would result in TWO FLIPS (down the columns, across the rows) and an all black result to show -2 x -3 = 6These models can also show INTEGER DIVISION BEYONDINTEGER OPERATIONS : BEYONDINTEGER OPERATIONS The next important phase is understanding sequences and patterns corresponding to a sequence of natural numbers. TOOTHPICK PATTERNS : TOOTHPICK PATTERNS Students learn to abstract using simple patterns TOOTHPICK PATTERNS : TOOTHPICK PATTERNS These “loop diagrams” help the students see the pattern here is 3n + 1: n = figure # B / R ALGEBRA PIECES These pieces are used for sequences with Natural Number domain : B / R ALGEBRA PIECES These pieces are used for sequences with Natural Number domain Black N, N ≥ 0Edge NRed -N, -N < 0Edge -N Pieces rotate ALGEBRA SQUARES : ALGEBRA SQUARES Black N2Red -N2Edge lengths match n stripsPieces rotate Patterns with Algebra Pieces : Patterns with Algebra Pieces Students learn to see the abstract pattern in sequences such as these Patterns with Algebra Pieces : Patterns with Algebra Pieces Working with Algebra PiecesMultiplying(N + 3)(N - 2) : Working with Algebra PiecesMultiplying(N + 3)(N - 2) First you set up the edges (N + 3)(N - 2) : (N + 3)(N - 2) Now you fill in according to the edge lengths FirstN x N = N2 (N + 3)(N - 2) : (N + 3)(N - 2) Inside3 x N = 3N OutsideN x -2 = -2N Last 3 x -2 = -6 (N + 3)(N - 2) : (N + 3)(N - 2) (N + 3)(N - 2) = N2 - 2N + 3N - 6 = N2 + N - 6 (N + 3)(N - 2) : (N + 3)(N - 2) This is an excellent method for students to use to understand algebraic partial products Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = Subtract 4N from both sets: same as adding -4n Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = Subtract -8 from both sets Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Students now try to factor by forming a rectangleNote the constant partial product will always be all black or all red = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Thus, there must be 2 n strips by 1 n strip to create a 2 black square blockTake away all NV=0 Black/Red pairs = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Thus, there must be 2 n strips by 1 n strip to create a 2 black square blockTake away all NV=0 Black/Red pairs = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Form a rectangle that makes sense = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Lay in edge pieces = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Measure the edge sets = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = 0 (N - 2)(N - 1) = 0 (N - 2) = 0, N = 2or (N - 1) = 0, N = 1 This last example; using natural number domain for the solutions, was clearly contrived. : This last example; using natural number domain for the solutions, was clearly contrived. In fact, the curriculum extends to using neutral pieces (white) to represent x and -x allowing them to extend to integer domain and connect all of this work to graphing in the “usual” way. : In fact, the curriculum extends to using neutral pieces (white) to represent x and -x allowing them to extend to integer domain and connect all of this work to graphing in the “usual” way. Materials : Materials Math in the Mind’s Eye Lesson Plans:Math Learning Center Burton: SabbaticalClassroom use modules Packets for today:“Advanced Practice” : Packets for today:“Advanced Practice” Integer work stands alone Algebraic work; quality exploration provides solid foundation You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
visual aSGuest800 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 62 Category: Science & Tech.. License: All Rights Reserved Like it (0) Dislike it (0) Added: October 13, 2008 This Presentation is Public Favorites: 0 Presentation Description No description available. Comments Posting comment... Premium member Presentation Transcript Northwest Two Year CollegeMathematics Conference 2006Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations : Northwest Two Year CollegeMathematics Conference 2006Using Visual Algebra Pieces to Model Algebraic Expressions and Solve Equations Dr. Laurie BurtonMathematics DepartmentWestern Oregon Universitywww.wou.edu/~burtonl Slide 2: These ideas useALGEBRA PIECES and the MATH IN THE MIND’S EYE curriculum developed at Portland State University (see handout for access) What are ALGEBRA PIECES? : What are ALGEBRA PIECES? The first pieces are BLACK AND RED TILES which model integers: Black Square = 1 Red Square = -1 INTEGER OPERATIONS Addition : INTEGER OPERATIONS Addition 2 + 3 5 black total = 5 INTEGER OPERATIONS Addition : INTEGER OPERATIONS Addition -2 + -3 5 red total = -5 INTEGER OPERATIONS Addition : INTEGER OPERATIONS Addition -2 + 3 Black/Red pair: Net Value (NV) = 0Total NV = 1 INTEGER OPERATIONSSubtraction : INTEGER OPERATIONSSubtraction 2 - 3 Take Away?? Still Net Value: 2 INTEGER OPERATIONSSubtraction : INTEGER OPERATIONSSubtraction 2 - 3 Net Value: 2 2 - 3 = -1 You can see that all integer subtraction models may be solved by simply added B/R--Net Value 0 pairs until you have the correct amount of black or red tiles to subtract. : You can see that all integer subtraction models may be solved by simply added B/R--Net Value 0 pairs until you have the correct amount of black or red tiles to subtract. This is excellent for understanding “subtracting a negative is equivalent to adding a positive.” : This is excellent for understanding “subtracting a negative is equivalent to adding a positive.” INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication 2 x 3 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication 2 x 3 Net Value = 62 x 3 = 6 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication -2 x 3 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication -2 x 3 INTEGER OPERATIONSMultiplication : INTEGER OPERATIONSMultiplication -2 x 3 Net Value = -6-2 x 3 = -6 Slide 16: -2 x -3 would result in TWO FLIPS (down the columns, across the rows) and an all black result to show -2 x -3 = 6These models can also show INTEGER DIVISION BEYONDINTEGER OPERATIONS : BEYONDINTEGER OPERATIONS The next important phase is understanding sequences and patterns corresponding to a sequence of natural numbers. TOOTHPICK PATTERNS : TOOTHPICK PATTERNS Students learn to abstract using simple patterns TOOTHPICK PATTERNS : TOOTHPICK PATTERNS These “loop diagrams” help the students see the pattern here is 3n + 1: n = figure # B / R ALGEBRA PIECES These pieces are used for sequences with Natural Number domain : B / R ALGEBRA PIECES These pieces are used for sequences with Natural Number domain Black N, N ≥ 0Edge NRed -N, -N < 0Edge -N Pieces rotate ALGEBRA SQUARES : ALGEBRA SQUARES Black N2Red -N2Edge lengths match n stripsPieces rotate Patterns with Algebra Pieces : Patterns with Algebra Pieces Students learn to see the abstract pattern in sequences such as these Patterns with Algebra Pieces : Patterns with Algebra Pieces Working with Algebra PiecesMultiplying(N + 3)(N - 2) : Working with Algebra PiecesMultiplying(N + 3)(N - 2) First you set up the edges (N + 3)(N - 2) : (N + 3)(N - 2) Now you fill in according to the edge lengths FirstN x N = N2 (N + 3)(N - 2) : (N + 3)(N - 2) Inside3 x N = 3N OutsideN x -2 = -2N Last 3 x -2 = -6 (N + 3)(N - 2) : (N + 3)(N - 2) (N + 3)(N - 2) = N2 - 2N + 3N - 6 = N2 + N - 6 (N + 3)(N - 2) : (N + 3)(N - 2) This is an excellent method for students to use to understand algebraic partial products Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = Subtract 4N from both sets: same as adding -4n Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = Subtract -8 from both sets Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Students now try to factor by forming a rectangleNote the constant partial product will always be all black or all red = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Thus, there must be 2 n strips by 1 n strip to create a 2 black square blockTake away all NV=0 Black/Red pairs = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Thus, there must be 2 n strips by 1 n strip to create a 2 black square blockTake away all NV=0 Black/Red pairs = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Form a rectangle that makes sense = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Lay in edge pieces = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? Measure the edge sets = 0 Solving Equations N2 + N - 6 = 4N - 8? : Solving Equations N2 + N - 6 = 4N - 8? = 0 (N - 2)(N - 1) = 0 (N - 2) = 0, N = 2or (N - 1) = 0, N = 1 This last example; using natural number domain for the solutions, was clearly contrived. : This last example; using natural number domain for the solutions, was clearly contrived. In fact, the curriculum extends to using neutral pieces (white) to represent x and -x allowing them to extend to integer domain and connect all of this work to graphing in the “usual” way. : In fact, the curriculum extends to using neutral pieces (white) to represent x and -x allowing them to extend to integer domain and connect all of this work to graphing in the “usual” way. Materials : Materials Math in the Mind’s Eye Lesson Plans:Math Learning Center Burton: SabbaticalClassroom use modules Packets for today:“Advanced Practice” : Packets for today:“Advanced Practice” Integer work stands alone Algebraic work; quality exploration provides solid foundation