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Geometric Models for Algebraic Concepts Gregg Velatini Dianna Spence GCTM Conference October 16, 2014

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POLYNOMIALS WITH ALGEBRA TILES

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Algebra Tiles: The Basics Algebra Tiles: The Basics 1, x, x 2 x + 3 3x

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Like Terms, Distributive Property Like Terms, Distributive Property 3x+2 3(x+2)

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Multiplying Binomials Multiplying Binomials (x+2)(x+3) (2x+1)(x+4) 2x 2 + 9x + 4 x 2 + 5x + 6

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Two Variables Two Variables (xy) (x+1)(y+2) xy + 2x + y + 2

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Products and Square Products and Square Products (x + y) 2 (2x+3)(y+1) 2xy + 2x + 3y + 3x 2 + 2xy + y 2

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More Squares More Squares (x + 6) 2 x 2 + 12x + 36

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More On Squares More On Squares Is the quantity (x 2 + 6x +3) a perfect square?

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Completing the Square Completing the Square Add units as necessary to complete the square.

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Completing the Square Completing the Square (x 2 + 6x +3) +6 is a perfect square (x 2 + 6x +9)

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Completing the Square Completing the Square (x 2 + 6x +3) +6 is a perfect square (x 2 + 6x +9) (x + 3)

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MIXTURE PROBLEMS WITH BAR MODELS

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2 liters of 30% acid are mixed with 1 liter of 60% acid. What is the resulting acid concentration? = 2 liters1 liter3 liters +

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2 liters of 30% acid are mixed with 1 liter of 60% acid. What is the resulting acid concentration? = 2 liters1 liter3 liters + 30 %60 % ? %

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2 liters of 30% acid are mixed with 1 liter of 60% acid. What is the resulting acid concentration? = 2 liters1 liter3 liters + 30 %60 % ? % The final concentration is 40% acid

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A recipe requires mixing 1 oz of 20% alcohol with 2 oz of 80% alcohol and 5 oz of orange juice. What is the resulting alcohol concentration? += 1 oz2 oz8 oz 20 %80 % ? % 18/80 = 22 1 / 2 % The final concentration is 22 1/2 % alcohol + 5 oz 0 %

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What amount and concentration of acid solution must be added to 2 gal of 30% acid solution in order to get 5 gal of 60% acid solution? = 2 gallons 3 gallons5 gallons + 30 %? % 60 % 3 gallons of 80% acid must be added.

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A paint maker receives an order for pink paint that is 40 % red and 60 % white paint. He has on hand several one gallon cans of dark pink, which is 70% red, and light pink that is 30% red. How much of the light and dark pink paint should he mix? Assume that he can only mix whole gallons of each color. = ? gallons + 30 %70 % 40 % Prom Blush Deep RosePerfect Mauve 50% is TOO strong

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A paint maker receives an order for pink paint that is 40 % red and 60 % white paint. He has on hand several one gallon cans of dark pink, which is 70% red, and light pink that is 30% red. How much of the light and dark pink paint should he mix? Assume that he can only mix whole gallons of each color. = ? gallons + 30 %70 % 40 % Prom Blush Deep RosePerfect Mauve 13/30 43.3% is TOO strong

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A paint maker receives an order for pink paint that is 40 % red and 60 % white paint. He has on hand several one gallon cans of dark pink, which is 70% red, and light pink that is 30% red. How much of the light and dark pink paint should he mix? Assume that he can only mix whole gallons of each color. = ? gallons + 30 %70 % 40 % Prom Blush Deep RosePerfect Mauve

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A paint maker receives an order for pink paint that is 40 % red and 60 % white paint. He has on hand several one gallon cans of dark pink, which is 70% red, and light pink that is 30% red. How much of the light and dark pink paint should he mix? Assume that he can only mix whole gallons of each color. = ? gallons + 30 %70 % 40 % Prom Blush Deep RosePerfect Mauve 40% is Just Right

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A paint maker receives an order for pink paint that is 40 % red and 60 % white paint. He has on hand several one gallon cans of dark pink, which is 70% red, and light pink that is 30% red. How much of the light and dark pink paint should he mix? Assume that he can only mix whole gallons of each color. = 3 gallons 1 gallon4 gallons + 30 %70 % 40 % Prom Blush Deep RosePerfect Mauve

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WORK RATE PROBLEMS WITH PATTERN BLOCKS

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= = = 1 / 2 1 / 3 1 / 6 Pattern Block Conventions 1 1/4 1/4 1 / 12

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Sample Problem Joe and Matt start a landscaping business together. Homes in their neighborhood have similarly-sized lawns. Typically, Joe can mow a lawn and trim all the shrubs in 3 hours. Matt usually needs 2 hours to do the same job. They decide to work together on 5 lawns. How long should it take them to finish?

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Rate Representation Joe: 3 hours for 1 lawn Matt: 2 hours for 1 lawn Joe Matt Hour:123

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Visualizing the Problem Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 Lawns 23 456

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Variations Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 Lawns 23 456

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Combining Rates Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:123 Lawns 465

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Variations Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 23 Lawns 456

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Revisiting the Algebra: Rates Joe: 3 hours for 1 lawn Matt: 2 hours for 1 lawn Joe Matt Hour:123 Joes rate: R J = 1 / 3 Matts rate: R M = 1 / 2

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Revisiting: Combined Rates Joe Matt 1 Hour Joe and Matt combined: Hourly rate is R = R J + R M = 5 / 6

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Revisiting: Setup and Solution At 5 / 6 lawns per hour, how many hours for 5 lawns? Hr:1 2 Lawns (R J + R M )h = 5 5 / 6 h = 5 h = 6

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A Twist Sue can paint a mailbox in 2 hours. It takes Bill 3 hours to paint the same mailbox. How long will it take them to paint three of the mailboxes working together?

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A Twist Sue can paint a mailbox in 2 hours. It takes Bill 3 hours to paint the same mailbox. How long will it take them to paint three of the mailboxes working together? Bill: 3 hours for 1 mailbox Sue: 2 hours for 1 mailbox Bill Sue Hour:123

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What now? Bill and Sue together: How long to finish 3 mailboxes? Bill Sue Hour:123 Mailboxes ? 12 3 3 / 5 hours or 3 hours, 36 min

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Try Another A pro cyclist can complete a race in 2 hours. A teacher takes 4 hours to complete the same race. If they share a tandem bike, how long will it take them to complete the race pedaling together?

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Try Another A pro cyclist can complete a race in 2 hours. A teacher takes 4 hours to complete the same race. If they share a tandem bike, how long will it take them to complete the race pedaling together? One hour = 20

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Try Another A pro cyclist can complete a race in 2 hours. A teacher takes 4 hours to complete the same race. If they share a tandem bike, how long will it take them to complete the race pedaling together? One hour = So + = 1 hour, 20 min 20

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Extending the Reasoning Maria and Dusti are decorating the gym with helium balloons. Maria can inflate and tie off 2 balloons every 3 minutes. Dusti requires 2 minutes to finish 1 balloon. Working together, how long will it take them have a batch of 35 balloons ready?

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Rate Setup Maria: 2 balloons every 3 minutes Dusti: 2 minutes for 1 balloon. Maria Dusti Minute:123

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From Concrete to Abstract Maria Dusti Minute: 132 456 Goal: 35 balloons Rate: 1 1 / 6 per minute 6 min 7 balloons 30 min 35 balloons 7 / 6 m = 35 m = 30 minutes

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DECIMAL MULTIPLICATION WITH BASE 10 BLOCKS

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Base 10 Blocks Revisited Use the flat as 1 (one whole). 1 1 / 10 1 / 100 0.1 0.01

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Base 10 Blocks Revisited 2.36

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Whole Number Multiplication 2 3

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Whole Number Mixed Number 2 2.5

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Whole Number Mixed Number 2 1.7

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Mixed Number Mixed Number 1.2 1.3

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Mixed Number Mixed Number 1.4 2.3

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Whole Number Proper Fraction 2 0.6

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Mixed Number Proper Fraction 1.3 0.6

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Mixed Number Proper Fraction 1.3 0.6

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Mixed Number Proper Fraction 1.3 0.6

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Proper Fraction Proper Fraction 0.4 0.6

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Proper Fraction Proper Fraction 0.4 0.6

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Proper Fraction Proper Fraction 0.4 0.6