Geometric Models for Algebraic Concepts Gregg Velatini Dianna Spence GCTM Conference October 16,...
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Geometric Models for Algebraic Concepts Gregg Velatini Dianna Spence GCTM Conference October 16, 2014
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Transcript of Geometric Models for Algebraic Concepts Gregg Velatini Dianna Spence GCTM Conference October 16,...
- Slide 1
- Slide 2
- Geometric Models for Algebraic Concepts Gregg Velatini Dianna Spence GCTM Conference October 16, 2014
- Slide 3
- POLYNOMIALS WITH ALGEBRA TILES
- Slide 4
- Algebra Tiles: The Basics Algebra Tiles: The Basics 1, x, x 2 x + 3 3x
- Slide 5
- Like Terms, Distributive Property Like Terms, Distributive Property 3x+2 3(x+2)
- Slide 6
- Multiplying Binomials Multiplying Binomials (x+2)(x+3) (2x+1)(x+4) 2x 2 + 9x + 4 x 2 + 5x + 6
- Slide 7
- Two Variables Two Variables (xy) (x+1)(y+2) xy + 2x + y + 2
- Slide 8
- 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
- Slide 10
- 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
- Slide 15
- 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
- Slide 18
- 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.
- Slide 20
- 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
- Slide 21
- 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
- Slide 22
- 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
- Slide 23
- 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
- Slide 24
- 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
- Slide 25
- 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?
- Slide 28
- 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
- Slide 30
- Variations Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 Lawns 23 456
- Slide 31
- Combining Rates Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:123 Lawns 465
- Slide 32
- Variations Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 23 Lawns 456
- Slide 33
- 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
- Slide 34
- Revisiting: Combined Rates Joe Matt 1 Hour Joe and Matt combined: Hourly rate is R = R J + R M = 5 / 6
- Slide 35
- 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
- Slide 36
- 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?
- Slide 37
- 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
- Slide 38
- 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
- Slide 39
- 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?
- Slide 40
- 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
- Slide 41
- 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
- Slide 42
- 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?
- Slide 43
- Rate Setup Maria: 2 balloons every 3 minutes Dusti: 2 minutes for 1 balloon. Maria Dusti Minute:123
- Slide 44
- 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
- Slide 45
- DECIMAL MULTIPLICATION WITH BASE 10 BLOCKS
- Slide 46
- Base 10 Blocks Revisited Use the flat as 1 (one whole). 1 1 / 10 1 / 100 0.1 0.01
- Slide 47
- 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
- Slide 59
- Proper Fraction Proper Fraction 0.4 0.6