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

POLYNOMIALS WITH ALGEBRA TILES

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

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

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

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

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

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

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

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

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

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

MIXTURE PROBLEMS WITH BAR MODELS

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

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 % ? %

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

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 %

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.

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 Rose”“Perfect Mauve” “50% is TOO strong”

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 Rose”“Perfect Mauve” “13/30 ≈ 43.3% is TOO strong”

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 Rose”“Perfect Mauve”

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 Rose”“Perfect Mauve” “40% is Just Right”

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 Rose”“Perfect Mauve”

WORK RATE PROBLEMS WITH PATTERN BLOCKS

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

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?

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

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

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

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

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

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

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

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

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?

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

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

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?

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

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

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?

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

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

DECIMAL MULTIPLICATION WITH BASE 10 BLOCKS

Base 10 Blocks Revisited Use the “flat” as 1 (one whole). 1 1 / 10 1 /

Base 10 Blocks Revisited 2.36

Whole Number Multiplication 2  3

Whole Number  Mixed Number 2  2.5

Whole Number  Mixed Number 2  1.7

Mixed Number  Mixed Number 1.2  1.3

Mixed Number  Mixed Number 1.4  2.3

Whole Number  Proper Fraction 2  0.6

Mixed Number  Proper Fraction 1.3  0.6

Mixed Number  Proper Fraction 1.3  0.6

Mixed Number  Proper Fraction 1.3  0.6

Proper Fraction  Proper Fraction 0.4  0.6

Proper Fraction  Proper Fraction 0.4  0.6

Proper Fraction  Proper Fraction 0.4  0.6