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
Slide 9
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?
Slide 11
Completing the Square Completing the Square Add units as
necessary to complete the square.
Slide 12
Completing the Square Completing the Square (x 2 + 6x +3) +6 is
a perfect square (x 2 + 6x +9)
Slide 13
Completing the Square Completing the Square (x 2 + 6x +3) +6 is
a perfect square (x 2 + 6x +9) (x + 3)
Slide 14
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
+
Slide 16
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 % ? %
Slide 17
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 %
Slide 19
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
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
Slide 29
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