Transcript Animated Slide Show

Geometric Models
for Algebraic Concepts
Gregg Velatini
Dianna Spence
GCTM Conference
October 16, 2014
POLYNOMIALS
WITH
ALGEBRA TILES
Algebra Tiles: The Basics
1, x, x2
x+3
3x
Like Terms, Distributive Property
3x+2
3(x+2)
Multiplying Binomials
(x+2)(x+3)
x2 + 5x + 6
(2x+1)(x+4)
2x2 + 9x + 4
Two Variables
(xy)
(x+1)(y+2)
xy + 2x + y + 2
Products and Square Products
(2x+3)(y+1)
(x + y)2
2xy + 2x + 3y + 3
x2 + 2xy + y2
More Squares
(x + 6)2
x2 + 12x + 36
More On Squares
Is the quantity (x2 + 6x +3) a
perfect square?
Completing the Square
Add units as necessary to
“complete the square”.
Completing the Square
(x2 + 6x +3) +6 is a perfect square
(x2 + 6x +9)
Completing the Square
(x + 3)
(x2 + 6x +3) +6 is a perfect square
(x2 + 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 liters
1 liter
+
3 liters
=
2 liters of 30% acid are mixed with 1 liter of 60% acid.
What is the resulting acid concentration?
2 liters
30 %
1 liter
+
60 %
3 liters
=
?
%
2 liters of 30% acid are mixed with 1 liter of 60% acid.
What is the resulting acid concentration?
2 liters
30 %
1 liter
+
60 %
3 liters
=
?
%
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 oz
20 %
2 oz
+
80 %
5 oz
+
0%
18/80 = 22 1/2 %
The final concentration is 22 1/2 % alcohol
8 oz
=
?
%
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
30 %
3 gallons
+
?%
5 gallons
=
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.
“Prom Blush”
“Deep Rose”
“Perfect Mauve”
? gallons
? gallons
? gallons
30 %
+
70 %
=
40 %
“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.
“Prom Blush”
“Deep Rose”
“Perfect Mauve”
? gallons
? gallons
? gallons
30 %
+
70 %
=
40 %
“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.
“Prom Blush”
“Deep Rose”
“Perfect Mauve”
? gallons
? gallons
? gallons
30 %
+
70 %
=
40 %
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.
“Prom Blush”
“Deep Rose”
“Perfect Mauve”
? gallons
? gallons
? gallons
30 %
+
70 %
=
40 %
“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.
“Prom Blush”
“Deep Rose”
“Perfect Mauve”
3 gallons
1 gallon
4 gallons
30 %
+
70 %
=
40 %
WORK RATE PROBLEMS
WITH
PATTERN BLOCKS
Pattern Block Conventions
1
1/
=
=
1/
2
=
1/
3
1/
6
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:
1
2
3
Visualizing the Problem
Joe
Joe & Matt together: How long to finish 5 lawns?
Lawns
Matt
Hour:
1
2
3
4
5
6
Variations
Joe
Joe & Matt together: How long to finish 5 lawns?
Lawns
Matt
Hour:
1
2
3
4
5
6
Combining Rates
Joe
Joe & Matt together: How long to finish 5 lawns?
Lawns
Matt
Hour:
1
2
3
4
5
6
Variations
Joe
Joe & Matt together: How long to finish 5 lawns?
Lawns
Matt
Hour:
1
2
3
4
5
6
Revisiting the Algebra: Rates
Joe: 3 hours for 1 lawn
Joe’s rate: RJ= 1/3
Matt: 2 hours for 1 lawn Matt’s rate: RM = 1/2
Joe
Matt
Hour:
1
2
3
Revisiting: Combined Rates
Joe
Joe and Matt combined:
Matt
1 Hour
Hourly rate is
R = RJ + RM = 5/6
Revisiting: Setup and Solution
At 5/6 lawns per hour, how many hours for 5 lawns?
Lawns
…
Hr: 1
2
(RJ + RM)h = 5
5/ h = 5
6
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:
1
2
3
What now?
Bill and Sue together: How long to finish 3 mailboxes?
Mailboxes
Bill
Sue
12
Hour:
1
2
3 3/5 hours
or
3 hours, 36 min
3
?
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?
20
=
One hour
So…
+
= 1 hour, 20 min
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:
1
2
3
From Concrete to Abstract
Maria
Dusti
Minute:
1
2
3
Goal: 35 balloons
Rate: 11/6 per minute
6 min  7 balloons
30 min  35
balloons
7/
m = 35
m = 30 minutes
6
4
5
6
DECIMAL
MULTIPLICATION
WITH
BASE 10 BLOCKS
Base 10 Blocks Revisited

Use the “flat” as 1 (one whole).
1
1/
10
0.1
1/
100
0.01
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