Transition Algebra Manuel Navarro and Dirk Hodges

Austin, TXMay 6th and 7th , 2011

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

10. “Baseball is ninety percent mental and the other half is physical.” (percentage)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

9. “Half the lies they tell about me aren't true.” (fractions)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

8. “I knew I was going to take the wrong train, so I left early.” (distance formula)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

7. “You better cut the pizza in four pieces because I'm not hungry enough to eat six.” (fractions)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

6. “I take a two hour nap, from one o'clock to four.” (subtraction)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

5. “90% of the putts that are short don't go in.” (percentage)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

4. “You give 100 percent in the first half of the game, and if that isn't enough in the second half you give what's left.” (percentage)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

3. “The towels were so thick there I could hardly close my suitcase.” (Volume)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

2. “A nickel ain't worth a dime anymore.” (value of money)

TOP TEN QUOTES FROM THAT GREAT MATHEMATICIAN,

YOGI BERRA

1. “He hits from both sides of the plate. He's amphibious.”

(okay it’s a biology question)

WHY DO ADULT LEARNERS STRUGGLE

WITH “TRANSITIONAL”

ALGEBRA?

Adult learners are encouraged to learn math

from a “hands on, real world” perspective -

Concrete learning.

But when they show up in the college classroom, they

encounter the world of Abstract learning – rules,

facts, theorems…

Strategy

Fluency and Fluidity

Teach fluidity between the world of the abstract and

the concrete, not just fluencyin one.

Fluency = Accuracy with little effort

Fluidity = Ability to change with ease

8 Simple Rules

to exist in the fluid world of solving

equations. 

1st Simple Rule

 The goal of algebra is to

find the value of an unknown.

X must be…

Isolated – like Beyonce’ (single ladies)

Positive – not like Country and Western music.

The answer needs to look like this:

x

Not like this:

2x or - x

2nd Simple rule

 A prerequisite for learning algebra is understanding

balance.

4x = 8

4x4 = 8

4x4

= 84

x = 2

3rd Simple ruleDistributive Property

 

If there are parentheses in an equation, then multiply across the parentheses

distributing evenly.

Using the Distributive Property

a.) 2(x + y)

= 2x + 2y

Example: Find each product by using the distributive property to remove the parentheses.

b.) 7(x + 2y – 5z) = 7x + 14y – 35z

c.) – 4(3a – 3b – 10c) = – 12a + 12b + 40c

a.) 2(x + y)

b.) 7(x + 2y – 5z) c.) – 4(3a – 3b – 10c)

4th Simple ruleAddition Property of Equality

 

Collect like terms by moving variables to one side of the fence

and non-variables to the other side (making sure to change the

sign if they cross over).

 

Much like corralling cattle and sheep.

3x = 6

 3x +2 = x -4 -2-x

Combining Like Terms

6x2 + 7x2

19xy – 30xy

13xy2 – 7x2y

13x2

– 11xy

Can’t be combined (since the terms are not like terms)

Terms Before Combining After Combining Terms

Combining Like Terms

We cannot combine a chicken and a goat and create a

Chickengoat

Combining Like Terms

Or a

Zonkey??

5th Simple ruleMultiplication Property of Equality

 

Divide (or multiply) both sides of an equation to get the variable

isolated and positive.

6x = 18

6 6

x = 3

6th Simple ruleMultiplication Property of Inequality

 

If the problem is an inequality, then all rules apply with 2

exceptions:1)When dividing by a negative

number the direction of the arrow must be changed.

2)When there is a compound inequality each set of terms must

be applied.

-6x > 18

-6 -6

x < 3

0< 20 8<-4x 20

-20 < -4x < -12

-4 -4 -4

5 > x > 3

7th Simple ruleSystems of equations

 

If problem has 2 variables then must substitute the value of one of the variables (which is many

times a polynomial) for the other.

Solve the following system using the substitution method.3x – y = 6 and – 4x + 2y = –8

Solving the first equation for y, 3x – y = 6

–y = –3x + 6 y = 3x – 6 Multiply both sides by – 1.)

Substitute this value for y in the second equation. –4x + 2y = –8 –4x + 2(3x – 6) = –8 Replace y with result from first equation. –4x + 6x – 12 = –8 Use the distributive property.

2x – 12 = –8 Simplify the left side.

2x = 4 Move 12 to the other side and change sign.

x = 2 Divide both sides by 2.

The Substitution Method

Continued.

Example:

Substitute x = 2 into the first equation solved for y.

y = 3x – 6 = 3(2) – 6 = 6 – 6 = 0

Our computations have produced the point (2, 0).

Check the point in the original equations.

First equation,

3x – y = 6

3(2) – 0 = 6 true

Second equation,

–4x + 2y = –8

–4(2) + 2(0) = –8 true

The solution of the system is (2, 0).

The Substitution Method

Example continued:

8th Simple ruleSquare Root Property

 

If problem has a quadratic (square) then utilize square root.

Square Root Property

Square Root PropertyIf b is a real number and a2 = b, then

ba

Solve x2 = 49

2x

Solve (y – 3)2 = 4

Solve 2x2 = 4

x2 = 2

749 x

y = 3 2

y = 1 or 5

243 y

Square Root PropertyExample:

Solve (x + 2)2 = 25

x = 2 ± 5

x = 2 + 5 or x = 2 – 5

x = 3 or x = 7

5252 x

Square Root Property

Example:

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