Algebra

Math 8May 2013

A brain teaser Think of a number. Add three. Find the square of the result. Subtract nine. Divide by the original number. Subtract 6. Write down your result.

A brain teaser We can solve the brain teaser with algebra: x (the number) x + 3 (the number + 3) (x + 3)2 = x2 + 6x + 9 (the number + 3 squared) x2 + 6x + 9 – 9 = x2 + 6x (the square minus 9) (x2 + 6x) ÷ x = x + 6 (divide by your original number) (x+ 6) – 6 = x (subtract 6) x (your original number!!)

Outcomes B14 add and subtract algebraic terms

concretely, pictorially, and symbolically to solve simple algebraic problems

B15 explore addition and subtraction of polynomial expressions, concretely and pictorially

B16 demonstrate an understanding of multiplication of a polynomial by a scalar, concretely, pictorially, and symbolically

A little vocabulary…Variable: a symbol that represents an unknown value.

For example : x, y, a, b, r, h , etc.

Coefficient: A number that preceeds a variable in an equation. The coefficient is multiplied by the variable.

For example, we can write 3 times w like 3w. 3 is the coefficient.

A little vocabulary…

Constant: A quantity that does not change. For example: ½ , -3, 6,

Term: A term consists of a constant, a variable, a coefficient that multiplies the variable, or two or more variables that are multiplied.

For example: 3, 4x, 4xy2, and x3 are all terms.

A little vocabulary…

An Algebraic Expression: An expression with one or more terms. There is no answer for the equation.

Example: 3x – 2-4x3 – 2y + 3xy -5

A little vocabulary…

An Algebraic Equation: Two separate expressions separated by and equal sign.

Example: 3x – 2 = 6-4x3 – 2y + 3xy -5 = 4x – 3y

Pause for Reflection 4x2 – 3x = 1 Expression or equation?

3y2 + 5x – 7z + 32c -2 Expression or equation?

3y2 + 5x – 7z + 32c -2 Combine terms?

3x – 5y3 – 8 Identify the coefficients

3x – 5y3 – 8 Identify the variables

3x – – 8 Identify the constants

A bit more vocabularyA monomial: An algebraic expression that contains

only one term. Example: x

A binomial: An algebraic expression that contains two terms.

Example: 3x – 2

A polynomial: An algebraic expression with more than two terms.

Example: 5x2 + 2x + 3

Even more vocabulary

Like terms: Terms with different coefficients but the same variable.

Examples: 3x and 2x are like terms4x and 5y are not like terms.x2 and x are not like terms.

ALGETILES1 -1

x -x

x

x

x2 -x2

y -y

y

y

y2 -y2

x

y

xy -xy

Algetiles are manipulatives that we use to represent terms concretely.

Represent the expressions with Algetiles

How can we represent the following expression with Algetiles:

x2 + 4x + 3

x

x

x2

x

x

x

x1

1

1

Practice Represent the following expressions with

Algetiles.

2x + 3-x2 +(-2)2x2 + 3x + 1x2 + (-2x) + 1

Practice 2x + 3 2x2 + 3x + 1

-x2 + (– 2) x2 + (-2x) + 1

x

x

1

1

1

x

x

x2x

x

x2 xxx

1

-x2

-1

-1

x

x

x2 -x -x 1

Can Terms be Combined? Well… that depends If we know that a certain distance is 2m and another

is 30 cm, can we say that the distance is 2 + 30 = 30 units?

NO – We must convert the distances to the same unit of measurement 2m + 0.3 m = 2.3 m

Can we add 2 tens and 5 units to get 7? NO - 2 tens plus 5 units = 25 Can we add 5 apples and 3 oranges to get 8

appanges? NO! We can only combine the same types of objects. (like terms)

Combining like terms If the terms have the same variable and the same exponent, then

we can combine them.

But, if the exponent is different, we cannot combine the terms.

x

x

x2

x

x

x

x

x

2x 3x

+ = x

x

x

x

x

x

x

2xx2

+ = x

x

x2

x

x

x2 2x+

5x

Combining like terms If the the term is positive or negative it does not affect like terms.

x

x

x2 x

x

x2 -x2

Zero Property We can combine a positive term and a

negative term to make zero.

x

x

x2 -x2+ = 0

x -x+ = 0

1 -1+ = 0

Combining like terms to simplify

x

x

x2x

x

x2 xxx -x -x 1

y

y

y2y

y

y2

y

y

y2

1

1

1

-1

-1

-x2 -x2

Regroup the like terms to make zeros if possible.

Combining like terms to simplify

x

x

x2

x

x

x2

xx

x -x-x

1

y

y

y2

y

y

y2

y

y

y2

1

1

1

-1

-1

-x2

-x2

Find the zeros and regroup what is left.

3y2 + x + 2