Using Variables

Topics for Today:

Our Algebra lesson focused on some of the language of Algebra and how words can be translated into algebraic expressions.

Image Credit: https://www.pinterest.com/pin/651473902320727044/

Vocabulary:  variable, algebraic expression, equation, open sentence

Sections Covered in Textbook:

1-1: Using Variables (Pages 4-8)

Resources & Tutorials:

1) What are numerical and algebraic expressions?
2) Translating mathematical expressions

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Dividing Polynomials

Topics for Today:

It may have been some time since you had to perform long division, and long division is a multi-step process like so many we have seen in Algebra.  If you follow the procedure, you will arrive at your answer (quotient).  The process for dividing polynomials is similar to long division of constants.

You should recall from earlier mathematics courses that the process for long division is as follows:

1. Figure out how many whole times the divisor divides into the dividend and place that number on top of the divisor.
2. Multiply this number by the divisor, and subtract this number from the dividend.
3. Bring down the next number in the divisor. 
4. Repeat until you have no more numbers in the dividend to bring down. 
5. If your final subtraction problem results in "0", you have no remainder; otherwise, your remainder is a part of a whole and should be represented as a fraction with the remainder number in the numerator, and the divisor in the denominator.  

---> When dividing polynomials by a binomial, we will look to the variable part of the binomial to make our decision on what divides into the dividend.  

Image Source: https://mathbitsnotebook.com/Algebra2/Polynomials/POPolyDivide.html

Sections Covered in Textbook:

12-5: Dividing Polynomials (pages 662-666)

Resources & Tutorials:

1) Review of Long Division
2) Dividing Polynomials (long division)
3) Cool Math - Dividing Polynomials Examples (not a video)

Multiplying and Dividing Rational Expressions

Topics for Today:

Rational expressions can be multiplied or divided just like regular fractions. Recall from yesterday's lesson that a rational expression is just a fraction with polynomials in the numerator and denominator.  As with dividing regular fractions, when we divide rational expressions, we must multiply by the opposite of the divisor (invert and multiply, or as some of you like to say, keep, change, flip!)

We should always focus on taking out common factors as soon as we can.  This process helps to ensure that our eventual answer is in simplest terms.

Sections Covered in Textbook:

12-4: Multiplying and Dividing Rational Expressions (pages 657-661)

Resources & Tutorials:

1) Multiply and simplify rational expressions
2) How to divide rational expressions
3) List of More videos for multiplying and dividing rational expressions

Simplifying Rational Expressions

Topics for Today:

We will now encounter polynomials in our fractions.  A rational expression is just a ratio (fraction) with polynomials in the numerator and denominator.  When we want to simplify these fractions, we follow the same rules as regular fractions: we need to divide common factors from the numerator and denominator.  To simplify, we need to look at the greatest common factor (GCF) as well as other factoring tools.  We will factor both the numerator and denominator, and then see if we have any common factors that simplify to 1.

Sections Covered in Textbook:

12-3: Simplifying Rational Expressions (pages 652-656)

Resources & Tutorials:

1) What is a rational expression?
2) Simplify Rational Expressions by factoring
3) Simplifying Rational Expressions by using opposite binomials

Inverse Variation

Topics for Today:

Inverse variation is another relationship between the x and y variables.  Inverse variation is defined by the relationship:

xy = k where k ≠ 0

As with direct variation, k is our constant of variation.  The shape of the inverse variation graphs are much different from what we've seen so far.  These graphs are a curved shape, and the larger the constants of variation, the further it moves from the origin.  There are boundaries with these functions that will be discussed in your Algebra II course.

Vocabulary:  constant of variation, inverse variation

Sections Covered in Textbook:

12-1: Inverse Variation (pages 636-642)

Resources & Tutorials:

1) What is inverse variation? 
2) How do you use the formula for inverse variation to write an equation?

Direct Variation

Topics for Today:

Although we discussed direct variation several months ago, as we discuss related topics, I felt it was a good idea to revisit this topic.  Direct variation refers to how two variables are related to each other.  In algebraic terms, a function in the form of y = kx, where k ≠ 0, is a direct variation.

This function is similar to our slope-intercept form of a line (y = mx +b).

For direct variations, there is no y-intercept, and all of these functions must pass through the origin (0, 0).  We are effectively dealing with part of our slope-intercept form, y = mx.

For direct variations, we use the variable "k" to represent the slope, which is also our constant of variation.

Vocabulary:  direct variation, constant of variation

Sections Covered in Textbook:

5-5: Direct Variation (pages 261-266)

Resources & Tutorials:

1) What is the formula for direct variation?
2) What is the constant of variation?
3) How do you use the formula for direct variation?
4) Direct Variation Class Notes

5) Lego Prices Desmos Activity - Class Code U5Q99S