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
4) Multiplying and Dividing Rational Expressions Class Notes
5) List of Review Topics for Final Exam

Assignments:

1) Multiplying and Dividing Rational Expressions Worksheet
2) Exam Review on Delta Math

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
4) Simplifying Rational Expressions Class Notes
5) List of Exam Review Topics

Assignments:

1) Simplifying Rational Expressions Worksheet
2) Review for your Exam on Delta Math

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?
3) Class Notes on Inverse Variation

Assignments:

1) Inverse Variation Worksheet
2) Exam Review topics
3) Delta Math Exam Review

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

Assignments:

1) Lego Prices Desmos Activity - Class Code Z36 QVP
2) Direct Variation Worksheet
3) Work on Exam Review

Scatter Plots - Day 2

Topics for Today:

We covered a few more examples of scatterplots and line of best fit and completed a Desmos activity to see how they are used with real data.


(Graphic Source: https://www.learnbyexample.org/r-scatter-plot-ggplot2/)

Sections Covered in Textbook:

1-9: Graphing Data on the Coordinate Plane (pages 59-64)
6-6: Scatter Plots and Equations of Lines (pages 318-323)

Resources & Tutorials:

1) Desmos activity from class:  Scatterplots (class code: S5M 4BC)

Assignments:

1) Desmos Activity: Dapper Cats (Class Code: YT3 JTY)
2) Exam Review Worksheet

Scatter Plots - Day 1

Topics for Today:

A scatter plot is a graph that relates two sets of data.
To make a scatter plot, plot the two groups of data as ordered pairs.

Most scatter plots are in the first quadrant of a coordinate plane because the data are usually positive numbers.


You can use scatter plots to look for trends in the data.  Three scatter plots below show the types of relationships two sets of data may have:

Sections Covered in Textbook:

1-9: Graphing Data on the Coordinate Plane (pages 59-64)
6-6: Scatter Plots and Equations of Lines (pages 318-323)

Resources & Tutorials:

1) What is a scatter plot?
2) How do you make a scatter plot?
3) List of videos discussing scatter plots and correlation.

4) Scatter Plots Class Notes

Assignments:

1)  Line of Best Fit Homework Packet
2)  Chapter 11 Test Corrections Due Wednesday

Chapter 11 Review

Topics for Today:

We have finished the topics for Chapter 11.  We will not be discussing graphing square roots (Algebra II) nor Trigonometric Ratios (Geometry/Trig).   The top concepts from our chapter include:

• Simplifying Radicals (3 Conditions)
• No perfect square factors under the radical
• No fractions under the radical
• No radicals in the denominator of a fraction
• The Pythagorean Theorem
• Distance Formula
• Midpoint Formula
• Simplifying radicals by adding and subtracting
• Simplifying radicals by multiplying and dividing
• Rationalizing denominators
• Conjugates
• Solving Radical Equations
• Squaring both sides of an equation
• Looking for and excluding extraneous solutions
• Equations with no real solution

Sections Covered in Textbook:

Chapter 11 (Sections 11-1 through 11-5 - pages 578-612)

Resources & Tutorials:

1) See Blog Entries for April 23rd through May 5th
2) List of Chapter 11 Review Topics
2) Chapter 11 Review - Answers

Assignments:

1) Chapter 11 Review worksheet

2) Delta Math Review assignment 
2) Chapter 11 Test tomorrow

Solving Radical Equations

Topics for Today:

We added to our equation solving tools today by working with equations containing radicals.  To solve these equations, we must isolate the variable on one side of the equation.  Once we do that, we can "undo" taking a square root by squaring both sides.  We must be careful when squaring equations so that our process does not result in extraneous (extra) solutions.  It's always best to check our solutions to make sure they satisfy the original equation.  As with many other equation types, we may have a situation where our equation has no solutions.  In Algebra I, we do not work with imaginary numbers (in our class they are the square roots of negative numbers), so if we encounter any of these, our equation has no real solution.

Vocabulary: radical equation, extraneous solution

Sections Covered in Textbook:

11-5: Solving Radical Equations (pages 607-612)

Resources & Tutorials:

1) How do you solve radical equations?
2) How to solve a radical equation with a binomial radicand.

Assignments:

1) Solving Radical Equations Worksheet
2) Chapter 11 Test Thursday

Conjugates and Other Roots

Topics for Today:

We finished our discussion of operations with radical expressions today with a method to simplify fractions with radical operations in the denominator.  We discussed the topic of conjugates to rationalize denominators that fall into this category.

We also discussed different roots other than square roots, and how to find them.

Vocabulary:  conjugate, cube root

Sections Covered in Textbook:

11-4:  Operations with Radical Expressions (pages 600-605)
**Other Root Functions are not in our book.

Resources & Tutorials:

1) Divide by Conjugate Method
2) Math is Fun: Cubes and Cube Roots (not a video).
3) How do you find the cube root of a perfect cube? 
4) Fourth Roots

Assignments:

1) Simplifying Radicals and Other Roots worksheet
2) Chapter 11 Test on Thursday