Scatterplots - 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 code: 63WGDD

Scatterplots 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

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)
4) Dividing Polynomials Class Notes

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

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

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

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

Chapter 11 Review Day

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 20th through May 3rd

2) Chapter 11 Review Sheet
2) Chapter 11 Review - Answers

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.

3)  Solving Radical Equations Class Notes

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

5) Conjugates and Other Roots Class Notes