Inequalities and Their Graphs

Topics for Today:

We moved on to Chapter 3 today.  Our discussion has moved from equations where both sides are equal, to inequalities where one side is larger or smaller than the other.  We also discussed the difference between solutions where the endpoint is included vs excluded and explored the graphs of inequalities.  A solution to an inequality is any value that will make the inequality true.  Inequalities differ from equations because inequalities often have infinite solutions that are bound by a particular value whereas equations typically have a finite solution set.

Graphic Credit: https://www.katesmathlessons.com/graphing-inequalities.html

Vocabulary: inequality, solution to an inequality

Sections Covered in Textbook:

3-1: Inequalities and Their Graphs (pages 134-138)

Resources & Tutorials:

1) What is an Inequality?
2) How Do You Graph Inequalities?
    (This video also includes infinite sets which we did not discuss.)

Literal Equations

Topics for Today:

Today we continued our exploration of equations, but focused on literal equations and formulas.  Literal equations are just equations that have more than one variable.  Formulas are mathematical or scientific facts, rules, or relationships expressed with mathematical symbols.  Students have been using formulas for much of their mathematics studies, although they may not be aware.  Earlier in this chapter, we used the formula for distance (d=rt) and also perimeter of a rectangle {P=2(l+w)}.

We can use our knowledge of solving equations to move variables around in literal equations or formulas, to solve for a particular variable.  The properties of equality (addition, subtraction, multiplication, division and distributive) still apply here in these examples.

The first step in any solving equation problem it to identify the variable that you are solving for.  Once that is done, we follow the reverse order of operations to isolate the variable, and follow the same steps we used for solving multi-step equations.  As a reminder, here are those steps again:

Vocabulary: formula, literal equation

Sections Covered in Textbook:

2-6: Formulas (pages 111-114)

Resources & Tutorials:

1) What is a literal equation?
2) How do you solve a formula for a variable?
3) Summary of solving literal equations.

Equations and Problem Solving Part 2 - Uniform Motion

Topics for Today:

We continued our discussion about problem solving by investigating uniform motion problems.   Most students are familiar with the basic formula for motion: 
distance = rate * time, or in algebraic terms, d = rt.

Uniform motion problems fall into three main categories: same-direction travel, round-trip travel, or opposite-direction travel.  Depending upon what the problem is asking, we'll combine our problem data in different ways to find our answer, but in each case, we will still apply the general formula (d=rt) to set up our problem.

Drawing diagrams to help picture what is going on in the problem is another helpful strategy.  Using a table to solve problems helps organize all the supporting data, and provides a systematic way to solve more complex problems.  Students are encouraged to use a table and to draw a picture of what is going on in the question to help better understand what is being asked.

Sections Covered in Textbook:

2-5: Equations and Problem-Solving Part 2 (pages 103-110)

Resources & Tutorials:

1)  How to Solve Opposite-Directions problem.
2)  How to Solve Same-Directions problem.
3)  How to Solve Round-Trip Travel Problems.

Equations and Problem-Solving Part 1

Topics for Today:

Now that we know how to solve all types of equations, we will use that knowledge to solve story problems.  There are many different types of story problems, but most of them can be categorized into one of several categories.  Today's lesson will focus on problem solving involving defining variables in terms of each other,  consecutive integers (which also involves defining variables in terms of another), and break-even problems.

Vocabulary: consecutive, break-even

Sections Covered in Textbook:

2-5: Equations and Problem Solving (pages 103 - 110)
We will continue working on this topic tomorrow with more examples.

Resources & Tutorials:

1) Solving Break-Even Problems
2) How to find the Break-Even Point
3) Solving Word Problems with Consecutive Integers

Solving Equations with Variables on Both Sides

Topics for Today:

Our discussion about solving equations moved on to situations where there are variables on both sides of an equation.  The basic properties of equality still apply when dealing with variable terms.  We used the addition, subtraction, multiplication, and division properties of equality to get variables on one side, and constants on the other side of the equation.  For equations with variables on both sides, it is possible for the equation to have infinitely many solutions (identity) or no solution at all.  Students will need to be on the lookout for these special cases, which show themselves is interesting mathematical ways.

Sections Covered in Textbook:

2-4: Equations with Variables on Both Sides (pages 96-100)

Resources & Tutorials:

1) Solving Equations with Variables on Both Sides
2) Solving Equations with Variables on Both Sides and Grouping Symbols

Solving Multi-Step Equations

Topics for Today:

We moved on from two-step to multi-step equation solving.  Today's lesson focused on simplifying with grouping symbols, and multiplying through to eliminate fractions and decimals to make the equation easier to solve.

Sections Covered in Textbook:

2-3: Solving Multi-Step Equations (pages 88 - 93)

Resources & Tutorials:

1) Solving Multi-Step Equations using Distributive Property
2) Solving Multi-Step Equations by Clearing Fractions

Solving Two-Step Equations

Topics for Today:

Our discussion about solving equations moved on to solving 2-step equations.  The order of operations still figures in to this process, although since we are undoing operations, we go in reverse.

Sections Covered in Textbook:

2-2: Solving Two-Step Equations (pages 81- 86)

Resources & Tutorials:

1) How do you solve a 2-step equation?
2) Math Antics - How to solve 2-step equations.

Solving One-Step Equations

Topics for Today:

Today we reviewed solving one-step algebraic equations.  We discussed what inverse operations are, and also defined the term solution.  Students are reminded that all they are accountable for all the mathematics that came before this class, as well as any new learned material.  Mathematics is a cumulative subject, and the skills built in the past will continue to be used to solve new problems.  That means that we will continue to integrate fraction and decimal operations into our problem solving.  Equation operations will include all types of real numbers.

Sections Covered in Textbook:

2-1: Solving One-Step Equations (pages 74-79)

Resources & Tutorials:

1) Math Antics - How to Solve a 1-step Equation.
2) How to solve an equation using addition.
3) How to solve an equation using subtraction.
4) How to solve an equation using multiplication.
5) How to solve an equation using division.
6) How to solve a decimal equation using subtraction.
7) How to solve an equation when multiplying fractions. 

Chapter 1 Review

Topics for Today:

Today we played a game of Money Grab to finish up our review of Chapter 1.  Students should make sure their binders are fully organized prior to joining class on Thursday.  Binders will be collected at the beginning of class.  For a list of topics that need to be in the notebook, open the Chapter 1 Table of Contents Google Docs that I have shared with you from the Chapter 1 folder.  

Topics covered in Chapter 1:  Using variables, order of operations, classifications of real numbers, real number operations (adding, subtracting, multiplying, and dividing), as well as the properties of equality.  

Sections Covered in Textbook:

Chapter 1:  Tools of Algebra (pages 4-58) - Omit section 1-9

Resources & Tutorials:

Properties of Real Numbers Days 1 & 2

Topics for Today:

Today we laid the groundwork for all the properties of real numbers, and we'll be taking two days to discuss and master them.  The properties are a set of truths that act as rules to allow us to rewrite numbers and equations to make them easier to solve.  We also use the properties to justify the different steps we take to simplify expressions and solve equations.  Students also made a set of notecards that can be used during any quiz or test this year.

This is the last section of this chapter.  I have posted a Chapter 1 Review on Delta Math for those who want more practice.  This assignment is optional but highly recommended.  

Sections Covered in Textbook:

1-8: Properties of Real Numbers (pages 54 - 57)

Resources & Tutorials:

1)  Video page for the addition, subtraction, multiplication and division properties.
2)  What is the substitution property of equality?
3)  Summary sheet of Properties from Math-Aids.

The Distributive Property

Topics for Today:

Today we discussed the distributive property  of multiplication over addition and subtraction.  We also used the distributive property to multiply large numbers using mental math.  We also discussed how to combine algebraic like terms, and expanded our algebra vocabulary.  We revisited translating phrases into algebraic expressions with the addition of the distributive property.

Vocabulary: constant, like terms, distributive property, coefficient, term

Sections Covered in Textbook:

1-7: The Distributive Property (pages 47 - 52)

Resources & Tutorials:

1) What is the distributive property?

Multiplying and Dividing Real Numbers

Topics for Today:

Earlier this week we took our first quiz of the year.  How did it go?  Did you properly prepare?  If not, what improvements can you make for the future?  Students will be allowed and expected to complete quiz corrections for partial credit.  Quizzes are meant as assessment and learning tools.  Students should always be asking, "How can I do better?".

We moved on to multiplying and dividing real numbers today.

Some basic sign rules for multiplication and division:

+ · + = +      + / + = +
+ · - = -      + / - = -
- · + = -      - / + = -
- · - = +      - / - = +

Vocabulary: Identity property of Multiplication, Multiplication Property of Zero, Multiplication Property of -1, Inverse Property of Multiplication, multiplicative inverse, reciprocal

Sections Covered in Textbook:

1-6: Multiplying and Dividing Real Numbers (pages 37 - 44)

Resources & Tutorials:

1) How do you multiply and divide numbers with different signs?
2) What are Multiplicative Inverses?
3) What is a reciprocal?

Subtracting Real Numbers

Topics for Today:

We continued our discussion of adding real numbers (both positive and negative) using a number line and integer chips.  Our rules for addition are:

• If the signs are the same, add and keep the sign.
• If the signs are different, subtract and keep the sign of the number with the larger absolute value.  

Stated another way...

Subtraction is actually the addition of opposites!  

For subtraction problems, first change to addition by adding the opposite, then follow your addition rules!  

                  6 – 10 =

              6 + (-10) =  

Subtract 6 from 10 and keep the negative sign because |-10| > |6|

Therefore, the answer is – 4. 

Sections Covered in Textbook:

1-4: Adding Real Numbers (pages 24-31)
1-5: Subtracting Real Numbers (pages 32-36)

Resources & Tutorials:

1) How do you add two negative numbers?
2) Rules for Adding Integers
3)  How to rewrite a subtraction problem as addition

Adding Real Numbers

Topics for Today:

We used models (number line and integer chips) and rules to add real numbers today.  We talked about the identity property of addition and also talked about inverses.  We also discussed how to use a number line to represent addition problems of both positive and negative integers.  


Vocabulary: identity property of equality, additive inverse, inverse property of addition

Sections Covered in Textbook:

1-4: Adding Real Numbers (pages 24-31)

Resources & Tutorials:

1) How do you add two negative numbers?
2) Rules for Adding Integers
3) What is the opposite of a number?
4) What are identities?

Exploring Real Numbers

Topics for Today:

We are still working on the language of Algebra and using precise words.  Today we discussed the different classifications for numbers and what sets various types of numbers belong to (Natural, Whole, Integers, Rational, Irrational, and Real).  We also reviewed the meaning of absolute value.

Vocabulary: natural numbers, whole numbers, integers, rational number, irrational number, real numbers, inequality, opposites, absolute value

Sections Covered in Textbook:

1-3: Exploring Real Numbers (Pages 17-23)

Resources & Tutorials:

1) What is a real number?
2) What is a rational number?
3) What are natural and whole numbers?

Exponents and Order of Operations

Topics for Today:

As with any language, algebra has some basic rules that must be followed to arrive at a correct answer.  Sometimes there are multiple ways to express an answer or number so we need a set of rules to follow so we always arrive at the same answer for any one particular problem.  The order of operations is a set of rules that tells problem solvers which operations to perform and in what order.

Vocabulary: simplify, exponent, base, power, order of operations (PEMDAS), evaluate 

Sections Covered in Textbook:

1-2: Exponents and Order of Operations (pages 9 - 16)

Resources & Tutorials:

1) What is the order of operations?
2) How do you use the order of operations

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

Welcome Back, Falcons!

Hey Everyone!

I'm looking forward to the start of another amazing year at Field School.

Please bookmark this site as it will be your one-stop location for anything you need for Algebra I class this year!

GO FALCONS!

Every boy known, every boy a brother 

                     

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

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.

Operations with 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

Operations with Radicals Part 1

Topics for Today:

Radicals have some similar properties as variables when we manage them in equations and expressions.  Just like variables, we can only combine radicals that are like each other.  When we combine or take away (add or subtract) radicals, we may only do so if our radicals are like each other.

We can only combine like radicals, and sometimes we need to simplify first, and then we may have like radicals that we can combine.  

The distributive property also works with radicals, including double distributing (otherwise known as FOIL).  

Vocabulary: like radicals, unlike radicals

Sections Covered in Textbook:

11-4: Operations with Radical Expressions (pages 600-606)

Resources & Tutorials:

1) How to add radicals together with like radicands?
2) How do you subtract radicals with like radicands? 
3) How do you subtract radicals with different radicands? 
4) How to use the distributive property with radicals?
5) How to "FOIL" with radicals
6) Divide by Conjugate Method (will do tomorrow)

Distance and Midpoint Formulas

Topics for Today:

We continued with applications of square roots today and how it applies to geometric concepts.  The distance formula can be used to find the length of any line segment that is plotted on a coordinate plane.  The distance formula is a direct application of the Pythagorean Theorem.

The midpoint formula is another geometric concept.  The midpoint of a line segment divides that segment exactly in half.  To find the midpoint of a line segment, we are basically taking the average of the coordinates of the endpoints.  

Vocabulary:  distance formula, midpoint, midpoint formula

Sections Covered in Textbook:

11-3: The Distance and Midpoint Formulas (pages 591-597)

Resources & Tutorials:

1) What is the distance formula?
2) What is the midpoint formula? 
3) How to find the coordinate of a midpoint given endpoints.

The Pythagorean Theorem

Topics for Today:

A special relationship exists with the lengths of the sides of a right triangle.  A famous Greek mathematician and philosopher by the name of Pythagoras proved its existence many years ago, although there is evidence that the ancient Babylonians knew of the relationship many centuries before.

The theorem states that if you have a right triangle (a triangle with one 90-degree angle), that the sum of the squares of its sides is equal to the square of the hypotenuse (the longest side).

Vocabulary: hypotenuse, leg, Pythagorean Theorem

Sections Covered in Textbook:

11-2: The Pythagorean Theorem (pages 584-590)

Resources & Tutorials:

1) What is the Pythagorean Theorem?
2) If you have the sides of a triangle, how can you tell if it's a right triangle?
3) Math is Fun - Pythagorean Triples

Simplifying Radicals Parts 1 & 2

Topics for Today:

We began our unit on radical expressions and equations today with an exploration of the process of simplifying radicals.  Just like other mathematical expressions, we have rules for what constitutes a radical in "simplest" form.  We will be spending two class periods learning about simplifying radicals. 

Like other algebraic concepts, there are properties that apply to radicals.

Vocabulary:  radical expression, rationalize

Sections Covered in Textbook:

11-1:  Simplifying Radicals (pages 578-583)

Resources & Tutorials:

1) What is the product property of square roots?
2) How do you use the product property of radicals to simplify a radical?
3) How do you multiply radicals?

Vertex Form of a Parabola

Topics for Today:

Today we explored the vertex form of a quadratic function.  Just like linear functions that have multiple forms that are each useful for certain things (slope-intercept, standard, point-slope), quadratic functions also have multiple forms (standard and vertex) that are used for different purposes.  Up to this point we have only used standard form.  

The vertex form of a parabola is very useful because it is very easy to locate the parabola's vertex, and when exploring families of graphs it is easy to see how translations (vertical and horizontal shift as well as vertical shrink or stretch) change the size and location of the graph.  

Sections Covered in Textbook:

Concepts pulled from outside materials

Resources & Tutorials:

1) How do you convert a quadratic equation from vertex form to standard form?

2) Finding the vertex of a parabola in standard form