Division Properties of Exponents

Topics for Today:

Today we tackled the last property of exponents that deals with division.

When dividing powers with the same base, we can simply subtract the exponents.  When dividing monomials, we must match up like bases with each other, and deal with them separately.

Sections Covered in Textbook:

8-5: Division Property of Exponents (pages 417-423)

Resources & Tutorials:

1) What's the quotients of powers rule?
2) How do you divide monomials using the quotients of powers rule?
     (*This video leaves a negative exponent - that is NOT simplest form!*)

3) Division Properties of Exponents Class Notes

More Multiplication Properties of Exponents

Topics for Today:

Today we reviewed the topics relating to exponents and exponent rules, including scientific notation.  We expanded our topic of multiplication of powers to include raising a power to a power, as well as taking a monomial to a power.  When a monomial (a number, a variable, or a product of a number and variable - this also includes whole number exponents) is raised to a power, each element of that product must be raised to that power.

Sections Covered in Textbook:

8-4: More Multiplication Properties of Exponents (pages 411 - 415)

Resources & Tutorials:

1) What the power of a power rule?
2) How do you take a monomial to a power?
3) More on the power of a product rule.

4) More on Multiplication Properties of Exponents Class Notes

Multiplication Properties of Exponents

Topics for Today:

Today we discussed how to manage multiplying powers with the same base.  We looked at several examples as well as explored how to multiply numbers together that are in scientific notation.  In summary, when multiplying powers with the same base, just keep the base and add the exponents together.  This process works for both positive and negative exponents.

Graphic Credit: https://www.onlinemath4all.com/multiply-powers.html

Sections Covered in Textbook:

8-3: Multiplication Properties of Exponents (pages 405-410)

Resources & Tutorials:

1) What is the product of powers rule? 
2) How do you find the product of powers? 
3) How do you multiply numbers using scientific notation? 

4) Multiplication Properties of Exponents Class Notes

Scientific Notation

Topics for Today:

Today we used exponents in a practical way when we learned about scientific notation.  Scientific notation is simply a way to write very large and very small numbers that follow a few rules.

Simply stated, scientific notation is the product of a number and a power of 10 that follows the format: 

a x 10ⁿ  where n is an integer and 1 ≤ a < 10

Image credit: https://pt.slideshare.net/jessicagarcia62/compute-with-scientific-notation/6?smtNoRedir=1

Vocabulary:  scientific notation

Sections Covered in Textbook:

8-2: Scientific Notation (pages 400-404)

Resources & Tutorials:

1) What's scientific notation? 
2)  How do you convert decimal notation to scientific notation? 
3) How do you convert from scientific notation to decimal notation? 
4) How do you order numbers in scientific notation? 

5) Scientific Notation Class Notes

Zero and Negative Exponents

Topics for Today:

We began a discussion about powers, bases and exponents today, and focused on bases with a zero exponent as well as negative exponents.

Summary

• Any non-zero number raised to the zero power equals one!
• Negative exponents are fractions.  If a factor is moved up or down in a fraction, the sign of the exponent is changed.  

Sections Covered in Textbook:

8-1: Zero and Negative Exponents (pages 394-399)

Resources & Tutorials:

1) What do you do with a zero exponent? 
2) What do you do with a negative exponent?

3) Zero and Negative Exponents Class notes

Applications of Systems Part 2

Topics for Today:

Yesterday we began to tackle applications of systems of equations.  Basically, we are going to be solving story problems that have two unknowns, requiring us to write two equations to solve them.  These types of problems can be categorized and patterns emerge as we see more and more of these types of problems.  

We did a Desmos activity today that helped us with building equations and solving them.  

Sections Covered in Textbook:

7-4: Applications of Linear Systems (pages 362-368)

Resources & Tutorials:

1) How do you solve a word problem using two equations? 
2) Simple word problem resulting in two equations (not a video)

3) Desmos Activity on Systems of Equations (Login with Google)

Applications of Systems

Topics for Today:

One of the things that is most annoying about Algebra I is the focus on the processes and procedures for solving equations, inequalities, and problems.  Most of the time we are focused on process rather than application, but this foundational toolset is critical to solving problems requiring higher thinking and reasoning.

Today we used our knowledge of solving systems of equations to solve some real-world problems.  Typically students lack confidence when solving story problems, although it is these very problems where we get to use all the skills we have been building.  As I continually reinforce to our students, mastering Algebra requires repetition and practice, like any other skill we hope to master.  The only way to become competent and confident solving story problems is to do them -- LOTS of them.

The main thing to remember when solving the linear systems we have been working on is that if we have two variables, we will need two equations to solve.  The same would be true for three variables (a topic for Algebra II where you need three equations).

For these story problems, first, we must identify and define our variables.   Second, we will analyze the given information and write our equations based upon the given information.  Once we have our equations, we can determine the best method to solve the system.  Finally, we must look at the question that was asked and make sure that our solution answers the question, that we have the correct units, and that our answer makes sense.

Many of these story problems follow a pattern, and identifying the pattern makes the problem easier.  For this topic, we normally have several patterns to choose from:  mixtures, distance-rate-time (these can come in many forms, and can deal with things like water and wind currents that speed up or slow down the traveler), and break-even.

Sections Covered in Textbook:

7-4: Applications of Linear Systems (pages 362-368)

Resources & Tutorials:

1) How do you solve a word problem using two equations? 
2) Simple word problem resulting in two equations (not a video)
3) Applications of Linear Systems Class Notes

Systems of Linear Inequalities

Topics for Today:

We expanded our discussions about linear inequalities and systems to include the topic of systems of linear inequalities.  We discovered in our lesson yesterday that linear inequalities include all the points on one side of a border.  When we combine two linear inequalities, we are going to look for where both overlap.  The only way to represent this overlap region is by graphing.  (Recall that we discussed and practiced three different ways of solving linear systems - graphing, substitution method, and elimination method.)

The solution to the system

y < 2x + 1 and

y > 1/2 x -3 

looks like this:

The red region represents the overlap,

and therefore the solution to the system.

Vocabulary:  system of linear inequalities, solution of a system of linear inequalities

Sections Covered in Textbook:

7-6: Systems of Linear Inequalities (pages 377-384)

Resources & Tutorials:

1) What is a system of inequalities? 
2) How do you solve a system of inequalities by graphing?

3) Systems of Linear Inequalities Class Notes

Linear Inequalities

Topics for Today:

We are still finding our way around the coordinate plane, and today we discussed how to graph the solution to a linear inequality.  A linear inequality describes a region of the coordinate plane that has a boundary line.  The solution to a linear inequality are all the coordinate points that make a linear inequality true.  The solutions to inequalities contain infinitely many more solutions than that of equations, and the same is true for linear inequalities.

Solutions to linear inequalities involve graphs.  The process for graphing linear inequalities is very similar to graphing linear equations with a few additional details.  The basic process for graphing linear inequalities is

1. Treat the inequality just like an equation.  Use the equation to graph the boundary line.
2. Determine if the boundary line is a part of the solution
• For equations that are strictly greater or less than (>  or <), the boundary is NOT included and should therefore be a dashed line.
• For equations that are greater than or equal to or less than or equal to (≥ or ≤) the boundary line IS included and should therefore be drawn as a solid line.  
3. Next, determine which side of the line the solution points fall.  The best way to accomplish this is to pick a point on either side, and test the inequality for truth.  The point that generates a true statement is on the side of the line with the solution.
4. Once the correct side of the boundary is found, shade this region to indicate where the solutions are.  

Vocabulary: linear inequality, solution of an inequality

Sections Covered in Textbook:

7-5: Linear Inequalities (pages 370-375)

Resources & Tutorials:

1. What is a linear inequality?
2. How do you figure out if the boundary is part of the graph of the inequality?
3. Linear Inequalities Class Notes

Solving Systems Using Elimination

Topics for Today:

We are still working on systems of linear equations.  Today, we discussed elimination method, and with a system of two equations, this method is really the preferred one.

Steps for Solving Using Elimination Method

1. In your original system, make sure both equations are in the same form (standard form works best!).  Line your equations up so the variables are aligned in columns.  
2. Determine which variable should be eliminated.  Look for matching numbers and opposite signs or create them using multiplication.  You may have to multiply both equations so that you can eliminate one variable.  
3. Eliminate the chosen variable.  Solve for the other variable.  
4. Take the value you found in Step 3 and substitute it into one of the original equations to solve for the other variable. 
5. Identify your solution – it will be an ordered pair!
6. Check both original equations with the solution you found.  

 Graphic Credit: http://www.hotelsrate.org/solving-systems-of-linear-equations-by-elimination-examples/

Vocabulary:  elimination method

Sections Covered in Textbook:

7-3: Solving Systems Using Elimination (pages 353-359)

Resources & Tutorials:

1. How do you solve a system of equations using the elimination by addition method? 
2. How do you solve a system of equations using the elimination by multiplication method?
3. What's another way of solving a system of equations using the elimination by multiplication method?
4. Solving Systems Using Elimination Class Notes