Operations with Radical Expressions - April 30th

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).  

Finally, we discussed how to manage fractions that have binomials in the denominator that contain radicals.  We can multiply by the conjugate, which results in the difference of squares and the removal of the radical.   (*We did not get to this concept today - we will tackle it on Monday.)

Vocabulary: like radicals, unlike radicals, conjugate 

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 on Monday)
7) Video recording of today's class (see Google Classroom for password).

Assignments:

1) Operations with Radical Expressions (no conjugates)

The Distance and Midpoint Formulas - April 29th

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.
4) Distance and Midpoint Formulas Class Notes
5) Video recording of today's class. (see Google Classroom for password).

Assignments:

1) Distance and Midpoint Formulas Worksheet
2) Chapter 10 Test Corrections due Thursday

Pythagorean Theorem - April 28th

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
4) Pythagorean Theorem Class Notes
5) Recording of Today's Class (see Google Classroom for Password)

Assignments:

1) Pythagorean Theorem Worksheet
2) Chapter 10 Test Corrections due Thursday

Simplifying Radicals Part 2 - April 27th

Topics for Today:

Today we worked more on simplifying radicals, with an emphasis on division.  We also defined what it means for radicals to be in simplest form.

Summary:  Radicals in Simplest Form.

A radical expression is in simplest form when all 3 conditions are true:

1. The radicand has no perfect square factors (other than 1).
2. The radicand has no fractions.
3. The denominator of a fraction had no radical.  

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?
4) Simplifying Radicals Part 2 Class notes.
5) Video Recap of Today's class. (see Google Classroom for Password)

Assignments:

1) Simplifying Radicals Part 2 Worksheet
2) Chapter 10 Test corrections due by Thursday

Simplifying Radicals - April 23rd

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.

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?
4) Video recap of today's class (see Google Classroom for password)

Assignments:

1) Simplifying Radicals Part 1 Worksheet
2) Chapter 10 Test Corrections due by Tuesday

Chapter 10 Test - April 22nd

Topics for Today:

We used our class time today to begin work on the Chapter 10 Test.

Sections Covered in Textbook:

No new sections covered today.

Resources & Tutorials:

None.

Assignments:

Chapter 10 Test (download from Google Classroom) due by 8pm tonight.  The method of turning is up to you:  you may scan and submit via Google Classroom, or you can simply scan and email to me (ariccabona @ fieldschoolcv.net or chickabona @ gmail.com - remove spaces first).  Please make sure that you look at your scanned copy before submitting to make sure it is legible and that your writing is dark enough for me to read.  I will grade and return tests as soon as all have been submitted to me. 

Chapter 10 Review Day 2 - April 21st

Topics for Today:

Concepts from Chapter 10 were reviewed for a second day.  Students should be reminded of the following concepts for their test:

• Axis of symmetry and vertex of a parabola
• Finding and estimating square roots
• Using the quadratic formula
• Using the discriminant
• Graphing parabolas using the vertex, y-intercept, and roots
• Solving quadratic equations using factoring, the quadratic formula, and completing the square

Sections Covered in Textbook:

Chapter 10 All but 10-9:  (Pages 510-558)

Resources & Tutorials:

See blog entries for Chapter 10 (dates March 30th - April 16th)

Assignments:

Study for Chapter 10 Test by completing Delta Math Chapter 10 Review and completing the Chapter 10 Review Sheet.  Here are the Review Sheet answers if you need them.  The test will be posted by 11:45am tomorrow.  Print and bring a copy to class with you.  Tests will be due by 8pm on Wednesday night.  

Chapter 10 Review Day 1 - April 20th

Topics for Today:

Today we began our review of Chapter 10.  This chapter was all about quadratic equations and their graphs.  When we "solve" quadratic equations, we are looking for the points where our graph either crosses or touches the x-axis.  If our graph does not touch the x-axis, we still have solutions, but they are imaginary (we'll save that topic for Algebra II, but students should know they exist!).

We used 4 methods to solve quadratic equations:

1. Graphing (using axis of symmetry, vertex, and y-intercept)
2. Factoring (not always possible)
3. Quadratic Formula (works for every equation)
4. Completing the square

We also worked with square roots (do you know your perfect squares???), as well as the discriminant of the quadratic formula to determine the types of solutions we have.  

We also approximated the square root values of irrational numbers based upon which perfect squares the irrational number fell between.  

We will spend 2 days of class time reviewing for our test on Wednesday.  

Sections Covered in Textbook:

Chapter 10 All but 10-9:  (Pages 510-558)

Resources & Tutorials:

See blog entries for Chapter 10 (dates March 30th - April 16th)

Assignments:

1) Delta Math Chapter 10 Review
2) Chapter 10 Review Sheet

Completing the Square - April 16th

Topics for Today:

Today we explored the final way to solve quadratic equations: completing the square.  We can apply our knowledge of perfect square trinomials to set our equations up so that When we take an equation of x^2+bx+c=0  and apply algebraic properties including our perfect square trinomial pattern to solve it, we call this process “completing the square”.

We complete the square to solve so that we are able to take the square root of each side of the equation to produce our solutions.  (So far we have used factoring and the quadratic formula to solve these equations).

Here is an example of completing the square:

Sections Covered in Textbook:

10-6: Completing the Square (pages 541-546)

Resources & Tutorials:

1) Solve by completing the square
2) How to use a shortcut to factor a perfect square trinomial
3) Completing the Square Class Notes
4) Video recording of today's class - see Google Classroom for the password.

Assignments:

1) Completing the Square Worksheet
**Planning ahead - Chapter 10 test next week - you will have a Delta Math assignment as part of your review.  Once posted, you should work on this prior to Monday's class.

Using the Discriminant - April 15th

Topics for Today:

The quadratic formula can be used to find the solutions of any quadratic equation that is in standard form.  There is a piece of the formula called the discriminant that is very useful to determine the types of solutions that our equation will have.   Additionally, we can tell if our equation is easily factorable by looking at the discriminant.  If the discriminant is rational, we have an easily factorable equation.

Sections Covered in Textbook:

10-8: Using the Discriminant (pages 554-558)

Resources & Tutorials:

1) What is the discriminant?
2) How do you use the discriminant to find out the number of solutions?
3) Video Recap of today's lesson - see Google Classroom for password

Assignments:

Using the Discriminant Worksheet

Using the Quadratic Formula - April 14th

Topics for Today:

One method that can be used to solve any quadratic equation is the quadratic formula.  The quadratic formula uses the coefficients from the equation to find the values for x when y is zero.  It is highly recommended that students MEMORIZE the quadratic formula.  The quadratic formula works even when we don't have real solutions (yes, there is such a thing as an imaginary number - stay tuned - you'll become very familiar with imaginary numbers in Algebra II). 

Vocabulary: quadratic formula

Sections Covered in Textbook:

10-2: Using the Quadratic Formula (pages 547-553)

Resources & Tutorials:

1) What is the quadratic formula?
2) How do you solve a quadratic equation using the quadratic formula?
3)  Quadratic Formula class notes
4) Link to today's recorded class (see Google Classroom for password).

Assignments:

Using the Quadratic Formula Worksheet

Factoring to Solve Quadratic Equations - April 13th

Topics for Today:

All of the work we have done on factoring has led to today's topic of solving quadratic equations by factoring.  We talked about the zero-product property (when multiplying, if one factor is zero, then the equation equals zero), and how we use it to find our solutions (also called roots or zeroes). 

An example of an equation that requires several steps to solve is included here:

Vocabulary:  zero-product property

Sections Covered in Textbook:

10-5: Factoring to Solve Quadratic Equations (pages 536-540)

Resources & Tutorials:

1) Video recap of Factoring to Solve Quadratic Equations
2) Notes from Factoring to Solve Quadratic Equations Presentation

Assignments:

Factoring to Solve Quadratic Equations Worksheet

Solving Quadratic Equations - April 2nd

Topics for Today:

Solving quadratic equations was the topic of the day.  We solved these equations by graphing and by using algebra.  For quadratic equations, we have three possibilities for our solutions:  we may have two solutions, one solution, or no REAL solutions.  The rules of algebra still apply when solving numerically - whatever we do to one side of the equation, must also be done to the other side to keep the truth of the equals sign.  Students were also reminded that squaring and taking the square root are inverse operations.

Sections Covered in Textbook:

10-4: Solving Quadratic Equations (pages 529-534)

Resources & Tutorials:

1)  Video Recap of Today's Lesson
2)  Notes from Quadratic Equations presentation.

Assignments:

1) Solving Quadratic Equations Worksheet
2) Complete Delta Math - Quadratic Equations and Functions Practice Set 1
3) Practice your Perfect Squares and Square Roots on Quizlet

Finding and Estimating Square Roots - April 1st

Topics for Today:

Today we discussed perfect squares and square roots.  Squaring and taking the square root are inverse operations.  Students will be asked to memorize the common perfect squares, and there is a Quizlet set that should hopefully make learning them fun.

Vocabulary: square root, principal square root, negative square root, radical, radicand, perfect squares

Sections Covered in Textbook:

10-3: Finding and Estimating Square (pages 524-528)

Resources & Tutorials:

1) What is a perfect square?
2) How do you find the square root of a perfect square?
3) How do you find the square root of a fraction?
4) How do you estimate a square root of a non-perfect square?

Assignments:

1) Finding and Estimating Square Roots Worksheet
2) Practice the Quizlet Sets on Perfect Squares and Square Roots by Friday