Simplifying Radicals Parts 1 and 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 Quadratic Function

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

Completing the Square

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

Using the Discriminant

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 a perfect square, 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?

Using the Quadratic Formula

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?

Factoring to Solve Quadratic Equations

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) What is the zero-product property?

2) How do you use the zero product property to find solutions to an equation?

3) How do you solve a quadratic equation by factoring?

Solving Quadratic Equations

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)  How do you solve a quadratic equation with two solutions by graphing?

2)  How do you use the square root method to solve a quadratic equation with two solutions?

Finding and Estimating Square Roots

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?

Quadratic Functions

Topics for Today:

Quadratic functions are still the topic of the day.  Today we worked with the axis of symmetry and used it to find our vertex.  Because parabolas are symmetric, we are able to find points on one side of the axis of symmetry and reflect them to the other side of the axis of symmetry.  Once we have the vertex, and a few points on either side of the axis of symmetry, we can easily draw our parabola.

Sections Covered in Textbook:

10-2: Quadratic Functions (pages 517-523)

Resources & Tutorials:

1) How do you find the axis of symmetry?
2) Find the axis of symmetry and your vertex