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

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?