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?

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.

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 Radical Expressions 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).  

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 tomorrow.)

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)

The 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 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?