# Math

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## Math Challenges for the Critical Thinker by Mark Twain Media

Challenge students to think outside of the box! Supplement curriculum with creative math that will enhance students' abilities to problem solve, learn and apply strategies, and think critically. Answer keys are included.

## Math: Consider the vectors vector u = < 1,-1, 3 >, vetor v = < 1, k, k >, vector...

You need to consider the condition for the vectors `< 1,-1, 3 >,< 1, k, k >,< k, 1, k^2 >` to be linearly independent to form a basis on `R^3` , hence, you need to evaluate the following determinant, such that:

`Delta = [(1,-1,3),(1,k,k),(k,1,k^2)]`

The problem provides the information that the vectors form a basis on `R^3` , hence, `Delta != 0` , such that:

`Delta = k^3 + 3 - k^2 - 3k^2 - k + k^2`

`{(Delta = k^3 - 3k^2 - k + 3),(Delta != 0):} => k^3 - 3k^2 - k + 3 != 0`

`k^2(k - 3) - (k - 3) != 0 => (k - 3)(k^2 - 1) != 0`

`k - 3 != 0 => k != 3`

`k^2 - 1 != 0 => k^2 !=` ` => k != +-1`

Hence, evaluating k for the vectors to form a basis...

## Strategies for Teaching Mathematics

Build students' understanding of mathematical concepts. Choose from a wide range of easy-to-implement strategies and model lessons that enhance mathematics instruction. Topics include: using manipulatives developing mathematical vocabulary teaching procedures developing problem solving abilities using games assessing mathematics thinking Differentiation is included in every lesson as well as current research and background information. Includes Teacher Resource zip file with reproducibles including rubrics and assessments.

## Math: How do I determine if this equation is a linear function or a nonlinear function?

The easiest way I have for knowing the difference between linear and nonlinear is the exponent value on the variable x

It is important to understand the root word in linear.  It is LINE.  A straight line, no curves.

For example:

y = 2x - 3  This is linear because the exponent on x is one. Thus your slope is standard rise over run, like a stair step and simply goes up or down.

Y = x^2 + x + 4 is nonlinear.  When graphed it becomes a parabola, which looks like a hill on your graph.  This is because the exponent on the variable of x is more than one.

This pattern...

STRAND: Geometry BAND: Transformations- 8.G. 9 Aim: Students will be able to identify and describe reflections over a given line

## Calculating the Slope of a Line

The slope of a line or line segment is the value that describes its steepness, incline, or grade. Think of a skier or snowboarder going down a ski slope. The steeper the hill, the higher the value of the slope will be.

Key Points

• You’ll need to know the coordinates of two points on a line to calculate its slope.
• The slope of a horizontal line is zero because a horizontal line has no rise.
• The slope of a vertical line is undefined because change in x (Δx) is zero, and you can’t divide by zero.
• A line with a slope that goes up as it goes to the right on a graph will have a positive slope, while one that goes down to the...

A presentation of a quadratic formula in exercise form.

## Order of Operations in Mathematics

The order of operations is the set of rules that tells you which parts of an equation to work on in what order so you will get the correct result when solving it. The order you do things in is critical in order to get the right results—work in the wrong order and you will likely get an incorrect answer.

Mastering the order of operations is one of the most important and often used skills you will acquire in solving mathematical equations consistently and successfully. With practice, it will become second nature, so don’t worry if it seems a little confusing at first.

How to Remember the Order of Operations

## Pre-Algebra by Mark Twain Media

Strengthen problem-solving abilities, increase positive attitudes, and encourage in-depth study. Includes answer key.

## Mental Calculation Drills

Mental calculation drills for addition, subtraction, multiplication, and division. Read orally and have students solve. Great for mental mastery.

## What Are Exponents (Math)?

An exponent is a shorthand way of expressing a number multiplied by itself a given number of times. An exponent is made up of two numbers: the base, which is the number multiplied by itself; and the exponent, the smaller superscript number that tells you how many times the base is multiplied by itself. The exponent is written as a smaller superscript number next to the normal-sized base number or variable.

Essential Steps

• When multiplying two exponents with the same base, the result is the base to the power of the sum of the exponents.
• When you have an exponent to the...

## Writing Proportions in Math

A proportion is a comparison between two ratios that are equal to each other. You will normally use proportions to solve for a single unknown between the two ratios. Remember, a ratio compares two values in a single fraction, whereas a proportion compares two ratios.

Use cross-multiplication to solve proportion problems. Writing out the ratio in words when setting up the equation will help you get the correct answer. Always be careful not to mix up your numerators and denominators. Be sure to understand the difference between ratios and proportions and when to use each of them. Most proportion exercises you’ll encounter will come in the form of story problems, so be sure to double-check how you set up...

## What Is the Distance Formula (Math)?

Need help in understanding general math concepts? Download this free eNotes guide to learn about the distance formula.

The distance formula is an equation used to find the distance between any two points on a graph. The distance formula is derived directly from the Pythagorean theorem which tells you the length of the hypotenuse c if you know the length of each of the other two sides of a triangle.

Key Points

• You’ll need to know the full coordinates of two points on a graph to solve for the distance.
• Distance is always expressed as a positive number, even if some or all the coordinates of your points are found in the negative...

## Math: Consider the vectors vector u = < 1,-1, 3 >, vetor v = < 1, k, k >, vector...

You need to evaluate the cross product of vectors `bar v` and `bar w` , such that:

`bar v x bar w = [(bar i, bar j, bar k),(1,k,k),(k,1,k^2)]`

`bar v x bar w = k^3 bar i + bar k + k^2 bar j - k^2 bar k - k bar i - k^2 bar j`

`bar v x bar w = (k^3 - k) bar i + (k^2 - k^2) bar j + (1 - k^2) bar k`

`bar v x bar w = (k^3 - k) bar i + (1 - k^2) bar k`

You need to evaluate the cross product `bar u*(bar v x bar w)` , such that:

`bar u*(bar v x bar w) = (bar i - bar j + 3 bar k)((k^3 - k) bar i + (1 - k^2) bar k)`

`bar u*(bar v x bar w) = 1*(k^3 - k) - 1*0 + 3*(1 - k^2)`

`bar u*(bar v x bar w) = k^3 - k - 3(k - 1)(k + 1)`

`bar u*(bar v x bar w) = k(k - 1)(k + 1) - 3(k - 1)(k...

## Fraction (Negative Exponent Rule)

This document provides a rule for working with a negative exponent. When a number is a fraction, for example (3/5), the negative exponent rule will be applied. This document contains an example of working with the rule.

## Math: How do you solve and graph a simple y=mx+b equation?For example: y=5x+-6 I don't get...

You have to note that y=mx+b is a linear equation, so the graph is a straight line.

In order to build the graph, you have to plug in values for x, and find the values for y.

For tracing the line, it's necessary to have two points. That means having 4 coordinates. You know that a point is defined by its coordinate (x,y).

y=5x-6

For x = 0

y = 5*0 -6

y = -6

The point has the following coordinates: (0,-6)

For x = -1

y = 5*(-1) -6=-5-6=-11

The point has the following coordinates: (-1,-11)

Now, in order to graph, trace the system of coordinates, which means two rectangular axes. ox is perpendicular to oy. Choose an unit of measure (for example, if you're...

## Quiz: Factoring

Factor each expression.

## Math: We have two points A ( 3,4,-5 ) and B (7,-3,8) what are the parametric equation and...

You need to find the direction vector of the line AB, such that:

`bar (AB) = (x_B - x_A)bar i + (y_B - y_A)bar j + (z_B - z_A)bar k`

`bar (AB) = (7 - 3)bar i + (-3 - 4)bar j + (8 + 5)bar k`

`bar (AB) = 4bar i - 7bar j + 13bar k`

You need to find the parametric equations of the line AB, such that:

`x = x_A + (x_B - x_A)*t => x = 3 + 4t`

`y = y_A + (y_B - y_A)*t => y = 4 - 7t `

`z = z_A + (z_B - z_A)*t => z = -5 + 13t`

Hence, evaluating the parametric equations of the line AB, yields `x = 3 + 4t , y = 4 - 7t , z = -5 + 13t.`

## Math: Assume that the bird instinctively chooses a path that will minimize its energy...

First let's set up the problem

Let x be the distance from C to B.  Then 14-x is the distance from C to D.
So the bird flies sqrt(5^2+x^2) to reach C, and then 14-x from C to D along the shoreline.

a)  E = 1.1sqrt(25+x^2)+14-x

(dE)/(dx) = 1.1(2x)/(2sqrt(25+x^2))-x = (1.1x)/(sqrt(25+x^2))-1

At a critical point (dE)/(dx) = 0 so (1.1x)/(sqrt(25+x^2))-x =0

We need to solve for x.

`1.1x=sqrt(25+x^2)` , `1.21x^2=25+x^2` ,` 0.21x^2 = 25` , `x^2=25/0.21` , `x ~~10.911"km"`

Plugging this into E we get E=16.291

We should also check x=0 and x=14, x=0 we get E=19.5, and x=14 E=16.352 so the answer to a) is C should be 10.911km.

b) `E = Wsqrt(25+x^2) + L(14-x)`

So...

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