Display Title

Closed Captioned Video: Algebra Nspirations: Data Analysis and Probability

Closed Captioned Video: Algebra Nspirations: Data Analysis and Probability

What are the two meanings of statistics? What does it really mean that an event has a 50% probability of occurring? Why are data analysis and probability always taught together? Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video answers these questions and addresses fundamental concepts such as the law of large numbers and the notion of regression analysis. Both engaging investigations are based on true stories and real data, utilize different Nspire applications, and model the seamless connection among various problem representations. Concepts explored: statistics, data analysis, regression analysis.

Exponential Functions

Brief Review of Exponents

When a number is raised to a power, exponents are involved. Here are the key components of an exponential expression.

Exponents make it easier to write expressions that would otherwise be cumbersome to write. See how much easier it is to write 3¹⁰ instead of the expression on the right?

Exponents are a different form of writing numbers. With this new form come new properties that are unique to exponents. The next section goes over these properties, also known as the Laws of Exponents.

Laws of Exponents

The Laws of Exponents are a set of properties that pertain to all exponential expressions. Study the definition below.

Laws of Exponents.
The rules for simplifying exponents when multiplying or dividing expressions with a common base and different exponents.

To learn more about exponents, click on this link. It is a presentation that goes over the Laws of Exponents in more detail. To see examples that use the Laws of Exponents, click on this link.

Exponential Expressions

Example 1

What is the value of x?

In order to work with the Laws of Exponents, make sure the terms have the same base. Both 8 and 32 can be written as powers of 2:

8 equals 2 cubed and 32 equals 2 to the power of 5

Rewrite the original expression with these powers of 2:

With a common base, you can now solve for x:

Solving the equation 3 x = 5.

Example 2

If 4x - 5y = 10, what is the value of this expression?

16 to the power of x over 2 to the power of y

Find a common base for the numerator and denominator, then simplify the expression:

2 to the power of 4 x over 2 to the power of 5 y equals 2 to the power of 4 x minus 5 y

Do you notice that the exponent for 2 is the same as the left side of the linear equation? Replace the exponents

Solving an exponential equation.

Example 3

Rewrite this expression as an exponential expression with a power of 5.

First, write the numerical expression as an exponential expression:

fourth root of 125 equals 125 to the power of 1 fourth

Next, rewrite 125 as a power of 5:

Finally, rewrite the entire expression:

Exponential Functions

An exponential function is of this form:

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mo>•</mo><msup><mi>b</mi><mrow><mi>c</mi><mi>x</mi></mrow></msup></mstyle></math>","truncated":false}$

In this function the terms a, b, and c are numbers and x is the independent variable. Here’s one of the simplest exponential functions:

${"mathml":"<math style=\"font-family:stix;font-size:36px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"36px\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mn>2</mn><mi>x</mi></msup></mstyle></math>","truncated":false}$

This is the graph of the function.

Graph of an exponential function.

As you can see from the graph, the values of f(x) increase dramatically for small increases in input values of x. Here are some other examples of exponential functions.

Compound Interest

A special case involving exponential expressions is compound interest. What is compound interest? Usually, the context is of money earning interest in a bank account.

Here is the compound interest formula

The compound interest formula for the case where the compounding is non-continuous.

This is the formula to use when the compounding periods are non-continuous. Use this formula for continuous compounding:

The compound interest formula for the case where the compounding is continuous.

To see examples of calculating compound interest, click on this link.

Compound Interest

Example 1

A savings account earns interest according the formula below. What is the exponential function that models this growth? How much is in the savings account after five years?

“Each year the amount of the account increases by 2.5%. The initial investment was $1750.”

Use the compound interest placeholder formula:

A equals P open parentheses 1 plus r over n close parentheses to the power of n t

In this formula:

A = Current amount in savings

P = Initial investment

r = Interest rate

n = Number of compounding periods

t = Number of years

We know the following:

P = 1750

r = 0.025

n = 1

We can now generate the exponential function that models the growth:

We can use this model to find the amount in savings after 5 years:

Evaluating A of T equals 1750 times 1.025 to the power of t, for t = 5.

This is part of a collection of closed captioned videos for the Algebra Nspirations video series. To see the complete collection of the videos, click on this link.

Note: The download is Media4Math's guide to closed captioned videos.

Related Resources

To see additional resources on this topic, click on the Related Resources tab.

Closed Captioned Video Library

To see the complete collection of captioned videos, click on this link.

Video Transcripts

This video has a transcript available. To see the complete collection of video transcripts, click on this link.

Common Core Standards	CCSS.MATH.CONTENT.6.SP.B.5, CCSS.MATH.CONTENT.8.SP.A.1, CCSS.MATH.CONTENT.HSS.CP.A.1, CCSS.MATH.CONTENT.HSS.CP.A.2, CCSS.MATH.CONTENT.HSS.CP.A.3, CCSS.MATH.CONTENT.HSS.CP.A.4, CCSS.MATH.CONTENT.HSS.CP.A.5, CCSS.MATH.CONTENT.HSS.CP.B.6, CCSS.MATH.CONTENT.HSS.CP.B.7, CCSS.MATH.CONTENT.HSS.CP.B.8, CCSS.MATH.CONTENT.HSS.CP.B.9
Duration	33 minutes
Grade Range	6 - 9
Curriculum Nodes	Algebra • Probability and Data Analysis • Data Analysis • Data Gathering
Copyright Year	2020
Keywords	algebra, data analysis, probability, statistics, Closed Captioned Video

Closed Captioned Videos

Overview: Closed Captioned Videos

Media4Math has an extensive video library and these same videos are available in closed captioned format.

Using the Closed Captioned Videos

You will find the Closed Captioned Videos in the Content Showcase. From the main screen, click on the Content menu and from there select Content Showcase. In the content showcase, go to the Closed Captioned Videos collection.

You will see the complete collection of captioned videos. Here are some sample titles from this list.

To see the complete collection of videos go to this link: https://www.media4math.com/closed-captioned-videos

Percents: Fraction-Percent Conversion	Measures of Central Tendency: Weighted Mean	Percents: Calculating Tax
Integers: Variable Expressions with Integers	Anatomy of an Equation: Two-Step Equations 11	Integers: Dividing Integers
Slope Formula: Positive Slope, Coordinates in Quadrant IV	Geometry Applications: Transformations, Segment 1: Translations and Rotations.	Algebra Nspirations: Linear Functions, Segment 3
TI-Nspire Mini-Tutorial: Midpoint Formula Template	TI-Nspire Mini-Tutorial: Finding the Median of a Data List	Algebra Nspirations: Rational Functions and Expressions
Geometry Applications: Coordinate Geometry, Segment 1: Longitude and Latitude.	Anatomy of an Equation: Quadratic Equations 4	Geometry Applications: Points and Lines, Segment 1: Introduction
Place Value--Making Tens	Fractions: Multiplying Three Fractions	TI-Nspire Mini-Tutorial: Composite Functions, Linear to Polynomial
Algebra Applications: Data Analysis, Segment 2: What Is a Mortgage?	Algebra Applications: Logarithmic Functions, Segment 2: Hearing Loss	Algebra Applications: Linear Functions, Segment 1: Introduction
Geometry Applications: Polygons, Segment 3: Composite Figures	Anatomy of an Equation: One-Step Multiplication Equations	Algebra Nspirations: Solving Systems of Equations, Segment 1
Rational Numbers: Multiplying Rational Numbers	Algebra Tiles: Solving Two-Step Equations Using Algebra Tiles	Algebra Applications: Rational Functions, Segment 3: Hubble Telescope
Ratios, Proportions, and Percents: Calculating Percents	Ratios: Numerical Models for Ratios	Integers: Integers and Exponents
Measures of Central Tendency: Finding the Mean of a Data Set I	Desmos Geometry Exploration: Points	Algebra Applications: Exponential Functions, Segment 2: What Is an Earthquake?
TI-Nspire Mini-Tutorial: (CAS) Solving a Cubic Algebraically and Graphically	Algebra Applications: Systems of Equations	Slope Formula: Positive Slope, Coordinates in Quadrant II
Algebra Applications: Data Analysis, Segment 3: What Is a What is a Subprime Mortgage?	Algebra Nspirations: Exponents and Exponential Functions	Algebra Nspirations: Solving Systems of Equations, Segment 3
TI-Nspire Mini-Tutorial: Composite Functions, Linear to Rational	Geometry Applications: Transformations	Geometry Applications: 3D Geometry, Segment 2: Pyramids
The Distributive Property: a(x - b), all constants positive	One-Step Equations: Multiplication	Algebra Tiles: Modeling Quadratic Equations Using Algebra Tiles
Ratios and Rates: Rates from Data	TI-Nspire Mini-Tutorial: Composite Functions, Linear to Radical	Desmos Geometry Exploration: Lines, Rays, Segments
Anatomy of an Equation: Quadratic Equations 5	TI-Nspire Mini-Tutorial: Finding the Zeros of a Quadratic Function Using TRACE Feature	Algebra Applications: Rational Functions
The Distributive Property: a(x + b), all constants positive	Anatomy of an Equation: Two-Step Equations 7	TI-Nspire Mini-Tutorial: Creating a Random Number Table
Linear Functions: Negative Slope, Positive y-Intercept	Quadratics: Using the Discriminant to Test for Solutions to Quadratic Equations	Algebra Applications: Quadratic Functions, Segment 4: Medicine
Measures of Central Tendency: Finding the Median of a Data Set	Algebra Applications: Data Analysis, Segment 1: Introduction	Algebra Applications: Functions and Relations, Segment 4: Elliptical Paths
The Distributive Property: a(bx + c), a negative, b and c positive	Geometry Applications--What Are Prisms?	Measures of Central Tendency: The Mean and Normally Distributed Data
Algebra Applications: Systems of Equations, Segment 1: Profit and Loss	Desmos Geometry Exploration: Intersecting Lines and Planes	Place Value--Counting by 10's
Geometry Applications: Quadrilaterals, Segment 3: Parallelograms and Trapezoids	Place Value: Comparing and Ordering Decimals Using Place Value	Measures of Central Tendency: Mean of a Probability Distribution
Percents: The Percent One Number is of Another	Algebra Nspirations: Functions and Relations	TI-Nspire Mini-Tutorial: Exploring Quadratic Functions in Standard Form Using Sliders
Integers: Comparing and Ordering Integers	Algebra Applications: Quadratic Functions, Segment 1: Introduction	One-Step Equations: Subtraction
Fractions: Adding Fractions by Using Equivalent Fractions I	Percents: Applications of Percent -- Grade	Quadratics: Solving Quadratic Equations in Standard Form by Completing the Square
Percents: Percents and Multiples of Numbers	Algebra Nspirations: Inequalities	Geometry Applications--Antiprisms
Slope Formula: Negative Slope, Coordinates in Quadrants III and IV	TI-Nspire Mini-Tutorial: Finding the Intersection Point of Two Graphs	Fractions: Multiplying Two Fractions
TI-Nspire Mini-Tutorial: Finding the Zeros of a Quartic Function Using Function Intersection	TI-Nspire Mini-Tutorial: Matrix Addition and Multiplication (3 x 3 Matrices)	Fractions: Multiplying by Fractions Greater Than 1
Anatomy of an Equation: Two-Step Equations 10	Two-Step Equations: Multiplication and Subtraction	Anatomy of an Equation: One-Step Addition Equations
Fractions: Dividing Unit Fractions by Whole Numbers	Fractions: Adding Mixed Numbers by Using Equivalent Fractions II	Geometry Applications: 3D Geometry
Algebra Applications: Systems of Equations, Segment 3: Ballistic Missiles	Ratios: Equivalent Ratios	TI-Nspire Mini-Tutorial: (CAS) Adding and Subtracting Rational Expressions
TI-Nspire Mini-Tutorial: Graphing a Scatterplot Using a Graphs and Geometry Window	Place Value: Comparing and Ordering Whole Numbers Using Place Value	Algebra Tiles: Solving One-Step Equations Using Algebra Tiles
Anatomy of an Equation: Two-Step Equations 9	Ratios and Proportions: What Are Proportions?	Slope Formula: Negative Slope, Coordinates in Quadrant III
Fractions: Adding Fractions by Using Equivalent Fractions II	The Distributive Property: a(bx - c), all constants negative	Algebra Nspirations: Inequalities, Segment 3
Fractions: Dividing Two Fractions	TI-Nspire Mini-Tutorial: (CAS) Solving a Quadratic Algebraically and Graphically	Fractions: Subtracting Mixed Numbers by Using Equivalent Fractions I
Percents: Calculating Commissions and Tips	TI-Nspire Mini-Tutorial: Graphing a Linear Function	Geometry Applications: Polygons, Segment 1: Introduction
Linear Functions: Positive Slope, Zero y-Intercept	Algebra Nspirations: Variables and Equations, Segment 3	Fractions: Subtracting Mixed Numbers by Using Equivalent Fractions II
Geometry Applications: Transformations, Segment 3: Rotations, Reflections, and Symmetry.	Algebra Applications: Functions and Relations, Segment 2: Parabolic Paths	Algebra Nspirations: Data Analysis and Probability, Segment 1
Percents: Percent Increase	TI-Nspire Mini-Tutorial: Exploring Radical Function Graphs with Sliders	Anatomy of an Equation: Two-Step Equations 5
Algebra Tiles: Modeling One-Step Equations Using Algebra Tiles	TI-Nspire Mini-Tutorial: Graphing a Linear Inequality	Geometry Applications: Circles, Segment 3: Chords and Inscribed Angles
TI-Nspire Mini-Tutorial: Composite Functions, Linear to Trigonometric	TI-Nspire Mini-Tutorial: Using the Coordinate TRACE Feature	Algebra Applications: Quadratic Functions, Segment 2: Pyrotechnics
TI-Nspire Mini-Tutorial: Exploring Logarithmic Graphs with Sliders	Linear Functions: Positive Slope, Negative y-Intercept	TI-Nspire Mini-Tutorial: A Probability Simulation of Rolling Two Dice Multiple Times (with Histogram)
Algebra Applications: Exponential Functions	TI-Nspire Mini-Tutorial: (CAS) Finding the Real and Complex Roots of a Cubic	TI-Nspire Mini-Tutorial: Composite Functions, Linear to Exponential
Geometry Applications: Quadrilaterals, Segment 1: Introduction	Quadratics: Solving Quadratic Equations with One Solution	Measures of Central Tendency: Sample Mean
Place Value: Rounding Whole Numbers	Integers: Adding Integers	TI-Nspire Mini-Tutorial: Composite Functions: Linear to Quadratic
Anatomy of an Equation: Two-Step Equations 6	Algebra Applications: Inequalities, Segment 1: Hybrid Cars	Rational Numbers: Rational Numbers and Irrational Numbers
Algebra Applications: Variables and Equations, Segment 3: River Ratios	Ratios and Rates: Converting Measurement Units	TI-Nspire Mini-Tutorial: Composite Functions, Linear to Quartic
TI-Nspire Mini-Tutorial: Finding the Zeros of a Cubic Function Using Function Intersection	Counting On	Integers: Numerical Expressions with Integers
Anatomy of an Equation: Linear Equations in Standard Form to Slope-Intercept Form 2: Ax + By = -C	Linear Functions: Positive Slope, Positive y-Intercept	Rational Numbers: Dividing Rational Numbers

Display Title Closed Captioned Video: Algebra Nspirations: Data Analysis and Probability

Display Title

Closed Captioned Video: Algebra Nspirations: Data Analysis and Probability

Exponential Functions

Brief Review of Exponents

Laws of Exponents

Exponential Functions

Compound Interest

This is part of a collection of closed captioned videos for the Algebra Nspirations video series. To see the complete collection of the videos, click on this link.

Note: The download is Media4Math's guide to closed captioned videos.

Related Resources

To see additional resources on this topic, click on the Related Resources tab.

Closed Captioned Video Library

To see the complete collection of captioned videos, click on this link.

Video Transcripts

This video has a transcript available. To see the complete collection of video transcripts, click on this link.

Overview: Closed Captioned Videos

Using the Closed Captioned Videos

© Media4Math. All rights reserved

Display Title

Closed Captioned Video: Algebra Nspirations: Data Analysis and Probability