A Review of Algebra Fundamentals

In this lesson, you will review mathematical models that you were

introduced to in Algebra I.

Michigan Department of Education High School Algebra II Content Expectations

STANDARD A1: EXPRESSIONS, EQUATIONS AND INEQUALITIES

• A1.1 Construction, Interpretation, and Manipulation of Expressions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)

• A1.1.4 Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x – 1) (1 – x2 + 3))

• A1.1.5 Divide a polynomial by a monomial.• A1.2 Solutions of Equations and Inequalities (linear,

exponential, logarithmic, quadratic, power, polynomial, rational, and trigonometric)

• A1.2.5 Solve polynomial equations and equations involving rational expressions (e.g. solve -2x(x2 + 4x+3) = 0), and justify steps in the solution.

Michigan Department of Education High School Algebra II Content Expectations

STANDARD A3: MATHEMATICAL MODELING• A3.1 Models of Real-world Situations Using Families of

Functions (Example: An initial population of 300 people grows at 2% per year. What will the population be in 10 years?)

• A3.1.1 Identify the family of function best suited for modeling a given real-world situation (e.g., quadratic functions for motion of an object under the force of gravity; exponential functions for compound interest; trigonometric functions for periodic phenomena.)

• A3.1.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers.

• A3.1.3 Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled.

Here’s how it works.

• Individually, you will navigate through all examples and complete the assessment at the end of this lesson.

• Take as much time as you need to complete this lesson (feel free to reread, review, and take notes)

• It is also recommended that you have a calculator with you as you review more challenging concepts.

• Terms underlined in orange are links to additional resource pages.

• Please click to return to the first page, and use the arrow buttons to move forward and backward.

At the end of this lesson, you will:

Understand the relevance of mathematical applications in real life situations through examining real-life examples (i.e. price-demand model, compound-interest formula)

Define mathematical model and understand the process of examining a given situation or “real-world” problem by developing an equation, formula, table, or graph that correctly represents the main features of the proposed situation.

Learning the Lingo

• A mathematical model is a mathematical structure that approximates the important features of a situation.

• It may be an equation or a set of equations, a graph, table, chart, or any of several other similar mathematical models.

• The process of examining a given situation or “real-world” problem and then developing an equation, formula, table, or graph that correctly represents the main features of the situation is called mathematical modeling.

Learning the Lingo

• Causal models are based on the best information and theory currently available.

• Causal models allow predictions or “educated guesses” to be made that are often close to the actual result observed.

• Many models used in business, industry, and laboratory environments are not able to give definitive answers (i.e. weather forecast)

Learning the Lingo

• Descriptive models are models in which just a few simple variables can be easily measured.

• Example: The formula for the area of a rectangle, A=lw (A = area, l = length, w = width) models the area of a rectangle and can be used to calculate the area of a rectangular figure precisely.

Time to make your brain BIG!

EXAMPLE 1 A Price-Demand ModelSuppose a grocery store sells small bunches of flowers to its customers. The owner of the store gathers data over the course of a month comparing the demand for the flowers (based on the number of bunches sold) to the prices being charged for the flowers. If he sells 15 bunches of flowers when the price is $2 per bunch but only 10 bunches when the price is $4 a bunch, he can graph the information to model the price-demand relationship. Check it out.

“Real World” Math

EXAMPLE 1 A Price-Demand ModelThis graph shows us that, as the price increases, the demand for flowers decreases. This model can help the store owner set a reasonable price so that he will make a profit but also sell flowers before they wilt!

Formulas as Models

EXAMPLE 2 Distance Formula• The formula d = rt, where d = distance,

r = rate, t = time can be used to calculate distance.

• The Shearin family is traveling across the country to California for their summer vacation. If they plan to drive about 8 hours per day and can average 60 mph, how far can they travel per day?… good question…

Formulas as Models

• First, we need to recall the distance formula from the last page (d = rt)

• Next, we plug in, or substitute, our values, r = 60 mph and t = 8 hours

• d = (60 mph)(8 hours)

• d = 480 miles

Let’s break it down.

Now, let’s say the Shearin family decides to travel at least 600 miles per day, how long will they be traveling each day if they average 60 mph on the highway?

d = rt (distance formula)

600 mi = (60 mph)(t) (substitute given values)

(divide both sides by 60)

10 hr = t60600 = 60

60t

Check whatcha know.

You Try!• Using the distance formula, what is t (time, in

hours) when r = 50 mph and d = 125 mi?

A. 1.5 hours

B. 2.5 hours

C. 3.5 hours

Note: Use the back arrow to return to this screen if you select an incorrect answer.

Check whatcha know.

You Try!

• Using the distance formula, what is t (time, in hours) when r = 50 mph and d = 125 mi?

A. 1.5 hour

B. 2.5 hours

C. 3.5 hours

Sorry, recheck your math. Hint: The distance formula is d = rt

Check whatcha know.

You Try!

• Using the distance formula, what is t (time, in hours) when r = 50 mph and d = 125 mi?

A. 1.5 hours

B. 2.5 hours

C. 3.5 hours

Congratulations, You are correct!

Check whatcha know.

You Try!

• Using the distance formula, what is t (time, in hours) when r = 50 mph and d = 125 mi?

A. 1.5 hours

B. 2.5 hours

C. 3.5 hours

Sorry, recheck your math. Hint: The distance formula is d = rt

More Formulas as Models

• EXAMPLE 3 Perimeter• Perimeter is the total distance around the

outside of an object and can be calculated by adding together all lengths of that object.

• The formula for the perimeter of a rectangle is 2L + 2W = P.

• The perimeter of a rectangular dog pen is 40 feet. If the width is 6 feet, find the length of the pen.

Let’s break it down.

• EXAMPLE 3 Perimeter2L + 2W = P (perimeter

formula)

2L + 2(6 ft) = 40 ft (substitute given values)

2L + 12 = 40 (simplify)

2L = 28 (subtract 12 from both sides)

L = 14 ft (divide both sides by 2)

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

L = ?

W = 6 ft

P = 40 ft

More Formulas as Models

EXAMPLE 4 Formulas as ModelsFinancial planning involves putting money into sound investments that will pay dividends and interest over time. Compound interest helps our investments grow more quickly because interest is paid on interest plus the original principal (amount you started with).

The formula M= P(1+i)n, where P = principal, n = years, and i = interest rate per period, can be used to calculate the value of an investment at a given interest rate after a designated number of years.

Check it out!

This table shows the value of a $1000 investment compounded yearly at 10% for 50 years. As you can see, the growth is phenomenal between the 20th and 50th year of the investment. This is an example of exponential growth calculated using a formula.

Number of years

Compounded Yearly ($)

0 1,000.00

1 1,100.00

2 1,210.00

5 1,610.51

10 2,593.74

20 6,727.50

30 17,449.40

40 45,259.26

50 117,390.85

More Formulas as Models

• EXAMPLE 5 The Compound-Interest Formula

• Lucy deposits $2500 into a savings account that pays 4.5% interest compounded monthly. Find the value of her account after 5 years.

• The formula to calculate compound interest is:

where M = the maturity value of the account, P = $2500 (the principal/original amount), r = 4.5% or 0.045 (interest rate), n = 12 months (how often interest is compounded), and t = 5 years (length of time).

M= P (1+ nr ) nt

Let’s break it down.

• EXAMPLE 5 The Compound-Interest Formula(compound-interest formula)

M = 2500 (1 + ) (substitute values)

M = 2500 (1 + 0.00375)60 (simplify fraction and exponent)

M = $3,129.49 (round answer to the nearest penny)

M= P (1+ nr ) nt

120.045 12:5

Note: For exact results, do not round any part of the answer until you have completed the problem. Then round the final answer to the nearest penny.

Check whatcha know.

You Try!

• Using the compound-interest formula, what would the value of an account be with an original principal of $5000 with a 4.5% interest compounded monthly over 10 years (P = $5000, r = 4.5%, n = 12, and t = 10)?

A. $7,834.96 B. $7,945.96 C. $9,834.96

Note: Use the back arrow to return to this screen if you select an incorrect answer.

Check whatcha know

A. $7,834.96 B. $7,945.96 C. $9,834.96

You Try!

• Using the compound-interest formula, what would the value of an account be with an original principal of $5000 with a 4.5% interest compounded monthly over 10 years (P = $5000, r = 4.5%, n = 12, and t = 10)?

Congratulations, You are correct!

Check whatcha know.

You Try!

• Using the compound-interest formula, what would the value of an account be with an original principal of $5000 with a 4.5% interest compounded monthly over 10 years (P = $5000, r = 4.5%, n = 12, and t = 10)?

A. $7,834.96 B. $7,945.96 C. $9,834.96

Sorry, recheck your math. Note: 4.5% is 0.045 as a decimal.

Check whatcha know.

You Try!

• Using the compound-interest formula, what would the value of an account be with an original principal of $5000 with a 4.5% interest compounded monthly over 10 years (P = $5000, r = 4.5%, n = 12, and t = 10)?

A. $7,834.96 B. $7,945.96 C. $9,834.96

Sorry, recheck your math. Note: 4.5% is 0.045 as a decimal.

Let’s sum it all up!

• Take this opportunity to review previous pages or to check notes before you start your quiz.

• After you feel comfortable with the material, please click on the link below to start your quiz.

• Please note: you may refer to your notes for the compound-interest formula.

• Click here to take your Check Quiz. Good luck!

Note: If you have any additional comments or questions, please post them to Moodle.

Works and Links Consulted

"Causal Models - How to Structure, Represent and Communicate Them - a Knol by Paul Duignan, PhD." Knol - a Unit of Knowledge: Share What You Know, Publish Your Expertise. Web. 09 Aug. 2011. <http://knol.google.com/k/causal-models-how-to-structure-represent-and-communicate-them>.

"Mathematical Models." Math Is Fun - Maths Resources. Web. 09 Aug. 2011. <http://www.mathsisfun.com/algebra/mathematical-models.html>.

"Mathematical Model." Wikipedia, the Free Encyclopedia. Web. 09 Aug. 2011. <http://en.wikipedia.org/wiki/Mathematical_model>.

"MDE - Mathematics." SOM - State of Michigan. Web. 09 Aug. 2011. <http://www.michigan.gov/mde/0,1607,7-140-38924_41644_42668---,00.html>.

"Perimeter." Wikipedia, the Free Encyclopedia. Web. 09 Aug. 2011. <http://en.wikipedia.org/wiki/Perimeter>.

"STAIR Quiz | Poll Everywhere." Text Message (SMS) Polls and Voting, Audience Response System | Poll Everywhere. Web. 09 Aug. 2011. <http://www.polleverywhere.com/survey/BEqG3lxqr>.

Timmons, Daniel L., Catherine W. Johnson, and Sonya M. McCook. Fundamentals of Algebraic Modeling: an Introduction to Mathematical Modeling with Algebra and Statistics. Belmont, CA: Brooks/Cole, 2010. Print.