Sapere Aude

“Dare to Know”

Spot the Error

x = yx2 = (x*y)

x2 - y2 = (x*y) - y2

(x+y) (x-y) = y(x-y) (x+y) = yx+x = x2x = x2 = 1

Math Magic!• Start with any number.• Add 12.• Multiply by 4.• Subtract 8.• Divide by 2.• Subtract your original number. • Subtract 6.• Subtract your original number.• Divide by 2.

And the surprise answer is…

Why Learn Math?

• $$$ (Careers that make ‘bank’)• Pure enjoyment• Applicability in everyday life • Develop ability to think LOGICALLY!

Jordan Woy, a highly respected sports agent and a principal in the sports marketing/management firm of Schlegel Sports. Jordan has represented numerous high profile athletesHere is what Jordon had to say:

I think there are several reasons why so many athletes “go broke”. First, whether it is a lottery winner, an athlete or a star entertainer, if they are not equipped with the knowledge on how to make and save money they are in trouble. When they didn’t earn it through disciplined business practices and they don’t have those skills they usually go through it quickly. Most lottery winners or athletes make a great deal of money in a short period of time. They start spending it on things that only go down in value (cars, jewelry, partying, entourage, etc) and start to evaporate the money they do have. They can carry this off until they stop earning big money. This is when the trouble starts. It is hard to believe that MC Hammer, Mike Tyson, Evander Holyfield and now Ed McMahon are broke…

Most athletes play for four to ten years if they are lucky. After they pay taxes (can be 40 to 50%) and agent fees and buy their first homes, cars, outfits, jewelry (plus, cars, clothes and jewelry for friends and family), they are left with very little….

However, if athletes educate themselves, learn money management skills and make smart, safe investments along the way, they are usually in very good shape.

So what will we be learning in this class?

• The difference between ratios and proportions and their application

• A little geometry (you’ll be ahead of the high school game)

• Basic probability theory…FUN stuff! (you seriously will love it, or your money back guaranteed)

1. Use ratios and rates to solve real-life problems.

2. Solve proportions.

A ratio is the comparison of two numbers using division.

For example: Your school’s basketball team has won 7 games and lost 3 games. What is the ratio of wins to losses?

Because we are comparing wins to losses the first number in our ratio should be the number of wins and the second number is the number of losses.

The ratio is games won___________games lost

= 7 games_______3 games

= 7__3

Ratio

Comparing two things by division

A fraction is a type of ratio that compares the part to the whole

5 to 8

5:8

5/8

Stats from Last Night’s Game

GirlsTigrett 35, Rose Hill 32Scoring: T - Quinn Thomas 23, Sydni Spraggins 10, Beard 2; RH - Jada Perkins 10, Magee 7, Beard 5, Fuller 4, Holt 4, Bates 2. 3-pointers: T - Spraggins; RH - Beard. Halftime score: T 19-17. Record: RH 5-4.

BoysTigrett 52, Rose Hill 47Scoring: T - Jaylen Barford 26, Malik Hicks 13, Bond 8, Cross 3, Love 2; RH - Landon Simmons 13, Kendell Walker 11, Theus 7, Beauregard 8, Ferguson 4, Perkins 2, Beard 2. 3-pointers: RH - Simmons, Walker. Halftime score: RH 19-18. Record: RH 5-4.

In a ratio, if the numerator and denominator are measured in different units then the ratio is called a rate.

A unit rate is a rate per one given unit, like 60 miles per 1 hour.

Example: You can travel 120 miles on 6 gallons of gas. What is your fuel efficiency in miles per gallon?

Rate = 120 miles________6 gallons = ________20 miles

1 gallon

Your fuel efficiency is 20 miles per gallon.

Rate

Ratio of two quantities measured in different units.

A unit rate is a special rate where the denominator is 1.

60 miles per hour

20 miles for every gallon

60 minutes in one hour

Writing the units when comparing each unit of a rate is called unit analysis.

Example: How many minutes are in 5 hours?

To solve this problem we need a unit rate that relates minutes to hours. Because there are 60 minutes in an hour, the unit rate we choose is 60 minutes per hour.

5 hours • 60 minutes________1 hour

= 300 minutes

HomeworkPage 272-273

Ratios: 10, 11Unit Rate: 22, 23Equivalent Rate: 30, 31

Bell RingerGiven the following sets of points,

give the slope of each line:

Line 1: (0,4) and (9,12)Line 2: (-4,2) and (2, -3)Line 3: (3, 5) and (3,8)

Writing the units when comparing each unit of a rate is called unit analysis.

Example: How many minutes are in 5 hours?

To solve this problem we need a unit rate that relates minutes to hours. Because there are 60 minutes in an hour, the unit rate we choose is 60 minutes per hour.

5 hours • 60 minutes________1 hour

= 300 minutes

Bell RingerUse a ‘conversion factor’ to solve

the following conversion:

$43 = ? Dollars1 day 1 week

An equation in which two ratios are equal is called a proportion.

A proportion is written by setting two fractions EQUAL:

a___ ___=b

cd

Read IT Correctly spell and pronounce the term

Write IT

Write the definition of the term in their own words

Draw IT

Represent the term through examples and visuals

To solve problems which require the use of a proportion we can use one of two properties.

The reciprocal property of proportions.

If two ratios are equal, then their reciprocals are equal.

The cross product property of proportions.

The product of ad=bc.

Example:

x

35

3

5

355

3 x

Write the original proportion.

Use the reciprocal property.

35355

335

x Multiply both sides by 35 to isolate the variable, then simplify.

x21

Example:

9

62x

x 629

Write the original proportion.

Use the cross product property.

6

6

6

18 x Divide both sides by 6 to isolate the

variable, then simplify.

x3

If the average person lives for 75 years, how long would that be in seconds?

If the average person lives for 75 years, how long would that be in seconds?To solve this problem we need to convert 75 years to seconds. We can do this by breaking the problem down into smaller parts by converting years to days, days to hours, hours to minutes and minutes to seconds.

There are 365.25 days in one year, 24 hours in one day, 60 minutes in 1 hour, and 60 seconds in a minute.

minute 1

seconds 60

hour 1

minutes 60

day 1

hours 24

year 1

days 25.365years 75

Multiply the fractions, and use unit analysis to determine the correct units for the answer.

2366820000seconds

John constructs a scale model of a building. He says that 3/4th feet of height on the real building is 1/5th inches of height on the model.

What is the ratio between the height of the model and the height of the building?

If the model is 5 inches tall, how tall is the actual building in feet?

What is the ratio between the height of the model and the height of the building?

What two pieces of information does the problem give you to write a ratio?

For every 3/4th feet of height on the building…

the model has 1/5th inches of height.

Therefore the ratio of the height of the model to the height of the building is…

feet 4

3

inches 51

3

4

5

1

feet 15

inches 4 This is called a scale factor.

If the model is 5 inches tall, how tall is the actual building in feet? To find the actual height of the building, use the ratio

from the previous step to write a proportion to represent the question above.

x

inches 5

feet 15

inches 4

1554 x

4

75

4

4x

feet 75.18x

Use the cross product.

Isolate the variable, then simplify.

Don’t forget your units.

HomeworkPage 283

Comparing ratios to see if they are proportions: 12, 19Solving proportions: 20, 31Solving ‘two-step’ proportions: 37, 40

Optional (for string): 41