CCM1 Unit 5

Transcript CCM1 Unit 5

CCM1 Unit 5

Warm Up – January 23 Complete the student Information sheet on your desk.

What Do I Remember  Spend 15 minutes going though the “What Do I Remember” worksheet

What you should have remembered 

 Mean- average  Median- middle number  Describing Data- SOCS  Standard Deviation- How much variation exists from the average  Range-The difference between the highest and lowest number

 Box Plot- Use the 5 number summary  Dot Plot  Histogram-

 5 Number Summary- Minimum, Q1, Median, Q3, and Maximum  IQR- The difference between Q3 and Q1  Outliers- A number that is numerically distant from the rest of the data

 Categorical – data that can be separated by category EX) sorting students by the model of car they have  Quantitative – data that can be separated by numerical values EX) sorting students by the number of siblings they have

 Algebraic Expressions- Does not have an equal sign, You evaluate Algebraic Expressions  Algebraic Equations- has an equal sign  Order of Operations- (PEMDAS)  Parenthesis  Exponents  Multiplication/Division  Addition/Subtraction  Evaluate- means to solve

 Solving Equations- Follow reverse order of operations to get the variable by its self on one side of the equation  Whole Numbers- Includes zero and all the positives (no fractions or decimals)  Integer- Includes all negatives, zero, and all positives (No fractions or decimals)  Variable- a symbol that stands for a number  Ex) x – 2 = 10 X is the variable

 Subtracting Negatives- Keep, change, change  Multiplying Negatives     Neg *Neg = Pos Neg*Pos = Neg Pos * Neg = Neg Pos*Pos = Pos  Dividing Negatives     Neg /Neg = Pos Neg/Pos = Neg Pos /Neg = Neg Pos/Pos = Pos

 Distributive Property- Distribute to EVERYTHING inside the parenthesis  Constant- just a number, does not have a variable  Coefficient- The NUMBER in front of (multiplied by) a variable  Like Terms-Terms with the same exact variables  Substitution- Putting a number in place of a variable to simplify an expression  Equivalent expression- two expressions that have the same most simplified form  Solving Equation- Follow reverse order of operations to get the variable by its self on one side of the equation

Less Than

<

 Less than or equal to

≤

 Greater Than

>

 Greater Than or Equal To

≥

 Open Circle- Use for

<

>

 Closed Circle- Use for

≤

≥

 Solving Inequalities- Solve like you solve equations; Follow reverse order of operations

 Domain  Independent Variable  X’s  Goes on x-axis  Goes in L1 when finding line of best fit  Range  Dependent Variable  Y’s  Goes on y-axis  Goes in L2 when finding line of best fit

 Function Notation  f(x) =  Example: f(x)= 2x + 4  Now-Next Rule: A recursive rule found by relating a number to what comes before it.

 Written as   Ex) NEXT = NOW + 4 NEXT = NOW/2

 Direct Variation    Equation: y=kx K = constant of variation Goes through the origin on a graph  Finding slope through 2 ordered pairs:   (x 1 , y 1 ) (x 2 , y 2 )  Slope- Intercept Form: y = mx + b  M = slope; B = y-intercept  Point Slope: y – y 1  M = slope; X 1 , y 1 = m( x – x 1 ) are from the ordered pair  Standard Form: Ax + By = C  No Fractions, C is a constant

 Parallel Lines- never intersect, have the SAME SLOPE  Perpendicular Lines- Slopes are opposite reciprocal  + change to –  – change to +  Flip the fraction

Pay It Forward  http://www.youtube.com/watch?v=DvKAP HvCPS8  To 1:17  http://www.youtube.com/watch?v=vmboo 6cj8ds  From 3:10 to 4:39

receive a Pay It Forward good deed at each of the next several stages of the process? Stage 1: Stage 2: Stage 3: Stage 5: Stage 4: Stage 6: 2. What is your best guess for the number of people who would receive Pay It Forward good deeds at the tenth stage of the process?

3. Which of the graphs below do you think is most likely to represent the pattern by which the number of people receiving Pay It Forward good deeds increases as the process continues over time? Graph Number: ___________

4. Take your values for each stage (from #1) and graph the data. Hint: Your x-axis needs to be the stage number and your y-axis needs to be the number of people receiving Pay It Forward.

Mathematical terminology This graphs represents EXPONENTIAL growth.

These data sets display properties of EXPONENTIAL FUNCTIONS.

Warm Up – January 24 th

Exponent Rules Review Exponents are a “short-hand” way of multiplying the same quantity over and over.

Example: X 4 = (x)(x)(x)(x)

Try Some: Expand the following  4 3  Y 4  X 2 y 5  w 6 z 1

Using Exponents to simplify Write using exponents  X*x*x*x  2*2*2*2*x*x*x*y*y  3*3*3*4*4*4*4*x

Zero as an exponent  Anything with an exponent of zero equals 1.

 Ex x 0 = 1  6 0 =  Y 0 =

Negative Exponents  When you have a NEGATIVE exponent you turn it POSITIVE and FLIP it.

 EX x -3

Try Some

Multiplication  When multiplying like bases you ADD exponents  Ex: x 4 x 2

Try some!

 X 3 x 4  Y 3 x 4 y 7  z 3 y 2 x 5 z 5 y 6 x 10

Exponents of Exponents  When you have an exponent of an exponent you MULTIPLY  EX: (x 4 ) 3

Try Some!

 (x) 5  (x 2 y 4 ) 5  (2x 3 ) 6

Division  When you divide like bases you SUBTRACT exponents

Try Some

Growing Sequences 

Arithmetic Sequence

next by always adding (or subtracting) the same value : goes from one term to the 

Common Difference

: The number added (or subtracted) at each stage of an arithmetic sequence 

Initial Term

: Starting term For example, find the common difference and the next term of the following sequence:

3, 11, 19, 27, 35, . . .

Growing Sequences 

Geometric Sequence

same value : goes from one term to the next by always multiplying (or dividing) by the 

Common Ratio

: The number multiplied (or divided) at each stage of a geometric sequence Determine the common ratio r of the Brown Tree Snake Sequence.

1, 5, 25, 125, 625, . . .

Warm Up – January 25 th

The geometric sequence from the Brown Tree Snake problem (1, 5, 25, 125, 625 . . .) can be written in the form of a table, as shown below: The Brown Tree Snake was first introduced to Guam in year 0. At the end of year 1, five snakes were found; at the end of year 2, twenty-five snakes were discovered, and so on…

Since we now have a table of the information, a graph can be drawn, where the year is the independent variable (x) and the number of snakes is the dependent variable (y). See below:

The curved graph of this problem situation is known as an exponential growth function. An exponential growth function occurs when the common ratio r is greater than one.

Group Work  In your group’s complete the worksheet handed out.

Tuesday, January 29 th Warm Up # 4

Homework Check – 5.4

What is the difference between an arithmetic and geometric sequence?

What is the difference between a common ratio & a common difference? Which type of sequence does each go with?

The Ladybug Invasion As a biology project, Tamara is studying the growth of a ladybug population. She starts her experiment with 5 ladybugs. The next month she counts 15 ladybugs.

1) The ladybug population is growing arithmetically. How many beetles can Tamara expect to find after 2, 3, and 4 months? Write the sequence.

2) What is the common difference?

3) Now put the sequence into a table in the space below.

4) How long will it take the ladybug population to reach 200 if it is growing linearly?

5) Suppose the ladybug population is growing exponentially. How many beetles can Tamara expect to find after 2, 3, and 4 months? Write the sequence.

6) What is the common ratio?

7) Now put the sequence into a table in the space below.

8) How long will it take the ladybug population to reach 200 if it is growing exponentially?

9) Why does it take the ladybug population longer to reach 200 when it grows linearly?

10) Graph both tables on the designated graphs provided below. Be sure to label your axes Linear Growth Exponential Growth

Homework Worksheet 5.5

Wednesday January 30

Warm Up - # 5 Homework 5.5

One Grain of Rice

Exponential Functions An exponential function is a function with the general form y = ab x a ≠ 0 and b > 0, and b ≠ 1

A and B

A is your starting value B is growth factor X is time Y is the amount after time

Starting Value and Growth Factor Identify each starting value and the growth factor.

Y= 3(1/4) x 2.

Y= .5(3) x 3.

Y = (.85) x

Modeling Exponential Functions Suppose 20 rabbits are taken to an island. The rabbit population then triples every year. The function f(x) = 20 • 3 x where x is the number of years, models this situation. What does “a” represent in this problems? “b”?

How many rabbits would there be after 2 years?

Modeling Exponential Functions Suppose a Zombie virus has infected 20 people at our school. The number of zombies doubles every day. Write an equation that models this.

How many zombies are there after 5 days?

If there are 1800 students at Knightdale, how long will it take for the Zombie virus to infect the whole school?

3. A Bacteria culture doubles in size every hour. The culture starts at 150 cells. How many will there be after 24 hours? After 72 hours?

4. A population of 2500 triples in size every year. What will the population be in 30 years?

Warm up Homework

Remember Exponential Functions An exponential function is a function with the general form

y = ab

x a ≠ 0 and b > 0, and b ≠ 1 A is your starting value B is growth factor X is time Y is the amount after time

Intervals of Time In an exponential equation when your growth factor happens over a special interval of time, we must divide our x by that interval.

ab n x

n is your interval of time

Modeling Exponential Functions

Suppose a Zombie virus has infected 20 people at our school. The number of zombies doubles every 30 minutes. Write an equation that models this.

How many zombies are there after 5 hours?

 4. A population of 2500 triples in size every 10 years.  What will the population be in 30 years?

Friday February 1 st Compound Interest

Warm Up Homework

Review

Quiz

Compound Interest

Decimals as Percent's To Write a Decimal as a Percent: Move decimal place 2 places right.

Ex: .3434

1.736

.25

2.2798

Percent's as Decimals To Write a Percent as a Decimal: Move decimal place 2 places left.

Ex: 45% 100.5% 63.3% 5.64%

Compound Interest Compound Interest is another type exponential function: Here

ç è æ 1 + P = starting amount

r n

ö ÷ ø

R = rate (AS A DECIMAL!!!!) n = period T = time

Compound Interest Find the balance of a checking account that has $3,000 compounded annually at 14% for 4 years.

Compound Interest Find the balance of a checking account that has $500 compounded semiannually at 8% for 5 years.

Transformations

Warm Up Homework Compound Interest Practice Go over Review Sheet

Transformations Remember: Translations are moments of a function!

Complete Worksheet

Transformation Sum Up y = 2 x-h + k + h left - h right k down + k up

Part of an Exponential Graph Horizontal Asymptote: A horizontal line that the graph “flat lines” at Y-Intercept: Where the graph crosses the y-axis.

Identify Horizontal Asymptotes and y-intercept

Write an Equation if the Base = 2

Exponential Decay

Warm UP Homework Finish Translations

Drug Filtering Assume that your kidneys can filter out 25% of a drug in your blood every 4 hours. You take one 1000-milligram dose of the drug. Fill in the table showing the amount of the drug in your blood as a function of time. The first three data points are already completed. Round each value to the nearest milligram

+ Time since taking the drug (hrs)

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68

Amount of drug in your blood (mg)

1000 750 562

How many milligrams of the drug are in your blood after 2 days?

Will you ever completely remove the drug from your system? Explain your reasoning. 5.

A blood test is able to detect the presence of the drug if there is at least 0.1 mg in your blood. How many days will it take before the test will come back negative? Explain your answer.

Recall: y = a•b x Initial (starting) value = a Growth or Decay Factor = b x is the variable, so we change that value based on what we are looking for!

Remember that the growth or decay factor is related to how the quantities are changing. Growth: Doubling = 2, Tripling = 3. Decay: Losing half = Losing a third =

Exponential Decay

When b is between 0 and 1!

Growth : b is greater than 1

If the rate of increase or decrease is a percent:

we use a base of 1 + r for growth or 1 – r for decay

Ex 1.

Suppose the depreciation of a car is 15% each year?

A) Write a function to model the cost of a $25,000 car x years from now.

B) How much is the car worth in 5 years?

Ex 2:

Your parents increase your allowance by 20% each year. Suppose your current allowance is $40.

A) Write a function to model the cost of your allowance x years from now.

B) How much is your allowance the worth in 3 years?

Complete the 2 practice problems

1. Assume that your kidneys can filter out 10% of a drug in your blood every 6 hours. You take one 200-milligram dose of the drug. Fill in the table showing the amount of the drug in your blood as a function of time. The first two data points are already completed. Round each value to the nearest milligram.

+ TIME SINCE TAKING THE DRUG (HR)

0 6 12 18 24 30 36 42 48 54 60 66

AMOUNT OF DRUG IN YOUR BLOOD (MG)

200 180

A) How many milligrams of the drug are in your blood after 2 days?

B) A blood test is able to detect the presence of the drug if there is at least 0.1 mg in your blood. How many days will it take before the test will come back negative? Explain your answer.

2. Calculate the amount of drug remaining in the blood in the original lesson, but instead of taking just one dose of the drug, now take a new dose of 1000 mg every four hours. Assume the kidneys can still filter out 25% of the drug in your blood every four hours. Have students make a complete a table and graph of this situation.

+ TIME SINCE TAKING THE DRUG (HR)

0 4 8 12 16 20 24 28 32 36 40 44 48

AMOUNT OF DRUG IN YOUR BLOOD (MG)

1000 1750 2312

A) How do the results differ from the situation explored during the main lesson? Refer to the data table and graph to justify your response.

B) How many milligrams of the drug are in your blood after 2 days?

HW 5.10

Growth VS Decay and Percent Growth and Decay

Exponential Functions Remember!

An exponential function is a function with the general form

y = ab

x/n a ≠ 0 and b > 0, and b ≠ 1

A and B

A Starting Value B is direction Growth b > 1 Decay 0 < b < 1

Y-Intercept and Growth vs. Decay Identify each y-intercept and whether it is a growth or decay.

Y= 3(1/4) x 4.

Y = 300(1.3) x 2.

Y= .5(3) x 3.

Y = (.85) x 5.

Y = 4500(.4) x 6.

Y = 76(3/4) x

Increase and Decrease by percent  Exponential Models can also be used to show an increase or decrease by a percentage.

Increase and Decrease by percent The rate of increase or decrease is a percent, we use a change factor/base of 1 + r or 1 – r.

Growth b > 1 Change factor (1 + r) Decay .

0 < b < 1 Change Factor (1 - r)

Percent to Change Factor When something grows or decays by a percent, we have to add or subtract it from one to find b.

Increase of 25% 2. increase of 130% 2.

Decrease of 30% 4. Decrease of 80%

Growth Factor to Percent Find the percent increase or decease from the following exponential equations.

Y = 3(.5) x 2.

Y = 2(2.3) x 3.

Y = 0.5(1.25) x

Percent Increase and Decrease A dish has 212 bacteria in it. The population of bacteria will grow by 80% every day. How many bacteria will be present in 4 days?

Percent Increase and Decrease  The house down the street has termites in the porch. The exterminator estimated that there are about 800,000 termites eating at the porch. He said that the treatment he put on the wood would kill 40% of the termites every day.  How many termites will be eating at the porch in 3 days?