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Unit 1: Mathematical Reasoning,Unit 1: Mathematical Reasoning,Data Analysis , and Probabi l i tyData Analysis , and Probabi l i ty

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Student Journal

Version 4.1

CREDITS

Authors: Robert Balfanz, Dorothy Barry, Dennis Goyette, Danny Jones, Guy Lucas, Tracy

Morrison-Sweet, Maria Waltemeyer

Contributors: Vicki Hill, Donald Johnson, Kate Kritcher, Nancy Klais, Hsin-Jung Lin, Kwand Lang,

Song-Yi Lee, Richetta Lobban, Mary Maushard, Linda Muskauski, Vandana Palreddy,

Carol Parillo, Jennifer Prescott, Patrick Reed, Matthew Salgunik, Ann Smith, Dawne

Spangler, George Selden, Jerri Shertzer, Wayne Watson, Arlene Weisbach, Theodora

Wieland, Frederick Vincent, Math Teachers of Patterson High School

Graphic Design: Gregg M. Howell

© Copyright 2011, The Johns Hopkins University. All Rights Reserved.

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Table of Contents

Lesson Page

Lesson 1: Topics in Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Lesson 2: Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Lesson 3: Inductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

Lesson 4: Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

Lesson 5: Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Lesson 6: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51

1. Using the square grid paper on the next page, draw all possible rectangles for the numbersin your group's column:

Lesson 1

Johns Hopkins University | Talent Development Secondary | Lesson 1 1

Unit 1: Mathematical Reasoning, Data Analysis, & Probability

Rectangle with an area of 2:

Number theory is the study of the relationships and patterns among numbers.

Natural numbers are the counting numbers 1, 2, 3, 4, etc.

The concepts of prime numbers, factors, multiples, and divisibility are frequently used inalgebraic expressions. Understanding these concepts will help prepare you for advancedmathematics.

Discovery Activity 1

An expression is a series ofoperations with numbers

and/or variables.

Definition

Lesson 1: Topics in Number Theory

2 Lesson 1 | Johns Hopkins University | Talent Development Secondary

Unit 1: Mathematical Reasoning, Data Analysis, & ProbabilitySquare Grid Paper

2. Complete the table of rectangle dimensions below for your group's numbers.

3. Examine the rectangle dimensions. Look for patterns and make a statement about anypatterns you see.



4. The numbers 2, 3, 5, and 7 are examples of prime numbers. Complete the statement below.

A prime number is a number that:

5. The numbers 4, 6, and 12 are examples of composite numbers. Complete the statementbelow.

A composite number is a number that:

6. In the table below find all the factors of the numbers assigned to your group. You may wantto refer to the rectangles that you drew or the Table of Rectangle Dimensions.

The factors of a number arethose numbers that divide into

the number without aremainder.

For example, the factors of 20are 1, 2, 4, 5, 10,

and 20.

Definition



7. Circle the prime numbers in the list of numbers in Exercise 6.

One method to determine the prime factorization of a number is a tree diagram. For example,the prime factorization tree for 20 is shown below.

24a.

Every number has its own unique prime factorization. There are no two unique numbers withthe same prime factorization. This extremely important fact is called the FundamentalTheorem of Arithmetic.

38b. 60c.

The prime factorization of anumber is the number written as aproduct of only prime numbers.

For example, the primefactorization of 20 is

Definition

8. Find the prime factorization of each number below.

A prime number has only twofactors - itself and 1.

The number 1 is not a primenumber.

Definition




Unit 1: Mathematical Reasoning, Data Analysis, & ProbabilityIf the sum of all the factors of a number, other than thenumber itself, is less than the number, the number iscalled deficient.

If the sum of all the factors of a number, other than thenumber itself, is greater than the number, the number iscalled abundant.

If the sum of all the factors of a number, other than thenumber itself, is equal to the number, the number is calledperfect.

For example,

Definition

9. Now, it is your turn. Pick five different non-prime numbers and determine if they areperfect, deficient, or abundant.



Lesson 1Unit 1: Mathematical Reasoning, Data Analysis, & Probability

People use inductive reasoning to make a generalization by observing and recognizing apattern. Mathematicians call a generalization from inductive reasoning a conjecture. Forexample, if every weekday morning about 6:30 a.m. a garbage truck picks up garbage on yourstreet, you might conjecture that when you hear a garbage truck come down your street it mustbe 6:30 a.m.

Not all conjectures made by inductive reasoning can be proven true. A conjecture can be provedfalse by finding a counterexample.

1. What counterexample could prove the garbage truck conjecture false?

A counterexample is anexample which demonstrates

that a statement is false.

Definition

2. Study the pattern of numbers and the square of those numbers below. Can theconjecture, "A number is always less than or equal to the square of the number" beproven true or false? If false, give a counterexample.

3. Christian Goldbach was a mathematician in Moscow, Russia. He is best remembered for aconjecture he placed in a letter to Leonhard Euler in 1742.

Goldbach's Conjecture

Any even number greater than 2 canbe written as the sum of two primenumbers.

For example,

The sum is the result of addition.

a. Give five more examples of Goldbach's Conjecture.

b. Can you come up with a counterexample to Goldbach's Conjecture by finding an evennumber greater than 2 that cannot be written as the sum of two prime numbers?


8 Lesson 1 | Johns Hopkins University | Talent Development High Schools

Exercises


1. Give the prime factorization of each number below.

a. 32

b. 48

c. 100

d. 53

2. Determine if each number below is perfect, deficient, or abundant.

a. 24

b. 25

c. 28

3. A prime pair consists of two consecutive odd integers that are prime. For example, 11and 13 create a prime pair. Determine three more prime pairs.


4. Conjecture: The rule n2 + 1 always generates a prime number when n is even. For example, n2 + 1 is prime when n is 2, 4, or 10.

When , then , which is prime.When , then ,which is prime.When , , then , which is prime.

a. Give two other examples of the conjecture.

b. Give two counterexamples of the conjecture.

c. Based on your responses to part b above, is the conjecture true or false?

Definition

Pattern 2:421 divided by 2 has a remainder of 1421 divided by 3 has a remainder of 1421 divided by 4 has a remainder of 1421 divided by 5 has a remainder of 1

5. Pick one of the following patterns below. Study the pattern and make a conjecture. Trade your conjecture with a partner to see if he or she can find a counterexample to your conjecture.

Pattern 1:

A conjecture is ageneralization made as a

result of inductivereasoning.

Inductive reasoning is aprocess of observing data,recognizing patterns, and

making generalizations basedon observations.





Outcome SentencesOne thing I learned from today's lesson is ________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

I now know a natural number is ________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

Using the square grid and drawing in rectangles helped me to understand that ____________

_______________________________________________________________________________________

_______________________________________________________________________________________.

A conjecture can be used to _____________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

I still have difficulty understanding _____________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.



Discovery Activity

Lesson 2



1. Complete the pattern in the nine-column number chart.

2. Find three number patterns in the chart. Circle or mark each number pattern or write abrief statement about each pattern. What is unique about your patterns?

3. What is unique about the last column?

Lesson 2: Number Patterns


Unit 1: Mathematical Reasoning, Data Analysis, & Probability4. Complete the pattern in the six-column number chart.

5. Find three number patterns in the chart. Circle or mark each number pattern or write abrief statement about each pattern.



Unit 1: Mathematical Reasoning, Data Analysis, & Probability7. Complete the pattern in the seven-column number chart.

8. Find three number patterns in the chart. Circle or mark each of the number patterns orwrite a brief statement about each pattern.


10. If you were to create a similar table that had eight columns, what would be unique aboutthe last column?

11. Without using a calculator, determine the divisibility by 10.a. Is 34 divisible by 10?

b. Is 30 divisible by 10?

c. The test for divisibility by 10 is to look at the__________________ digit.

d. Products with a factor of 10 end in _________.


b. Is 25 divisible by 5?

c. Is 20 divisible by 5?

d. How would you test divisibility by 5? Explain.


b. What is the sum of the digits of 27?

c. Is the sum of the digits of 27 divisible by 3?

14. Determine if each number below is divisible by 10.a. 432

b. 560

c. 18,485


b. 720

c. 17,025


b. 831

c. 504



Definition

A number is divisible by asecond number if the secondnumber divides into the first

without a remainder.





17. Give one example of each divisibility rule.

a. Divisible by 1___________________________________________________________

b. Divisible by 2___________________________________________________________

c. Divisible by 3___________________________________________________________

d. Divisible by 4___________________________________________________________

e. Divisible by 5___________________________________________________________

f. Divisible by 6___________________________________________________________

g. Divisible by 7___________________________________________________________

h. Divisible by 8___________________________________________________________

i. Divisible by 9___________________________________________________________

j. Divisible by 10___________________________________________________________

k. Divisible by 11___________________________________________________________

Symbolize It

Lesson 2



A football teamhas 11 playerson the field at

a time.

Football Coach Perkins liked to have his team line up in four rows forwarm-up drills. He really liked it when each row had the same number ofathletes. However, this year when he had the team line up in four rows, hehad one extra athlete with no place to go.

1. Draw a sketch of how the team lined up in four rows with one extraathlete. How many total athletes did you place on the team?

2. Draw a sketch of another team with a different number of athletes lined up in four rowswith one extra athlete. How many total athletes did you place on this team?

3. List ten different-sized teams that could have four equal rows of athletes with one extraathlete.

4. Would all of the sizes in Exercise 3 create football teams that make sense? Why or why not?



Now let us see what a KWL for Prime Numbers would look like.

TOPIC: Prime NumbersWith a partner, fill out the “K” column with what you Know about prime numbers.

After you have completed the “K” column, enter two things in the “W” columnrepresenting two things you Want to know about prime numbers.



Math at Work

2 8Lesson 2

a. How would you describe these numbers?

b. What type of numbers are in the first row?

c. What does each column represent?

2. During the Third Century B.C., a Greek mathematician, Eratosthenes of Cyrene, developed a method of catching primes by eliminating composite numbers. This method is now known as the Sieve of Eratosthenes. It is still the most effective way of finding all very small primes (those less than 1,000,000).

1. Emily found a sheet of paper with these numbers written on it.

a. Use the Sieve of Eratosthenes method to find all prime numbers between 1 and 100 by using the following steps:

• Write down all the whole numbers from 1to 100.

• Cross out 1.• Circle 2, the first prime number, and cross

out all multiples of 2.• Circle 3, the next prime number, and cross

out all of its multiples.• Circle the next prime and cross out all of its

multiples.• Continue the process until you have found

all the primes from 1 to 100.

Hint: Use the divisibility rules.



b. List the primes between 1 and 100.

c. How many primes are there between 1 and 100?

d. Compare your answers.

On September 4, 2006 two professors at Central Missouri State Universitydiscovered the 44th Mersenne prime. The number they discovered is

232,582,657 –1 with 9,808,358 digits. In order to win $100,000.00 they needed to find a Mersenneprime that had 10,000,000 digits. They were just short.

You could be the next person to find the next Mersenne prime. All you need is the rightcomputer, a program from GIMPS: The Great Internet Mersenne Prime Search, a lot of timerunning the program, and perhaps some luck. Currently the Electronic Frontier Foundation isoffering the following awards.

• $50,000.00 to the first person or group who discovers a prime number with at least100,000,000 decimal digits.

Wouldn't you like to earn thousands of dollars using your computer's spare processor time?There are still an infinite number of primes to catch. Surf over to the GIMPS: The GreatInternet Mersenne Prime Search website at http://www.mersenne.org/prime.htm and join thesearch for the next record-setting, rare, and historic Mersenne prime!



CALCULATOR CHALLENGE

fyi

2. Determine the fourth Mersenne prime.

3. Is the number represented by the expression prime?

7 is a Mersenne prime because it is a prime number and

A Mersenne (Mehr-SENN) prime is a prime number that can be represented by theexpression 2n – 1.

For example,3 is a Mersenne prime because it is a prime number and 22 – 1 = 3.7 is a Mersenne prime because it is a prime number and 23 – 1 = 7.

1. Write a short explanation of why 11 is, or is not, a Mersenne prime.

As of August 2009, 47 Mersenne primes have been found.



Before you complete four outcome sentences, list three things you learned about primenumbers in the “L” column below.

Outcome SentencesI now see a relationship between ____________________________________________________ and

_______________________________________________________________________________________

_______________________________________________________________________________________

I am now able to ______________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

I discovered/rediscovered a way to _______________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

I can now explain ______________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________



Falafel (Middle Eastern)

Falafel is a mixtureof chick peas, spices,flour and water,which is deep-friedin oil. It is served ina pita bread withvegetables and tahina sauce. InIsrael, falafel is the Americanequivalent of the hamburger. It issold on every other corner, inrestaurants and at sidewalk stands.

Dolmades(Greek)

Dolmades aregrape leavesstuffed with

ground beef, rice, parsley, salt,pepper and egg whites.Dolmades are usually servedcold. When served warm,dolmades come with a saucemade of beaten eggs and lemonjuice.

Discovery ActivityAaron and five of his friends went to El Café International for lunch. Juan ate quesadillas,dolmades, and falafel and became ill. Jordan ate quesadillas but neither dolmades nor falafeland felt fine. Harvey ate dolmades and falafel but not quesadillas and became ill. Pepito wasnot hungry. He did not eat anything, and he felt fine. Jack ate quesadillas and falafel but notdolmades and became ill. Aaron ate dolmades and falafel but stayed away from the quesadillas.He also became sick.

1. Use the table below to organize your information and to determine the food that probablycaused the illness.

Make a conjecture as to what food caused the illness.

Quesadillas (Mexican)Quesadillas are a Mexican form of cheese/meat pie. A meatfilling of beef or chicken is prepared with onions and spices.The warm filling is then spread over flour tortillas, and bakedwith cheese for 5 to 7 minutes.


Lesson 3: Inductive Reasoning



You just used inductive reasoning to make a conjecture.

Scientists and mathematicians use inductive reasoning to make a generalization by observingdata and recognizing patterns. In mathematics, a generalization is called a conjecture. Amathematician tests a conjecture either by searching for a counterexample (a case that provesthe conjecture is wrong) or by deductive reasoning.

2. How could you prove your conjecture true?

3. Pick one person’s information to eliminate. Describe how eliminating that information maychange your conjecture on what food caused the illness.

4. You are flying to Austria. Your airplane makes a stop in Madrid. You desperately need touse the bathroom and you are extremely shy, so you won't ask anybody.

There are two doors that appear to be two bathrooms. The doors are marked Damas andCaballeros, but you do not know Spanish.

List at least two ways by which you would reach your conjecture.

Now, use inductive reasoning to determine the fractions that model the repeatingdecimals.

5. Study each pattern below.

a. b.

c. d.



6. The following is a real-world example of using inductive reasoning to arrive at a correctconclusion.

Gregg was shopping in the supermarket when he noticed a lot of people had docking stationsin their carts. He thought the docking stations were probably on sale. As he noticed evenmore people with them in their carts, he thought they must really be cheap! After finallyasking someone for the location of the docking station in the store, Gregg discovered thatthey were only $19.99, which was quite cheap, so he bought one!

Now you share a situation with a partner where you used inductive reasoning and arrivedat a correct conclusion.

7. The following is a real-world example of using inductive reasoning to arrive at an incorrectconclusion.

It is January, and for the last six days my neighbor, Mr. High, has had to defrost his carwhich made him late for work. He thought that because it is January and it is cold, he wouldhave to defrost his car every day. Mr. High set his alarm early on the seventh day, and tohis surprise there was no frost on his car. In this case, the weather can be unpredictable.

Now you share a situation with a partner where you used inductive reasoning and arrivedat an incorrect conclusion.



Now, you are going to use inductive reasoning to find patterns in different sequences. If youobserve carefully, you can find a pattern and make a conjecture for the next term.

8. Determine the next term in the pattern based on inductive reasoning.

a. 1, 10, 100, 1000,

b. 0, 10, 21, 33, 46, 60,

c.

9. Explain how you found the next term in the sequences.

10. Create the first five terms of a sequence. Give the sequence to a friend and ask that friendto find the next two terms in the sequence. Have your friend write the rule for theconjecture.

a. List the first five terms of a sequence.

b. What are the next two terms?

c. What is the rule for the conjecture?

In a sequence of numbers,each number is called a term.

For example, in the followingsequence of numbers, the third

term is 9 and the seventh term is 21.

3, 6, 9, 12, 15, 18, 21, 24, . . .

Definition

Symbolize It

Lesson 3



1. Draw the next two figures in the pattern.

1st 2nd 3rd 4th 5th

2. Underneath each figure, write the number of dots that make up the figure.

Work on Exercises 3 and 4 with your partner. Illustrate the triangular numbers.

3. What is the eighth triangular number?

4. What is the tenth triangular number?

5. One of the following rules can be used to find the nth triangular number. After completingExercises 5 a. and 5 b. below, circle the rule that generates triangular numbers.

Because this rule gives 3 and 8 as the first two terms of the sequence, it is not the rule fortriangular numbers because the first two terms should be 1 and 3.

a. Use the space below to check the second rule,

b. Use the space below to check the third rule, n(n – 2).

Rule 1: n(n + 2) Rule 2: Rule 3: n(n – 2)

Checking the First Rule: n(n + 2)• When n = 1, n(n + 2) = 3, because 1(1 + 2) = 3.

• When n = 2, n(n + 2) = 8, because 2(2 + 2) = 8.



6. Use the nth triangular number rule to find the 15th triangular, 30th triangular, and 100th

triangular numbers.


_______________________________________________________________________________________

_______________________________________________________________________________________

I now know that a triangular number is ________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

I used inductive reasoning today to _____________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

I still have difficulty understanding ______________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________

Discovery Activity

Lesson 4



Data analysis is a process that begins with a real-world question or problem and concludes withideas, solutions, and summaries. The process generally includes data collection, dataexploration, and graphing.

1. For example, how many students in your class ate breakfast this morning?

This type of data collection is a quick count. Another approach would be to create an elaboratesurvey and carry it out for several days. The questions you pose and the resources you havewill directly affect the complexity of the data collection, tabulation, and organization of theresults.

2. What would be the purpose of collecting this kind of data?

3. To whom might this information be important?

4. What are ways to display the data related to this question?

Lesson 4: Data Analysis


Tally charts are used to illustrate categories of data with no numerical ordering. Tally chartsallow you to organize and display the data at the same time you collect the data.

How Students Travel to School

5. Fill in the tally chart by collecting data from your class.

How Students Travel to School


a. What is the purpose of this chart?

b. Who might find this information important?

Frequency bar graphs can show how the numerical value of one category compares to thevalue of another category.

7. What can we conclude about the class shown above by lookingat the stack plot?



6. Complete the frequency bar graph that matches the tally chart data collected for Exercise 5.

Stack plots, or line plots, show an accurate count of data along one axis.



Graphs convey factual information, and they provide opportunities to make inferences thatmay not be directly observable in the graph. For example, we can conclude that students didnot take a train to school because zero students were represented in the "Other" category.

The difference between actual fact and inference is an important idea in graph constructionand science.

A histogram is a form of bar graph in which the categories are displayed at equal intervals(always the same size) along a numerical scale. The intervals cover all possible data.Consequently, there are no spaces between the bars of the histogram.

8. What does this histogram tell us about the hours students spent on homework?

9. What is the interval size of the categories?

10. How many students study between four and six hours?

11. How many students were surveyed?



12. Survey your class on the hours each student spends completing homework. Display thedata on the histogram.

13. What units did you use for the frequency axis?

14. Variables represent quantities that may take on many values. They may also stand forsomething that can change, such as the height of a teenager or the price of a soda.

What are the two variables in the histogram used in Exercise 12?



When graphing two-variable data such as temperatures that occur over time, it makes senseto represent the data in a way that shows both variables. A line graph allows you to displayboth variables at once. For example, the following line graph shows time and temperature asvariables for the change in temperature of a cup of hot chocolate. Note: 20O Celsius is roomtemperature, which is 68O Fahrenheit.

15. Write a short story in the space below that explains what happened to the cup of hotchocolate.

Two-variable information can also be organized in a table. For example, the table below showsthe average height of an American female over time.

Average Height of an American Female

Source: www.cdc.gov

Tables do not always clearly show the relationship between two variables. Patterns andtrends are often best observed when displayed in a graph.

16. Complete the line graph below with the data from the table above.



17. You can now locate ordered pairs and interpret data.

a. Find the point located at (8, 127).

b. What is the graph telling you about the averageheight of an American female at age 8?

c. What does the point (2, 84) represent?

d. What does the point (20, 163) represent?

18. We can also analyze changes.

a. The average height of American females increases by ______________ from ages 2 to 6.

b. The difference in height between the two ordered pairs (2, 84) and (6, 114) is ________.

19. By analyzing the graph, we can estimate ordered pairs that are not listed in the table.

a. What could be the average height of a 9-year-old American female?

b. The coordinates of that point are (9, _________________).

c. What would be the approximate age of an American female who had a height of 155centimeters?

d. The coordinates of that point are _______________________________________.



Ordered Pairs

A pair of numbers, such as ageand height, describing the

location of a point (x,y) on acoordinate plane is called an

ordered pair.

Definition



20. Compare the good quality graphs below with the poor quality graphs. Create a list of itemsthat classifies good quality graphs and poor quality graphs.

Good Quality Graphs Poor Quality Graphs

Good Qualities Poor Qualities

Distance Jet Traveled Over Time

Symbolize It

Lesson 4



1. Create a frequency bar graph using the following data.Number of Natural Satellites as of December 2007

* In August, 2006 Pluto was downgraded from a planet to a "dwarfplanet" by The International Astronomical Union (IAU); however, Plutostill has three planetary satellites orbiting in its atmosphere.

2. Write one observation from the frequency bar graph.



Math at Work

2 8Lesson 4

1. Your math teacher recorded the following 30 scores on the mid-term exam.

Score Number of Students100 195 390 585 480 475 670 365 060 255 150 1

a. Describe what type or types of graphs would lend themselves to display the data fromthe table.

b. Pick one of the graphs in 1a and sketch it below.

c. If you had to pick one score that would represent your class the best, which score wouldyou pick? Why? Study your graph to help make your decision.

d. If your teacher said that he or she forgot one score to place in the table, where would themost probable place be in the table that it would go?

Teacher Reference



2. Students in Mr. Reed's homeroom were asked about the number of pets they had. A stackplot of their answers is shown below.

a. How many students answered the survey?

b. How many students have more than three pets?

c. How many students have no pets?

d. What was the most common answer?

e. Using this graph, predict the number of people that possibly own only one pet out of allthe students at Mr. Reed’s school. The school has 800 students.

f. Explain what would be similar and different about this graph and a graph that would represent the whole school.


3. The concentration of carbon dioxide in the atmosphere has been rising. The oceans absorb alot of carbon dioxide, but the burning of fossil fuels and the clearing of forests could beincreasing the greenhouse gases. These gases may be the cause of Earth's changing climate,sometimes called global warming. The chemical formula for carbon dioxide is CO2.

a. Complete the table. (ppm means parts per million.)

b. Write a statement about any patterns you see in the graph.

Greenhouse gases arecomponents of the

atmosphere thatcontribute to the

greenhouse effect. Somegreenhouse gases occur

naturally in theatmosphere, whileothers result fromhuman activities.

Source: cdiac.ornl.gov





_______________________________________________________________________________________

_______________________________________________________________________________________.

I now know that collecting data ________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

I prefer to use _______________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

Working in groups helped me to understand that _________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

It makes more sense to _________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

It was difficult to _____________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.

It was easy to _________________________________________________________________________

_______________________________________________________________________________________

_______________________________________________________________________________________.


Discovery Activity


Graphs and tables give a lot of data. We can summarize that data by producing a few numbersthat describe the complete set. These numbers are called statistics. Statistics are used todescribe the center of the data and the spread of the data.

Three measures of central tendency are mode, median, and mean.

The mode is the value that occurs most frequently in the set of data. If two or more values arethe most frequent, there is more than one mode.

1. Consider Terri's math scores.

81, 62, 95, 65, 62, 73, 84, 65, 56, 65, 77, 82, 90

The mode of this set is:

2. Order Terri's math scores from least to greatest.

The median is the middle value in an ordered set of data. Half of the values are either aboveor below the median. If there is an even number of values, the median is the average of the twomiddle values.

3. The median of this set of Terri's math scores is:

The mean is the sum of all the values in the set of data divided by the total number of values.

4. The mean of this set of Terri's math scores is:

Terri took an additional math test.

5. Determine what this additional score could be if it did not affect the:

a. mean b. median c. mode

6. What would the score of the additional test need to be for Terri to have an 80% average?

a. If Terri received a 100% on the additional test, would she increase her letter grade froma “C” to a “B”?

Lesson 5: Descriptive Statistics



a. Determine the mean height of the class.

b. Determine the median.

c. Determine the mode.

d. Determine the maximum.

e. Determine the minimum.

f. Determine the range.

g. What does the mean of this data tell us?

h. What 5 heights could you add to the listwithout affecting the mean, median, ormode?

Three ways of measuring spread are the maximum, minimum, and range.

The maximum is the greatest number.

The minimum is the least number.

The range is a single number that tells the difference between the maximum and theminimum values.

7. Determine the maximum, minimum, and range of Terri's math scores.

1st Set of Terri's Scoresa. Maximum

b. Minimum

c. Range

8. As a class, create a list of the heights of all the students.

Symbolize It

Lesson 5



1. Recall the information your teacher recorded in Lesson 4 on the 30 scores of the mid-termexam.

Score Number of Students100 195 390 585 480 475 670 365 060 255 150 1

a. Graph the data with a graph of your choice below.

b. Determine the mean, median, and mode of the set of data. Place a mark on the graphto represent each value.

c. Describe which value of the mean, median, or mode you think represents the data thebest.


Math at Work

2 8

Source: www.census.gov

1. Determine the mean, median, and modeof the life expectancy data for thecountries given.

a. Mean

b. Median

c. Mode

2. What does the mean of the lifeexpectancy data tell us?

3. Determine the range of the lifeexpectancy data.

Female Life Expectancy at Birth as of 2008

To determine the median,the numbers first must be

placed in order from least togreatest.


County Life Expectancy at Birth

Andorra 87

Argentina 80

Australia 84

Brazil 77

Burundi 53

Canada 84

Chile 81

Colombia 77

Cuba 80

France 84

Germany 82

Hungary 78

Italy 83

Japan 86

Kenya 57

Mexico 79

Netherlands 82

Nigeria 49

Peru 72

Poland 80

Puerto Rico 83

Russia 73

United Kingdom 82

United States 81

Venezuela 77



4. Jauntea's math class created a stack plot of the number of siblings of each member of the class. What do the four x's above the 3 represent?

7. Is it possible to add two new pieces of data to change the mode? Explain your answer below.

5. Determine the mean, median, and mode of the data.

a. Mean

b. Median

c. Mode

6. What does the mean of the stack plot data tell us?




Outcome SentencesI can now describe the difference between the mean and the median which is, _______________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

We used statistics in our class to ________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

To find the median, you must first _______________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

I still have difficulty understanding ______________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

Setting the Stage

Lesson 6



We all have an intuitive concept of chance. Some events, when compared to others, have abetter, worse, or the same chance of occurring.

1. Categorize each event as definite, impossible, or perhaps.

a. It will snow tomorrow.

b. All living things die.

c. Some students will be absent tomorrow.

d. Jim will not have a birthday this year.

e. Plants are good mathematicians.

Common sense tells us that if an event will never happen its probability will be zero. Forexample, the probability of pulling a red marble out of a bag full of only blue marbles is zero. Inmathematics, the statement would look like which is read "The probability of a redmarble is zero."

An event that will definitely occur has a probability of one. For example, the probability ofpulling a blue marble out of a bag of only blue marbles is one. The mathematical statementwould look like .

Events in the "perhaps" category have probabilities between zero and one. For example, weoften hear on a TV weather report that there is a 25% chance of rain. The mathematicalstatement could look like:

You can express probability values in at least four different ways. You can use percents,decimals, fractions, or ratios.

2. Write each probability with two other methods.

a. A newborn has a 1 in 2 change of being a girl.

b. There is a 30% chance that it will snow tonight.

Lesson 6: Probability




1. Conduct one of the following experiments to determine the experimental probability.

a. Roll two dice together 50 times and record how many times the sum of the dice is 7.

b. Record how many times you make a free throw out of ten shots.

c. Flip a coin 20 times and record the number of times it lands with heads up.

The sample space is a list ofall possible outcomes of an

event.

Definition

Theoretical probability is the number of outcomes that you would expect to occur dividedby the total number of outcomes. A formula for theoretical probability is:

For example, the probability of getting tails by flipping one coin is 50%.

A collection of all possible outcomes is called a sample space.

The formula for theoretical probability can be used for flipping coins as long as we believe thecoins are fair. A fair coin is equally likely to land on heads or tails when tossed.

2. Determine the theoretical probability of obtaining a 5 when rolling one die. List thesample space and circle the desired outcome.

Three approaches to probability are intuitive, experimental, and theoretical.

Experimental probability is determined by the results of an experiment. For example,Camille threw 100 shots in a row from the free-throw line and made 67 baskets. Herexperimental probability of making a shot from the free-throw line is or 67%.

The two outcomes of the coin are listed below with the desired outcome circled.

T H

Sample space:

Lesson 6



On the first toss, the coin will either be heads or tails. On the second toss, the coin can landheads or tails regardless of the outcome of the first toss. The sample space has four outcomes:HH, HT, TH, or TT. If you desire to get two tails, the number of outcomes would be one out ofa sample space of four. The probability would be 25%.

For more complicated sample spaces, we can construct a tree diagram. For example, the treediagram below shows the sample space for flipping a coin twice.

3. What is the probability of getting two heads when flipping a coin twice?

4. What is the probability of getting at least one tail when flipping a coin twice?

5. What is the probability of not getting a tail when you flip a coin twice?

Symbolize It

Lesson 6



For Exercises 7 and 8, write your own probabilities based on the tickets drawn from a hat.

Tickets and Ticket Numbers

Randomly, one ticket will be drawn from a hat. Find each probability.

1.

2.

3.

4.

5.

6.

7.

8.



Math at Work

2 8

1. Are theoretical and experimental probabilities different? Explain why or why not.

2. If the probability of an event is 75%, what is the probability that the event will not happen?Explain your answer.

3. The cafeteria staff handed out brown bag lunches. There are an equal number of bagscontaining chicken, tuna fish, ham & cheese, and roast beef sandwiches. At random youselect a bag.

a. What is the probability that your bag contains a chicken sandwich?

b. What is the probability that your bag contains either a tuna fish or a ham & cheese sandwich?

4. Determine the probability of rolling a sum of 9 with two dice. In the space below, use atree diagram to draw the sample space of rolling two dice.

b. What is the number of possible outcomes?

c. Multiply the number of choices for each item of clothing.

d. Compare the two numbers. What do youobserve?



Definition

a. In the space below, complete the tree diagram for the possible outcomes of completeoutfits you could wear.

5. You will be spending the weekend at a friend’s house. You are packing the following items:

Pants Tops Shoes Possible Outcomes

Fundamental CountingPrinciple

If one event occurs m ways and asecond event occurs n ways, thetotal number of both ways to occuris mn. For example, if you have 5choices for a shirt and 3 choices forpants, then you have choices for a pant-and-shirt outfit.



6. An identification card has a combination of six letters and numbers. The first threepositions must be non-zero whole numbers. The last three positions must be letters. Boththe letters and numbers can repeat.

Use the fundamental counting principle and a calculator to determine how many differentidentifications cards can be produced.

Examples of identification cards:

7. How many difference results are possible if you deal two cards from a 52–card deck?Remember, that after you have dealt the first card, there are only 51 cards left.

CHALLENGE1. For Exercise 6, how many different ID cards could be produced if both letters and numbers

could not repeat?

2. What is the probability of dealing four consecutive Kings from a 52-card deck?



Outcome SentencesI now understand how to ________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

I can now describe the difference between experimental probability and theoreticalprobability, which is ___________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

Probability can be used to _______________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

I still need more help in understanding __________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

Math at Work

2 8





Note TakingName: __________________________________Class: __________Period: __________Topic: ___________________________________Objective: ________________________________

WHAT DO I KNOW

WHAT DO I WANT TO KNOW

WHAT DID I LEARN


ResourcesThe authors and contributors of Transition to Advanced Mathematics gratefully acknowledges thefollowing resources:

Boyer, Carl B. A History of Mathematics, Second Edition. New York: John Wiley & Sons, Inc., 1991.

Donovan, Suzanne M.; Bransford, John D. How Students Learn Mathematics in the Classroom. Washington, DC: The National Academies Press. 2005.

Dricoll, Mark. Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann, 1999.

Eves, Howard. An Introduction to the History of Mathematics (5th Edition) Philadelphia, PA: Saunders College Publishing, 1983.

Harmin, Merrill. Inspiring Active Learning: A Handbook for Teachers. Alexandria, VA: Association for Supervision and Curriculum Development, 1994.

Harmin, Merrill. Strategies to Inspire Active Learning. White Plains, NY: Inspiring Strategies Institute. 1998.

Hoffman, Mark S, ed. The World Almanac and Book of Facts 1992. New York, NY: World Almanac. 1992.

Hur, Meir Ben. Investigating the Big Ideas of Arithmetic. Glencoe, Illinois: International Renewal Institute, Inc. 2005.

Hur, Meir Ben. Making Algebra Accessible to All. Glencoe, Illinois: International Renewal Institute, Inc. 2005.

Kagan, Spencer. Cooperative Learning. San Clemente, CA: Resources for Teachers. 1994.

Karush, William. Webster's New World Dictionary of Mathematics. New York: Simon & Schuster. 1989.

Lamon, Susan, J. Teaching Fractions and Ratios for Understanding. Mahway, NJ: Lawrence Erlbaum Associates, Publishers: 2005.

Lehrer, Richard; Chazan, Daniel. Designing Learning Environments for Developing Understanding of Geometry and Space. Mahway, NJ: Lawrence Erlbaum Associates, Publishers: 1998.

McIntosh, Alistair, Barbara Reys, and Robert Reys. Number Sense: Simple Effective Number Sense Experiences. Parsippany, New Jersey: Dale Seymour Publications. 1997.

McTighe, Jay; Wiggins, Grant. Understanding by Design. Alexandria, VA: Association for Supervision and Curriculum Development. 2004.

Marzano, Robert J. Building Background Knowledge for Academic Achievement. Alexandria, VA: Association for Supervision and Curriculum Development. 2004.

Marzano, Robert J.; Pickering, Debra J.; Jane E. Pollock. Classroom Instruction that Works. Alexandria, VA: Association for Supervision and Curriculum Development. 2001.

National Council of Teachers of Mathematics, The. Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics. 2000.

National Research Council. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press. 2001.

Ogle, D.M. (1986, February). "K-W-L: A Teaching Model That Develops Active Reading of Expository Text." The Reading Teacher, 39(6), 564-570.

Payne, Ruby L. A Framework for Understanding Poverty. Highlands, TX: aha! Process,Inc. 1996.

Payne, Ruby L. Learning Structures. Highlands, TX: aha! Process,Inc. 1998.

Posamentier, Alfred S.; Hauptman, Herbert A. 101+ Great Ideas for Introducing Key Concepts in Mathematics.Thousand Oaks, CA: Corwin Press. 2006.

Sharron, Howard; Coulter, Martha. Changing Children’s Minds. Imaginative Minds. 1993.

Van de Walle, Jon A. Elementary and Middle School Mathematics: Teaching Developmentally (4th Edition). New York: Addison Wesley Longman, Inc. 2001.

The authors and contributors of Transition to Advanced Mathematics gratefully acknowledge thefollowing internet resources: http://seer.cancer.gov; http://cdiac.ornl.gov/; www.census.gov; www.mersenne.org;www.nssdc.gsfc.nasa.gov


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Unit I Cover SJ Unit I Cover SJ - tdschools.org€¦ · 1. Using the square grid paper on the next page, draw all possible rectangles for the numbers in your group's column: Lesson