Comparing+and+Scaling

Proma Roy

Essential Question:
Which comparing strategies work best in which situations?

Notes:
proportion - statement about equivalent ratios or fractions (helps calculate one of the missing numbers)

To choose the number of delegates from each region, you can compare the population of each region to the population of the U.S: __population of the region__ population of the US

__population of region__ equals __delegates from region__ (unknown) population of US equals total delegates

To figure out how many delegates should be chosen from a given region, you need to solve the corresponding proportion. For example: To find the number of conference delegates who should come from the South Atlantic region you need to solve the proportion __45,000,000__ equals __delegates from South Atlantic region__ 250,000,000 equals 1000 Using what you know about equivalent fractions, you could write: __45,000,000__ equals __180__ 250,000,000 equals 1000 And that means that the South Atlantic region should have had 180 delegates.
 * The population in 1990 of the South Atlantic region of the US was about 45 million people
 * The total population of the United States then was about 250 million people
 * The total number of delegates would be 1000

Problem 6.3 Selecting Delegates
A. //How many of the 1000 delegates should be chosen from each of the nine geographic regions?// Total Population of each region (in 1000s): Total Population of the United States (in 1000s): 248,710 Total number of delegates: 1000 To find out the number of delegates needed for each region: To check: 53 + 151 + 169 + 71 + 175 + 61 + 107 + 55 + 157 equals 999 ; 1000 - 999 equals 1, so there's one empty seat Number of delegates needed for each region:
 * New England: 13,207
 * Middle Atlantic: 37,602
 * East North Central: 42,009
 * West North Central: 17,660
 * South Atlantic: 43,567
 * East South Central: 15,176
 * West South Central: 26,703
 * Mountain: 13,659
 * Pacific: 39,127
 * 248,710 divided 1000 equals 248.71
 * New England: 13,207 divided by 248.71 equals 53
 * Middle Atlantic: 37,602 divided by 248.71 equals 151
 * East North Central: 42,009 divided by 248.71 equals 169
 * West North Central: 17,660 divided by 248.71 equals 71
 * South Atlantic: 43,567 divided by 248.71 equals 175
 * East South Central: 15,176 divided by 248.71 equals 61
 * West South Central: 26,703 divided by 248.71 equals 107
 * Mountain: 13,659 divided by 248.71 equals 55
 * Pacific: 39,127 divided by 248.71 equals 157
 * New England: 53
 * Middle Atlantic: 151
 * East North Central: 169
 * West North Central: 71
 * South Atlantic: 175
 * East South Central: 61
 * West South Central: 107
 * Mountain: 55
 * Pacific: 157

B. //How many of the 1000 delegates should be from metropolitan areas, and how many should be from rural areas?// Metropolitan areas (in 1000s): 192,726 out of 248,710 ( 192,726 divided by 248.71 equals 775 ) Rural areas (in 1000s): 55,984 out of 248,710 ( 55,984 divided by 248.71 equals 225 ) Check: 1000 - 775 equals 225 775 of the 1000 delegates should be from metropolitan areas, and 225 should be from rural areas. C. //How many of the delegates should be of Hispanic origin?// Hispanic: 22,354 ( 22,354 divided by 248.71 equals 90 ) 90 of the delegates should be of Hispanic origin. D. //Four racial groups are named in the data: white; black; Native American-Eskimo-Aleut; and Asia-Pacific Islander. How many of the total delegates should represent each of these races? How many should represent the category “all other races” (which is not mentioned in the data)?// 802 delegates should represent the white race, 120 should represent the black race, 8 should represent the Native American-Eskimo-Aleut race, 29 should represent the Asia-Pacific Islanders, and 39 should represent the "all other races" category.
 * White: 199,686 ( 199,686 divided by 248.71 equals 802 )
 * Black: 29,986 ( 29,986 divided by 248.71 equals 120 )
 * Native American-Eskimo-Aleut: 1959 ( 1959 divided by 248.71 equals 8 )
 * Asia-Pacific Islander: 7274 ( 7274 divided by 248.71 equals 29 )
 * All Other Races: ( 199,686 + 29,986 + 1959 + 7274 equals 238,905 ; 248,710 – 238,905 equals 9805 -> all other races) 9805 ( 9805 divided by 248.71 equals 39 )
 * Check: 802 + 120 + 9 + 29 + 39 equals 999

E. //Use your answers to A-D to help you develop a plan for selecting the delegates. Describe your plan in a report that you could submit to the conference organizers.// To select the delegates, the total number of people in each ethnic group or race group and the total number of people in the metropolitan and rural areas are important. Another important number is the quotient total number of the US divided by 1000, because that is the number of delegates allowed. You can use the quotient to divide the number of people in any group by to see how many of the delegates should be from that group. To select the delegates, the conference organizers should use the information about how many delegates per region, total number per state, and then going into details about how many out of those states have what types of background or race.


 * Follow-up**

Homework
Collected: Assigned: Ace 6: 4, 9**, 12**

= =
 * Jula DeCosse and Proma Roy**

**Essential Question:**

 * Which comparing strategies work best in which situations?**

**Problem 6.1 Scaling Up or Down**

 * How big was T. Rex compared to a "larger than average" human being?**

T-Rex, the object of all children's nightmares, the prehistoric monster in the closet. Before dinosaurs became extict, T-rex were the most fierce and feared of them all. However what would have made them so fearful? Of course, one of the tihngs that could have made them so fearful, to us humans anyways, would probably have to be the size. A tyrannosaurus rex was at least 6 meters tall, which is about 3 times larger than a larger than average human, who are only about 2 meters tall. The incisors were about 15 cm long, while an average human's incisor is only about 1 cm long, this makes a tyrannosaurus rex's teeth about 15 times larger than ours! But those two facts are nothing compared to their weight. An average T-rex weighed about 8,100 kilograms, while a larger than average human weights about 90 kilgrams. The scale factor of that is 90, so a T-rex would weigh about 90 times more than a human. A larger human skull can be about 20 centimeters long, while an uncovered T-rex skull is 1 meter long, which means it's head is 5 times bigger than a large human's. So this giant reptile, this "king tyrant lizard" was huge, and we are certainly lucky that we don't have them today.
 * T. Rex ||  || "Larger than average" human being ||   ||
 * Weight -> || 8100 kilograms || Weight -> || 90 kilograms ||
 * Height -> || 6 meters || Height -> || 2 meters ||
 * Skulls -> || 1 meter || Skulls -> || 20 centimeters ||
 * Incisors -> || 15 centimeters long || Incisors -> || 1 centimeter ||

**Follow-up**
1. If an infant T. rex was the same height as the human described in the problem, the scale factor between a grown T-rex and the infant T-rex would be 15.

2. The incisors of this young T-rex would be 5 cm long (3 / 15 is 5).

3. The skull of this young T-rex would be 33.3 cm (100 / 3 is 33.3)

**Problem 6.2 Using Rules of Thumb**
A. //It takes about 100 maples trees to make 25 gallons of maple syrup.// If Mr. Paulo made syrup from all of his sugar maple trees and ended up with 16 gallons of syrup, he had about 64 maple trees. ( 100 divided by 25 is 4 ; 5 gallons per tree -> unit rate ; 4 x 16 is 64)

B. //A 5-minute shower requires about 18 gallons of water.// You use about 2.16 gallons in an 8-minute shower. You would use 788.4 gallons if you take 8-minute shower every day for a year. ( 5 minutes -> 18 gallons ; unit rate is 0.27 gallon per minutes ; 8 minutes x 0.27 is 2.16 gallons ; 2.16 x 365 days is 788.4 )

C. //A double-spaced page of text contains about 250 words if it is printed in Times with a font size of 12, and about 330 words if it is printed in Times with a font size of 10.// Jeremy printed his term paper in 10-point Times, and the paper came to 15 double-spaced pages, which means he had 4950 words ( 15 times 330 is 4950 ).

D. Jogging burns about 100 calories per mile. If Elizabeth jogs at a rate of 4.5 miles per hour, it will take her 2 hours 42 minutes to burn off the 1200-calorie lunch she ate ( 1200 divided by 100 is 12 miles -> she needs to jog 12 miles ; 12 divided by 4.5 is 2.7 hours is 2:42 )

Follow-up

Homework
Collected: Assigned: Ace 6: 1, 3, 11, 14

Jula DeCosse

Notes -
How are rates, ratios, and proportions related and how are they different?

Problem 5.4 -

A. Which state, North Dakota or South Dakota has a greater population density? N. Dakota - 638,000 people in 68,994 square miles of land

Anuron Mitra

Problem 5.1 Estiamating the Size of a Crowd
In this problem there is a pciture of a crowd, portrayed in the way of dots. Each dot represents a person. We have to estimate the number of people in this crowd and explain the methd that we use. This is the method that I used. Method: Random Sampling Step 1: Divide the picture into cm square. The crowd is 126 cm². - Done Step 2: Label the columns with number on the top. Example underneath. - Done Step 3: Label the rows with alphabets to the side. Example underneath. - Done Step 4: Take 10 squares and count dots in each: - Done A1 = 17 dots B2 = 13 dots C3 = 17 dots D4 = 15 dots E5 = 15 dots F6 = 17 dots I14 = 16 dots H13 = 15 dots G12 = 16 dots I1 = 17 dots Average = 15.8 Approximate number of people in the crowd is 15.8 x 126 which is 1991 Step 5: Find the average of the 10 samples. - Done Step 6: Now multiply the average by the area of the squares of the whole crowd to get an approximate number of people in the crowd. - Done After carrying out the random sampling method, I arrived at an estimate that 1991 people attended the rally.

5.1 Follow-Up
In your group, discuss ways your method might lead to a poor estimate of the crowd size? A way the random sampling method might lead to a poor estimate is if, instead of taking a large number of samples and finding the average (by doing this you are coser to the actual number), you just take one or two or three samples and then multiply it by the area. The number that results would be quite far from the actual. The more random smaples you have the better. In this way you are getting closer to the actual than just taking a very small amount of samples.

Homework
Assigned: Problem 5.1 and follow-up ACE 5(pages 59-63):# 3 and 10

Collected:

Anant Dalela

Problem 4.2 (Using Unit Rates)
Madeline’s car went 580 miles with 19 gallons of gasoline. Luis’s car went 452 miles with 15.5 gallons of gasoline.

A. Luis’s car’s mileage 452 (miles) / 15.5 (gallons) 29 miles / gallon Madeline’s car’s mileage 580 (miles) / 19 (gallons) 31 miles / gallon Therefore, Madeline’s car is more fuel-efficient (she had better mileage, i.e. her car went more miles per gallon)

B. Gallons of gas 0 1 2 3 4 5 6 7 8 Miles in Madeline’s car 0 31 62 93 124 155 186 217 248 Miles in Luis’s car 0 29 58 87 116 145 174 203 232

C. Luis’s car (rule) - m = 29g Madeline’s car (rule) - m = 31g

D. (i) Luis’s car Number of miles that would be covered by using: 9.5 gallons of gas - 9.5 X 29 = 275.5 miles 15.5 gallons of gas -15.5 X 29 = 449.5 miles 19 gallons of gas - 19 X 29 = 551 miles 23.8 gallons of gas – 23.8 X 29 = 690.2 miles 100 gallons of gas – 100 X 29 = 290 miles 150 gallons of gas – 150 X 29 = 4350 miles

(ii) Madeline’s car Number of miles that would be covered by using: 9.5 gallons of gas - 9.5 X 31 = 294.5 miles 15.5 gallons of gas -15.5 X 31 = 480.5 miles 19 gallons of gas - 19 X 31 = 589 miles 23.8 gallons of gas – 23.8 X 31 = 737.8 miles 100 gallons of gas – 100 X 31 = 310 miles 150 gallons of gas – 150 X 31 = 4650 miles

Follow Up

1. 2. Both graphs are similar as: - They show the same type of data (but for different cars) - Both lines (connecting the points) are increasing at constant rates The graphs are different as: - The line showing Madeline’s car has a steeper line, thus we can conclude that her car has better mileage, and that the number of miles per gallon (for her car) is higher I also observed that there was a definite pattern by which the difference in the number of miles for between their cars increased their cars, always by 2, with one in each direction (higher and lower), probably due to the initial difference in their mileage (31 - 29 =2).

3. As explained earlier, the main factor that makes the two graphs different is the difference in the miles per gallon in the cars. Because of this, Madeline’s graph is steeper.

Homework
Assigned: ACE 4: 1, 4-6**,10 and 11

Meshan Perera December,10,07**

Notes:
E.Q.: How can I use a pattern to make a prediction? You can use patterns when things are obvious so it helps you give tips. Louis's Car: 15.5 gallons **from Denver to Monument Park and then back. Madeline's Car:** 19 gallons **from Denver to Pritchett and back.**

**Problem 4.1: Comparing Fuel Economy**
Louis's Car: 452/15.5~ 29 So, that proves the Madeline's car is more fuel efficient because it travels more than Louis's car by 2 gallons so it is very close but still Madeline's car is a bit more efficient.**
 * Madeline's Car: 580/19 ~ 31

**Problem 4.1 Follow Up**

 * I think that percents wouldn't be so use full in this problem because these type of numbers aren't that easy to make into percents so you can use ratios, but I am not saying can't use percents but for me it wouldn't be so helpful .**

**Homework: ACE 4: 1,4-6(**),10-11
Colleted : NONE Meshan Perera

Notes:
E.Q.: How can I use a proportion to increase or decrease the size of an image or quantity? By making equivalent ratios (scaled up 1cup/2 cup = 20cups/40cups) Mix A 2 cups concentrate 3 cups cold water 1 cup concentrate 4 cups cold water 4 cups concentrate 8 cups cold water 3 cups concentrate 5 cups cold water Percents= total amount divided by total
 * Mix B**
 * Mix C**
 * Mix D**

Problem 3.1: Mixing Juice
A. I think that Mix A (2 cups concentrate, 3 cups cold water) is the most "orangey" because 2/5 is concentrate which isn't more than half but it is very close compared to the other mixes, there would be 40% of the drink in orange juice which is good. B. I think mix B(1 cup concentrate ,4 cups cold water) is the least orangey because there is only 20% of orange juice in it and 1/5 of it is concentrate. C. ??

Follow Up Problem 3.1
1. I didn't really use ratios in this problem because i mainly used percents, but you can use ratios by dividing the fractions and making them into ratios for example for every 2 cups of concentrate you should put 3 cups of cold water if u want a very orangey juice.
 * 2.** ??

Collected : NONE =Comparing and Scaling= Journal and Homework Record with vocabulary 2.2 EQ: How can we use percentages to compare different sets of data?

Table for 2.1 "Think about This!" (Total of 5 boys and 4 girls)


 * || Girls || Boys ||
 * Soccer || 3 || 4 ||
 * Cricket || 0 || 5 ||
 * Dodgeball || 4 || 5 ||
 * Badminton || 1 || 4 ||
 * Bicycling || 3 || 5 ||
 * Camping || 0 || 0 ||
 * Exercise Walking || 4 || 4 ||
 * Fishing || 0 || 0 ||
 * Swimming || 4 || 5 ||

Proma Roy

Homework : ACE: 9-15, 16
Collected: Assigned: Problems 2.1 and 2.2 + ACE: 9-15, 16 **+ Mathematical Reflections (p. 25)**

Homework (Heading 3) Collected: (normal) Assigned: (normal)

2.2 Comparing Your Class to the Nation
Suppose you were asked to write a news story about the popularity of camping in the United States based on the data in the table. A. What headline would you use for yo

Jula

1.3 - Getting the message across
Suppose you were asked to write a news story about the popularity of camping in the United States based on the data in the table. A. What headline would you use for your story? What would your first sentence be? My headline for this story would be //Camping : popularity and//. And my first sentence would be "Did you know that 89,762,000 people go camping, in the United States alone? I chose these because I think that a straight forward title would grip someones attention, and that having a big number always attracts someone. I think that it is just a habit to think that if that many people are doing it, it must be fun. B. Write five statements you could use in your story to compare the popularity of camping among people in the three age groups. In each statement, be clear about which groups you are comparing. Your comparisons should be specific and based on mathematics.

- 21, 304,000 people go camping from the ages twelve to seventeen. 41,808,000 people however go camping from the ages twenty – five to thirty – four. Which shows that the age group twenty – five to thirty – four has about 1.96 more times people than the age group twelve to seventeen. - 15,968,000 people do NOT go camping twice a year form the age group twelve to seventeen, while the age group eighteen to twenty – four has 21,883,000 people who do not go camping twice a year. So, 75 % from the ages twelve to seventeen, do not go camping twice a year, as opposed to the 82 % of people from the ages eighteen to twenty – four. - 4,767,000 people in the age group eighteen to twenty – four go camping twice a year while 10,000,000 people from the age group twenty – five to thirty – four go camping twice a year. - 82 % of people from the age groups eighteen to twenty – four do not go camping, but only 76 % of people from the age group twenty – five to thirty – four do not go camping twice a year. - In total 89,762,000 people go camping twice a year, but only 20,103,000 out of those go camping twice a year.

1.3 Follow up -
According to the data, what percent of people from the age 12 - 34 go camping at least twice a year? 22 % of people from ages 12 - 34 go camping twice a year. 21,304,000 + 26,650,000 + 41,808,000 = 89,762,000 total who camp 5,336,000 + 4,767,000 + 10,000,000 = 20,103,000 total who camp twice a year 20,103,000 / 89,762,000 = 22.3, round to the nearest ones place 22.3 = 22 %

Homework : ACE 1 : 7 - 11
Collected: Assigned: Problem 2.1, ACE 2 : 1- 8 (1 & 2), 17 - 22a

Homework (Heading 3) Collected: (normal) Assigned: (normal)

Anuron

**Notes**
a table (allows us to compare a lot of data) graphs (helps us to see patterns in data) percentages ratio fraction differences (subtraction) using words**
 * __EQ - What methods are there for comparing things?__

**1.2 Targeting an Audience**
I would use statement 6, '40% of the students prefer radio to television,' because first of all this statement is very crafty as it is not giving the number of people surveyed and the amount of people who like television. Instead it's just saying the amount of people who like radio and what's more is that it's turning this amount into a percent, which is easy to remember, and also saying that this percent of people like radio than television.
 * A. Read the statements below about how Neilson students prefer to spend their evenings. Tell whether each statement accurately reports the results of the survey. Explain your answers?**
 * 1) **Statement 1: This statement does accurately report the survey results because the statement reports the survey results of T.V as 6 out of 10, which is equivalent to the real survey results 60 out of 100.**
 * 2) **Statement 2: This statement does accurately report the survey results because the statement repots the survey results as a ratio of 4 to 6, which is equivalent to the real survey result’s ratio: 40 to 60.**
 * 3) **Statement 3: This statement does accurately report the survey results because it says that the difference between the results of the television to radio is 20, and that is by how much the television people outnumber radio people and to confirm this, I did the necessary calculations which is 60(amount of television people) – 40(amount of radio people) =20, and so the television people //do// outnumber the radio people by 20.**
 * 4) **Statement 4: This statement does accurately report the results because the ratio of the real results of television people who outnumber radio people is 60 to 40, while the ratio given in the statement, 3 to 2, is really the most simplified form of the ratio 60 to 40.**
 * 5) **Statement 5: This statement does accurately report the survey results because the statement says that you have to multiply 1.5 to the number of radio people to get the number of television people and once you do the necessary calculations i.e. 40x1.5=60, then you know that multiplying 1.5 to the radio people //does// give you the amount of television people.**
 * 6) **Statement 6: This statement does accurately report the survey results because the amount of people who like radio is 40, and if you convert 40 to a percent i.e. 40 upon 100 x 100 = 40%, then you //do// get 40%**
 * 7) **Statement 7: This statement does accurately report the survey results because the fraction of television people given in this statement is 3 by 5, while the real fraction of television people is 60 by 100. 60 by 100 and 3 by 5 are equivalent fractions and so they are same and 3 by 5 is a right way to show how many students prefer television.**
 * B. If you were writing a paper to convince local merchants that they would reach more students by advertising on the radio than on television, which statement from above would you use? Why?

C. Imagine that you are the advertising director for a television station in the town where Neilson is located. You have been asked to prepare a report for a meeting between your ad department and a large local skateboard manufacturer. Which accurate statement from above would you use to try to convince the manufacturer to advertise on your station? Why? I would use statement 1, '6 out of 10 students prefer television to radio,' because this statement clearly shows that the majority of 10 students like television, and the people who like radio are a minority in this situation. In other words, the statement is clearly showing that out of 10 students 6 like television and the difference, that is the radio people, is obviously smaller.**

**1.2 Follow-Up**
1. For each statement in part A in the problem, write a similar statement about your class data.** Similarities-**
 * After conducting a survey of the 9 people in our class we found out that 3 people prefer watching T.V. in the evenings while 6 people prefer the Internet in the evenings.
 * **Statement 1: 2 out of 3 students prefer the Internet to the T.V**
 * **Statement 2: Students prefer the T.V to the Internet by a ratio of 1 to 2.**
 * **Statement 3: Students who prefer the Internet outnumber those who prefer the T.V by 3.**
 * **Statement 4: Students who prefer the Internet outnumber those who prefer the T.V by a ratio of 2 to 1.**
 * **Statement 5: The number of students who prefer the Internet is 2 times the number who prefer the T.V**
 * **Statement 6: 33.3% of the students prefer the T.V to the Internet.**
 * **Statement 7: 2/3 of the students prefer the Internet to the T.V.**
 * 2. In what ways is your class data similar to the Neilson data? In what ways is your data different?
 * **Our class data follows the same format of writing as in the Neilson data. For example: One of the statements of the Neilson data is, '6 out of 10 students prefer television to radio,' and one of our class data's statement is '2 out of 3 students prefer the Internet to the T.V.' Both the statements talk about the larger survey result value i.e. Neilson talks about the television people who had more value and our class talks about the Internet preference as it has more value**
 * **Our class data has the same number format of writing as in the Neilson data. For example: One of the statements of the Neilson data is, 'Students who prefer television outnumber those who prefer radio by 20,' and one of our class data's statement is 'Students who prefer the Internet outnumber those who prefer the T.V. by 3.' Both the statements show the relationship as the difference between the greater survey value and the smaller survey value i.e. Neilson subtracts the radio people from the television people as the television people are the greater survey result and our class subtracts the T.V people from the Internet people as the Internet people are the greater survey result.**
 * Differences-**
 * **Our class data and Neilson's data have different numbers.**
 * **Our class data has television being less popular other than Neilson's, which has television more popular.**

**Homework**
ACE 1(pages10-14):#7-11 Collected:
 * Assigned:Problems 1.2 and 1.3 with follow-ups

Belle**

**Notes**
a table (allows us to compare a lot of data) graphs (helps us to see patterns in data) percentages ratio fraction differences (subtraction)
 * __EQ - What methods are there for comparing things?__

In this problem, there are four advertisements that show different ways we can compare data.**

**1.1 Writing Ads**

 * 1.1 A.** Describe what you think each of the four statements means. Explain how each shows a comparison. Be sure to tell //what// is being compared and //how// it is being compared.

Ad #1. I think this ad shows that only 1/3 of the people that participated liked Cola Nola. Ad #2. This ad shows the actual number of people that like either one of the brands. 17,139 people like Bolda Cola, and 11,426 people like Cola Nola. Ad #3. This is the difference in numbers between Bolda Cola and Cola Nola. (17,139 - 11,426 = 5,713) It shows that 5,713 people like Bolda Cola more than Cola Nola. Ad #4. This one shows the percentage of people in the total participants that like Bolda Cola.


 * 1.1 B.** Is it possible that all four advertising claims are based on the same survey data? Explain your answer.

Yes, they are based on the same survey data because: If I divide 17,139 by 3 (ratio, Ad #1) I get 5,713 (Ad #3). Ad #2 shows the data. We can base all our other Ads on this one. Ad #4 shows the percent, we can figure this out by adding 17,139 and 11,426 (equals 28,565) and multiply by 0.6 (60% as a decimal) and get 17,139, which clearly proves that 60% of the participants like Bolda Cola.


 * 1.1 C.** Which comparison is the most accurate way to report the survey data? Why?

I think Ad #2 is the most accurate because it shows the actual numbers of the participants. If I said Ad #1,3,4, we wouldn't know what number all of it is being based on.


 * 1.1 D.** Which comparison do you think would be the most effective advertisement for Bolda Cola? Why?

I think the one with the percentage is most effective because it is a small number. When people see it, they realise it is more than 50%, which is good and people can easily imagine how much that is. I think it is nice and simple to uunderstand. Ad #1 shows very "small" numbers, so when people see the 3 to 2 ratio, they will think its not a lot.

Homework
1.1 Writing Ads (page 6), no follow-up ACE 1, Page 10-11, questions 1-4

Collected: Assigned: