algebra 1 ms 1 cheongshim international academy 2016 ... · welcome to mr. joselson’s algebra...

Algebra 1

MS 1

Cheongshim International Academy２０１6

written by

Steven Joselson

www.mrjoselson.com [email protected]

CheongShim MS1 Math

Table of Contents

Syllabus

Page

Student Strategy Guide 1

Chapter 1 Data Analysis 4

Chapter 2 Number Systems 43

Chapter 3 Exploring Number Theory 62

Chapter 4 Variables and Equations 122

Chapter 5 Functions and Relations 215

Glossary 267Study Guide 272

Additional Blank Coordinate Planes for Graphing 285

Chapter 1 Data Analysis

1-1 Graphical Representation of Data

1-2 Measures of Central Tendency

1-3 Scatterplot – Line of Best Fit

Chapter 2 Number Systems

2-1 Number Systems ℕWℤℚℝ2-2 Integers

2-3 Absolute Value

2-4 Comparing Numbers

Chapter 3 Number Theory

3-1 Factors – Prime/Composite numbers

3-2 LCM,GCF,Multiples-exponent notation

3-3 Scientific notation

3-4 significant digits

3-5 Percent/Percent of Change, greatest possible error, percent error

3-6 Proportionality

Ext. Divisibility

Chapter 4 Variables and Equations

4-1 Order of Operations4-2 Like Terms4-3 Properties of Real Numbers4-4 Polynomials, factors, GCF 4-5 Verbal <--> Algebraic Expression/Equations4-6 Linear Equations4-7 Word Problems – mixture, integer, investment, uniform motion4-8 Inequalities in one variable (number line)

Chapter 5 Functions and Relations

5-1 Interpreting graphs

5-2 Coordinate plane

5-3 Graphs of Functions

5-4 Relations vs Functions

5-5 Function Values

5-6 Direct Variation

5-7 Inverse Variation

5-8 Set Builder and Interval Notations

5-9 Slope and intercepts of Lines

English – Korean vocabulary translations

Graphical Representation of Data

MS1 Syllabus

1 [email protected]

Welcome to Mr. Joselson’s Algebra class. Listed below are several ways to help you in better comprehending the concepts involved in our lessons. These are suggestions on how to take notes in class or organize your notes on your own.

1. Comparison and Contrast MatrixA Comparison and Contrast Matrix is a chart used to compare and contrast concepts in our lessons. The chart helps you visualize the different ways of performing a task; how concepts are similar or different, and allows you to choose a method that you find more to your liking. For other concepts it allows you to list the properties and see examples. The chart below compares and contrasts the two methods of solving quadratic equations.

Solving Quadratic Equations

2. Network TreeA Network Tree shows the relationship between a concept and the steps involved in solving the procedure used for the concept. The tree below shows the quadratic formula and that by examining the discriminant, you can determine the number of roots (answers) to the quadratic equation.

Factoring Quadratic Formula

Form ax bx c2 0+ + = ax bx c2 0+ + =

Method/Formula (x + m)(x + n) = 0m and n are real numbers

x b b aca

=- ± -2 4

2

Solution x + m =0, therefore x = ‐mand x + n =0, therefore x = ‐n

X equals the solutions given by this formula

Quadratic Formula

x b b aca

=- ± -2 4

2

Discriminant

D=b ac2 4-

2 real roots

No real roots

D>0

D=0

D<0

1 real root

2 [email protected]

3. Series of Events Chain

A Series of Events Chain is a listing of steps in a procedure. The chain below shows the steps in solving a linear equation.

Steps Example

If the check does not show a true statement, you should check your steps and find where you made an error.

3x + 5 = x – 7 ‐5 = ‐5 subtract 5 from both sides

Perform the opposite operation of the constant on the left side to both side of the equation.

3(‐6) + 5 = (‐6) – 7‐18 + 5 = ‐6 – 7

‐13 = ‐13✔Check you answer by substituting the solution in to the original equation

Solve the equation and list the answer

x = ‐6

2x = ‐12

∙

∙

multiply both sides by

Multiply by the reciprocal of the coefficient of the variable to both sides of the equation

3x = x – 12 ‐x = ‐xsubtract x from both sides

Perform the opposite operation of the variable on the right side to both sides of the equation

3 [email protected]

4. Note‐Taking using Labels and Notes (Concepts and explanations)

When you are reading the sections of the chapter, you should take notes. One method to take notes involves listing the concepts in the column on the left and your notes in the column on the left. These notes should include definitions, examples and cues to help you remember the concept.

These are four methods of numerous methods that can be used to better comprehend the lessons you will read in the textbook. Using all or some of these will help you do better as you read the sections and give you notes that will be easy to use to study for the exams.

Concepts NotesCounting Numbers

Whole Numbers

Integers

Rational Numbers

1,2,3,4……Does not include 0No fractionsNo decimalsNo negative numbers

0,1,2,3,4….No fractions or decimalsNo negative numbers

…‐2,‐1,0,1,2…..Both positive and negative numbers and 0No fractionsNo decimals

Includes positive and negative numbers and 0Includes fractionsIncludes all decimals that terminate or repeatDoes not include any number that can’t be represented as the quotient of two integers


4 [email protected]

Let's start the chapter with this "Getting to know your classmates" activity. Through the activity you will know your classmates.Fill in the blanks

1. Your favorite color2. Your favorite food3. What job you would like to

have after university or college4. Favorite toy/game5. Favorite sport or activity to

play or to watch6. Favorite school subject7. Favorite holiday or day of the

week8. Favorite tv show9. My mom or my dad’s job10. My favorite book, cartoon or

comic

This information can be collected an represented in many ways. Each group will collect the data for one of the categories and decide how to present it to the class.There are many ways to display data. Two of them are:

<Pie Chart or Circle Graph> <Bar Graph> Each section represents a Each bar represent an amount.percent of the whole.Types of Graphs

Graphical Representation of Data


5 [email protected]

Bar GraphsIn bar graphs are usually used to display "categorical data", that is data that fits into categories. For example suppose that I offered to buy donuts for six people and three said they wanted chocolate covered, 2 said plain and one said with icing sugar. I would present this in a bar garph as:

Histograms on the other hand are usually used to present "continuous data", that is data that represents measured quantity where the numbers usually take on any value in a certain range. A good example is weight. If you measure the weights of a group of adults you might get and numbers between say 90 pounds and 240 pounds. We usually report our weights as pounds or to the nearest half pound but we might do so to the nearest tenth of a pound or however accurate the scale is. The data would then be collected into categories to present a histogram. For example:

might be a histogram for heights (with the appropriate scale on the vertical axis). Here the data has been collected into categories of width 30 pounds.


6 [email protected]

The difference in the way that bar graphs and histograms are drawn is that the bars in bar graphs are usually separated where in histograms the bars are adjacent to each other. This is not always true however. Sometimes you see bar graphs with no spaces between the bars but histograms are never drawn with spaces between the bars.

Line Graphs

Comparing various sets of data can be complicated, but line graphs make it easy. The plotted peaks and dips on the grid allow you to monitor and compare improvement and decline. Line graphs are most often used by scientists, professionals and students.

Pictographs

People first encounter pictographs during childhood, at school, work, and all over magazines and on TV. These diagrams, which use small picture symbols to compare information, are a media favorite.


7 [email protected]

Pie Charts/Circle Graphs

Simple to make and simple to understand, a pie chart is a popular form of data comparison, consisting of a circle that is split into parts.

1

Which type of graph is shown at the left?

Circle Graph

Line Graph

Bar Graph

Picture Graph

2


Circle Graph

Line Graph

Bar Graph

Picture Graph


8 [email protected]

3


Circle Graph

Line Graph

Bar Graph

Picture Graph

4


Circle Graph

Line Graph

Bar Graph

Picture Graph

5


Circle Graph

Line Graph

Bar Graph

Picture Graph


9 [email protected]

Determine your height in cm:

Boy or Girl Height Boy or Girl Height

1. 15.

2. 16.

3. 17.

4. 18.

5. 19.

6. 20.

7. 21.

8. 22.

9. 23.

10. 24.

11. 25.

12. 26.

13. 27.

14. 28.

Measures of Central Tendency


10 [email protected]

Mean (평균)

The mean (평균）of data is the average of the data and is determined by dividing the total of the data by the number of data terms. Data is groups of information. If there is a data value that is much higher or lower than the average, it is called an outlier.

The mean in the table is cm.

Stem and Leaf plot (줄기와 잎 도표)

A Stem and Leaf plot (줄기와 잎 도표) is a away to organize data. The left column is the stem which is the most significant figures. The right column is the leaf. The leaf is a one-digit number which is the less significant figure. If the height is 157 cm, the 15 is the stem and the 7 is the leaf.

<Draw Stem and Leaf plot>

Key ____/___ = cm



Mode (최빈값)

Mode (최빈값) is the value occur-ring most often. Look in any leaf box for a number that shows up numerous times. If one number appears in a row more than any other number in any other row, it along with the stem is the mode. If several numbers show up the same number of multiple times, they are each the mode. There can be sev-eral modes. If none of the data appears more than other data, there is NO mode.10,10,10,20,20,20,30,30,30,40,40,40 No mode10,20,30,40,50,60 No Mode10,20,20,30,30,40,50 Modes 20,3010,10,20,20,30,30,30 Mode 30

The mode(s) in the stem and leaf is/are _____________________________.

Median (중앙값)

Median (중앙값) is the middle number of an ordered list. If there are an even number of terms, there will be two middle numbers. The median is the mean of those two numbers.

The median of the data is _________________.

Which should we use to describe the data ???Mean, median and mode are called the Measures of the Central Tendency.



Measure of Central Tendency

The measure of central tendency is the average or expected value of the data set.

The BEST measure of central tendency: The mean is often used to describe a data set (when there is no outlier). When there is an outlier, the median is used. When the data is nonnumeric (not numbers) or when choosing the most popular item, the mode is used.

Range (범위)

The range (범위)is the difference (subtraction) between the greatest and least (highest and lowest) data values. This tells you the spread of the data.A double stem and leaf plot organizes separate data. We can regroup the data based on separating the heights of the boys and girls.

Key ___/__ = Leaf(girls) Stem Leaf (boys)

We could then find the mean, median, and mode for the girls and the mean, median, and mode for the boys.



Box and Whisker Plot

A box and whisker plot shows how the data is spread out. The lower and upper quartiles (separates the data into fourths) need to be determined. The lower quartile is the median of the lower half of the data while the upper quartile is the median of the upper half of the data. Arrange the data in an organized list from lowest to highest. Determine the median. The lower quartile (1st quartile) is the median from the lowest number up to but not including the median of all the data. The upper quartile (3rd quartile) is the median from the highest number to the number just above the median of all the data. Using five numbers (lowest number, lower quartile, median, upper quartile, highest number), plot them on the grid below. Draw a box from one quartile to the other and then draw line segments (whiskers) from the quartile to the greatest and smallest data.lowest number: ______lower (1st) quartile: ______ median: ______upper (3rd) quartile: ______ highest number: ______

The difference of the 3rd quartile and 1st quartile is called the Interquartile Range. When a number is more than 1.5 times the interquartile range below quartile 1 or above quartile 3, it is an outlier.



Outlier – an outlier is part of the data which is more than 1.5 times the interquartile range below quartile 1 or above quartile 3.

If the data is 1,9,13,13,15,17,19,23,23,39

the median is 16quartile 1 is 13quartile 3 is 23

the Interquartile range (IQR) is 23-13=10

1.5 x the interquartile range is 15

Any number which is 13-15=-2, -2 or more is an outlier or 23+15=38, 38 or more is an outlier. So 39 is an outlier. We mark outliers on a box and whisker plot with an asterisk *

We include the outliers when we are calculating the mean, median, mode and range.



Frequency/ Frequency Table

The frequency of a particular data value is the number of times the data value occurs.A frequency table is made by arranging the data values in ascending order (from smallest to largest) with their corresponding frequencies.

If the heights of the students in the class are: 142, 144, 144, 152, 156, 156, 156, 162, 163, 167, 167, 167, 167, 167, 168, 172, 172, 174, 175, we can make a frequency table to show the data.

The frequency of a group or class interval is the number of data values that are in the range specified by that group interval.

Height (cm) Tally Frequency

142 | 1

144 || 2

152 | 1

156 ||| 3

162 | 1

163 | 1

167 |||| 5

168 | 1

172 || 2

174 | 1

175 | 1

Range of Height (cm) Tally Frequency

140-149 ||| 3

150-159 |||| 4

160-169 ||||||| 8

170-179 |||| 4

Sum = 19



Using the Double Stem and Leaf Plot, for the data determine:

Boys Girls

Mean ____________ Mean ____________

Median ____________ Median ___________

Mode ____________ Mode _____________

Range _____________ Range _____________

Box and Whisker Plot (make a separate plot for the boys and the girls)

Create a frequency table for the boys and one for the girls. Height (cm) Tally Frequency Height (cm) Tally Frequency



The Mean

1. The mean of a group of 5 numbers is 7. What might the numbers be?

The Median

2. The median of a group of 7 numbers is 10. What might the numbers be?

The Mean and Median

3. Describe a sample for which the median is 3 more than the mean.

Mean is Equal to the Median

4. Five students took a mathematics test and four of those students re-ceived marks of 5, 6, 8 and 9 out of a possible 10. Try to work out the score of the fifth student if the mean mark is equal to the median mark.

The Score

5. Five students took a geography test and four of those students re-ceived marks of 5, 6, 8 and 8 out of a possible 15. Try to work out the score of the fifth student if:가． the mean is equal to the mode.

나． the mean is greater than the median.

다． the mean is greater than 8.



Adding a Number to Each Score

6. Find the effect on the mean when a set of scores has 4 added to each score.

Each Score Doubled

7. Find the effect on the mean when a set of scores has each score doubled.

Each Score Halved

8. Find the effect on the mean when a set of scores has each score halved.

Percentage Increase

9. Find the effect on the mean when a set of scores has each score increased by 20%.



Effect on the Median and Mode

10. Find the effect on:i. the median

when a set of scores has:가. 5 added to each score

나. each score doubled

다. each score halved

라. each score increased by 50%

ii. the mode

가. 5 added to each score

나. each score doubled

다. each score halved

라. each score increased by 50%

11. Calculate the mean, median and mode for each of the following가) 2, 8, 7, 8, 5, 8, 10, 5

나) 18, 22, 22, 17, 30, 18, 12

다) 5, 5, 5, 5, 10



12) The mean score on a set of 20 tests is 75. What is the sum of the scores?

13) The tram at a ski area has a capacity of 50 people with a load weight limit of 7500 pounds. What is the mean weight of the passengers if the tram is loaded to capacity?

14) The names and ages for each person in a family of five is as follows:

a.) What is the mean age?

b.) Find the mean in 5 years from now (everyone is 5 years older). Hint: Make a new chart which shows each person’s age 5 years

from now.

c.) Find the mean in 10 years from now.

d.) Describe the relationship among the answers you found in parts a-c. Do you see a pattern? Describe it.

16) Suppose you own a hat shop and decide to order hats in only one size for this year. To decide which size to order, you look at last year’s sales figures (records), which are itemized according to size. Should you find the mean, median or mode for the data to decide what size hat to order? Explain your answer.

Name Don Jane Kirk Jean ScottAge 40 36 8 6 2



17) The following table gives the annual salaries of the 40 dancers of a certain dance troupe.

To find the mean annual salary for the troupe you would fill in the table, add up the salaries and divide by the number of dancers.

sum $1,651,000 and divide by the number of dancers (40) to get the mean salary of $41,275.

a) Find the median salary.

b) Find the mode.

18) If Sabin travelled 45 miles in 90 minutes, what was the mean speed?

19) The youngest person at Google is 17 and the range of the ages at Google is 27 years. How old is the oldest person at Google?

Salary Number of Dancers

Salary x # of Dancers

$18,000 2 $36,000$22,000 4$26,000 4$35,000 3$38,000 12$44,000 8$50,000 4$80,000 2$150,000 1



20) Choose the set of numbers that fits the descriptions given in each of the following:

a. Choose the set that has a mean of 6 and a range of 6Set A: 3, 5, 7, 9Set B: 2, 4, 6, 8Set C: 2, 3, 4, 15

b. Choose the set that has a mean of 11, a median of 11, and a mode of 11

Set A: 9, 10, 10, 11, 12, 12, 13Set B: 10, 11, 11, 11, 11, 11, 14Set C: 9, 11, 11, 11, 11, 12, 12

c. Choose the set that has a mean of 3, a median of 3 and has no mode

Set A: 0, 2.5, 6.5Set B: 1, 3, 3, 5Set C: 1, 2, 4, 5

21) To receive a grade of “A” in a class, Seoungmin needs at least a mean of 90 on the 5 exams. The grades on the first 4 exams were 84, 95, 86 and 94. What is the minimum score needed on the fifth exam to earn an A in the class. Seungmin needs a 90 or more (no rounding up).

22) The mean of 5 numbers is 6. If one of the 5 numbers is removed, the mean becomes 7. What is the value of the number that was removed?

23) Seungwoo’s median score on 3 tests was 90. His mean score was 92 and his range was 6. What were his 3 test scores?



Make a frequency table for each data set.

24) The following are the ounces in canned goods of 18 cans: 11, 15, 16, 16, 14, 15, 10, 8, 8, 12, 15, 16, 8, 15, 8, 14, 10, 10

25) Make a grouped frequency table for the following temperatures:75, 78, 99, 88, 74, 89, 61, 91, 84, 78, 75, 92, 88, 67, 82, 85

26) Construct a box and whisker plot for the following mileages per gallon of various cars:22 18 14 28 30 12 38 22 30 39 20 18 14 16 10Be sure to draw a number line with an appropriate scale. Be neat.Min:______ Q1:______ Q2(Med):______ Q3:______Max:______ Range:________ IQR:________

Size (ounces) Frequency

Class Frequency60-6970-79



Stem-and-Leaf Plots & Histograms (a bar graph with quantitative values for the x-axis)

1. The number of floors of each building in a particular city is recorded. Make a stem-and-leaf plot of the data.

2. Find the least value, greatest value, mean, median, mode, and range of the data set.

3. Find the least value, greatest value, mean, median, mode, and range of the data in the stem-and-leaf plot.

Number of Floors24 38 46 16 28 31 12 27 32 49 33 12 38 34 48 22 36 29 47 4149 30 21 17 40 13 32 15 31 21 13 16 43 33 30 25 28 29 13 11

Stems Leaves2 2 4 8 8 8 9 93 0 2 2 4 6 6 84 1 1 4 65 0 2

Key: 5|2 means 52

Stems Leaves6 3 6 7 2 9 8 4 7 7 9 0 0 1 1 3 3 3 Key: 6|3 means 63



4. Daniel and 7 of his classmates took the ACT test, a college entrance exam. The student’s test scores were as follows: 26, 28, 32, 19, 23, 24, and 31. Using the data provided, make a stem-and-leaf plot.

5. Use the data in the table to make a stem-and-leaf plot.

The table shows the scores that 20 students received on a test. 6. Use the data to make a frequency table showing the students scores.

7 Use the frequency table to make a histogram.

Hundreds of Acres Lost in Forest Fires70 45 60 15 20 23 1765 57 48 12 27 101 19

Students Test Scores75 82 83 90 7793 85 88 72 7481 88 76 79 8386 94 78 82 80

Number of Sit-ups Students Can Do in 1 MinuteNumberFrequency 10 8 9 6 2



A student tracks the number of questions on each of her quizzes. The following table shows how many questions were on each quiz.

8. Based on the information, complete the following frequency table.

9. Use the frequency table to make a histogram.

10. Use the frequency table to make a histogram.

5 9 14 18 1619 7 6 7 312 10 8 17 17

Number 1–5 6–10 11–15 16–20Frequency

Summer Camp Attendance Last WeekAges 0–5 6–11 12–17 18–23

Frequency 15 25 17 19

Number of Jobs by AgeAge 16–25 26–35 36–45 46–55 56–65Frequency 20 25 31 15 9



1.

Key: 1 | 1 means 11

2. The least stem and least leaf give the least value: 22.The greatest stem and greatest leaf give the greatest value: 52.

Use the data values to find the mean: The median is the average of the middle values: 33.The mode is the number that occurs most often: 28.The range is the difference between the greatest and least value: 30.

3. The least stem and least leaf give the least value, 63.The greatest stem and greatest leaf give the greatest value, 93.To find the mean, you must add up all the values and divide by the number of values in the plot. (63 + 66 + 72 ... + 93) ÷ 14 = 84.21The median is the average of the two middle values in the table, 88.5.To find the mode, look for the number that occurs most often in a row of leaves. Then identify its stem. The mode is 93.The range is the difference between the greatest & least value, 93–63=30

4.Place the numbers in order from least to greatest. Group the numbers by their tens digit. The stem column should contain the tens digit of each group in ascending order. Each row of leaves should contain the units digit of each number in that group in ascending order.

Number of FloorsStems Leaves

1 1 2 2 3 3 3 5 62 1 1 2 4 5 7 8 83 0 0 1 1 2 2 3 34 0 1 3 6 7 8 9 9

ACT Scores Stem Leaves 1 9 2 3 4 6 8 3 1 2 Key: 1|9 means 19



5.

6. Step 1 Choose equal intervals.Step 2 Find the number of data values in each interval. Write these numbers in the “Frequency” row.

7.

Hundreds of Acres Lost in ForestFires Stem Leaves 1 2 5 7 9 2 0 3 7 3 4 5 8 5 7 6 0 5 7 0 8 9 10 1 Key: 1|5 means 15

0-9 10-19 20-29

Freq

uenc

y

Number of S it-Ups30-39 40+

123456789

10111213



8.

9.

10.

Number 1–5 6–10 11–15 16–20Frequency 2 6 2 5

Num

ber

of C

hild

ren

Ages of Children

0–5 6–11 12–17 18–23

Summer Camp Attendance

4

8

12

16

20

24

Num

ber

of J

obs

Ages of Employees

16–25 26–35 36–45 46–55 56–65

Jobs by Age

4

8

12

16

20

24

28

32



A Scatterplot is a graph that relates two groups of data. Plot the data as points on a graph.

The chart below shows some temperatures in Seoul and Gapyeong on the same days.

A scatterplot can show a correlation (relationship) if the data is general going in the same direction.

A scatter plot is a graph of a set of data pairs (x, y). If y tends to increase as x increases, then the data have a positive correlation. If y tends to decrease as x increases, then the data have a negative correlation. If the points show no obvious pattern, then the data have approximately no correlation.

Seoul 0 4 5 11 15 22 26

Gapyeong -2 1 3 7 12 20 22

가평

서울



A correlation coefficient, denoted by r, is a number from -1 to 1 that measures how well a line fits a set of data pairs (x, y). If r is near 1, the points lie close to a line with positive slope. If r is near -1, the points lie close to a line with negative slope. If r is near 0, the points do not lie close to any line.

TELEPHONES Describe the correlation shown by each scatter plot.

Cellular Phone Subscribers and Cellular Service Regions, 1995–2003Cellular Phone Subscribers and Corded Phone Sales, 1995–2003



For each scatter plot, (a) tell whether the data have a positive correlation, a negative correlation, or no correlation, and (b) tell whether the correlation coefficient is closest to -1, 0, or 1.

Determine whether the correlation coefficient for the data is closest to -1, 0, or 1.

-1 0 1

Tell whether the data have a positive correlation, a negative correlation, or no correlation.

4. 5. 6.

Tell whether the correlation coefficient for the data is closest to -1, 0, or 1.

7. 8. 9.



Line of Best Fit

A line of best fit (or "trend" line) is a straight line that best represents the data on a scatter plot. This line may pass through some of the points, none of the points, or all of the points. One way to draw the line of best fit is with a ruler or straight edge. Try and place the ruler so it goes through as many points as possible while having a similar number of points above and below the line. If there are only a few outliers (points far away from the others), they can be ignored. If there are a lot of outliers, there might not be a correlation between the points and a line of best fit may not be possible.



1. Plot the points and create a line of best fit.

2. The table below gives the number of hours spent studying for a science exam (x) and the final exam grade (y).

Predict the exam grade of a student who studied for 6 hours.

2 6 5 8 11 10 120 4 5 6 7 8 10

X 2 5 1 0 4 2 3Y 77 92 70 63 90 75 84



1. Record the distance your frog jumps in centimeters.

2. Order the following from least to greatest length.

Total Distance:

Official Distance:

3. Fill in the chart.

The Celebrated Jumping Frog



Find the mean, median, mode and range for each data set. 1） 13, 13, 10, 8, 7, 6, 4, 5 Mean:__________________ Median:_________________ Mode: _________________ Range:__________________ 2） 20, 30, 35, 24, 36, 47, 48 Mean:__________________ Median:_________________ Mode: _________________ Range:__________________

3. Which set of data was used to make the stem-and-leaf plot?

Stem Leaf1 42 4 5 3 6

a. 14, 245, 36 c. 14, 24, 25, 36b. 4, 4, 5, 6 d. 1.4, 2.4, 2.5, 3.6

key 1/4=1.4



Use the data below for problems 4 – 7.Number of Hours Spent Sleeping at Night

4. Make a frequency table for the data above. 5. Find the measures of central tendency for the data. Mean:___________________ Median:__________________ Mode:___________________ Range:___________________ 6. Which measure of central tendency do you think gives the best indication of the number of hours a person spends sleeping each night? Explain. 7. Suppose another person was surveyed who said that he spends 3 hours sleeping at night. How would this affect the mean, median, mode and range?

5 8 6 7 49 8 7 5 98 10 7 7 86 8 8 7 89 8 7 5 910 7 8 8 6



Vocabulary ActivityAcross4. graph that relates two groups of data7. a graph arranged by quartiles9. middle number of an ordered list10. value occurring most often11. difference between the highest and lowest data valueDown1. the number of times a data value occurs2. data much higher or lower than the average3. average of the data5. a correlation when both sets of data increase at the same time6. each fourth of the data8. a line that best represents the data

Chapter 1 Review


1. Determine the range of the sets of data and find the mean of the ranges.{1,2,3,5,7,8,11,14}{4,5,7,7,8,9,9,12}

2. Determine the mean of the sets of data and find the mean of the means.{1,2,3,5,7,8,11,14}{4,5,7,7,8,9,9,12}

3. Determine the range of the sets of data and find the median of the ranges.{1,2,3,5,7,8,11,14}{4,5,7,7,8,9,9,12}

4. Determine the mean of the sets of data and find the median of the means.{1,2,3,5,7,8,11,14}{4,5,7,7,8,9,9,12}

5. Determine the range of the sets of data and find the range of the ranges.{1,2,3,5,7,8,11,14}{4,5,7,7,8,9,9,12}

6. Determine the mean of the sets of data and find the range of the means.{1,2,3,5,7,8,11,14}{4,5,7,7,8,9,9,12}

Chapter 1 Review


7. Determine the best measure of central tendency of the data.{2,3,5,6,8,12,13}

8. Determine the best measure of central tendency of the data.{1,8,15,22,25,28,32,39,45}

9. Determine the best measure of central tendency of the data.{10,40,50,60,70,80}

10. Determine the best measure of central tendency of the data.{40,50,51,52,55,57,59,60}

11. Determine the range and mode of the data. {2,3,5,6,8,12,13}

12. Determine the range and mode of the data. {21,8,15,22,2,28,22,39,35}

13. Determine the range and mode of the data. {10,10,50,50,70,70}

14. Determine the range and mode of the data. {84,58,53,52,45,58,59,62}

15. If 4 scores (with a maximum score of 10) are 5, 7, 8, 10, determine the 5th score if the mean and the median are equal.

Chapter 1 Review


16. If 4 scores (with a maximum score of 10) are 3, 5, 8, 10, determine the 5th score if the median and the mode are equal.

17. If 4 scores (with a maximum score of 10) are 5, 7, 8, 10, determine the 5th score if the mean is greater than the median.

18. Create a box and whisker plot for the data {10,15,20,25,26,28,30,35,36,38,40,45,50}

19. Create a box and whisker plot for the data {2,4,5,6,8,10,12,14,22,24,29,33,35,38}

20. Create a stem and leaf diagram for the data {1.2, 1.4, 1.6, 1.7, 2.1, 2.3, 2.5, 2.9, 3.0, 3.1, 3.1, 3.1, 3.3, 3.4, 3.4}

Chapter 1 Review


21. Create a stem and leaf diagram for the data{8, 10, 13, 17, 17, 19, 21, 21, 21, 21, 23, 25, 29, 31, 32, 32, 34, 38, 40}

22. Create a frequency table for the data{8, 19, 24, 11, 17, 29, 21, 24, 21, 21, 23, 25, 24, 31, 32, 32, 24, 38, 42, 24, 10}

23. Create a frequency table for the data {1,2,6,3,5,2,1,4,4,2,1,4,3,2,4,1}



Numbers can be classified into sets called number systems.Natural (counting) Numbers (자연수)

ℕ={1,2,3,4,5,….}Whole Numbers (음이 아닌 정수 or 0 + 자연수)

W={0,1,2,3,4,5,6 ……..}Integers (정수) [whole numbers and their opposites]

ℤ={ ….-3,-2,-1,0,1,2,3…..}Rational Numbers (유리수) ℚ={all #s which can be expressed as

, b≠0}. The ratio of two integers Includes fractions, integers, whole numbers, natural numbers and decimals that repeat (3.3333333…) or terminate (4.25).

Real Numbers (실수) ℝ={all positive and negative numbers and zero}



Irrational Numbers (무리수) J ={any number that cannot be

represented as }

Note: many different symbols are used for Irrational numbers including K, ℚ’ and ℝ-ℚ

Depending on the problem, you will be told what number system you are using. If you are not told, unless it is obvious, use the Real number system.

If you are determining the number of people, you would use natural numbers since you cannot have negative or fractional numbers of people. For determining the number of pounds of bananas you can buy, you would use rational numbers since you can buy part or partial pounds (obviously only positive answers would be practical.)

Negative numbers

towards -∞

Positive numbers

towards +∞



1. Which set of numbers would you use (you want to choose the smallest set that would include all possible choices) for:a) the number of students who got 100 on the midterm exam?b) your height?c) the temperature reported in the newspaper?

2. Name the set(s) of numbers that each number is a member of:ex. 0 [whole, integer, rational, real]a) 0.5b) 6c) -3.14d) 2.1111...e) f) -

3. Is the statement true or false. If false, give a counterexample (a counterexample is a specific instance of the falsity).a) All negative numbers are integersb) is a real numberc) Every multiple of 5 is an odd numberd) When you divide two integers the result is another integer.

4.



Opposite

Two numbers that are the same distance from zero on a number line are opposites. The opposite of 5 is -5. The opposite of -3 is 3.

What is the opposite of 0 ?The way to write the opposite is to use - in front of a number or value.

- (3)= -3, - (-3) = 3

That gives us three uses of the “-” symbol.- is used to show a number is ___________.- is used between numbers to perform __________.- is used to denote (show) the _____ of a number.

Negative(음수) versus Minus(뺄셈)minus means subtraction. It does not mean a number is to the left of zero on the number line.

−5 is pronounced "negative five" rather than "minus five"; but "negative" is usually taught as the only correct reading. Textbooks encourage −x to be read as "the opposite of x" or even "the additive inverse of x" to avoid giving the impression that −x is necessarily negative. (http://en.wikipedia.org/wiki/Minus)

Remember - just putting a negative sign in front of a number doesn't make it negative. If the number was already negative then you just made it positive!



Additive Inverse (덧셈에 대한 역원)

The number which is added to another number to obtain a sum of 0. A number and its opposite are additive inverses of each other.

1) If 2 is the opposite of a number, what is the number?

2) If x is the opposite of a number, what is the number?

3) If a number is positive, what is the sum of the opposite of the number and the absolute value of the number?

4) If a number is negative, what is the sum of the opposite of the number and the absolute value of the number?

5) What is the additive inverse of:ㄱ) 2ㄴ) -3ㄷ) 1ㄹ) 0ㅁ) xㅂ) -y



Perform the indicated operation:



Absolute Value (절댓값)the distance of a number from 0. The absolute value of both 2 and -2 is 2 because they are both 2 away from 0 on the number line. Absolute value is denoted (shown) by placing a number or value between two vertical lines.

|-3| = 3, |4| = 4 |-1.34| = 1.34

i f ≥

i f

To force a number to be negative, you can write|. This takes the number, makes it positive, and then negates it (negates means makes negative).Perform operations between the absolute value lines then make the solution positive.|6 - 5| = |1| = 1|-4 +3| = |-1| = 1|-3 - 2| = |-5| = 5|-3 ∙ 2| = |-6| = 6|-6 ÷ (-2)| = |3| = 3



Find each absolute value:1. |-5| =2. |4.54| =3. |-| =4.

=

5. =

Speed is the absolute value of velocity. Speed is only related to the rate of movement. Velocity includes both direction and speed. Positive velocities are used for the direction forward or upwards while negative velocities are used for backwards or downward motion.

Comparing NumbersIn comparing two or more numbers, inequality symbols are used < is less than> is greater thanor an equality symbol is used.= is equal to

Other inequality symbols that are not normally used in number comparisons. They are used in algebraic equations or inequalities:≤ is than less or equal to≥ is greater than or equal to≠ is not equal to



A number line can be used to compare numbers. A number to the left of another number is less than the number to its right or a number on the right is greater than a number on its left.

-5 < -1 < 0 < 8 or 8 > 0 > -1 > -5

In comparing fractions, obtain a common denominator or change them to decimals.Compare

to

or

Therefore,

or

The answer should be in the same form as the question. Even if you convert to decimals or get a common denominator, you should use the same numbers given in the question.

You can order fractions by comparing their location on the number line. Numbers are listed from left to right as least to greatest (smallest to largest) or from right to left as greatest to lowest (largest to smallest).



Jihan owes $3 to his friend since he borrowed $1 on three different nights to buy ice cream after dinner. Eunji owes $10 to her friends because she buys brownies which are more expensive. Taewook has $4 in his pocket because he never buys anything. He just asks everyone for some of their snacks.

Eunji says she has the most because -$10 has two digits and -$3 and $4 are one digit numbers. (The 10 and 3 are negative since Jihan and Eunji owe the money to someone). Who has the most money (insert inequality symbols):

1. Eunji Jihan Taewook 2. Jihan Eunji3. Taewook Jihan4. Taewook Eunji



1. List three numbers that are real numbers, rational numbers, and also integers.

2. List three other numbers that are real numbers, integers and whole numbers.

3. List three other numbers that are real numbers and rational numbers.

4. Explain when you would use each number system (counting numbers, whole numbers, integers, rational numbers, real numbers). Give an example of a situation where you would use each one.

※(5~9) Which number systems do all of the following numbers belong to:

5. 3, 5, 7, 11, 200

6. 1.3, -6.5, 0

7. 0, 11, 111, 1111

8.

9.

※(10~21) Compare the following numbers by writing an inequality or equality symbol between the terms (use <, =, or >)

10. |4| -4

11. |-4| -4

12. |4| |-4|

13. -3 |3|

14. |a| |-a|



15. 0.5

16. |-4| 3

17. |-4| |3|

18. -|-4| |3|

19. -(-3) -|-3|

20. -|a| -a when a>0

21. -|a| -a when a<0

22. Why is it necessary to state whether a is positive or negative in questions 20 and 21?

※(23~26) Compare the following numbers by placing them in order from least to greatest.

23. 3, -6, 0, 0.3, -0.2

24.

25. 3.14, , 3,

,

26. 1.6, 1.61, 1.601, 1.611

※(27~32) Simplify each expression.

27. |3 – 5|

28. |5 – 3|

29. |3| - |5|

30. | -24 | + 4

31. | -24 + 4 |

32. |-24| + |4|



33. Explain the difference between taking the sum of two absolute values and the absolute values of a sum (like questions 32 and 31).

※(34~37) Simplify.

34. |5 ∙(-4) |

35. | (-5) ∙(-4) |

36. | (-5) ∙ (-4) |

37. | 5 ∙ 4|

38. What conclusion can you make about absolute value and multiplication based on the previous four problems.

※(39~40) Determine if the following statements are always, sometimes, or never true.

39. The sum of the absolute values of two numbers is equal to the absolute value of the sum of the two numbers.

40. The product of the absolute values of two numbers is equal to the absolute value of the product of the two numbers.

41. Give an example of when absolute value is used outside of the classroom.

42. If Yesung plots two points on a number line is it always possible to find a point between them. Explain.



Combine five numbers using the four arithmetic operations (+, –, ×, ÷) to arrive at a

"target" number. Use the numbers 1–10 only.

Rules to play.

w You use all the numbers.

w You do not have to use them in the same order that they appear.

w You do not have to use all four arithmetic operations.

w You can use parentheses.

w You cannot connect two digits to form a 2-digit number (concatenating digits is not

allowed.)

Here is an example of the game.

1. List solutions you made and compare with solutions made by others.

2. Explain what problem-solving strategies you used while playing Primary Krypto.

Your Five Numbers

16 2 2 3 19Target Number

2



Across8. all numbers that can be expressed as a fraction9. a rectangular arrangement of dataDown1. counting numbers2. two numbers the same distance from zero3. a constant4. all positive and negative numbers and zero5. the plural of matrix6. the distance a number is from zero7. positive and negative whole numbers



Negative Numbers in YOUR Life

Mathematics is a world of its own that can be used to MODEL things in the real world,

but, like any model, is not IDENTICAL to that real world. Negative numbers are part of

the "model world", not the real world.

There are some situations where negative numbers make sense. In the case of

temperatures, negative values are possible. Most thermometers are really number lines

that stand upright. The numbers can be thought of as temperature changes. Positive

numbers make the temperature indicator rise. Negative numbers make the temperature

indicator fall below zero.

In the case of money, a negative number can be interpreted according to the given

situation. If you are just spending money from your banking account, a negative balance

means that you are overdrawn--but it does still have meaning, because you now owe

that much money to the bank.

ACTIVITY

I. Make a situation or a story where negative numbers are involved.

II. What jobs use negative numbers?

For example, a stock market broker would have to deal with negative numbers.

A weather man would deal with negative numbers. What other jobs?


1. Arrange the numbers I tell you so the sum of all sides of the triangle are equal2. Each number may only be used once 3. Find all the solutions - there are more than one4. Rotating the numbers in the same order is the same solution.5. Use the large circles for trial and error and write correct answers in the other



Factors (약수) Factors are the terms that when multiplied together result in the original term, expression or number.

12 = 4 * 3 (4 and 3 are factors)12 = 6 * 2 (6 and 2 are factors)

All the numbers that can be multiplied to obtain a numbers are the factors of that number. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

A number is factored when it is written as a product of two or more natural numbers.

A natural number is divisible by another natural number if it is divided evenly. 10 is divisible by 2 since the quotient (the solution to a division problem) is another natural number. 10 ÷ 2 = 5. 10 is not divisible by 3 since the quotient is not a natural number. The quotient is

.

What type of number is ?

1. What are all the factors of 16?

2. What are the factors of 10?

3. What are the factors of 23?



Prime number (소수)

A natural number is a prime (소수) number if it has exactly two factors, 1 and the number itself.

13 is prime - it is only divisible by 13 and 1. 2 is the smallest prime number.

Why is 2 the smallest prime natural number?

Other prime numbers include 3, 5, and 7. There is an infinite number of prime numbers.

composite number (합성수)

If a natural number is not prime it is composite and has three or more factors.

What number is neither prime nor composite?

Prime factorization (소인수분해)

Prime factorization (소인수분해) is factoring a number as the product (solution to a multiplication problem) of only prime numbers.

12 = 2*2*3 = 2²*3

Since there are several identical prime factors, we write that factor as a power with the exponent being the number of times that factor occurs.

2³ means 2 occurs 3 times.



1. What is the prime factorization of 16?

2. What is the prime factorization of 10?

3. What is the prime factorization of 23?True or False4. Thirty-nine is a composite number?

5. Twenty-nine is a prime number?

6. Nineteen is a composite number?

Circle the prime numbers : 5, 12, 21, 11, 8, 2, 1, 17, 27

Power (거듭제곱)

A power (거듭제곱) has two parts - the exponent (지수) and the base (밑). The number that is multiplied several times is the base and the amount of times it gets multiplied is the exponent.

means 4 gets multiplied with itself 7 times. 4*4*4*4*4*4*4. 4 is the base and 7 is the exponent. The term is called a power.

Variable (변수)

A variable (변수) which is a letter that represents a number can be written in a power. is y*y*y*y*y.

To factor 6, the number which is called a coefficient (계수) since it is multiplied with the variable and the variable are factored separately. 6 = 3*2 =y*y so the factors of 6 are 2*3*y*y



To determine the prime factors of a number, write the number and divide by a prime number for which that the number is divisible. Continue doing that with the quotient until the new quotient is a prime number. All the numbers on the side are the prime factors

Determine the prime factors of 48

Select a prime number that goes into 48 evenly. Since it is even, I choose 2. I divide by 2 and get 24 which I write below the 48. I select another prime number and again I choose 2 since it is even. I obtain 12 and continue. I end up with 3 a prime number. So, my prime factors are 2*2*2*2*3 (expanded form) which we write as (exponent form) which is equal to 48.

Other Types of Numbers and Primes:

Mersenne’s Prime: A prime number of the form where n must be prime. Examples include 3, 7, 31, 127.

Perfect number: Any positive integer that is equal to the sum of its distinct proper factors (factors other than the number itself).Example: 6 (proper factors: 1,2,3; 1+2+3=6) 28 (proper factors: 1,2,4,7,14; 1+2+4+7+14=28)

2 48

2 24

2 12

2 6

3


Determine the prime factorization of: (write your answers in exponent form)1. 18

2. 72

3. 29

4. 100

True or False

5. The prime factorization of 45 is ∙ ?

6. The prime factorization of 63 is ∙ ?

Write the prime factorization in expanded form (No 지수)

7.

8. -30

Evaluate the expressions

9. ∙∙

10. ∙∙∙∙

11. If Goeun doubles an odd number, does the list of all possible factors double? What if Gyuri doubles an even number?


12. Christian Goldbach made a conjecture (a proposal believed to be true but not proven) that every even natural number greater than 2 is the sum of two prime numbers. List the even numbers from 20 to 50 and write the two prime numbers that add up to them.

13. Goldbach also made a conjecture that every integer greater than 5 can be written as the sum of three primes. List the integers from 7 to 14 and write them as the sum of three prime numbers.

14. Goldbach also states that every odd number larger than 5 can be found as the sum of a prime number and twice a prime number. 7=2(2)+3Find the sums for the odd numbers larger than 7 up to and including 19.

15. Twin primes are prime numbers whose difference is 2 such as 3 and 5, 5 and 7, and 11 and 13. What are the next 5 pairs of twin prime numbers.

16. Every even integer can be expressed as the difference between two primes. List the even numbers from 2 to 12 and the two primes who difference equals that number.

17. A prime triple is a list of three different prime numbers for which the last and first number has a difference of 6 such as 101, 103, and 107. List three other sets of prime triples.

Sieve of Eratosthenes Sieve of Eratosthenes Sieve of Eratosthenes


How to find prime numbers?This is used to find all the prime numbers less than or equal to a given integer by Eratosthenes' method. It is called Sieve of Eratosthenes.

Prime numbers

Let’s find all the prime numbers less than or equal to 60, proceeding following

steps.

Step1. Initially, let equal 2, the first prime number.

Step2. Starting from , count up in increments of and mark each of these numbers greater than itself in the list. These numbers will be 2, 3, 4, etc.; note that some of them may have already been marked.

Step3. Find the first number greater than in the list that is not marked. If there was no such number, stop. Otherwise, let now equal this number (which is the next prime), and repeat from step 3.

When the algorithm terminates, all the numbers in the list that are not marked are prime.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

1. Find all the prime numbers less than or equal to 120, using Sieve of

Eratosthenes.

2. Discuss with your partner if there is an easier method to determine if 131 is prime number.

The number of factorThe number of factorThe number of factor


How to find how many factors are in a number?Let’s find the number of factors of 144.Step1. Calculate the Prime Factorization of 144

= ×××××

Step2. Find the number of factor of

the number of factor of : 5

Step3. Fill in the box to find the factors of

the number of factors of : 5

the number of factors of 3:

the number of factors of :

Step4. Find all the factors of

the number of factor of : 5



How many factors does 1125 have?

How many factors does ＊ have? Why? ( : prime numbers)

1 2

3

×

1 2

1 2

3

×

×



Common Factor (공약수)

When you have two numbers and determine their factors, the numbers that are common to both of them are called Common Factors.

10 and 25 have common factors10 has factors of : 1, 2, 5, 1025 has factors of : 1, 5, 25

The common factors are: 1 and 5

Relatively prime (서로소)

Two numbers are relatively prime [also called co-primes] (서로소) if their GCF is 1.

Greatest Common Factor (최대공약수)

The Greatest Common Factor (GCF) is the largest of the common factors. The greatest common factor must the no larger than the smaller of the two numbers.

If the numbers were 10 and 20, the factors are:10 - 1, 2, 5, 1020 - 1, 2, 4, 5, 10, 20

The GCF is 10.

When variables are involved, the greatest common factor is the term with the lowest exponent for each variable.

2 ) 10 20 5 ) 5 10 1 2GCF : ×



The GCF of and .

Why is this??

The factors of are x*x*x*x*y*y*yThe factors of are x*x*y*y*y*y*yThe factors in common are x*x*y*y*y which is .

For terms involving both numbers and variables - called variables with coefficients, find the GCF of the coefficients and the GCF of the variables and combine.

The GCF of 6xy and 14 is 2xy.

6xy = 2*3*x*y = 2*7*x*x*yThe 2, one of the x and the y are common to both.

What is the greatest common factor of:

1. 20, 34

2. 30, 40

3. 27, 45

4. 18, 36



Write two different numbers whose GCF is:

5. 6

6. 9

7. 2

8. 10

Determine the area and perimeter of the following rectangles.

9. 7 10. 5

3 5

11) Are the area and perimeter for problem 9 relatively prime (서로소)? Explain your answer.

12) Are the area and perimeter for problem 10 relatively prime (서로소)? Explain your answer.

13. Find the GCF of and .



Least Common Multiple (최소공배수)

A number that is a multiple of two numbers is called a common multiple. The smallest of them is the Least Common Multiple (LCM).

Every number has an infinite number of multiples. There is only one LCM for any group of numbers. The highest it can be is the product of the numbers although it is often lower.

6 - 6, 12, 18, 24, 30, 36, 42, 48, 54, ...9 - 9, 18, 27, 36, 45, 54, 63, 72, 81, ...

The common multiples are 18, 36, 54, ...

The LCM is 18. (Note - the product of 9 and 6 is 54 and 54 is a common multiple but not the least common multiple.

To find the LCM, determine the prime factors of the numbers. Multiply all the numbers that are not in common by the ones that are common.

not common common6 = 2 * 39 = 3 * 3

2 * 3 * 3 = 18

non-common

common



You only multiply by the common number once. If our numbers are 8 and 12 -8 = 2 * 2 * 212= 3 * 2 * 2

The non-common terms are 2*3 and the common terms are 2*2. Multiply 2*3*2*2 = 24.

Verify -8 - 8, 16, 24, 32, 40, 48,... 12 - 12, 24, 36, 48, ....24 is the lowest number they have in common.

Variables are also factors.

The LCM of and

The common factors are The non common factors are * 28a = or * = ∙∙

What are the least common multiples of:

1. 9, 15

2. 12, 42

3. 4x, 14y

4.

5.

6. Find the LCM of and .

2) 8 12 2) 4 6 2 3

GCF is ×

LCM is ×××



Summary

The GCF of x and y where x >y is less than or equal to y.

x y The GCF of 20 and 4 is 4 The GCF of 20 and 6 is 2

The LCM of x and y where x >y is greater than or equal to x.

x y The LCM of 20 and 4 is 20 The LCM of 20 and 6 is 60


1. Which two numbers are relatively prime? ① 13 and 39 ② 7 and 49 ③ 9 and 42 ④ 21 and 78 　　　⑤　9 and 56

2. A prime number is any natural number that ① is not divisible by 2 ② has three or more factors ③ has exactly two factors, itself and 1 ④ is not an even number ⑤ has many factors but none are repeated

3. Keuho has a box of dominos which measure 4 cm by 14 cm. Without overlapping any dominos or cutting any of them, what is the least number of dominos Keuho needs to form a square region. Drawing a diagram to may be helpful.

4. Nahyeon notices that a traffic light turns red every 6 minutes. Another traffic light turns red every 8 minutes. A third one turns red every 10 minutes. At 10 AM, the three traffic lights turn red at the same time. When is the next time the three lights will turn red at the same time?

5. There are two stacks of boxes. Each stack has boxes of the same size. The first stack has boxes that are all 8 inches tall. The second stack has boxes which are 14 inches tall. What is the least number of boxes of each size to have stacks of the same height?

6. For exercise 5 did we need to determine the LCM or the GCF to figure out the solution?


7. If two numbers are relatively prime, how would you determine their LCM?

8. If one number is a multiple of another number, what is their LCM?

9. Find two prime numbers whose LCM is 38.

10. Find two different composite numbers whose LCM is 24.

11. Find two pairs of different even numbers whose LCM is 12.

12. What is the prime factorization of 126? ① ∙∙ ② ∙∙

③ ∙∙∙ ④ ∙

⑤ ∙∙∙

13. Which statements are false? ① The GCF of 8 and 12 is 2. ② The LCM of 8 and 12 is 24. ③ 4 is a prime factor of both 8 and 12 ④ 1 is a prime number ⑤ 5 is the LCM of 25 and 35

14. Two toy electric trains travel around two oval tracks that are the same size, one track above the other. If one train passes the station every 20 seconds and the other train passes its station directly above the other station every 8 seconds, when will the two trains pass the stations at the same time?


Find the Greatest Common Factor (GCF) by playing a card game with your partner.

1. Each person has the cards on which are written the numbers 2, 3, 5, 7. (There are three cards for each number).

2. Share a multiplication board which has boxes to put cards on. 3. Put a card on a box by turns not GCF is over 31.

For example, the game is going on above. If A puts 2 or 7 on the last box, the GCF will be over 31. Therefore, A will lose.

4. A winner gets the point as many as the Least Common Factor.

5. The final winner will be a person who gets 300 point first.

6. If the game is not done after putting cards on all the boxes, lay a new card on the top of a card which has been put on the board.

Do you have any idea to calculate GCF, LCM more quickly? If so, share the idea with your partner after playing the game.

A

B

× × × × ×

× × × × ×

2 3 5 3 3 ?

32 2 7 5 ?



Scientific notation (과학적 기수법)Scientific notation is a way to represent very large or small numbers without having to use many digits. The number is written as × , where is number whose absolute value is greater than or equal to 1 and less than 10 and n is an integer.

Instead of writing 2,500,000,000, pronounced two billion five hundred million, it can be written as × and 0.0000045 pronounced 45 ten millionths can be written as ×.

The above chart is called a place value table that shows the value of each place of a number written in decimal form. It also shows the exponential equivalent and the fractional value.

In referring to Korean money, 100,000 won should be thought of as 10 units of ten thousand won or shib man won (십만원) since 10,000 is an important counter for Korean won. Amounts of Korean money greater than 10,000, references the 10,000 won (만원) unit.



3,400 is written in scientific notation as ×. You can look at 3.4 and move the decimal place 3

to the right. and fill the empty loops with zeros to get 3400.

× is written as 0.00134 in decimal form (the "normal" way to write numbers.) We can

take 1.34 and move the decimal place 3 to the left

which would be .00134 and it is customary to put a digit to the left of the decimal point so a 0 is placed as 0.00134 .

Negative exponents represent smaller numbers while positive exponents represent larger numbers.

Decimal Form Scientific Notation 4,500,000 ×

5,000 ×

0.0000123 ×



Rewrite the numbers in proper scientific notation:

1. 430

2. 3,400,000

3. 0.0000045

4. ×

5. ×

Rewrite the numbers in decimal form:

6. ×

7. ×

8. ×

9. ×

10. ×



To multiply two numbers written in scientific notation, multiple the coefficient () terms and then add the exponents of the other terms (powers). If the new coefficient is ten or greater, you must regroup to be in proper form.

× ∙× ×

× ∙× ×

×

× ∙× × ×

Rule for Multiplication - When you multiply numbers with scientific notation, multiply the coefficients together and add the exponents. The base will remain 10.

Rule for Division - When dividing with scientific notation, divide the coefficients and subtract the exponents. The base will remain 10.

Multiply and write the answers in both scientific notation and decimal forms.1. ×∙×

2. ×∙×

3. ×∙×

4. ×∙×

5. × ÷ ×

6. × ÷ ×



You may add or subtract numbers in scientific notation only if they have the same power.

× × ×

If not, you must either convert to standard form, add, and then convert back, or rewrite the number so the exponents match (by changing it from proper scientific notation form but leaving it in exponential form.) × ×

× × ×

1. × ×

2. × ×

3. × ×

4. × ×

5. × ×

6. × ×



Significant Digits (유효숫자)

What time is it?Someone might say “1:30” or “1:28” or “1:27:55”. Each is appropriate for a different situation. In math we describe a value as having a certain number of “significant digits.” The # of significant digits in a value includes all digits that are certain and one that is uncertain.“1:30” likely has 2, 1:28 has 3, 1:27:55 has 5 significant digits.

It is better to represent the exact number 100 as × since that shows that the number has been rounded to the nearest ones place or is exact. Alternatively you can underline the position of the last significant digit. e.g. 100. This is especially useful when doing a long calculation or for recording experimental results.

Don’t round your answer until the last step in a calculation.

Which digits are significant?

1. Digits from 1-9 are always significant. 2. Zeros between two other significant digits are

always significant 3. One or more additional zeros to the right of

both the decimal place and another significant digit are significant.

4. Zeros used solely for spacing the decimal point (placeholders) are not significant.



Adding with Significant DigitsHow far is it from Seoul to room 331? then to 334?Adding a value that is much smaller than the last significant digit of another value is irrelevant. When adding or subtracting, the # of significant digits is determined by the significant digit furthest to the left when #s are aligned according to their decimal.

RULE: When adding or subtracting your answer can only show as many decimal places as the measurement having the fewest number of decimal places.

Multiplication and DivisionDetermining significant digits for questions involving multiplication and division is slightly different. For these problems, your answer will have the same number of significant digits as the value with the fewest number of significant digits.a) 608.3 x 3.45 b) 4.8 ÷ 3923.45 has 3 significant digits, so will the answer. 608.3 x 3.45 = 2098.635 = 2.10x103

4.8 has 2 significant digits, so will the answer.4.8 ÷ 392 = 0.012245 = 0.012 or 1.2x10-2

RULE: When multiplying or dividing, your answer may only show as many significant digits as the multiplied or divided measurement showing the least number of significant digits.



Unit conversions

Sometimes it is more convenient to express a value in different units.

When units change, the number of significant digits must remain the same.

1.23m = 123 cm = 1230mm = 0.00123km

Notice that these all have 3 significant digits. This should make sense mathematically since you are multiplying or dividing by a term that has fewer (1) significant digits. [multiplying by 10, 100, 0.001...]

×

Simplify making sure you pay attention to the significant digits.

1. 4.60 + 3 =

2. 0.008 + 0.05 =

3. 22.4420 + 56.981 =

4. 13.7 x 2.5 =

5. 0.00003 x 727 =



6. 5003 / 3.781 =

7. 89 / 9.0 =

Identify the number of significant digits show in each of the following examples.

8. 400

9. 200.0

10. 0.0001

11. 218


Change the following numbers to proper scientific notation 1. 65.7

2. 0.00545

3. 22,450,000

Change the following numbers to standard notation 4. 8.85 x

5. 1.847 x

6. 3.400 x

Determine the solution and make sure the answer is in proper scientific notation form.

7. (4.42 x ) x (4 x )

8. (3 x ) x (7 x )

9. (9.2 x ) ÷ (6.3 x )

10. (2.4 x ) ÷ (5.49 x )

11. (6.423 x ) + (4.123 x )


12. (4.54 x ) - (1.01 x )

How many digits are significant?

13. 320

14. 0.00530

15. 22,568

16. 4755.50

Determine the solution using significant digits.

17. 200 - 87.3 = 18. 67.5 - 0.009 =

19. 71.86 - 13.1 =

20. 200 x 3.58 =

21. 5000 / 40 =

22. What is the ratio of the Milky Way's radius to our solar system's radius given that, the distance from Pluto to the sun is 5.9× meters and the Milky Way's radius is 3.9×10 meters? Make sure your answer follows the rules of significant digits.


23. Using the following data:

the volume of a sphere =

Radius of the earth ≈ 6 × 10 mMass of earth ≈ 6 × 10 kg

Find the density of the earth in 　 kg/m (Note: this method will only

yield an approximate value for the density of the earth). Make sure your answer follows the rules of significant digits.



Percent (백분율)

Percent is a representation of a part of a whole. 100% is the whole or equal to 1. 50% is equal to 1/2 and 200% is twice as much.

If you have 12 t-shirts and 3 of them are black, what percentage are black? You create a fraction,

and change it into decimal form and

multiply by 100 to put it in percent form.

∙

Fraction Decimal Percent

0.25 25%

2.4 240%

16.7% or 17%



Write the fraction as a decimal and a percent:

1.

2.

3. What percent is shaded?

4. What percent is area 1?



Percent of Change

Percent of Change is the ratio of the amount of change to the original amount which is then expressed as a percent. A percent increase occurs when the value increases from its original amount. A percent decrease occurs when the value decreases from its original amount.

If the price of a candy bar went from ￦500 to ￦600, you can calculate the percent of change :

or

∙

∙ ∙

∙

∙

So there was a 20% change and since the price went up, it is a 20% increase.

What would the percent of change be if the price of the candy bar went from ￦600 to ￦500? Is it a 20% decrease since we just switched the numbers around?

Calculate the percent change.

1. 270 is increased to 1134

2. 14 is increased to 70



3. 374 is decreased to 187

4. 150 is decreased to 39

5. In the 1949 census, which was the first official census, the Korean population was calculated to be about 20,000,000. According to the census of 1985 the population was about 40,000,000 people. What is the percent of change?

6. In December 1987, one US dollar was equal to 1950 Korean won. In 1998, the average value of the US dollar was equal to 1404 won. What was the percent of change. In November 2009, the average value was 1160 won. What was the percent of change from 1987 to 2009 and 1998 to 2009?


1. Seven out of every ten boys play football. What percent play football?

2. Eighteen out of every twenty girls like to dance. What percent do not like to dance?

3. About what percent is each section (1,2,3,4)?

Determine the percent change:

4. ￦2000 to ￦3500

5. 10 pounds to 8 pounds

6. 16 cm to 22 cm

7. ￦10,000 to ￦9000

8. ￦9000 to ￦10,000

9. The sneakers DoHyun wanted went on sale and were ￦24,000. Last week they were ￦30,000. What was the percent change?

10. Athletes gets their strength measured. An athletes increase in elbow strength went from 90kg to 125kg. What was the percent increase in the athlete's strength?



Students often assume that each measurement they make in the laboratory is correct and accurate. They also usually assume the values they derive (obtain) through experimentation are very accurate. There are many sources of error that prevent students form being as accurate as they would like to be. Percent error calculations are used to determine how close to the true values, or how accurate their experimental values or data calculations actually are.

The last time you measured something you probably measured it to the nearest centimeter or millimeter.

Greatest possible error (오차의 한계)

The greatest possible error in a measurement is one half of that measuring unit.

If you measured your height as 172 cm, it could have been in the range of 171.5 cm to 172.5 cm. Otherwise you would have rounded to a different number. So the greatest possible error is 0.5 cm.

If several things are measured for the same problem, the number of significant digits should be the same.



What is the greatest possible error:

1. 10. m

2. 32 cm

3. 1.3 mm

4. 0.32 grams

5. 120 km

Percent error (백분율 오차)

The percent error is a ratio between the greatest possible error and the measurement we obtained.

For practice problem 5, we determined the greatest possible error was 5 km on the measurement of 120km. The percent error is

. The percent error is

about 4%.

percent error =

Determine the percent error of ：１．　　

m.

2. 32 cm.



If you were measuring the area of a room and you measured 10m × 8m, then since 8 is rounded to the nearest one’s digit, 10 would be assumed to have the same precision or accuracy so it would also be rounded to the nearest one’s digit.

Since the greatest possible error is 0.5m, we would calculate the minimum area as having lengths of 0.5 less than our measurement and the maximum area as 0.5 greater than our measurement.

To determine the percent error when the measurements are multi-dimensional (more than one measurement like area and volume), you determine the ratio of the greater difference between the measurement and the maximum or minimum, and the measurement.

percent error =

9.25 is the greater difference. The percent error of the room measurement is

×

Length Width Area Difference

Measured 10 8 80

Minimum 9.5 7.5 71.25 8.75

Maximum 10.5 8.5 89.25 10.75

Length Width AreaDifference with

measured value

Measured 10 8 80

Minimum 9.5 7.5 71.25 8.75

Maximum 10.5 8.5 89.25 9.25



In contrast: if the problem was 10m × 8m instead of 10m × 8m, the calculations would be:

(this example is unlikely to happen since we would measure both to the same degree of precision. Since we measured the 8m to the one’s digit so the 10m should also be measured to the one’s digit)

The minimum area of the room is 37.5 and the maximum area is 127.5 based on our measurements showing an area of 80.

The percent error would be 59.375%. Again, this is an unlikely answer and you will probably never see a problem like this.

Find the minimum and maximum areas and the percent error for these measurements:

1. 3cm x 5cm

2. 13m x 15m

3. 10km x 30km

Length Width AreaDifference with

measured value

Measured 10 8 80

Minimum 5 7.5 37.5 42.5

Maximum 15 8.5 127.5 47.5


Determine the greatest possible error:1. 14 m

2. 6.3 cm

3. 12.13 g

4. The price of the Samsung notebook LeeJune wanted dropped from ￦1,500,000 to ￦1,350,000. What was the price decrease and percent of change?

5. Minkyu is a philatelist (stamp collector). The value of his collection changed from ￦5,000,000 to ￦6,000,000. What is the percent of change?

6. Minsol is a numismatist (coin collector). The value of the collection changed from ￦4,000,000 to ￦5,000,000. What is the percent of change?

Find the minimum and maximum areas and determine the percent error:

7. Minju measures her bedroom after going over a friend's house and seeing her awesome bedroom that has a full size video game and a large plasma screen tv on the wall. Minju's bedroom measured 4m x 3.2m.

8. Minji measured the cover to a box. She recorded the measurements of 1m x 95cm.

9. Seohyun measures a package and it is 12cm x 8cm x 10cm.



Rate (비율)

A rate is a comparison of two quantities with different units of measure and is often shown as a fraction or as x per y.

41 gallons per 3 hours or

30km per gallon or

￦8000 per hour or ￦

When the denominator is 1, the rate is called a unit rate because it shows the rate for 1 unit.

Ratio (비(比))

A ratio is a comparison of two numbers with the same units of measure and is represented as a fraction or described as x to y.

or 1 to 2

My weight is 72 kg. Younghyun's weight is 54kg.

In simplest form,

or 4 to 3.



Write whether the quotient is a rate or ratio by writing the word RATE or RATIO.

1.

2.

3.

4. 60 km per hour

Write as a ratio and simplify

5.

6.

7. secmin

8. Which is a better buy? 6 apples for ￦1180 or 10 apples for ￦1800.

9. Determine the unit rate for ￦200,000 for 25 hours of work.

10. If $1 from the United States is equal to ￦1100, how many won could you get for $40? How many dollars would you get with ￦400,000?



11. In a baseball field, the infield portion called the baseball diamond is 90 feet by 90 feet and the entire field is 270 feet by 270 feet, what is the ratio of the baseball diamond's area to the area of the entire baseball field?



Proportion (비례식)

A proportion is an equation that has two ratios that are equal to each other.

is read as " a is to b as c is to d". If you

know three parts of a proportion, you can determine the 4th part. The a and d parts are called the extremes (외항) while the b and c parts are called the means (내항).

The cross product property also know as the means-extremes theorem lets us know the product of the means is equal to the product of the extremes.

→

To determine the value of a missing term, cross multiply and determine the missing value to make the products equal.



Which pair of numbers are the means? Which pair are the extremes?

1.

Determine whether the proportions are true or false.

2.

3.

4.

Determine the missing value.

5.

6.

7.

8.

9. Sunmin and Seonghwa competed in a 20 km boat race. Sunmin traveled at a rate of 4km every 25 minutes. Seonghwa traveled at a rate of 3km every 30 minutes.

a. How many minutes did it take each person to complete the race? How many hours?

b. How fast in km per hour were the boats traveling?

c. Who won the race?



Similarity (닮음)

Two triangles are similar if the angles have the same measure and the lengths of the sides are in proportion.

These two triangles have angles which have the same measure but different lengths. Notice the related sides have lengths that are in proportion. The lengths of the sides of the triangle on the right are 1/2 the lengths of the triangle on the left.

All the corresponding pairs of sides are in proportion.

Determine the length of the sides with lengths labeled x and y.

8

4

2

4510

56°

45°

79°

45°

56°

79°


A typical heartbeat rate (pulse) is 72 beats per minute. When you exercise, this rate increases.

1. Assume you have a typical heartbeat. If you take your pulse for 10 seconds, how many beats would you feel?

Hint: min

sec

Don’t forget to change minutes to seconds or seconds to minutes so the units match!

2. After running for 30 minutes, you take your pulse for 6 seconds and feel 12 beats. What is your heartbeat in beats per minute?

3. After playing football for 45 minutes during lunch, you take your pulse for 12 seconds and feel 26 beats. What is your heartrate in beats per minute?

4. After jumping rope for 20 minutes, you take your pulse for 15 seconds and feel 30 beats. What is your heartrate in beats per minute?

5. If you sleep 250,000 hours by the time you are age 70, about how many hours will you have slept by age 30?

6. Soomin read 40 pages of a book in 50 minutes. How many pages should she be able to read in 80 minutes?


7. Yeonjae found out that after working for 9 months she had earned 6 days of vacation time. How many days will she have earned after working for two years?

8. A car travels 125 km in 3 hours. How far would it travel in 5 hours?

9. Yuchan asks the seventh-grade students to name their favorite flavor of ice cream. Yuchan finds that 3 out of every 8 students prefer strawberry to vanilla. If there are 160 students, how many prefer strawberry?

10. A store makes a profit of W15,000,000 for every 300 coats that they sell. If they make a profit of W25,000,000, how many coats did they sell?

11. A recipe calls for 2 eggs for every 5 cups of flour. A local chef will use 35 cups of flour, how many eggs must he have?

12. If a watch is losing 1.5 minutes for every 12 hours that go by, how many minutes will it lose in one week?

13. Five out of every seven households have cable TV. If 42,000 households in a certain city have a TV, how many do not have cable TV?


The following is the article about storm damage in the US. After reading it, answer the question below.1.26 Millions Powerless, 24 Dead

After Storm

by The Associated Press

WASHINGTON July 3, 2012

WASHINGTON (AP)- Utility crews struggled

to catch up with a backlog of millions of

people without electricity for a fourth hot

day Tuesday as frustration grew and

authorities feared the toll of 24 storm deaths

could rise because of stifling conditions and

generator fumes.

Power was back for more than a million

customers but lights— and air-conditioning — were still out for about 1.26 million homes

and businesses in seven states and the

District of Columbia. The damage was done

by powerful wind storms that swept from the

Midwest to the Mid-Atlantic late Friday,

toppling trees and branches into power lines

and knocking out big transmission towers and

electrical substations.

...<omitted>...

Some people said the destruction over the

weekend was reminiscent of that caused by

Tropical Storm Isabel in 2003 and Hurricane

Irene in 2011.

Last year, it took Baltimore Gas and

Electric company eight and a half days to

restore service to all 752,633 customers who

lost power during Hurricane Irene. This time,

the power company initially confronted more

than 600,000 people without power. It said

restoration efforts will extend into the

weekend.

...<omitted>...

Baltimore Gas and Electric said in a letter

posted on its website that it would take

hundreds of thousands of man-hours to clear

debris and work through outages. Crews were

working around the clock in 16-hour shifts.

<omitted>...

1. Find true values (exact) and approximate values (rounded) in the article.

2. After choosing one of the approximate values from the article, suppose that the value is rounded to the nearest followings. Make a chart using the chosen number.

hundreds thousands

significant digits

(significant digits)×

The greatest possible error

3. After finding any article which has true values and approximate values, make a chart like above.


<attach article>

1. Find true values and approximate values in the article.

2. After choosing one of the approximate values from the article, suppose that the value is rounded to the nearest place value. Make a chart using the chosen number.

tens tenths

significant digits

(significant digits)×

The greatest possible error


Divisibility Rules

2- If the last digit is even, the number is divisible by 2.

3- If the sum of the digits is divisible by 3, the number is also.

4- If the last two digits form a number divisible by 4, the number is also.

5- If the last digit is a 5 or a 0, the number is divisible by 5.

6- If the number is divisible by both 3 and 2, it is also divisible by 6.

7- Take the last digit, double it, and subtract it from the rest of the number; if the result is divisible by 7 (including 0), then the number is also. Repeat for larger numbers.

8- If the last three digits form a number divisible by 8, then so is the whole number.

9- If the sum of the digits is divisible by 9, the number is also.

10- If the number ends in 0, it is divisible by 10.

11- Alternately add and subtract the digits from left to right. (You can think of the first digit as being 'added' to zero.) If the result (including 0) is divisible by 11, the number is also.Example: to see whether 365,167,484 is divisible by 11, start by subtracting:[0+]3-6+5-1+6-7+4-8+4=0; therefore 365,167,484 is divisible by 11.

12- If the number is divisible by both 3 and 4, it is also divisible by 12.

Background

When describing the reasoning behind these shortcuts, consider numbers of up to four digits, such as 6745. However, we will not use a specific number, instead we will write the number as

1000*a + 100*b + 10*c + d.

A specific number like 6745 is obtained when a=6, b=7, c=4, and d=5.


Let's figure out why these work...

Divisibility by 2

(WORDS) If the ones digit is evenly divisible by 2, the entire number is divisible by 2

(ALGEBRAICALLY) a 4 digit number abcd = 1000*a + 100*b + 10*c +d=2(500*a+50*b+5*c) + d. Since 2(500*a+50*b+5*c) is divisible by 2, 2(500*a+50*b+5*c) ÷2 =(500*a+50*b+5*c) then if d (the ones digit) is divisible by 2, the entire number is divisible by 2.

[specific example – 4532 = 4*1000+5*100+3*10+2 = 2(4*500+5*50+3*5) + 2 clearly the first part is divisible by 2 since one factor is two, the other is (4*500+5*50+3*5). Then if the last digit, 2 is divisible by 2, the entire number is divisible by 2.]

Let's work on trying to prove the divisibility rules for 3, 4, 5, 7, 8, 9 and 10.

Use the divisibility tests to determine whether the numbers are divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12.

1. 30

2. 192

3. 2160

4. 12,321

5. 51,840


6, Find the smallest natural number divisible by 2, 3, 4, 5, 6, 8, and 9.

7. Yoonsae is part of a team that is planting a forest. Yoonsae has to plant the trees in a rectangular region. Any rectangle can be used as long as the area is 1600 square meters. If the lengths in meters are natural numbers, what are the possible dimensions of the region? (Dimensions for a 10 square meter region would be 1 x 10, 2 x 5, 5 x 2, 10 x 1. Do not multiply!!)

8. Yunhye needs to construct a fence around the rectangular region in question 7. What is the smallest perimeter (둘레)?

9. Write the rule in words for each divisibility test for the number 3. Then algebraically prove the rule.



1) What sets of numbers are included in integers? in real numbers? in whole numbers? in counting numbers? in irrational numbers?

2) What are the prime factorizations of :a) 32 b) 77 c) 45 d) 111 e) 100 f) is 7*5-3 a correct prime factorization of 32?

3) Solve and make sure the answer is in scientific notation: a) ×× ×

b) × ×

9) Find the unit rates:

a)

b) min

c)

d)

4) Perform the indicated operation with your answer in terms of significant digitsa) 3.234 + 42.12

b) 32.2040 – 19.65940


c) 34.2 * 21

d) -3　＊　4.231

5) Determine the GCF and LCM of eacha) 10, 15

b) 32 and 15

c)

d)

e) 7, 11

6) Solve for the variable(s):

a)

b)

c)

d)

7) What is the percent of change?

a) 1000 won to 1500 won

b) 1500 won to 1000 won


8) Find the values of the variables and the ratio of the similar trianglesa)

b)

c) How do you know two triangles are similar?

9) Find the percent error when you measure

a) 3.2mm x 4.1mm

b) 11m x 16m

10) What is the greatest possible error when you measure

a) 10 m

b) 10. m

c) 11.25 m

11) Write the number:

a) Thirteen thousand six hundred fifty

b) twelve and three hundredths

12) Change to a percent: a) 4.5

b) 0.23

c)

3 4

6

10.5 y

z

5

z

y

49.5

0.9 8.1

100°100°

65° 65°15° 15°


d)

e)

13) 50 grams of kimchi requires 2 grams of pepper sauce. The cafeteria makes 650 grams of kimchi and the workers use twice as much pepper sauce as they are supposed to use. How many grams of pepper sauce did the cafeteria workers use?

121

1) integers includes whole and counting(natural) numbers; real numbers includes irrational, rational, integers, whole, and natural numbers; whole numbers include natural numbers; counting numbers only includes itself; irrational numbers only includes itself.

2) a) ×××× or b) 7 x 11 c) 3 x 3 x 5 d) ３ x 37 e) 2 x 2 x 5 x 5 or ×

f) no. prime factorization only includes multiplication of natural numbers.

3) a) × b) ×

4) a) b) min

c) not a rate! this is a ratio! d)

5) a) 45.35 b) 12.5446 c) 720 d) -10

6) a) 5, 30 b) 1; relatively prime, 480 c) d) e) 1, 77

7) a) a=9 b) b=

c) 50.4 d) any values where y is twice x

8) a) 50% increase b) 33% decrease (33.3% or 33 1/3%)

9) a) y=14 z=21 b) y=45 z= 5.5 c) Corresponding angles need to be equal, corresponding sides need to be proportional

10) a)

or b)

or

11) a) 5m b) 0.5m c) 0.005m

12) a) 13,650 b) 12.03

13) a) 450% b) 23% c) 33.3% or 33% d) 35% e) 126%

14) 52 grams (very spicy!)

Chapter 4 Variables and Exponents

122

A VariableA variable　 (변수） is a symbol

usually a letter that represents a number.

An Expression

An expression （수식）is a combination of number and operation symbols like 3 + 5, 9 - 5, 2 + 3 ∙ 8

An Algebraic Expression

An algebraic expression （대수적　 수식）is a combination of variables, constants, and operations symbols.

Examples of algebraic expressions:n +7 the sum of a number and seven

x-3 the difference of a number and three

5y the product of five and a number

the quotient of a number and eleven

An equation is two expressions or constants or one of each with an equal sign between them.

An algebraic equation（대수적　 방정식） is an equation which contains numbers, variables and operation symbols.


123

The value of the variable

The value of the variable(미지수의　 값）is the number that the variable represents. It is also called the solution　（해）.

If 2x=6 then x=3. 3 is the value of the variable x for this problem.

x can have a different value in another problem.

You can also find the value of an expression when you are given the value of the variable.

If z=3, find the value of 4z + 6 - zYou solve it by substituting 3 for z;4(3) + 6 - 3 = 12 +6 - 3 = 18 - 3 = 15

When you use formulas, each term in the formula is a variable. To find the area of a rectangle, we use the formula A = lw where A is the area, l is the length and w is the width.

Find the area of a rectangle :

1. when the length is 6m (meters) and the width is 8m.

2. when the length is 8cm and the width is 6cm.

3. when the length is 12ft and the width is 4.5ft.

4. when the length is 90m and the width is 0.5m.


124

Order of Operations（연산의　순서）

PEMDAS - Please Excuse My Dear Aunt Sally

P - ParenthesesE - ExponentsMD - Multiplication and Division

(from left to right)AS - Addition and Subtraction

(from left to right)

When solving a problem you must follow the order of operations or you might end up with an incorrect solution.

Simplify all the items inside parentheses (and you must follow the order of operations while simplifying). Parentheses are also called grouping symbols.

Then simplify all terms （항）which have exponents.

Then starting from the left and moving to the right perform all multiplication and division. Do whichever comes first. Division may be first.

Then starting from the left and moving to the right perform all addition and subtraction. Do whichever comes first. Subtraction may be first.


125

2 + (6 - 2 ∙ 4 +1) - 6 ÷ 3

Parentheses - (6 - 2 ∙ 4 +1) multiplication first

( 6 - 8 + 1) then from left to right addition and subtraction. SUBTRACTION comes first here.

(-2 + 1) = -12 +(-1) - 6 ÷ 3

Exponents 6 = 36

2 +(-1) - 36 ÷ 3

Next multiplication and division from left to right.

2 +(-1) - 12

Last, addition and subtraction from left to right.

1 - 12 = -11

2 +(6 - 2 ∙ 4 +1) - 6 ÷ 3 = -11

If you have a fraction, simplify the entire numerator (분자) and then simplify the entire denominator (분모) and then divide the results. You can simplify either first as long as you simplify them before dividing.


126

Evaluate the expression:1. ∙ ÷

2. ∙ ∙

3. ∙ ∙

4. ∙

÷∙

5. ÷ ∙ ÷

6. ÷ ∙ ÷

7. ÷

8. ÷


127

Distance = rate ∙ time Distance = speed ∙ time

Find the distance when:

1. the rate of speed is 4 km/h and the time is 2.5 hours.

2. the rate is 8 miles/h and the time is hours.

Use the formula F =

C +32 to determine the temperature in degrees

Fahrenheit when you are given the temperature in Celsius degrees.

3. 20℃

4. 28℃

5. 11℃

6. CSIA wants to have a football field that is 110 meters long and 50 meters wide. How many square meters of artificial turf are needed for an awesome football field?

mph = miles per hour

m/h = miles per hour

km/h = kilometers per hour


128

7. Write an algebraic equation to represent the perimeter of a triangle whose sides have lengths of a, b, and c.

8. If each side in problem 7 was increased by 1, how does the perimeter change?

9. The circumference of a circle is equal to twice the measurement of the radius of the circle times . Write an equation to represent the circumference (원주) of a circle who radius has a measurement of r.

10. Using the formula from question 9, what is the circumference of a circle which has a radius of:

a. 2

b. 1

c. 3

11. When the radius increases by 1, how does the circumference of the circle change?


129

12. If the radius is doubled, what happens to the circumference?

Simplify the expressions:13. ÷∙

14. ∙ ∙

15, ÷∙

16. ∙÷

17. ÷∙

∙

18. Eunjin decides to build a swimming pool with a fence around it. The pool will be 3 meters by 4 meters. There will be a rectangular 1 meter cement area between the pool and the fence. Draw a diagram and determine the cost of the fencing material if it costs $5 per meter for the fence.


130

19. Chaeyoung likes to go to the horse show. The horses perform in a circular arena. The horses get ready by trotting around the perimeter of the arena 10 times. If the diameter of the area is 26 meters, how far do the horses trot during the time they are getting ready (warming up)? Hint: Use the formula from problem 9.

20. If the width of a rectangle is twice the length, find the perimeter of the rectangle when the width is 10 cm.

Insert grouping symbols (parentheses) to make these equations true.21. ÷ = 12

22. ∙

23. ÷ ∙ ∙

24. ∙


131

Term（항）- a number, a variable or the product of a number and one or more variables ( 2, 3x, -7x)

Constant （상수）- a term without a variable (3, -7, 101)

Coefficient (계수）- the numerical factor of a term (6, -3y)

Like Terms （동류항）- have exactly the same variable factors 　(3x and -2x, xy and 4xy)

Not Like Terms (unlike terms) (8x and 7y; 5y and 2)

Combining Like Terms -

Add or subtract the coefficients 3xy + 7xy = (3+7)xy = 10xy

4d - 7d = (4-7)d = -3d

5x + 3y + 6x = (5+6)x + 3y = 11x + 3yThe 3y is not a Like Term

and remains separate.


132

Add or subtract by combining like terms.1.

2. 5x + 2xy - 3y + 2x + xy

3. 8w + 12 - 4z + 4

4. yz - 2yz +3y + 6 – yz

　　　　　Properties　（성질）

Distributive Property （분배법칙）- To distribute means to give

something to each member of the group. When I distribute prints, I give one to everyone in the class.

5(y + 4) - a number in front of parentheses means multiply it. So we distribute the factor 5 to each term of the sum.5(y+4) = 5(y)+ 5(4) = 5y +20

a(b +c) = a(b) +a(c) = ab +aca(b - c) = a(b) - a(c) = ab - ac

It can also be written with the factor after the parentheses.(b +c)a = b(a) +c(a) = ba + ca(b - c)a = b(a) - c(a) = ba – ca

We also call 2x+4y=2(x+2y) distributive property,


133

1. 3(x + 4)

2. -2(y + 5)

3. 2x(x + 2)

4. -2(z - 5)

5. 2w(3x + 4y)

6. 2x + 3 y

Find the area by multiplying the lengths of the sides.

7. The large rectangle is my property in Florida. The rectangle inside is my house. I want to hire a lawn service and they want to know the area of the lawn. My house measures x by x-3 . Find the area of my lawn　 and driveway.

2x

x + 6

Mr Joselson's

housex

x-3


134

MORE PropertiesProperties of Real NumbersFor every real number a, b, and c:

Commutative Property of Addition（덧셈의　교환법칙）

a + b = b + a example 3 +7 = 7 +3 For addition you can change the order you add the terms.

Commutative Property of Multiplication （곱셈의　교환법칙）

a∙b = b∙a example 3∙7 = 7∙3For multiplication you can change the order you multiply the terms.

Associative Property of Addition （덧셈의　결합법칙）

(a + b) + c = a + (b + c) example (6 + 4) + 5 = 6 + (4 +5)For addition you can rearrange the parentheses to add the terms.

Associative Property of Multiplication

（곱셈의　결합법칙）(a ∙ b) ∙ c = a ∙ (b ∙ c)

example (6 ∙ 4) ∙5 =6 ∙(4 ∙ 5)For multiplication you can rearrange the parentheses to multiply the terms.


135

Identity Property of Addition (０ is　덧셈의　항등원） a +0 = a

example 3 + 0 = 3The sum of a number and 0 is the number.

Identity Property of Multiplication（１ is　곱셈의　항등원）

a∙1 = a example ∙

When you multiply a number by 1, the product is the original number.

Inverse Property of Addition

（－a is　덧셈의　역원）a +(-a) =0

example 3 +(-3) = 0Adding a number and it's opposite results in 0.

Inverse Property of Multiplication（1/a is　곱셈의　역원）

∙

example ∙

Multiplying a number by its reciprocal results in 1.

Substitution Property

When you perform a calculation like addition or subtraction and substitute the solution (sum or difference)(3 + 2) + x = 5 + x


136

Multiplication Property of Zero

For every real number, x, multiplying by 0 gives a product of 0. ∙ example ∙

Multiplication Property of –1

For every real number x, multiplying by -1 gives a product whose value is the opposite of the original number. ∙ example -1∙-2 = 2Note:The complete name of Distributive Property is:Distributive Property of Multiplication over Addition (Subtraction).

a(b+c)= ab+acDistributive Property of Multiplication over Addition

a(b-c)= ab-acDistributive Property of Multiplication over Subtraction

Closure Property 닫혀있다

A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set. If an element outside the set is produced, then the operation is not closed.Example: If you add two even numbers (from the set of even numbers), is the sum even?


137

Checking: 10 + 12 = 22 Yes, 22 is even. 6 + 8 = 14 Yes, 14 is even. 2 + 100 = 102 Yes, 102 is even.

Since the sum (the answer) is always even, the set of even numbers is closed under the operation of addition.

If you divide two even numbers (from the set of even numbers), is the quotient (the answer) even?

Checking:12 / 6 = 2 Yes, 2 is even.24 / 2 = 12 Yes, 12 is even. 100 / 4 = 25 NO, 25 is not even!

When you find even ONE example that does not work, the set is not closed under that operation. The even numbers are not closed under division.

The set of round smiley faces


138

For #1-13, name the property being used in the question.

1. ∙ ∙∙

2. ∙

3. 9(-2) = -2(9)

4. 4(6 - 2) = 4(6) - 4(2)

5. -2 +2 = 0

6.

7. 2 + (-3) = (-3) +2

8. -7 + (2 + 9) = (-7 + 2) + 9

9. ∙

10. ∙

11. ∙

12. ∙

13. 12- 6 - x = 6 - x

14. Simplify 3(6x - 4)

15. Simplify -2(x - 7 + y)


139

Monomial （단항식） a number, variable or the product of them 　　 (½ , -5, , , )

Binomial （이항식）an expression containing two terms or monomials 　 ( , , )

Trinomial （삼항식） an expression containing 3 terms ()

Polynomial (다항식)A Polynomial is an expression containing one monomial or sums or differences of monomials.

Degree of polynomial (다항식의 차수)The degree of a polynomial　 is the highest exponent. If there are two or more variables in a term, add the exponents together. degree 3

degree 6 (2+4) degree 1

4 degree 0 ()


140

Standard form of a polynomial is when the exponents are decreasing.

example

We classify polynomials using the numbers of terms.

Monomial - 1 termBinomial - 2 termsTrinomial - 3 terms

A combination of a monomial, binomial, or trinomial or more than three terms are referred to as polynomials.

We can also classify them using the degree of the polynomial.

Polynomial

Number

of

terms

Name using

number of

terms

DegreeName using

degree

14 1 monomial 0 constant-2x+61 2 binomial 1 linear

3 trinomial 2 quadratic

1 monomial 3 cubic

2 binomial 44th degree

(quartic)

4 polynomial 55th degree

(quintic)


141

Classify the polynomials by the number of terms and by the degree.

1. -13x

2. 72x

3.

4.

5.

6. -12.6

7.


142

Adding and Subtracting Polynomials

The simplest way is to line up the terms so the terms of the same degrees are lined up. Then simply add or subtract.

To add

line up the columns in table form.

=

To subtract

line up the columns in table form.

=

constant

+4

+ +2

+6

constant

+4

- +2

+2


143

1.

2.

3. Subtract the second polynomial from the first.

4.

5.

6. 3x + 5

x+3

Find the area of the shaded region if the large rectangle has dimensions of 3x + 5 by x + 3 and the smaller rectangle has dimensions of x+4 by x.

x + 4

x


144

Deductive Reasoning (연역적 추리)

Deductive Reasoning (연역적　추리）is the process of reasoning logically from given facts to a conclusion（결론）.

Inductive Reasoning (귀납적 추리)

Inductive Reasoning　（귀납적　 추리） is making conclusions based on patterns you observe. The conclusion you reach is called a conjecture (추측）


145

The first three questions illustrate deductive reasoning.

Name the property that allows us to perform the step.

1.a

b.

c.

d.

2.a.

b.

c.

d.

3.a.

b.

c.

d.

e.

f. = 8x + 12

For #4-9, name the property identified:4. 13 + x + 18 = 13 + (x + 18)

5. 10 + 0 +(y+2) = 10 + (y+2)

6. 3(z + 5) = 3(z) +3(5)


146

7. 3 + m + 2 = m + 3 +2

8. ∙

9 3 + x = 3 + 1∙x

10. Simplify 2 - 3(x + 1)

11. Simplify 5 + (2-x) + 4(x+1)

12a. What is the next number in the pattern?

1, 3, 6, 10,

The answer you gave is your conjecture. You reached a conclusion based on the pattern you observed. (Inductive Reasoning)

12b. 1, 2, 4,


147

Classify the polynomials by type based on the degree and number of terms.

13.

14.

15.

16. 14x + 2

17.

Perform the indicated operations.18.

19.

Find the area of the shaded region.20. 4x - 5

2x+4 x+1

2x


148

Factoring is simplifying polynomials into a product of simpler numbers, usually prime numbers. All numbers which divide evenly into a number are factors of that number. All polynomials which divide evenly into another polynomial are factors of that polynomial.

When you have two numbers and determine their factors, the numbers that are common to both of them are called common factors　（공약수）.10 and 25 have common factors10 has factors of : 1, 2, 5, 1025 has factors of : 1, 5, 25The common factors are: 1and 5

The Greatest Common Factor (GCF) （최대공약수）is the largest of the common factors. The greatest common factor must be no larger than the smaller of the two numbers.If the numbers were 10 and 20, the factors are:10 - 1, 2, 5, 1020 - 1, 2, 4, 5, 10, 20The GCF is 10.


149

When variables are involved, the greatest common factor is the variable term with the lowest exponent for each variable.

The GCF of and

Why is this??The factors of is ∙∙∙∙∙∙

The factors of are ∙∙∙∙∙∙

The factors in common are ∙∙∙∙ which is .

For terms involving both numbers and variables - called variables with coefficients, find the GCF of the coefficients and the GCF of the variables and combine.

The GCF of and is . = ∙∙∙

= ∙∙∙∙

The 2, one of the x and the y are common to both.

What is the GCF of , , and ?

So if we want to factor (remove) the GCF from 3x3 + 12x2 - 15x, we find the GCF and place it in front of the trinomial as a product. We write the remaining factors in the parentheses:

______ ( )


150

Like shown in the example above, what is the GCF and the other factor of:

1. 15w + 21

2.

3.

4.


151

Multiplying a Monomial by a PolynomialWe use distributive property.

We multiply the term outside the parentheses by

each term inside the parentheses.

Multiply

1.

2. (x+10)3x

3. -5a(a-1)

4. 4y2(9y3 + 8y2 - 11)

5. (x + 1) – x(x2 - 1)


152

What is the GCF and the other factor of:

1. 14w + 35

2.

3. 8x3 + 12x2 - 52x

4. 15b3 + 45b - 30

Multiply

5. 9m(2m +3)

6. (2x+5)6x

7. -2a(3-2a)

8. 2y2(7y3 + 6y2 - 14)


153

9. ( + 1) –

10.

11. -3 + 2


154

Verbal Expressions

Basic Number OperationsSum （합）– answer to an addition problem

Difference　（차） – answer to a subtraction problem

Product （곱）– answer to a multiplication problem

Quotient （몫） - answer to a division problem

Addition sum, more than, plus, increased by

Subtraction difference, minus, less than, decreased by

(4 less than a number )

Multiplication product, times, multiplied by

Division quotient, divided by

Changing verbal expressions to algebraic expressions:The sum of 6 and a number 6 + xFive more than a number y +5

A number plus 11 z +11


155

The difference of 3 and a number3 – a

The difference of a number and 3b – 3

Four less than a number c – 4less than is the exception when the order is switched from the expression

11 minus a number 　　11 - d

The product of 2 and a number 2e

Six times a number 6f

A number multiplied by 4 4g

The quotient of a number and 5

Twenty divided by a number

Six more than eight times a number

6 +8 ∙ n = 6+8n

Three less than the sum of 4 and a number

sum of - tells us we must add before doing other operations (4+x)

less than - lets us know to subtract something from another term

three than less - 3 (not 3 -)

(4+x) - 3


156

When changing from a verbal expression or equation to an algebraic expression or equation, the operation goes between the terms (replaces the word “and”).

The difference between 3 and 5 is 3-5 =-2

ORDER MATTERS

A number decreased by the sum of 4 and the square of another number

another number - more than one variable involved in this statementSquare - to the second power. 제곱 b - (4 +)

When we use the words sum, difference, product or quotient as part of an expression, it tells us we must do that operation first and therefore need parentheses around it.


157

Translate the verbal expressions to algebraic expressions

1. 5 more than a number

2. Half of a number

3. 5 plus the quotient of 3 and a number.

4. 7 more than twice a number.

5. 8 decreased by four times a number.

6. Three times the difference of a number and 12.


158

An equation （방정식）is formed when an equal sign is placed between two expressions. The two expressions are the left side and right side.

is means equals =

Eleven more than a number is 23n + 11 = 23

34 decreased by a number is 2234 - x = 22

5 is 6 less than a number 5 = y - 6

Half of the sum of 8 and a number is 7.

Four times the difference of 3 and a number is 8.

Haeun and two friends bought a pizza. Haeun paid 3000 won for an equal share of the pizza. Write an equation to determine the total cost of the pizza.


159

(Write an equation)1. The product of 15 and a number is 3.

2. The quotient of two different numbers is 11.

3. Haein has a rare coin collection. Haein’s favorite coin will be 45 years old in only 3 years. Write an equation to determine the age of the coin now, with c being the current age of the coin.

4. The perimeter (sum of the lengths of the sides) of a quadrilateral (a 4 sided polygon) is 112. Write an equation to determine the perimeter of the quadrilateral. 3x + 6

12 13

6y


160

Solving equations:

Many problems can be solved without much effort.x + 10 = 14

You know the solution is x=4 but what are the appropriate steps to solve any problem.

We think of an equations as having two sides (each side is an expression) that are in balance. If we change one side, we must do the same thing to the other side.

If we have x +10 =14

and we want to know the value of x, we can subtract 10 from each side to "solve" (isolate and obtain a value) for x.

x + 10 = 14 - 10 = -10 x = 4


161

There are several common methods of solving an equation with one variable.

1. Add the same number to both sides of the equation.2. Subtract the same number from both sides of the equation.3. Multiply the same number to both sides of the equation.4. Divide both sides of the equation by the same number.5. Interchange (swap) the two sides.

Inverse operations – operations that undo one another (addition- subtraction, multiplication – division)

adding subtractingmultiplying dividingare each inverse operations of each other.

In the previous example, x + 10 =14, we are adding 10 to the variable. To isolate (get alone) the variable, we perform inverse (opposite) operations.

If I have 5 of something and add 4 to it, I can return to my original amount by subtracting 4, the inverse of adding 4. 5 + 4 - 4 = 5


162

Steps to solve an algebraic equation:

Steps: Simplify both sides of the equation - combine like terms.

Do the opposite of variable on the right side with addition or subtraction

Do the opposite of constant (number) on the left side with addition or subtraction

Do the inverse of the multiplication or division of the variable being solved.

This process will always get the variable on the left and the solution on the right. You can do it other ways and get the same solution.One Step solutions

x - 4 = 8 + 4 = +4 add 4 to both sides x = 12

x + 4 = 8 -4 =-4 subtract 4 from both sides x = 12

-2x = 10 (divide by -2, or multiply

∙

∙

by the reciprocal - )

x = -5

∙∙

(multiply both sides by 3)

15


163

Show your steps and Check!

1. x +3 = 2

2. x - = 3

3. 2.5x = 15

4.

Multistep problems

Simplify each side by performing any mathematical

operation shown and combine like terms so there

is at most one variable term and one constant

term on each side of the equal sign.

Then do the inverse operations to get the variable

to one side and the constant to the other.

Lastly, do the inverse of multiplication or division

to get rid of the coefficient (number multiplied

with) the variable.


164

2(x + 3) - 4 = x + 4 + 2x

simplify each side, deal with the left side and right

side separately

2x + 6 -4 3x + 42x + 2 3x + 42x + 2 = 3x + 4 -2 = -2 2x = 3x + 2-3x = -3x -1x = 2 x = -2

3x - 2(x+1) = 13 – 3x

3x -2x -2 = 13 –3x

x - 2 = 13 –3x + 2 = +2

x = 15 - 3x+3x = +3x 4x = 15

∙

∙

x =

each side is simplified

constants are now only on the rightvariable on one side, constant on the otherlast step was to divide by –1

Distributive propertycombine like terms (Substitution)inverse operation of constant

inverse operation of variablemultiply by the reciprocal

Simplified fraction, mixed number or decimal


165

Checking your solution

Substitute the value of the variable back into the

original equation.

Simplify each side. DO NOT add or subtract from

side to side.

3x – 2(x+1) = 13 – 3x

3(3.75)-2(3.75+1) = 13-3(3.75)

11.25-2(4.75) = 13-11.2511.25-9.5 = 1.75

1.75 =1.75 ✓

(Solve and CHECK)

1. 3(2x - 3) + 2 = 2x +5

2. 4 - 3(3x + 3) = -2(2 – x)


166

(Solve by showing your steps and CHECK)1. x +8 = 12

2. x - 4 = -6

3. x + 3 = -4

4. x - 9.7 = 11.2

5. x + = 4

6. 3x = -18

7. 4x = 18

8.

9.


167

10.

11. 2x + 3 = 11

12. 3x - 4 = 11

13. 3(x+1) = 18

14.

15. 3(2x + 4) - 2(x+1)= 3x-11

16.


168

17.

18. 6x + 7 = 8x – 13

19. 28 – 2.2y = 11.6y + 262.6

20. 7 – 3x = x – 4(2 + x)

21. -8(4 + 9x) = 7(-2 – 11x)

22. 6x + 4 = 2x + 2

23. Use the fact that vertical angles are congruent to write an equation and solve it.


169

Identity (항등식)

Identity （항등식）– the solution is a true statement without a variable like 4=4 (not like x=3 as an answer); every value is a solution – There are infinitely many solutions （무수히　많은　해）.2(x+3)=2x+6, when you solve you get 0=0Any value for x is a solution.

No Solution （해가　없다） - 2x=2x+1, when you solve you get a false statement, 0=1When all the variables are eliminated and you get a false statement there is no solution.

Literal Equations

An equation involving more than one variable.

A=½bh

You can solve for a different variable.

Solving for h (getting h by itself)

A=½bh (multiply both sides by 2)

2A=2(½bh),

2A=bh (divide both side by b)

or


170

1. Solve for w;

2. Solve for z; x + y +2z =11

3. Solve for x;

4. Solve for s;

5. 2(-2x+3) - 4x = 4x + 6

6. 3x + 2(x-1) = 5(x +32)

7.


171

1. Area of a triangleSolve for h:

2. Simple InterestSolve for rA = P + Prt

3. Simple InterestSolve for PA = P + Prt

4. Area of a trapezoidSolve for

5. Find the height of a triangle when the Area is 20 and the length of the base is 3.


172

6. 14-(2w+5) = -2w+9

7. 9x + 3x -10 = 3 (3x + x)

8.

9. 5c - 2(c + 2) = -(2c +15)

10.

11. Hyungyu and Hoejung left the starting line at the same time. Hyungyu crossed the finish line in 30 minutes and averaged 12 kilometers per hour. Hoejung crossed the finish line 5 minutes later. Since they traveled the same distance their speeds multiplied by their times should be equal. Find Hoejung's speed.


173

12. 15x – 3 = 3(5x – 1)

13. 18x + 2x – 8 = 4(5x + 1)

14. 5x - 9 = 5 + 5x

15. 4(x - 9) = –36 + 4x

16. 7x - 9 = –4 – 7x

17. 3(x - 1) - x = 3 + 2(x – 3)

18. 6(n - 11) = 12 + 4(2n – 3)


174

Word problems are some of the tougher problems in Algebra. Understanding the English words can often be more challenging than the problem itself.

Consecutive Integers are integers which are one apart. 3, 4, and 5 are three consecutive integers. Odd and even consecutive integers are two apart. -4, -2, 0, 2 are consecutive even integers while -11, -9, -7 are consecutive odd integers. Whether they are odd or even, they are still two apart.

Consecutive integers are often defined as:x = 1st integerx + 1 = 1st consecutive integerx + 2 = 2nd consecutive integerorx - 1 = 1st integerx = 1st consecutive integerx + 1 = 2nd consecutive integer

Consecutive Even integers are often defined as:x = 1st even integerx +2 = 1st consecutive even integerx +4 =2nd consecutive even integeror2x - 2 = 1st integer2x = 1st consecutive even integer2x + 2 = 2nd consecutive even integeror2x = 1st even integer2x +2 = 1st consecutive even integer2x +4 =2nd consecutive even integer

Consecutive odd integers are often defined as:x = 1st odd integerx + 2 = 1st consecutive integerx + 4 = 2nd consecutive integeror2x - 1 = 1st integer2x+1 = 1st consecutive odd integer2x+3 = 2nd consecutive odd integer


175

The sum of three consecutive integers is 150.

To solve it algebraically, we first define the variables.

Let x = 1st consecutive integerLet x+1 = 2nd consecutive integerLet x + 2 = 3rd consecutive integer

x + (x+1) +(x+2) =1503x + 3 = 150 -3 = -33x = 147

x = 49

the three numbers are

x, x+1, x+249, 50, 51

check 49+50+51 = 150

150 = 150 ✔

1.The sum of two consecutive even integers is 118.a. Define the variables

b. Write an equation with variables on one side.

c. Solve the equation.d. What are the two integers?


176

An object that moves at a

constant speed is said to be in uniform motion　

（등속　운동）.

or gives the formula where

distance (d) equals the product of time (t) with

the rate(r) of speed (s).

It is often helpful to make a table or chart.

Same Direction travel

Jeongin leaves on a train from a train station

at 4PM. It travels at an average rate of 45

km/h (kilometers per hour). A high-speed

train leaves the same station an hour later at

an average speed of 60 km/h. GungGwon

leaves on a second train that follows the same

path as the first train on a parallel track.

When will the second train catch up to the

first train?

Let t = time train 1 traveled

Let t-1 = time traveled by train 2

(it traveled one less hour)

Since we want to know when they have

traveled the same distance, we know the

distances will be equal.

45t = 60(t-1), t = 4, so the second train will

catch the first train 4 hrs. after the first train

left, at 8 PM.

Rate/speed time distance

Train 1 45 t 45t

Train 2 60 t-1 60(t-1)

거리

속력 시간


177

(Same Direction Travel)

Soyoon is moving to a new home. A moving

van leaves Soyoon’s house traveling at an

average rate of 40 km/h. Soyoon’s family

leaves the house 30 minutes later following the

same route in their car. They traveled at an

average rate of 60 km/h. Although it is a 2

day trip, Soyoon wants to get a game from the

moving van. Determine when the car will

catch the van by filling in the table and

solving, so he can get a game and a math

book to do Mr. Joselson's homework.

This problem was an example of two objects

traveling in the same direction.

You can use the table to define the variable or

use “Let t = time the family takes to catch up

to the van”

Be Specific!

Rate/Speed Time Distance

M o v i n g

Van

F a m i l y

car


178

Round Trip - there and back

Donghan wants to go to Chuncheon to buy a

new computer at Emart. Because of traffic,

she averages only 30 km/h. On the way home

there is less traffic and the average speed is

50 km/h. The total travel time is 3 hours.

How long does it take her to drive back from

the computer store.

Let t = time to the store or use a table

30t = 50(3-t)

30t = 150 - 50t

80t = 150

t =

BUT we want the time to return home so

.

Rate or

speedTime Distance

There =

to the store30 t 30t

Back =

return home50 3-t 50(3-t)


179

(Round Trip Travel)

Minseo takes a plane from Incheon to Taiwan

traveling at a rate of 320 km/h. The airplane

returns at an average rate of 280 km/h. The

total travel time is 12 hours. Determine the

travel time each way and the distance from

Incheon to Taiwan.


180

Opposite direction travel

Mincheol and Sanghyuk leave their homes

traveling in opposite directions on a straight

road. Mincheol drives 15 km/h faster than

Sanghyuk. After three hours, they are 225 km

apart. Find the rate of speed of each of them.

starting point

Mincheol Sanghyuk

They leave the same time and each have a

rate and a time. If you add up their distances,

you will get the total distance.

Mincheol's distance= rt = (r+15)3

Sanghyuk's distance = rt = r(3)

3(r+15) +3r = 225 (total distance)

3r + 45 + 3r = 225

6r + 45 = 225

6r = 180

r=30

Mincheol's rate 45 km/h

Sanghyuk's rate 30 km/h


181

(Opposite Direction Travel)

Eunwoo takes a bus and Junghyun takes a car

from the same place and traveled in opposite

directions. If the bus is traveling at 50 km/h

and the car is traveling at 55 km/h, in how

many hours will they be 210 km apart?


182

Different speeds based on the current of the

water or the speed of the wind.

With the current, against the current, with the

wind, against the wind, upstream, downstream

– these words represent problems when there

is a speed of the wind or water that is

changes the speed of the person or object

depending on the direction.

With the current, downstream, with the wind,

downwind all represent faster speeds since

the speed is being increased by the wind or

current of the water.

Against the wind, upwind, upstream, against

the current all represent the speed being

slowed by the wind or current of the water.

If the speed of the plane is 500 km/h,

downwind = 500+w and

upwind =500-w

where w= speed of the wind


183

WIND problem

Jiyeon flies 400 miles on a plane travelling

with the wind and 300 miles when the plane is

travelling against the wind in the same length

of time. If the speed of the wind is 22mph,

what is the speed of the plane in still air?

Still air is air that is unaffected by the wind.

=

400(s-22)=300(s+22)

4s-88=66+3s

s=154

The speed of the plane in still air is 154 miles

per hour.

Speed Time Distance

With the

Winds+22

400

A ga i n s t

the winds-22

300


184

(Define variable, solve and answer the question)

1. Chaebin took a bus traveling at an average rate of 50 kilometers per

hour and made the trip to town in 6 hours. If the bus had traveled at 45

kilometers per hour, how many more minutes would it have taken to make

the trip?

2. Keungyoung takes the bus to Everland for a fun day. There are so

many people going that there is a lot of traffic on the roads. The bus is

only able to travel 40 km/h. After spending 12 hours there, Keungyoung

goes home. Fortunately there is very little traffic and the bus is able to

travel at 90km/h. The total travel time was 5 hours. How long did it

take each direction?

3. Sohyun took a drive to town at an average rate of 40 km/h. In the

evening, Sohyun drove back at 30 km/h. If she spent a total of 7 hours

traveling, what is the distance traveled by her?


185

4. Min travels on a commercial jet 1620 miles in the same amount of

time it takes a corporate jet to travel 1260 miles. The rate of the

commercial jet is 120 mph greater than the rate of the corporate jet.

Find the rate of each plane.

5. Gahyeon drives 32 km by car and then flies an additional 576 km by

plane. The rate of the plane was nine times the rate of the car. The

total time of the trip was 3 hours. Find the rate of the plane.

6. Juhyeon started to walk around a track that is 1500 meters around.

Juhyeon is walking at 3 km/h. Himchan starts 5 minutes later at a light

jog of 6 km/h. How long will it take Himchan to catch up to Juhyeon?


186

7. A helicopter leaves Gimpo Airport and flies north at 180 km/h.

Twenty minutes later a plane leaves the airport and follows the helicopter

at 330 km/h. How long does it take the plane to catch up to the

helicopter?

8. A small plane leaves an airport and flies north at 250 km/h. A jet

leaves the airport 30 minutes later and follows the small plane at 375

km/h. How long does it take the jet to overtake the small plane?

9. Yuri got in a car that started out from Seoul toward Gapyeong at the

rate of 60 km/h. Beomyoung got in a second car leaving from the same

point 2 hours later and drove along the same route at 75 km/h. How

long did it take Beomyoung's car to catch up to Yuri's car?


187

10. Somin gets on a tourist bus leaving Seoul at 1:00 PM for Busan.

Exactly 24 minutes later, Yubin gets in a truck going in the same

direction. The tourist bus moves at a steady 60 km/h. The truck travels

at 80km/h. How long did it take the truck to overtake the tourist bus?

11. Exactly 20 minutes after Jinil left home, Gahee set out to overtake

her starting at the same place. Jinil drove at 48 km/h and Gahee drove

at 54 km/h. How long did it take Gahee to overtake Jinil?

12. Jihyeon drove from home to Seoul at 75 km/h. When Jihyeon

returned, the traffic was heavier and Jihyeon drove at 50 km/hour. If it

took 1 hour longer to return than to go to Seoul, how long did it take

Jihyeon to drive home?


188

13. It takes a plane 1 hour less to fly from Seoul to Taipei at 600km/h

than it does to return at 450 km/h. How far apart are the two cities?

14. It takes a plane 2 hours less to fly from Seoul to Taipei at 600km/h

than it does to return at 450 km/h. How far apart are the two cities?

15. A car started out from CSIA toward Busan at the rate of 48 km/h. A

second car left from the same point 2 hours later and drove along the

same route at 60 km/h. How long did it take the second car to overtake

the first car?


189

16. One car travels 50 km/h and another one travels 55 km/h. If they

start from the same place and travel in the same direction, after how

many hours will the faster car be 35 km ahead of the slower car?

17. One car travels 62 km/h and another one travels 48 km/h. If they

start from the same place at the same time and travel in the same

direction, after how many hours will the faster car be 42 km ahead of the

slower car?

18. Two freight trains started at the same time from towns 564 km apart

and met in six hours. The average rate of one train was 14 km/h faster

than that of the other train. Find the rate of each train.


190

19. Two passenger trains started at the same time from towns 608 km

apart and met in 4 hours. The rate of one train was 8 km/h slower than

that of the other. Find the rate of each train.

20. Donggeon left Gapyeong on a bicycle riding at an average of 8 km/h

five hours before Donghyun left by automobile from the same starting

point. Donghyun overtook Donggeon in exactly one hour. At what

average rate was Donghyun traveling?

21. Minkyoung left camp on a bicycle at noon and rode at an average

rate of 10km/h. Seungyoon left camp in a van at 1:30 P.M. and overtook

Minkyoung in 30 minutes. At what average rate was Seungyoon traveling

in the van? Hint: How long did the van travel?


191

22. Two freight trains started at the same time from towns 448 km apart

and met in 8 hours. The average rate of one train was 19 km/h faster

than that of the other train. Find the rate of each train.

23. Two passenger trains started at the same time from towns 288 km

apart and met in 3 hours. The rate of one train was 6 km/h slower than

that of the other. Find the rate of each train.

*24. A plane can fly 300 miles downwind in the same time as it can

travel 210 miles upwind. Find the speed of the wind if the plane can fly

225 mph in still air.


192

Remember to define the variables, set up the equations, solve and check.

25. Find two consecutive integers whose sum is 45.

26. Find three consecutive even integers whose sum is 72.

27. Find two consecutive odd integers whose sum is –88.

28. Find four consecutive odd integers whose sum is 56.


193

29. Find two consecutive even integers such that the sum of the larger

and twice the smaller is 62.

30. Find three consecutive integers such that the sum of twice the

smallest and 3 times the largest is 126.

31. Seven times a number is equal to 12 more than 3 times the number.

Find the number.

32. Ten more than 6 times a number is 4 less than 4 times the number.

33. The second of two numbers is 4 times the first. Their sum is 50.

Find the numbers.


194

34. The second of two numbers is 5 more than twice the first. Their

sum is 80. Find the numbers.

35. The perimeter of a rectangle is 24 inches. Find the dimensions if its

length is 3 inches greater than its width.

36. Find the measures of the angles of a triangle if the angles are

represented by x, 4x, and 4x.

37. The perimeter of a triangle is 51 centimeters. The lengths of its

sides are consecutive odd integers. Find the lengths of all three sides.

38. Two-thirds times a number plus 7 equals 7 minus the number. Find

the number.


195

39. A family drove to a beach resort at an average speed of 55 mph and

later returned over the same road at an average speed of 65 mph. Find

the distance to the resort if the total driving time was 12 hours.

*40. A cargo ship traveling with the current traveled 180 miles in 9

hours. On its return voyage traveling against the current, it took 18 hours

to travel the same distance. The speed of the current is 5 mph. Find the

speed of the cargo ship.

41. Yujung mixes a 15% mercury solution with an 85% mercury solution.

How much of each solution should she to make 2.5 liters of a solution

that is 50% mercury?


196

42. Two cyclists start at the same time from opposite ends of a course

that is 42 miles long. One cyclist is riding at 16 mph and the second is

riding at 12 mph. How long after they began will they meet?

43. A car and a bus set out at 2 pm from the same spot, headed in the

same direction. The average speed of the car is twice the average speed

of the bus. After 2 hours, the car is 68 miles ahead of the bus. Find the

rate of the bus and the car.

44. A pilot flew from one city to another city averaging 150 mph. Later,

it flew back to the first city averaging 100 mph. The total flying time was

5 hours. How far apart are the two cities?


197

45. A 555-mile, 5-hour plane trip was flown at two speeds. For the

first part of the trip, the average speed was 105 mph. Then the tailwind

picked up, and the remainder of the trip was flown at an average speed

of 115 mph. For how long did the plane fly at each speed?

*46. A boat travels for three hours with a current of 3 mph and then

returns the same distance against the current in four hours. What is the

boat's speed in calm water? How far did the boat travel one way?


198

Simple Interest Problems

（단순이자율　문제）

There are two types of interest used in “real

life.” Simple interest and compound interest.

Simple interest is money paid only on the

original investment. Compound interest is

money paid on the original investment and also

money paid on the interest earned the previous

year. For the word problems in this section,

we will deal with simple interest only.

(Information only) Compound Interest is

where A=total amount,

P=principal, r=interest rate as a decimal,

n=number of periods of compounding per year

and t=number of years.

The necessary principles used here are:

Simple interest is a percentage of an amount.

I = Prt, Interest = Principal times Rate times

Time

The method to follow here is to add up all the

interest earned from the separate accounts and

set it equal to the total interest earned.


199

Yujung invested $3000. She invested part at

8% and the rest at 7

%. The total return

(interest earned) for one year was $231. How

much was invested at each rate.

I=Prt

The sum of the interest from each investment

equals the total interest.

0.08(3000-x) +0.075x = 231

80(3000-x) + 75x= 231,000

240,000 - 80x + 75x = 231,000

5x = 9,000

x = 1,800

The total amount invested at 8% is $1200

(3000-1800) and the total amount invested at

7.5% is $1800.

Principal rate time interest

Amount invested

at 8%3000-x 0.08 1 1(0.08)(3000-x)

Amount Invested

at 7.5%x 0.075 1 1(0.075)(x)


200

1. Eunji invested $18,750, part at 12% and

the remainder at 10% simple interest. If the

total yearly interest from these investments

was $2117, find the amount invested at each

rate.

2. Jaekyoung invested money at 8% and 12%

simple interest. $3000 more is invested at 8%

than at 12%. If the total combined yearly

interest from the investments was $760, how

much was invested at each rate?


201

Mixture Problems　（혼합물　문제）

Suppose a container contains a 120 gallon

mixture of water and red dye. If 15% of the

solution is red dye, then the amount of red

dye present in the container is 15% of 120 =

0.15 * 120 = 18 gal.

We will be interested in mixing certain

solutions to obtain a desired solution. The

main principles used here are

(A) Amount of substance in mix = quantity (R)

times the percent of the mixture solution (m)

and then add the amounts of substance in each

container to get the desired amount in the final

mixture.

How many liters of a 50% alcohol solution

must be added with 80 liters of a 20%

alcohol solution to make a 40% alcohol

solution?

0.5x +16 = 0.4x +3.2

0.1x = 16, x= 160

m R A50% 0.5 x 0.5x20% 0.2 80 1640% 0.4 x+80 0.4x+3.2

beaker method:

draw below


202

1. How much candy worth 32 cents a pound

must be mixed with candy costing 25 cents

per pound to create 35 pounds of mixed candy

selling at 30 cents per pound?

* Adding pure contents

The original concentration of water is 100% –

20% = 80%

The resulted concentration of water is 100%–

30% = 70%

The water evaporated is 100% water, which is

1 as a decimal.

Change all the percents to decimals.

Let x=amount of water evaporated. The result

would be 20–x.

*1. Jaehyuk has 20 ounces of a 20% of salt

solution. How much water should he evaporate

to make it a 30% solution?

original removed resultconcentration 0.80 1 0.7amount 20 x 20-xproduct 0.8(20) 1x 0.7(20-x)


203

Define variable or make a table and solve. Use only one variable!

1. Jisu invests $6,000 at an annual simple interest rate of 14%. How

much additional money must she invest at a rate of 10% so that the total

interest earned is 12% of the total investment? (solve like a mixture

problem)

2. Pinto beans cost $0.69 per kg and kidney beans cost $0.89 per kg. A

100 kg mixture is valued at $81 (the price per kilogram is $0.81). How

many kilograms of each kind of bean is in the mixture?

3. A market sells coffee from Brazil for $3.00 per pound and coffee from

Columbia for $3.80 per pound. How many pounds of each should be used

in order to sell a blend of 125 pounds for $3.30 per pound.


204

Define variable or make a table and solve. Use only one variable!4. How many pounds of chocolate worth $1.20 a pound must be mixed

with 10 pounds of chocolate worth 90 cents a pound to produce a mixture

worth $1.00 a pound?

5. Gisuram has $50,000 to invest. Fund Y yields 14% interest. Fund X

yields 6% interest. She earns $4,500 in interest income this year. How

much should Gisuram put in each fund?

6. An investment of $3,000 is made at an annual simple interest rate of

5%. How much additional money must be invested at an annual simple

interest rate of 9% so that the total annual interest earned is 7.5% of the

total investment? Hint: Treat this more as a mixture problem.


205


7. Jiyi has $6,000 to invest into two simple interest accounts. The annual

simple interest rate on one account is 9%; on the second account, the

annual simple interest rate is 6%. How much should she invest in each

account so that both accounts earn the same amount of annual interest?

8. Hyunhee deposited an amount of money into a high-yield mutual fund

that returns a 9% annual simple interest rate. A second deposit, $2,500

more than the first, was placed in a certificate of deposit that returns a

5% annual simple interest rate. The total interest earned on both

investments for one year was $475. How much money did Hyunhee

deposit in the mutual fund?

9. Hyeseo invests $22,500 into two accounts. The first account has an

annual simple interest rate of 1.5%. The other account has an annual

simple interest rate of 3%. If Hyeseo receives the same amount of

interest from each account, how much did Hyeseo invest in each account?


206


10. Sehwan works in a lab. Ayeon needs a 15% acid solution for a

certain test, but the supplier only ships a 10% solution and a 30%

solution. Rather than pay the hefty surcharge to have the supplier make a

15% solution, she decides to mix 10% solution with 30% solution, to make

a 15% solution. Gunheui needs 10 liters of the 15% acid solution. How

many liters of 10% solution and 30% solution should be used?

11. How many liters of a 70% alcohol solution must be added to 50

liters of a 40% alcohol solution to produce a 50% alcohol solution?

12. Two hundred liters of a punch that contains 35% fruit juice is mixed

with 300 liters (L) of another punch. The resulting fruit punch is 20% fruit

juice. Find the percent of fruit juice in the 300 liters of punch.


207


*13. Ten grams of sugar are added to a 40-g serving of a breakfast

cereal that is 30% sugar. What is the percent concentration of sugar in

the resulting mixture?

*14. Hyerin has 20 ounces of a 20% of salt solution, How much salt

should she add to make it a 25% solution?

*15 Two bond funds pay interest at rates that differ by 2%. Money

invested for one year in the first fund earns $315 interest. The same

amount invested in the second fund earns $385. Find the lower rate of

interest.

original added resultconcentrationamountproduct


208

An Inequality （부등식）compares the values of

two expressions.

Inequalities

a < b a is less than b

≤ a is less than or equal to b

a > b a is greater than b

≥ a is greater than or equal to b

a ¹

b a is not equal to b

Inequalities are solved the same way equations

are solved.

Simplify both sides of the equation

Do opposite of variable on the right side

with addition or subtraction

Do opposite of constant (number) on the left

side with addition or subtraction

Do inverse of the multiplication or division

of the variable to solve

The only difference is the last step. If you

multiply or divide by a negative, you change

the direction of the inequality symbol.

WHY?????


209

x > 5 greater than or less than is a hollow point

x≤ -2 with the or equal to, makes it a solid point.

Important Words Sample

Sentence

Equivalent Form Translation

is at least Bill is at least

21 years old.

Bill's age is greater than or equal to 21

b >21

is at most At most 5 s t u d e n t s dropped the course

5 or fewer students dropped the course

n <5

cannot exceed e a r n i n g s cannot exceed $1200.

earnings must be less than or equal to $1200

r <1200

must exceed The speed must exceed 15 km/h

The speed is greater than 15 km/h

s > 15

is less than

is up to

Spot's weight is less than 50 kilograms

- w < 50

is more than,

is greater than

Busan is more than 200 km away.

- d > 200

is between The film was between 90 and 100 minutes long.

- 90 < t < 100


210

Solving Inequalities:

Solve and graph

1. 5x + 10 > 20

-10 = -10

5x >10

x > 2

2. 6 - 4x ≤ 22

-6 = -6

-4x ≤ 16

≥

x≥ -4

Solve and graph

1. ≤

2.

3. 3(4-6x) > 36


211

Verbal Inequalities

An amusement park charges \5000 for

admission and \1000 for each ride. You go

to the park with \16,000. Write an inequality

to represent the possible number of rides you

can go on and solve it to determine the

maximum number of rides you can ride.

5000 +1000r ≤ 16,000

-5000 = -5000

1000r ≤ 11000

r ≤ 11

The maximum number of rides is 11.

Yesung and Gunho decide to order a pizza.

They have \12,000 to spend. A large cheese

pizza costs \8000 plus \600 for each

additional topping. Write an inequality to

represent this situation and solve it to see the

maximum number of toppings they can order.


212

Define variable and check. Use only one variable!

1. Five chapter exams are given and an 80% must be achieved in order

to pass the course. One student has completed the first four exams with

scores of 71, 84, 79, and 81. Write an inequality to find the minimal score

the student can make on the fifth exam in order to pass the course.

2. CSIA is having a fall carnival. Admission into the carnival is \3000

and each game inside the carnival costs \800. Write an inequality that

represents the possible number of games that can be played having

\10,000. What is the maximum number of games that can be played?

3. Dongyoon and Minji attend a championship baseball game. Between

innings they decide to go to the snack stand. They have \15000 and

find that sodas are \2000 and that hot dogs are \2000. Write and solve

an inequality that models the maximum number of items that can be

bought.


213

Define variable and check. Use only one variable!4. The length of Yeowon's rectangular dining room is 12 ft. If the

area of the room is at least 96 square feet, write and solve an inequality

to determine what the smallest width the room could have?

5. Yedam earned \4000 and \5500 in interest the last 2 years. How

much interest must Yedam earn this year so that her average earnings

over the three year period is more than \6000?

6. Yoonho has a 95 and a 99 on his first two math tests. What must

Yoonho get on the third test to have at least a 90 test average.

Represent the problem with an inequality and solve.

7.　The high temperatures for the last two days were 28° and 15°. What

must the high temperature for the next day be in order for the average

temperature for the 3-day period to be more than 19° ?


214

Translate each verbal inequality into an algebra inequality and solve and

graph on a number line.

8. One-half of m is greater than zero

9. The opposite of k is greater than 2.

10. The reciprocal of t is less than 0.

11. The product of t and 5 is at most -6.


215

Being able to interpret a graph is very important. In business, you are often supplied with graphs and charts or are requested to make them.

Pay attention to the and -axes and how they are defined.

Labeling the Axes (plural of Axis) is the independent variable, often time while is the dependent variable and the value of is based on the value of .


216

Look at this speed time graph. Try to work out what this motor bike is doing at each of the points A, B, C and D.

Answer) A – Accelerating hardB – Still accelerated but less hardC – Constant speed of 75mphD – Slowing down at a constant rate

1. Suppose you pour water into each container below at a steady rate. Match each container with the graph that shows the change in the height of the liquid in the container over time. Explain your choices.


217

2. Decide what could be taking place in these graphs. Label each section.

(a) Bank Account

(b) Hair Length

3. Write a title, label the sections of the graph, label the x and y axes and write a story about what might be occurring.

Balance

(Amount

of

money)

Time

Length

Time


218

Drawing graphs of real-life problems1. Choose a suitable scale for each axis2. Decide how many points to plot3. Draw the graph with suitable accuracy4. Provide a title and label the axis

For example, you go to 삼겹살 restaurant from home. Graph this situation by the elapsed time and the distance from home.

Create a graph for the following scenario.1. You leave your house to visit a friend. On the way, you stop at the local shop

for 5 minutes to buy some sweets. You arrive at your friend's house which is

1.5km from your home and 15 minutes after you left home. You stay with your

friend for 2 hours before leaving. It takes you 20 minutes to get home.

2. Your height off the ground on a roller coaster ride lasting 10 minutes.

3. Your movement for a day at the mall.

4. Your pulse rate as you watch a scary movie.


219

Two number lines that intersect at right angles form a coordinate plane（좌표평면).　The horizontal axis is the -axis（축）. The vertical axis is the -axis（축）. The axes (plural of axis) interest at the origin （원점）which we label point (0,0).The plane is divided into four sections which are called quadrants　（사분면）.

An ordered pair（순서쌍）of numbers identifies the location of a point. These numbers are the coordinates of the point on the graph.

The point on the graph has coordinates (-5, 2). The -5 is the -coordinate or abscissa(좌표）and the 2 is called the -coordinate or ordinate(좌표). The -coordinate directs you to go left or right a certain number while the -coordinate lets you know how far to move up or down. A positive number move to the right on the -axis or up on the -axis while a negative number is left on the -axis and down on the -axis.


220

On the coordinate plane below,

1. Plot the points and label with the letter. A(3, 5) B(5, -2) C(-3, -4) D(-4, -3)

2. Label the quadrants (I, II, III, IV or 1,2,3,4)

3. What conjecture can you make about the coordinates of points in quadrant I? quadrant II? III and IV?

-10 -5 5 10

4

2

-2

-4

-6


221

Connecting two points forms a line

segment (선분）which is a part of a line with endpoints at each end. A line（직선） continues on forever in both directions. You can find the point halfway between the endpoints called the midpoint.

The midpoint　（중점） is determined by finding the mean of the x-coordinates and the mean of the y-coordinates.

The midpoint formula : The midpoint of

and is

. It is a point

halfway between the two points.

The midpoint between the segment whose endpoints are (1,4) and (3,9):

Find the midpoint of the segments with endpoints of:1. (2, 4), (3, -4)

2. (0, 7), (-12, 7)

3. (-4, -6), (-2, -12)


222

1. Plot the following points and label with the letter: A(3, 1), B(-2, -2), C(- 4, -1), D(1, -2), E(-1,0), F(0, 3), G(0, -4), H(4, 0), I(0, 0)

2. Find the midpoint of the line segments with the following endpoints.(1) (6, 0), (-6, 0)

(2) (0, 3), (4, -3)

(3) (5, 7), (11, 3)

(4) (-4, -2), (-8, -8)

(5) (2, -1), (-2, 4)

(6) (0, 4), (-4, -2)


223

Function rule（함수식）shows how variables are related. It is just the equation of the function!!

Table（표）shows the specific input and output (x and y values)

Graph gives a visual picture of a function

Continuous data （연속데이터）- any value between numbers has meaning (a line or where points are connected) (Are there values between x terms on the scale of the graph??) Walking to some place is a continuous graph because time continues.

Discrete data　（이산형데이터）- only certain values have meanings (only plot points without connecting). The number of cans of soda on different shelves is discrete because there are only whole numbers of cans.

Graphing Functions : Pick any values of x. § For a line - use 3 points


224

Graphing a linear function :

Step 1. Make a table.

Step 2. Plot the points on the table. Remember

is the same as .

Step 3. Draw a line through the points. You just

drew the graph of .


225

1. Make a table and graph

2. Make a table and a graph of

3. Make a table and a graph of


226

4. Make a table and a graph of y=4x+2

5. If we graph the cost of the number of pounds of bananas we buy is it discrete or continuous?

6. If we graph the number of goals a player scored each game, is it discrete or continuous?


227

A relation(관계) is a set of ordered pairs.｛(1, -3), (2, -3), (3, 0), (4, 2), (5, -1)}

Domain (정의역) is all the (or first) values [Input, Independent Variable]Domain = { 1, 2, 3, 4, 5}

Range (치역) is all the (or second) values [Output, Dependent Variable ]Range = {-3, -1, 0, 2}

Mapping matches each member of the domain with a member of the range. For the relation{ (1, -3), (2, -3), (3, 0), (4, 2), (5, -1) }, the mapping is shown as follows.

A function （함수）　is a relation that assigns exactly one output or range value for each input or domain value. Each value corresponds with exactly one value. Y-values can be repeated. X-values can not.{ (1,2), (2,2) } is a function, but { (1,2), (1,3) } is NOT a function.


228

The Codomain (공역) is the set of

values that could possibly come out (target space).

The Codomain is actually part of the definition of

the function while the Range is the set of values

that actually do come out.

You can define a function with a domain

and codomain of integers (because you say so).

But by thinking about it you can see that the range

(actual output values) would be just the even

integers.

So the codomain is integers (you defined it that

way), but the range is even integers.

How to determine whether a relation is a function

or not : If you map a relation and each member of

the domain has only one arrow coming from it, it

is a function.

For example, (1,2), (2,2) is a function

(1,2), (1,3) is NOT a function.


229

Vertical Line test If after you graph a relation, you

can draw any vertical line through the graph and it

only passes through one point, the relation is a

function. The graph in (I) is a function, but the

graph in (II) is not a function.

한 x값에 대해 두 가지 이상의 y값이 존재하냐 안하냐를 판별하는 테스트

(I)

(II)

Determine whether the relation is a function. State

the Domain and Range.

1. {(1,5),(2,６),(3,3),(4,3),(２,5)}

2.

Input Value 0 1 2 3 4 5

Output Value 0 1 2 3 4 5


230

1. For a relation {(1,3), (2,3), (5,4), (5,9)}(1) Map the relation.

(2) What is the domain?

(3) What is the range?

(4) Is the relation a function?

(5) How can you tell from the map?

2. For a relation { (-3,-1), (0,3), (2,4), (3,9), (4,3) }(1) Map the relation

(2) What is the domain?

(3) What is the range?

(4) Is the relation a function?

(5) How can you tell from the map?

3. Circle the relations which are functions. (1) (2) (3) (4)

5

4

2

-2

-4

-10 -5 5 10

4

2

-2

-4

4

2

-2

-4

5

4

2

-2

-4


231

Globalization refers to the integration among societies and economies across the globe. Globalization has led to social, economic, technical, cultural and ecological interdependence among nations. Especially, international economic relations play an important role in the growth of economies across the world. For economic relations between nations to be successful, a number of conditions need to be fulfilled. Unrestricted movement of goods and services, flow of capital, mobility of workforce, and reduction of regulatory obstacles need to be ensured for successful economic relations.

ACTIVITY

Here is the excerpt of an article about international economic relations.

How rising food prices are impacting the world

....

Yet 2012 is providing a window into how far the world has come in dealing with the

fluctuations of the weather and the interdependence of world food supplies -- and

how far it still needs to go. The pressures to improve the system will only increase.

The planet is growing hotter and drier, and the global population stands at 7 billion

and counting.

Long gone are the days when the world's population was subject solely to the whims

of local crops and climate. The green revolution of the 20th century, which generated

spectacular increases in grain yields, and the advent of cheap shipping created a

global food market that's transformed the lives of hundreds of millions of people.

But it also means that a bad year for farmers in the United States or Russia or

Australia can ripple out quickly and become a disastrous year for consumers in Egypt

or Indonesia.

......

<Sept 24. 2012, wfmj.com>

1. Make a relation among the nations in the article related by food imports and

exports.

2. What can you tell from the relation you made about international relations?


232

Function Notation（함수　표기법） Another way to write an equation instead of is function notation. The output () is called pronounced of .So when we choose 1 for , we write , so . The coordinate is (1,5). This way, we can make a table of values for any function.

For a function , complete the table. The left column is the -values (the domain). The middle column is to calculate the function. The right column is the output or the range.

Function Rule You can write a function rule from a table of values or a graph. Analyze the table of values or the coordinates and see how you can get from the 1st term (x coordinate) to the second term ( or ). Sometimes it is multiplying or dividing or adding or subtracting. Other times it is a combination of the two.

-1 2(-1) +8 6

0

2

10


233

The function rule is written by determining the

slope which is the change in the y-value ∆ divided by the change in the x-value ∆.

)0()()( fxxxfxf +

DD

=

For a function given by the table,

2 4

1 2

1 2

the slope is the change in divided by the change in is

.

from the first line of the table. So, the function rule of from this table is

.

Write a Function Rule for each situation. Determine if it is Discrete or Continuous.1. The total cost T(x) of x pounds of coffee beans if each pound costs ３000￦.

2. The total weight W(x) of x iPod when each iPod weighs 2.６ pounds.

Δ means the change

(increase or decrease).

.

　　　　


234

For 1-3, does this data or graph make a function? Write the domain and range1.

2.

For 3-4, evaluate the function at the given x-values.3. for

4. for

For 5-8, write the equation of the function represented in the tables.5. 6.

x y0 01 42 83 12

x y0 11 32 53 7

7. 8.





x y1 0-1 -42 23 4

x y0 41 62 83 10


235

For 9-10, write a function rule to model each situation.9. The total cost t(k) of kimchi when each gram costs ₩200.

10. The area of a square when we know the length of a side is s.

11. What are the advantages of using a function rule rather than having a table of values?

12. Write a function rule that models a real world situation. Explain what the input value and output values represent and what the domain and range can be.

✩13. Your cell phone plan is ₩12,000 per month for the first 300 minutes and unlimited text messaging. For each minute over the ３00 min, there is a ₩４0 fee. Write a function rule for the monthly cost, C(m) of your cell phone bill based on the number of minutes, m you use.


236

A direct variation （정비례　관계）is a

situation in which two quantities such as hours and pay, or distance and time change at a constant rate. The ratio between the quantities is

constant. For example, if your hourly rate doubles, your total pay also doubles (even though the number of hours worked remains the same). where is your total pay, is your hourly rate and is the number of hours worked.

For any given problem

any y divided by

its related is equal to any other divided by its related .

A mechanic who is paid hourly knows that working longer means making more money. That's because his pay varies directly as the number of hours worked. As his hours increase, so does the amount of his paycheck.

A function in the form , where ≠, is a direct variation. is the coefficient of , also called the constant of variation. The variables and are said to vary directly with each other. is read as varies directly with . 　In

America it is .


237

If you know a direct variation exists, you can write the equation given one point, (2, 4). ;

So, is the equation.

To solve direct variation problems, you will need to be able to solve one-step equations, linear equations and sometimes literal equations.

Suppose varies directly as , and when . Find when . When we see the phrase “ varies directly as ” that tells us when should be using the equation . The next piece of information tells use that and . We will use this information to find the value of .

divide both sides by 4

Now use the fact that to find the value of that the problem is asking for.

Solving direct variation by proportion:

Since then each

for the same direct

variation problem will be equal. For the same problem in the previous example, suppose varies directly as , and when .


238

However, if I want to write the equation of variation, I still need to determine which is

and the equation would be .

Your distance from lightning varies directly with

the time it takes you to hear thunder. If you hear

thunder 10 seconds after you see lightning, you are

about 2 miles from the lightning. Write an

equation for the relationship between time and

distance.

Solution) The distance varies directly with the time.

Define variables first.

Let = the number of seconds between your

seeing lightning and your hearing thunder.

Let = your distance in miles from the lightning.

since ,

This equation relates the time (x seconds) it takes

you to hear the thunder to the distance (y in miles)

you are from the lightning.


239

1. Suppose m varies directly as p, and m = 22

when p = 8. Find p when m = 90.

2. Suppose v varies directly as g, and g = 70 when

v = 30. Find v when g = 9.

3. Suppose x varies directly as y, and x = 8

when y = 10. Find x when y = 16.

4. Suppose y varies directly as x, and y = 9 when

x = 3/2. Find y when x = 1.

5. Suppose c varies directly as b, and c = 81 when

b = 10. Determine the constant of variation and

write the equation of direct variation.

To determine if an equation is a direct variation,

solve for a variable to determine if the equation is

of the form . If it is, the equation is a

direct variation, otherwise it is not.

Is a direct variation?

If we solve for , we get

or if we solve

for we get

and since there are no

constant terms, this is a direct variation.

Direct variation is a line passing through the

original but does not include a horizontal or

vertical line which may also pass through the

origin.


240

Is the equation a direct variation?1.

2.

3.

4.

For problems 5-7, determine whether the

relation is a function. State the Domain and

Range.

5. {(1,5),(2,６),(3,3),(4,3),(２,5)}

6. {(1,1),(2,1),(3,1),(0,0)}

7.

8. Is this graph a function?

I n p u t

Value0 1 2 3 4 5

Output

Value0 1 2 3 4 5


241

Determine for 1-6 if the equations are direct variation?1. 2. 3.

4. 5. 6.

For 7-10, write an equation of the direct variation that includes the given point.7. (2, 5) 8. (-3, 4)

9. (8, -4) 10. (-4, -12)

11. The perimeter of a square varies directly with the length of one side of the square. Define the variables and write a direct variation to model the relationship.

For 12-15, does y vary directly with x? If so, write the equation.12. 13.

1 32 63 9

1 42 53 6

14. 15.

-1 4-2 8-3 -12

-1 -12 28 8

For 16-18, state True of False:16. The graph of a direct variation may pass through (5,10)

17. The graph of a direct variation may pass through (0,10)

18. If you double a value of a direct variation, the y-value also doubles.

Chapter 5 – Functions and Relations

242

19. The amount of blood in a person varies directly with the persons weight. A person who weighs about 77kg has about 5 liters. Determine the constant of variation. Write the equation of variation. Estimate the number of liters of blood in your body.

For 20-23, the ordered pairs are from the same direct variation. Determine the missing value.20. (2, 6), (6, y) 21. (3.5, 11), (14, y)

22. (x, 4), (3, 12) 23. (2, y), (3, 6)

24. A car gets 40 km per liter of gasoline. The distance, d, number of km travelled varies directly with the number of liters l of gas used.a. If gas costs ￦1800 per liter, write a function relating the cost c for l liters of gas. Is this direct variation?¶b. Write a direct variation relating the cost of gas to the km traveled.

25. Minhyoung got a job working at a cinema. The charts shows the earnings for different weeks. If Minhyoung works 40 hours over vacation how much will Minhyoung earn?Hours Works Money Earned in Won

12 138,000

19 218,500

31 356,500


243


244

An inverse variation (반비례　관계）is a

situation in which one quantity increases while

another quantity decreases.

When the ratio of one variable to the

reciprocal of the other is constant (when the

product of the two variables is constant), one

of them is said to vary inversely as the other;

that is, when

,

or xy = a, y is said to vary inversely as x. A

is the constant of variation for the inverse

variation.

Yeonhee took a watermelon to a picnic and

knows that each person will receive more

watermelon if there are fewer attendees, but

each person will receive less watermelon if

there are more attendees. That's because the

amount of watermelon for each person varies

inversely as the number of attendees. The

more people, the less each person gets.

Yewon is a truck driver and knows that driving

at 75 miles per hour will get the truck to the

destination faster than driving at 65 mph,

because time is inversely proportional to

speed. As Yewon's speed increases, the travel

time decreases.


245

If y varies inversely with x and y=5 when x=3, write the equation of the inverse variation.xy = a(3)(5) = aa=15

xy = 15 or

Hungry? How about some pizza???

If a pizza costs 10,000 won and Yujeong and Yulle split it, it costs 5000 won each. If Joonho and Yeowon also split it with them, it costs 2500 won each.x = the number of people, y = price they paya = constant of variation = price of pizzaInverse variation is xy = a or y = a/x(5000)(2)=10000 and

(2500)(4)=10000With inverse variation, the x and y variables always have a product of a constant, a.xy =10000Since xy=a for all x, y pairs in any inverse variation, then every =.For this problem 4(2500)=2(5000). For 6 people

4(2500)=6(y) y=1666


246

1a. Determine the constant, a of inverse variation when x = 2 and y = 6 and write the equation of inverse variation.

1b. What about when x = 3 and y = 4?

2. With an inverse variation one point on a graph is (6,3). What is the y coordinate of a point with x=2? (2, y).

3. The time and speed of your trip vary inversely. You drive to school for 1 hour at 90km/h. If you travel at 60 km/h, how long will it take you?


247

Write an equation of the inverse variation that includes the given point.

1. (2, 5)

2. (-3, 4)

3. (8, -4)

4. (-4, -12)

5a. The cost of gasoline for a trip from CSIA to Seoul is \40,000.

Hyoseo and Moogun want to split the cost so they pay \20,000. If

Geonhye decides to go, how much will each pay?

5b. Is this an example of inverse variation?

5c. What is the equation?

For problems 6 – 9, does y vary directly or inversely with x? If it does,

then write the equation of variation.

6 7

8 9

x y2 64 1210 30

x y2 304 1510 6

x y3 126 69 4

x y3 249 812 6


248

10. If you double the x-value of an inverse variation, the y-value also

doubles. True or False.

11. What happens to the y-value of an inverse variation when the

x-value triples?

12. Boyle's Law states that volume V varies inversely with pressure P

for any gas at a constant temperature in an enclosed space. Suppose a

gas at a constant temperature occupies 15.3 liters at a pressure of 40

milliliters of mercury. Write the equation that models this situation.

13. Jokingly there is a "First Law of Air Travel" that states the distance

to your connecting gate is directly proportional to the amount of luggage

you are carrying and inversely proportional to the amount of time you

have to reach your connecting flight."

Will the distance to your gate be greater or less for this trip than your

last trip if:

a. You have more luggage.

b. You have less time to make your flight.

c. You have less luggage.

d. Try and write the equation of this situation.


249

You can determine if an equation is a direct or

inverse variation from the data by determining

if x times y is a constant or y divided by x is

a constant for all the values.

Inverse Variation

Direct Variation

(0,0) would give an undefined fraction which

also allows us to know it might be direct

variation.

If you have data - a table of values or

coordinates, if the product of y and x is the

same number (k), it is inverse variation. If the

quotient of y and x is the same number k, it is

direct variation.

x y xy-2 4 -82 -4 -84 -2 -8

6

-8

x y

-1 -4 41 4 42 8 43 12 4


250

Is this an example of direct or inverse variation?

1. Buying bananas at 2000 won per pound.

2. Determining the length and width or rectangles which have an area of

10.

What type of variation is this?

3. The amount of rocket fuel and the distance the rocket travels.

4. The mass, m, and volume, v, of a substance are related by the

equation 2v=m, where 2 is the density of the substance.

5. The time, t, spent waiting in line for the most popular roller coaster

at Lotte World, and the number of people, n, in the line are related by

the equation t=an.

¶6. The time, T minutes, taken for a stadium to empty varies directly as

the number of spectators , S, and inversely as the number of open exits,

E.

(a) Write down a relationship connecting T, S and E.

It takes 12 minutes for a stadium to empty when there are 20,000

spectators and 20 open exits.

(b) How long does it take the stadium to empty when there are 36,000

spectators and 24 open exits?


251

The graph of a direct variation, y=ax is a line that passes through the

origin.

a > 0 a < 0

The graph of an inverse variation,

or is called a hyperbola.

You can graph it by plotting points.

or

or

x y-4 -0.75-3 -1-2 -1.5-1 -30 Undefined

1 32 1.53 14 0.75

Graph

and


252


253

Direct Variation

1. If 8 oranges cost $10.40, how many oranges can be bought for $33.80?

2. If 18 dolls cost $630, how many dolls can be bought for $455?

3. If a man earns $805 per week, in how many days will he earn $1840?

4. If car covers 102 km with 6.8 liters of gas, how much distance will it

cover with 24.2 liters of gas?

5. The cost of 16 bags of washing powder, each weighing 1.5kg, is $672.

Find the cost of 18 bags, each weighing 2kg.

6. If 3 people can weave 168 shawls in 14 days, how many shawls will

be woven by 8 people in 5 days?

7. If the cost of transporting 160 kg of goods for 125km is RS 60, what

will be the cost of transporting 200 kg of goods for 400 km?

8. If the wages of 12 workers for 5 days are $7500, find the wages of 17

workers for 6 days.


254

Inverse Variation

1. If 32 men can reap a field in 15 days, in how many days can 20 men

reap the same field?

2. 12 men can dig a pond in 8 days. How many men can dig it in 6

days?

3. A hostel has enough food for 125 students for 16 days. How long

will the food last if 75 more students join them?

4. 8 taps having the same rate of flow, fill a tank in 27 minutes. If two

taps go out of order, how long will the remaining taps take to fill the

tank?

5. If 5 men working 6 hours a day can reap a field in 20 days, in how

many days will 15 men reap the field if they work for 8 hours a day?

6. If 270 kg of corn would feed 42 horses for 21 days, for how many

days would 360 kg of it feed 21 horses?


255

7. Five machines, when operated for 9 hours each day, can harvest a

farm in 16 days. How many days would 8 machines take to harvest the

same farm, if each machine is now operated for 10 hours each day?

☆8. 6 men, working 8 hours a day, earn $ 8400 per week. What will be

the earning per week of 9 men who work for 6 hours a day?

☆9. A fort had enough food for 80 soldiers for 60 days. How long

would the food last if 20 more soldiers join after 15 days?

Answers to Direct Variation

1) 26 2) 13 3) 16 4) 363

5) 1008 6) 160 7) 240 8) 12750

Answers to Inverse Variation

1) 24 2) 16 3) 10 4) 36 5) 5

6) 56 7) 9 8) $9450 9) 51


256

Set Builder Notation is a shorthand used to

define sets, often sets that have an infinite number of

elements.

The set {x | x > 0 } is read aloud, "the set of all x

such that x is greater than 0." It is sometimes written

with a colon : instead of a |

{ formula for elements | restrictions }

{x:x≠4}The set of all real numbers except 4

{x|x<5} The set of all real numbers less than 5

{|x is a real number} The set of all real numbers

greater than or equal to 0

{|x∈ℝ} is the same as the one above.

{2n+1: n is an integer} The set of all odd integers.

{...-3, -1, 1, 3, 5,...}


257

Interval Notation is notation that represents the

endpoints of a line as an interval. Parentheses and

brackets are used to show whether the endpoints

are included or excluded on the graph. A

parenthesis shows that point is not included and a

bracket shows the point is included.

(2, 5)

1 and 4 are not included in the graph

[-1, 8)

-1 is included while 8 is not included.

Sometimes a bracket and parenthesis are used instead of solid (closed, filled, shaded) or hollow (open, empty, unshaded) points.

The above data can be written in interval and set notation forms.

Set Builder Notation {x| -7≤x≤-1 ∪ x>5}Interval Notation [-7, -1] ∪ (5, ∞)

Since infinity, ∞ is not an exact point, it always has a parenthesis and not a bracket.

If the arrow on the graph points to the left, the graph goes towards -∞ . We distinguish +∞ from -∞ due to the direction.

Summary of mathematical symbols

< less than

< less than or equal

> greater than

> greater than or equal to

( or ) does not include the actual point

[ or ] does include the actual point

∪ union or “or”

∩ disjunction or “and” (mutually exclusive)


258

Write the data that is graphed on the number line using both set builder and interval notations.

1.

2.

3.

Write the following in both interval and set builder notation:

4. -3 ≤ x ≤ 2 or 3 ≤ x < 8

(Hint: or means ∪ , union

and means ∩ , intersection)

5. "All numbers from negative six up to and including negative one."

6. x ≠ 2.

7. -1 > x or 2 < x < 4

8. What is the interval of the graph between the number line graphs of

these two intervals: [-7, 0] U (4, 8]?

(What is the missing interval that would connect the two segments without any duplicate points.)


259

The slope (기울기）of a non-vertical line is the

number of units the lines rises or falls for

each unit of horizontal change from left to

right.

As you may recall from the section on writing

a function rule, the function rule is written by

determining the slope which is the change in

the y-value ∆ divided by the change in

the x-value ∆.

∆

∆ is the slope. The change in the

y-value divided by the change in the x-value.

The slope of a non-vertical line passing

through the points and is

. It is okay to change the order

as long as the terms are in the same order in

the numerator and denominator.

.

In America, slope is referred to as m instead

of a.

)0()()( fxxxfxf +

DD

=


260

A line with a positive slope rises to the right.

A line with a negative slope falls to the right.

A line with a 0 slope or No slope is horizontal

y=a where a is a real number

A line with an undefined slope is vertical.

x=b where b is a real number

4

2

-2

-4

-6

-10 -5 5 10

4

2

-2

-4

-6

-10 -5 5 10

4

2

-2

-4

-6

-10 -5 5 10

4

2

-2

-4

-6

-10 -5 5 10


261

In real world problems, slope is referred to as

a constant rate of change or an average rate

of change. Common units of rate of change

are won per hour (pay for a job at Emart or

Lotteria) or kilometers per hour (rate of speed

for a trip).

Dahyeon is climbing up an 800 meter cliff. By

2 PM, Dahyeon has climbed 150 meters up the

cliff. By 6PM, Dahyeon has reached an

altitude of 290 meters. Find the average rate

of change in Dahyeon’s altitude during the 4

hours.

We can think of this as two coordinates on a

graph. With x being the time and y being the

height.

(2, 150), (6, 290)

a = 35 meters per hour

Dahyeon’s average climbing rate was 35

meters per hour.

From a graph we can determine the slope by

finding two points we can identify (passes

through corners so we know the points have

integers coordinates). We can then count up

and then right (or down and left) for a positive

slope or up and left (or down and right) for a

negative slope. The graph on the right is up

4 and over 2,

.


262

Practice:

Find the slope of the points.

1. (1,2), (3,6), (5,10)

2. (0,4), (6, 16)

3. (-3, 4), (4, -3)

4. (-2, -2), (4, -5)

5. Describe the slopes of the 4 lines.

a.

b.

c.

d.


263

Where a line crosses the axis is called an

intercept (절편）.

The x-intercept (x절편）is the point where the

line crosses the x-axis and is a point (x, 0)

since the y-coordinate is 0 on the x-axis.

The y-intercept（y절편）is the point where the

line crosses the y-axis and is a point (0, y) or

sometimes called (0,b) since b is the

abbreviation for the y-intercept in the equation

of a line.

Find the x- and y-intercepts for y=4x-1.

The y-intercept is found by letting 　x = 0.

y=4(0)-1 y = -1

We write the y-intercept as (0,-1)

This shows a secret we learn soon. The form

of y=ax+b is slope-intercept form and b is the

y-intercept.

The x-intercept is found by letting y = 0.

0=4x-1, 4x = 1, x=

We write the x-intercept as .


264

1. Determine the slope and intercepts from the graph.

Find the slope of the line passing through the points:

2. (-3,-2), (1,6)

3. (0,0), (-2, -5)

4. (2, 5), (2, 11)

5. (-2, 9), (11, 9)

4

2

-2

-4

-10 -5 5 10


265

Determine the x- and y-intercepts of the following lines:

6. 3x - 2y = 12

7. y = 5x + 3

8. 4x = 2y +3

Any two points on a line can be used to determine its slope. Any three

points on the same line or form the vertices of a triangle. Determine if

the 3 points form a line or a triangle.

9. (-2,1), (1,2), (3,3)

10. (-2,-4), (2, -2), (6,0)

11. At 1:05 Sihyun was in an airplane 4000 meters above the ground

and jumped out of the airplane wearing a parachute. At 1:15, Sihyun was

2600 meters above the ground. Determine the average rate of change in

meters per minute. What about meters per hour?

266

Write the data that is graphed on the number line using both set builder and interval notations.

1.

2.

3.

Write the following in both interval and set builder notation:4. -5 ≤ x ≤ 0 or x ≥ 4(Hint: or means ∪ , union

and means ∩ , intersection)

5. "All numbers from six up to and including twenty three."

6. x ≠ -4

7. 0 > x or 2 < x < 5

8. What is the interval of the graph between the number line graphs ofthese two intervals: [-7, -1] U (2, 11]?(What is the missing interval that would connect the two segments without any duplicate points.)

267

Math Glossary

Aabsolute value 절댓값

acute angle 예각

acute triangle 예각 삼각형

additive inverse 덧셈에 대한 역원

adjacent 인접한 (이웃하는)

algebraic equation 대수 방정식

algebraic expression 대수적 수식

alternate angle 엇각

altitude (다각형의) 높이

angle 각

arc 호

area 면적

arithmetic mean 산술평균

arithmetic progression 등차수열

arithmetic sequence 등차수열

associative law 결합법칙

associative property 덧셈의 결합법칙

of addition

associative property 곱셈의 결합법칙

of multiplication

average speed 평균 속력

average velocity 평균 속도

axis 축

Bbar graph 막대그래프

base 밑변, (지수로그의) 밑

bisect 이등변

bisector 이등분선

binomial 이항식

bounded 유계의

Ccardinality 원소의 개수

center 중심

central angle 중심각

chord 현

circumference 원주

coefficient 계수

common difference 공차

common divisor 공약수

common measure 공약수

common multiple 공배수

common factor 공통인수 (공약수)

common ratio 공비

commutative law 교환 법칙

commutative property 덧셈의 교환법칙

of addition

commutative property 곱셈의 교환법칙

of multiplication

complement 여집합

complementary angle 여각

complex fraction 번분수

composite number 합성수

compound interest 복리 (이자문제)

conclusion 결론

congruity 합동

conjecture 추측, 가설

consecutive integers 연속된 정수들

constant 상수

constraints 제약조건

coordinate plane 좌표평면

corresponding angles 동위각

cost 원가, 가격, 정가

cube 세제곱, 입방체

268

Ddecimal 소수

decimal number 십진수

decimal fraction 소수 부분

decimal point 소수점

decrement 감소율, 감량

deductive reasoning 연역적 추론

degree of a polynomial 다항식의 차수

denominator 분모

diagonal (line) 대각선

diameter 직경(지름)

difference 차

digit 자릿수

direct proportion 정비례

direct variation 정비례 관계

discount 할인(액)

discrete data 이산형 데이터

distance 거리

distributive law 분배법칙

dividend 피제수, 나뉨수

divisible 나뉘어 떨어지는

divisor 제수

domain 정의역

Eelement 원소

empty set 공집합, Ø

equation 방정식

equidistant 같은 거리에 있는

equilateral triangle 정삼각형

exponent 지수

expression 수식

exterior angle 외각

evaluate 값을 구하다

even number 짝수

expansion 전개

Ffactor 인수

factorization 인수분해

fraction 분수

frequency (통계) 도수

function 함수

Ggeometric mean 기하 평균

geometric progression 등비수열

geometric sequence 등비수열

greatest common divisor 최대공약수

greatest common factor 최대공약수

greatest common measure 최대공약수

Hharmonic mean 조화평균

heptagon 칠각형

hexagon 육각형

horizontal 수평의

hundreds 백의 자리

hundreds place 백의 자리

hundredths 소수 둘째 자리

hypotenuse 빗변

Iidentity property of 덧셈의 항등원, 0

addition

identity property of 곱셈의 항등원, 1

multiplication

increment 증분, 증가량

indirectly proportional 반비례

269

inductive reasoning 귀납적 추론

inequality 부등식

infinite decimal 무한소수

infinite number of 해가 무수히 많다

solutions 해

integer 정수

intercept 절편

interest 이익

interior angle 대내각

intersection 교집합, 교점

inverse operations 역연산

inverse property of 곱셈의 역원

multiplication

inverse property of 덧셈의 역원

addition

inverse proportion 반비례

irrational numbers 무리수

inverse variation 역비례 관계

isosceles 이등변삼각형

Lleast common multiple 최소공배수

left side 좌변

legs (삼각형의) 밑변

like terms 동류항

line 직선

line segment 선분

linear inequalities 일차부등식

Mmaximum 최댓값

mean 평균

median 중간값

midpoint 중점

mode 모드, 최빈값, 최빈수

monomial 단항식

minimum 최솟값

midpoint 중점

multiple 배수

Nnatural number 자연수

negative integer 음의 정수

normal 법선

no solution 해가 없다. (불능)

null set 공집합, Ø

numerator 분자

Oobtuse triangle 둔각삼각형

obtuse angle 둔각

odd number 홀수

octagon 팔각형

one decimal place 소수 첫째 자리

order 차수

ordered pair 순서쌍

ordered set 순서집합

origin 원점

Pparallel 평행

parallel lines 평행선

parallelogram 평행사변형

parentheses 광호

pentagon 오각형

percent 퍼센트, 백분율

perimeter 둘레

perpendicular 수직

perpendicular lines 서로 수직인 직선

270

point 점

polygon 다각형

polynomial 다항식

positive integer 양의 정수

power 거듭제곱

power set 멱집합

price 매가(가격)

prime factorization 소인수분해

prime number 소수

product 곱

progression 수열

properties 성질

proportion 비례식

Qquadrilateral 사각형

quadrants 사분면

quotient 몫

Rradius (radii) 반경 (반지름)

range 치역 (범위)

rate 비율, 속도

rate of increase 증가율

ratio 비

rational numbers 유리수

ray 반직선

real numbers 실수

reciprocal 역수

rectangle 직사각형

regular polygon 정다각형

relatively prime 서로소

remainder 나머지

rhombus 마름모

right triangle 직각삼각형

right angle 직각

right side 우변

root (제곱)근

rounding off 반올림

Sscience triangle 부등변삼각형

sector 부채꼴

segment 선분

sequence 수열

set 집합

similarity 닮음

similar term 동류항

simple interest 단리문제

problems

simultaneous equations 연립방정식

slope 기울기

solution 해(답)

square 제곱, 정사각형

subset 부분집합

sum 합

supplementary angle 보각

system of linear 연립일차방정식

equations

system of linear 연립일차부등식

inequalities

Ttangent line 접선

tens place 십의 자리

tenths 소수 첫째 자리

term 항

thousands place 천의 자리

271

thousandths 소수 셋째 자리

transversal 횡단선

trapezoid 사다리꼴

triangle 삼각형

trinomial 삼항식

two decimal places 소수 둘째 자리

Uunion 합집합

universe 전체집합

unknown 미지수

Vvalue of the variable 미지수의 값

variable 변수

Venn diagram 벤 다이어그램

vertex (vertices) 꼭짓점

vertical 수직의

vertical angle 맞꼭지각

volume 부피

Wweight 무게

whole number 자연수와 0

Xx-axis x축

x-coordinate x좌표

x-intercept x절편

Yy-axis y축

y-coordinate y좌표

y-intercept y절편

272

Algebra 1 Study Guide

Divisibility Rules

2 - The last digit is even.

3 - The sum of the digits is divisible by 3.

4 - The last two digits form a number

divisible by 4.

5 - The last digit is a 5 or a 0.

6 - The number is divisible by both 3 and

2.

7 - Take the last digit, double it, and

subtract it from the rest of the number; if

the answer is divisible by 7 (including 0).

8 - The last three digits form a number

divisible by 8.

9 - The sum of the digits is divisible by 9.

10 - The number ends in 0.

Properties of Real Numbers

Commutative Property of Addition

x + y = y + x ; (3 + 6 + 7 = 6 + 3 + 7)

Changing the order of the terms

Commutative Property of Multiplication

xy = yx ; (3•x = x•3)

Associative Property of Addition

(x+ 3) + y = x + (3 + y)

Rearranging the parenthesis (order of operations)

Associative Property of Multiplication

(3 • 2) • y = 3 • (2 • y)

Distributive Property of Multiplication

3(x + 2) = 3x+6

The product of the term in front of the

parentheses and the terms inside

Additive Identity

a + 0 = a

The sum of 0 and any number is that same

number.

Additive Inverse

a + (-a) = 0

The sum of the opposite of a number and the

number is 0.

Multiplicative Identity

a • 1 = a

The product of a number and 1 is that same

number.

Multiplicative Inverse

11

=×a

a

The product of a number and its reciprocal is 1.

Multiplication Property of Zero

a • 0 = 0

The product of any number with zero is zero.

Multiplication Property of -1

a • -1 = -a

The product of a number and -1 is the opposite

of the number.

Measures of Central Tendency

Mean (average)– sum of all the data divide by

the number of data elements

nxxxx nn ++++ -121 ...

The mean is the best

representation of the data when there is no

outlier. An outlier is a number distant from the

other numbers.

Median – the middle data term when the data is

arranged from ascending or descending order

(from highest to lowest or lowest to highest).

If there is an even number of data terms, the

median is the mean of the middle two terms.

The median is the best representation of the

data when there is an outlier.

Mode – the term that occurs most often or

most frequently. The mode is the best

representation of the data when you asked for

the data which occurs most often or the data is

not numbers.

Range – gives you the measure of the distance

of the highest and lowest data values. Subtract

the lowest data value from the highest to find

the range.

273

Quartiles

1st – The median of the first half of the data

(between 1st number and median)=25th

percentile

2nd – The median of all the data = 50th

percentile

3rd quartile – the median of the second half of

the data (between the median and the last

number) = 75th percentile

With two points are P1 (x1,y1) and P2 (x2,y2) :

Slope a = 12

12

xxyy

--

Parallel Lines have the equal slopes

Perpendicular Lines have slopes which are

opposite reciprocals of each other.

Midpoint between two points

÷øö

çèæ ++

2,

21212 yyxx

Distance Between two points

d =

( ) ( )212

212 yyxx -+-

.

Pythagorean Theorem

c2 = a2 + b2

a c

b

The square of the hypotenuse of a right triangle

is equal to the sum of the squares of the legs.

Equations of Lines

Slope intercept form y = mx + b (y = ax + k)

where m or a is the slope of the line and b or k

is the y-intercept, the point where the line

crosses the y-axis.

Point-slope form y – y1 = a(x-x1) where a is the

slope and (x1,y1) is the point that is known.

Standard or General Form Ax + By = C or

Ax + By + C = 0 where A, B, C are integers

and A > 0.

Quadratic Formula For a quadratic equation in

standard form, ax + by + c = 0, the solutions

can be determined by using the formula

aacbbx

242 -±-

=

Special Cases of Multiplying/Factoring Polynomials

( ) 222 2 bababa ++=+ ( ) 222 2 bababa +-=-

( )( ) 22 bababa -=-+

Discriminant

acb 42 -

determines the number

of real solution. If

acb 42 -

<0, there are 0

real solutions;

acb 42 -

=0, there is 1 real

solution;

acb 42 -

>0, there are 2 real solutions

Factoring Rules

cbxx ++2

= (x + d)(x + e)

where d +e = b and de = c

If the c term is positive both d and e have the

same sign as b. If the c term is negative then

only one of d and e is negative.Commonly Used Formulas

D=st or D=rt

D = distance, s = speed and t=time. (r for

rate of speed is often used in the US)

I=Prt I = simple interest, P = principal(starting

amount), r=rate(as a decimal) t = time (in

years)

A=P(1+r) where A = total amount of money

after time t

Exponent Rules

baba xxx +=×

( )nnn xyyx =

( ) abba xx =

bab

a

xxx -=

n

n

n

yx

yx

÷÷ø

öççè

æ=

aa

xx 1

=-

,10 =x 000 =

xx =1

274

Extraneous Solution - a solution that does not

solve the original equation. If you solve an

equation, you must check to make sure the

solution actually works.

Scientific Notation – a method use to write

very large or small numbers. A number in

scientific notation is written as a x 10n where

101 <£ a

and n is an integerOrder of Operations

PEMDAS

Parentheses, when multiple parentheses, work

from inside ones (nested) first,

Exponents,

Multiplication and Division from Left to Right

(multiplication is not always first),

Addition and Subtraction from Left to Right

(addition is not always first).

Classifying Numbers

Natural Numbers (Counting Numbers) {1,2,3,…}

Whole Numbers {0,1,2,3,…}

Integers {….-2,-1,0,1,2,….}

Rational Numbers – any number that can be

represented as a fraction (includes terminating

and repeating decimals)

Irrational Numbers – Cannot be expressed as a

fraction such as or

Real Numbers – includes all numbers in the

groups above,Triangles

Scalene – no equal sides

Isosceles – two or more equal sides

Equilateral – 3 equal sides

Acute – all angles are less than 90 degrees

Right – one angle is exactly 90 degree

Obtuse – one angle is greater than 90 degrees

Arithmetic Sequence

A(n) = a + (n – 1)d

A(n) – nth term

a – first term

n – term number

d – common difference (difference from one

term to previous term)Angles

180 degrees -sum of the measures of the

angles in a triangle

180(n-2) – sum of the measures of the angles

of an n-sided polygon.

nn )2(180 -

measure of each angle of an

n-sided regular polygon.

Geometric Sequence

1)( -×= nranA

A(n) – nth term

a – first term

n – term number

r – common ratio

(the quotient between

one term and the

previous term)

Sets

275

Sets and Venn DiagramsA set is a list of objects in no particular order; they could be numbers, letters or even words. An ordered set has the list in order.A Venn diagram is a way of representing sets visually.

An object is an element （원소）of a set when it is contained in the set.

The universe （전체집합）　(usually represented as U) is a set containing

all possible elements, while the empty set or null set (공집합）

(represented as Ø or { }) is a set containing no elements.

The cardinality （원소의개수）of a set is the number of elements in the

set. A set with 3 elements has cardinality of 3, |P|=3，　n(P)=3

Example - we use whole numbers from 1 to 10.

We will define two sets taken from this group of numbers:

Set A-odd numbers={1 , 3 , 5 , 7 , 9}

Set B = the numbers which are 6 or more in the group={6 , 7 , 8 , 9 , 10}

Some numbers from our original group appear in both of these sets. Some

only appear in one of the sets. Some of the original numbers

don't appear in either of the two sets. We can represent these facts using

a Venn diagram.

Sets

276

The two large circles represent the two sets. The numbers which appear in both sets are 7 and 9. These will go in the

center section, because this is part of both circles.

The numbers 1, 3 and 5 still need to be put in Set A, but not in Set B,

so these go in the left section of the diagram.

Similarly, the numbers 6, 8 and 10 are in Set B, but not in Set A, so will

go in the right section of the diagram.

The numbers 2 and 4 are not in either set, so will go outside the two

circles in the area called the Universe.

Set A = {1,3,5,7,9} Set B = {6,7,8,9,10}

The final Venn diagram looks like this:

We can see that all ten original numbers appear in the diagram.

The numbers in the left circle which includes the middle section are

Set A { 1 , 3 , 5 , 7 , 9 }

The numbers in the right circle are Set B { 6 , 7 , 8 , 9 , 10 }

Intersection ∩ or ^

The intersection （교집합）of sets A and B is those elements which are in

set A and set B. The shaded area in the diagram represents the

intersection of sets A and B.

Union U

The union （합집합）of sets A and B is those elements which are

Sets

277

in set A or set B or both. The shaded area of the diagram shows

the union of A and B is below.

The difference of sets, A-B is the set of all elements of set A which do not belong to

set B

A-B is different from B-A. B-A has all the members of set B that are not in set A.

A=(A-B)∪(A∩B) and B=(B-A)∪(A∩B)

Example:

A={1,2,3,4,5,6} B={4,5,6,7,8}

Then A-B = {1,2,3} and B-A = {7,8}

which shows A-B ≠ B-A

Practice

Work out the answer to each of these questions

(a) Which numbers are in the union of A and B?

(b) Which numbers are in the intersection of A and B?

A B

2 5 3 7 9 -1 8 6 0 11 1 -3 14 19

Sets

278

The complement （여집합）of a set is the set containing all elements of

the universe which are not elements of the original set.

U = UniverseA = members of Set A

=everything in the universe that is not in Set A (sometimes shown as

).

A set is a subset （부분집합）of another when all the elements in the first set are contained in the second set A⊆B.

All sets are subsets of the universe.

All sets are subsets of themselves and the null set is also a subset of all sets. Two sets are equal if they are subsets of each other.

A proper subset （진부분집합）of a set is all subsets except the set itself A⊂B.

Any symbol with a diagonal line thru it changes it to "not." Instead of being equal to "=", it is not equal to "≠".

b∉{set of vowels} b is not an element in the set of vowels.

The power set （역집합）of a set S is the set of all subsets of S.

The cardinality of the power set S (when S is finite) is equal to , where n is the cardinality of S.

For set R = {1, 2, 3}.The subsets are { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.

Sets

279

The Power Set is the set of all subsets : {{ }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}.The proper subsets are { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}.

The cardinality of R is 3 (there are 3 elements in set R).

The cardinality of the power set is .

There are 8 members of the power set listed above.

Practice:1. The Universe U = {1,2,3,4,5,6,7} Set A = {1,3,5,7}

a. Draw a Venn Diagram showing the universe, set and the complement of the set.

b. What is the cardinality of A:

c. What is the cardinality of :

2. A = {2, 4, 5}

a. List all subsets:

b. List all proper subsets:

c. List the power set:

d: The cardinality of set A is :

3. Jinyong likes: Soccer, Basketball, Swimming, and Volleyball.

Mingi likes: Softball, Swimming, Yoga, and Soccer.

First, fill in the table below to represent the situation (put an 'O' if they

like the sport):

Then, create a Venn Diagram to represent the table:

Sport Jinyong MingiSoccerBasketball

Yoga

Swimming

Volleyball

Softball

Sets

280

4. Using the chart below, create a Venn Diagram.

5. The Universe is the set of integers between 1 and 20, inclusive. {1,2,3, ....19,20}.

Create a Venn Diagram by making two sets. The sets are odd numbers and numbers divisible by 5.

6. Consider the sets A = {1, 3, 5}, B ={3, 4, 5}, and U ={1, 2, 3, 4, 5}.

Create a Venn Diagram for this Universe

7. Shade in the area of the Venn Diagram that is represented by each choice.

a. A

b.∪

c. ∩

d. ∪∩

StudentBeen to

California

Been to New

York

Youngmin O

Yeajoon O

SeungWeon O O

Sekeun O

Sebin O

Jooyean O

Se-Yeon O O

Sets

281

e. ∩∩

f. ∩

g. ∩∪

Exercises:1. Out of forty students, 14 are taking English Composition and 29 are

taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class?

2. In a class of students every student participates in the soccer team or the debate team. 10 students only participate in the debate team, 31 students only participate in the soccer team and 12 participate in both teams. How many students are there in the class?

3. A survey of 60 people was taken and the following results were seen:12 customers drank only tea and coffee6 customers drank only juice29 customers drank tea2 customers drank only tea and juice10 customers drank tea, coffee and juice33 customers drank coffee1 customer drank only juice and coffee

A. How many customers drank juice?B. How many customers did not drink juice, tea or coffee?C. How many customers drank only coffee?D. How many customers drank only tea?

Sets

282

4. CSIA is taking three music groups to a festival.30 students sing in the choir

40 students play in the orchestra50 students march in the band15 students belong to the choir and the orchestra10 students belong to the orchestra and the band14 students belong to the choir and the band5 students belong to all three groupsHow many students will need seats on the bus?

5. Mr. Joselson gave his algebra students three optional problems (A,B and C) for extra credit and calculated:

54% of the students did problem A 45% did B

36% did C12% did A and B17% did B & C20% did a & C5% did all three

What percent of students did not turn in any of the problems?

6. Out of forty students, 16 are taking English Composition and 27 are taking Chemistry. If eight students are in both classes, how many students are in neither class? How many are in either class?

7. In a class of students every student participates in the winter festival or the fall festival. 10 students only participate in the fall festival, 18 students only participate in the winter festival and 12 participate in both festivals. How many students are there in the class?

Sets

283

8. A survey of 100 people was taken and the following results were seen:22 customers drank only tea and coffee8 customers drank only juice49 customers drank tea5 customers drank only tea and juice20 customers drank tea, coffee and juice53 customers drank coffee4 customer drank only juice and coffee

How many customers drank juice?How many customers did not drink juice, tea or coffee?How many customers drank only coffee?How many customers drank only tea?

9. CSIA is taking three music groups to a festival.30 students sing in the choir30 students play in the orchestra20 students march in the band15 students belong to the choir and the orchestra10 students belong to the orchestra and the band14 students belong to the choir and the band5 students belong to all three groupsHow many students will need seats on the bus?

10. Mr. Joselson gave his algebra students three optional problems (A ,B and C) for extra credit and calculated:44% of the students did problem A25% did B 39% did C16% did A and B 19% did B & C25% did A & C 15% did all three

What percent of students did not turn in any of the problems?

Sets

284

11. Draw a Universe containing the sets.U= {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}A= {1,3,5,7,8.9.11.13}B={2,4,5,6,8,10,13,15}C={1,2,3,4,5,10,14}

12. Determine the cardinality of the set. {1,2,3,5,6}

13. Determine the cardinality of the set. {-5,-4,-3,-2,0}

14. Write the power set. {1,2,5,6}

15. Write the power set. {1,6}

algebra 1 ms 1 cheongshim international academy 2016 ... · welcome to mr. joselson’s algebra...

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