ALGEBRA 1ALGEBRA 1

PREMIER CURRICULUM SERIESPREMIER CURRICULUM SERIESPREMIER CURRICULUM SERIESBased on the Sunshine State Standards for Secondary Education,

established by the State of Florida, Department of Education

Author: Bernice Stephens-AlleyneCopyright 2009

Revision Date:12/2009

Author: Bernice Stephens-AlleyneCopyright 2009

Revision Date:12/2009

I N S T R U C T I O N S

Welcome to your Continental Academy course. As you read through the text book you will see that it is made up of the individual lessons listed in the Course Outline. Each lesson is divided into various sub-topics. As you read through the material you will see certain important sentences and phrases that are highlighted in yellow (printing black & white appears as grey highlight.) Bold, blue print is used to emphasize topics such as names or historical events (it appears Bold when printed in black and white.) Important Information in tables and charts is highlighted for emphasis. At the end of each lesson are practice questions with answers. You will progress through this course one lesson at a time, at your own pace. First, study the lesson thoroughly. (You can print the entire text book or one lesson at a time to assist you in the study process.) Then, complete the lesson reviews printed at the end of the lesson and carefully check your answers. When you are ready, complete the 10-question lesson assignment at the www.ContinentalAcademy.net web site. (Remember, when you begin a lesson assignment, you may skip a question, but you must complete the 10 question lesson assignment in its entirety.) You will find notes online entitled “Things to Remember”, in the Textbook/Supplement portal which can be printed for your convenience. All lesson assignments are open-book. Continue working on the lessons at your own pace until you have finished all lesson assignments for this course. When you have completed and passed all lesson assignments for this course, complete the End of Course Examination on-line. Once you pass this exam, the average of your grades for all your lesson assignments for this course will determine your final course grade. If you need help understanding any part of the lesson, practice questions, or this procedure: Click on the “Send a Message to the Guidance Department” link at the top of the

right side of the home page

Type your question in the field provided

Then, click on the “Send” button

You will receive a response within ONE BUSINESS DAY

ALGEBRA 1

3

ALGEBRA 1

4

About the Author…

Bernice Stephens-Alleyne is a Trinidadian immigrant who is math-certified in Maryland and Florida. She started teaching school at the age of seventeen and taught within other professions as well. Ms Stephens-Alleyne received her initial teacher training in Trinidad, West Indies, at the then state-of-the-art Mausica Teachers College. Bernice continued her education at Howard University, American University, and Trinity University in Washington DC. She has taught every grade level including adults and college levels and considers her richest teaching moments at Trinity University in Washington DC. The most recent teaching experiences include teaching at-risk youth at an Alternative High School in Maryland and teaching teachers how to teach math at Trinity University. Other professional experiences have been as a business woman, an economist, union activist, and a television appearance on NBC Nightline with Ted Koppel. Ms Stephens-Alleyne comes from a large family whose members straddle many professions. She has two adult offspring and four grandchildren at the time of this writing. Bernice enjoys reading and learning about health issues and education theory and practice. Her hobbies include nature walks, jazz music, ballroom dancing, and clothing design. Creating unusual curricula that enhance the learning process is one of her many education projects; others include math art competitions, and actively campaigning against standardized testing. Ms Stephens-Alleyne enjoys living in the Miami, Florida area.

Mathematics: Algebra 1 by Bernice Stephens-Alleyne

Copyright 2005 Home School of America, Inc.

ALL RIGHTS RESERVED

For the Continental Academy Premiere Curriculum Series Course: 1200310

Published by

Continental Academy 3241 Executive Way Miramar, FL 33025

ALGEBRA 1

5

ALGEBRA I

TABLE OF CONTENTS

PAGE NUMBERS LESSON 1: PRETEST AND REVIEW ............................................................... 7 Lesson 1-A Pre-Test of Basics for Algebra Preparation .................................... Lesson 1-B Review of Major Algebra Concepts ................................................ LESSON 2: THE ALGEBRA-GEOMETRY CONNECTION ...............................15 Lesson 2-A Discover Formulas ......................................................................... Lesson 2-B Apply Formulas to Real-World Problems........................................ Lesson 2-C Mini Project ..................................................................................... Lesson 2-D Congruency and Similarity.............................................................. Lesson 2-E Standard and Metric Measurements............................................... LESSON 3: FOUNDATIONS OF GEOMETRY ..................................................39 Lesson 3-A Perpendicularity and Parallelism..................................................... Lesson 3-B Two-Column Proofs ........................................................................ Lesson 3-C Slope, Distance Formulas, and Mid-Point.......................................

LESSON 4: ALGEBRAIC THINKING ................................................................61 Lesson 4-A The Language of Algebra ............................................................... Lesson 4-B Interpreting Data from Graphs and Charts...................................... Lesson 4-C Simplify Polynomials....................................................................... Lesson 4-D Solve Problems Using Two First-Degree Equations ...................... Lesson 4-E Lesson review ................................................................................

LESSON 5: DATA ANALYSIS AND PROBABILITY.........................................83 Lesson 5-A Managing Information..................................................................... Lesson 5-B Probability....................................................................................... Lesson 5-C Lesson review and self-check......................................................... COURSE OBJECTIVES 99 EDITOR’S REMINDER: WITHIN IN EACH OF THE LESSONS THERE APPEARS SECTIONS OF QUESTIONS AND REVIEW FOR THE STUDENT TO COMPLETE. BY COMPLETING THESE VARIOUS EXERCISES, THE STUDENT WILL BE BETTER PREPARED FOR THE FINAL CLOSED BOOK COURSE EXAMINATION AND EACH OF THE 10 QUESTION LESSON ASSIGNMENTS.

ALGEBRA 1

6

ALGEBRA 1

7

LESSON 1

LESSON 1-A: PRE-TEST AND REVIEW OF BASICS FOR ALGEBRA PREPARATION SECTION 1: MATH VOCABULARY Write 2 words that can replace each word in the list

Word New Word New Word 1. add 2. subtract 3. multiply 4. divide

Circle the word or phrase that best matches the one in bold type 5. Terminates: ends, begins, difference, repeating 6. Evaluate: find the difference, find the solution, find the value of, simplify 7. Variable: any number, any letter, any answer, a letter that replaces a number 8. Divisor: a number you divide by, the number you are dividing, the answer to a division problem, fraction bar Write a definition for each word 9. Prime number: _______________________________________________________ 10. Factor: ____________________________________________________________ 11. Multiple: __________________________________________________________ 12. Numerator: ________________________________________________________ SECTION 2: PATTERNS AND SEQUENCES Choose the next number in each sequence: 1. 4, 6, 8, 10, ______________________ 2. 2, 3, 5, 7, 11, 13, _________________ 3. 7, 14, 21, 28, ____________________ 4. 1, 4, 9, 16, ______________________ 5. 5, 3, 1, -1, -3, ____________________ 6. 1, 1, 2, 3, 5, 8, 13, ________________ 7. 1, ½, ¼, 1/8, _____________________ 8. 1z, 2y, 3x, ______________________ 9. 1000, 5000, 250, _________________

ALGEBRA 1

8

10. 1.5, 3, 4.5, 6, ___________________ SECTION 3: TYPES OF NUMBERS and HODGE PODGE TYPES OF NUMBERS Give an example(s) of the number indicated: 1. Natural number: ______________________________________ 2. Integer: _____________________________________________ 3. Rational number: _____________________________________ 4. Irrational number: _____________________________________ 5. Repeating decimal: ____________________________________ 6. Terminating decimal: __________________________________ 7. Square number: _______________________________________ 8. Triangular number: ____________________________________ 9. Consecutive integers (at least 3): _________________________ 10. Consecutive odd integers (at least 3): _____________________ HODGE PODGE Select the best answer: 11. Fractions, decimals, and integers are examples of (a) whole numbers (b) rational numbers (c) natural numbers (d) none of the above 12. 100 is (a) odd (b) square (c) a fraction (d) a decimal 13. The absolute value of a number is always (a) positive (b) negative (c) zero (d) a fraction 14. Some integers are also (a) natural numbers (b) percents (c) fractions (d) decimals 15. Multiplying by a fraction (a) decreases the number (c) makes the answer a fraction (b) makes the number zero (d) increases the number 16. If ½ x = 10, then x = (a) 5 (b) 20 (c) .5 (d) 50 17. If the sum of the digits of a number is divisible by 3, that number is divisible by (a) 6 (b) 2 (c) 3 (d) 4 18. All multiples of 5 end in (a) 5 and 0 (b) 2 and 3 (c) 2 and 0 (d) 2 and 5

ALGEBRA 1

9

19. √16 = (a) 8 (b) 4 (c) 2 (d) 16 20. a/b is the correct way to write (a) the quotient of a and b (c) the difference between a and b (b) the product of a and b (d) the sum of a and b SECTION 4: ORGANIZING INFORMATION and PROBLEM SOLVING ORGANIZING INFORMATION Choose the best answer: 1. You are asked to find the mean, the median, and mode of this set of numbers: 12, 5, 15, 0, 3, 18, 37, 12, and 17. The best way to start is to (a) put the numbers in a table (c) put the numbers in order from least to greatest (b) make a list (d) put the numbers in order from greatest to least 2. Another name for the mean is (a) average (b) middle (c) end (d) sum 3. You can find the mean by (a) adding the numbers (c) multiplying the numbers (b) dividing the numbers (d) adding the numbers and then dividing by the number of numbers 4. The numbers in question 1 represent temperature in Fahrenheit for a city during winter. The best graph to draw would be (a) straight line (b) bar graph (c) circle graph (d) scatter plot 5. Study the numbers in the data. One can predict the temperature for the next day by finding (a) median (b) mode (c) mean (d) all of the above. PROBLEM SOLVING Single response 1. Draw 2 different rectangles to show combinations of the factors of 40. 2. A runner practiced his run everyday doing 20 laps around a field 18 feet long and 37 feet wide. What distance would he run after 5 days? 3. If this runner’s specialty was 440 yards, about how many times would he have to run around this field to practice for the 440? 4.New York is about 1,000 miles away from Florida. Express this distance in scientific notation.

ALGEBRA 1

10

5. Seven is left when 13 is subtracted from x. What is the value of x? 6. What is another name for the additive inverse? 7. The sum of two consecutive numbers is 15. What are the numbers? 8. What would be the cost of 4 ink cartridges at $40 each and 2 reams of paper at $7.99 each. Taxes are 7% (7 cents on each dollar). 9. What is 35% of $700? 10. If 60% of a number is 144, what is the number? SECTION 5: GEOMETRY Choose the best answer 1. These figures are (a) congruent (c) 3-dimensional (b) similar (d) none of the above 2. A cube is made up of (a) 4 faces (b) 6 faces (c) 2 faces (d) 0 faces 3. A tessellation can have (a) rotations (b) reflections (c) symmetry (d) all of the above 4. The angles of a triangle add up to (a) 360º (b) 90º (c) 100º (d) 180º 5. One of these sets is not congruent: (a) (b) (c) (d) 6. The angles of any quadrilateral measure 360º. If 3 of the angles in a 4-sided figure measure 50º, 110º, and 125º, what is the measure of the 4th angle? 7. This set is a dilation because one is (a) larger than the other (c) they are congruent (b) they are not congruent (d) they are polygons 8. The magic number can be obtained by (a) subtracting across (c) multiplying vertically (b) dividing down (d) adding vertically, horizontally, or diagonally

ALGEBRA 1

11

9. The Pythagorean Theorem works only with (a) triangles (b) acute angles (c) equilateral triangles (d) right triangles 10. The longest leg of a right triangle is called the (a) base (b) height (c) hypotenuse (d) slope

LESSON 1-B - REVIEW OF MAJOR CONCEPTS Number sense

1. A prime number has two factors, 1 and itself. 2. A composite number has more than 2 factors. 3. There are 25 primes in the first 100 counting numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23,

29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 4. Two is the only even prime. All other primes are odd.

The factors of 2 are 2 and 1; the factors of 7 are 7 and 1. They are primes and have only 2 factors.

5. A factor is a number that divides into another. 2 divides into 10, so 2 is a factor of 10. Other factors of 10 are 1, 5, and 10. 10 is therefore composite.

6. A multiple is many times another number. 6 is a multiple of 3. 7. One way to find the factors of a number is to know the rules of divisibility.

a) 2 will divide into any even number and all numbers ending in 2, 4, 6, 8, and 0 are even

b) If the sum of the digits of a number can be divided by 3, then that number can also be divided by 3. 234 can be divided by 3 because 2 + 3 + 4 = 9 and 9 ÷ 9 = 1

c) If a number can be divided by 2 and 3, it can be divided by 6 d) If the sum of the digits of a number can be divided by 9, then that number can

be divided by 9. 846 can be divided by 9 because 8 + 4 + 6 = 18 and 18 ÷ 9 = 2 e) A number ending in 5 or 0 can be divided by 5 f) A number ending in 0 is a multiple of 10 and a multiple of 2 and is therefore

divisible by 2 and 10. g) If the last 2 digits of a number can be divided by 4, that number can be divided

by 4. 932 ÷ 4 = 233 9. Decimals are either terminating or repeating. Repeating decimals have a bar drawn over the digits that repeat. 10. Natural numbers are 1, 2, 3, 4, etc. 11. Whole numbers are natural numbers and zero. 12. Integers are all whole positive and negative numbers and zero. 13. An irrational number is a number with a decimal that never ends and never repeats. Example √2 = 1.4142135… 14. A perfect square is the product of an integer times itself. Example 3 x 3 = 9 15. The top number in a fraction is the numerator; the bottom number in a fraction is the denominator. 16. The fraction bar means to divide. If we divide a smaller number by a larger one the answer is a fraction. 17. Percents, fractions, decimals, and ratios are different ways of expressing parts of a whole. They are fractional concepts. 18. Probability is a fractional way to represent the chances of success.

ALGEBRA 1

12

19. Probability can be expressed in any one of the 4 fractional formats. 20. Consecutive numbers are numbers in sequence. Organizing information

1. Information often comes in large groups of numbers. One can organize information by making tables, lists, graphs, or finding ways to make one number represent a large set of numbers.

2. We often use the mean, median, or mode (one number) to represent a large group of numbers.

3. The mean is the average of a set of data. We find the mean by adding all the numbers and then dividing by the number of numbers.

4. The median is the middle number in a set of numbers after arranging them in order from least to greatest (numerical order).

5. The mode is the number occurring most frequently. There can be more than one mode or no mode in a data set.

6. The mean, median, and the mode are called measures of central tendency because they describe how numbers are located in the middle of the data set.

7. The range is the difference between the largest and the smallest number in the set of data. The range is a measure of dispersion or a way to describe how far away the numbers are from the mean.

8. There are 9 different types of graphs: Line graph, bar graph, circle graph, pictograph, line plot, scatter plot, box and whisker plot, stem and leaf plot, and histogram.

9. Graphs serve different purposes and display data in special ways. Every time we use graphs or numbers to describe data we lose something from the original data.

10. Some graphical displays can be misleading. Inappropriate scales, intervals, or axes can lead to errors in the graph.

11. Many graphs are set up using an X or a Y axis. Usually the independent variable is placed on the X axis and the dependent variable on the y axis.

12. An independent variable is unaffected by other information in the graph. 13. The dependent variable will change according to the independent variable. 14. When the dependent variable is affected by the independent the graph might have

a functional relationship. 15. The vertical line test can demonstrate whether the graph is a function or not. 16. Graphs that are functions will have one and only one y value for every x value. 17. We use the coordinate plane to display information that is negative and positive. 18. The coordinate plane is divided into 4 quadrants which are numbered

counter-clockwise starting from the top right. 19. Where X and Y meet is the origin and has a zero value. 20. The X-axis is drawn horizontally and the Y-axis is drawn vertically. 21. Numbers to the right of zero on the X-axis are positive and numbers to the left of

zero on the X-axis are negative. 22. Numbers above zero on the Y-axis are positive while numbers below zero on the

Y-axis are negative. 23. Quadrant 1 has all positive values (+,+) and quadrant 3 has all negative values

(-, -).

ALGEBRA 1

13

24. In quadrant 2, the X value is negative and the Y value is positive (-, +) but in quadrant 4, the X value is positive and the Y value is negative (+, -).

25. The X value always comes first.

Measurement and basic geometry Formulas worth knowing: Perimeter: Perimeter means distance around. The simplest way to measure around an object is to add the measures on each side. We say perimeter for polygons and circumference for circles and circular objects. Perimeter is one-dimensional.

1. The perimeter of a rectangle = 2l + 2w or 2b + 2h; w = width; l = length; b = base; h = height. These are interchangeable terms.

2. The perimeter of a square = 4s because each side measures the same. Adding the same measure is the same as multiplying the same number several times.

3. The perimeter of any triangle = Sum of the sides. 4. The perimeter of an equilateral triangle = 3s (same reasoning as #2) 5. The perimeter of any regular polygon = sn where n = number of sides.

Area: Area means surface measure. Area is two-dimensional. Area is always square measure. That means one can measure the area by creating a number of squares and counting them. The area of a rectangle that is 2 units by 3 units can be drawn in square units: The rectangle is 6 square units. Units may be centimeters, feet, inches, or any other unit of measure.

6. Area of a rectangle = lw or bh 7. Area of a triangle = ½ (lw) or ½ (bh) Remember that a triangle is one half of a

rectangle. 8. Area of a trapezoid = ½ (b1

+ b2) h. Base 1 and base 2 represent the two parallel sides of a trapezoid. H is the height or the distance at right angles from the base.

Base 1 height Base 2

9. Area of any polygon = Area of a triangle times the number of triangles in that polygon = ½ (bh) n

10. The sum of the angles in a triangle = 180º 11. The sum of the angles in a quadrilateral = 360º 12. The sum of the angles in a pentagon = 540º 13. The number of triangles in any polygon = Number of side minus 2 = n – 2.

Explanation: Quadrilateral has 4 sides minus 2 = 4 – 2 = 2 triangles. 14. A right angle has 90º; an acute angle has < 90º; an obtuse angle has > 90º. 15. A right triangle has 1 angle that is 90º; an acute triangle has all 3 angles < 90º; an

obtuse triangle has 1 angle > 90º.

ALGEBRA 1

14

16. An equilateral triangle has 3 equal sides; an isosceles triangle has 2 equal sides; a scalene triangle has 0 equal sides.

17. Triangles are classified according to the measure of their angles or the measure of their sides.

18. It is not possible to have a right obtuse triangle, a scalene isosceles triangle, or a right equilateral triangle.

19. Possible triangle combinations are: Right scalene, right isosceles, acute isosceles, obtuse isosceles, obtuse scalene and obtuse isosceles.

20. A straight angle measures 180º or two right angles. Algebraic thinking Different ways to say the same thing

ADD SUBTRACT MULTIPLY DIVIDE Increase Decrease Product Share

Total Minus Times Among More Take away Twice Quotient

More than Less Thrice Divisor Sum Less than Square Dividend

Altogether Remainder Cube Into In all Left over Exponent How many times

1. A variable is a letter used in place of a number. 2. Evaluate means to find the value of. 3. Solution is the number that solves an equation. The solution replaces the variable to

make the statement true. 4. An expression contains terms but no equal sign. 5. An equation contains terms and has an equal sign. 6. We simplify an expression but solve an equation. 7. Examples of ways to convert words to algebraic expressions or equations:

a) 2b means two times b where b is an unknown number b) 2a + 5 means add 5 after multiplying 2 by a c) x2 means multiply x by x d) a minus b means to subtract b from a e) a/b means a divided by b f) The sum of x and y means to write an expression that shows the operation:

x + y g) The multiplication sign may not be used in algebra. Sometimes the parentheses

means to multiply or a dot may be used h) 2(b + h) means to add h to b, then multiply the answer by 2. The order of

operations is always in effect. SOLVE INSIDE THE PARENTHESES FIRST. i) Another way to write example h is 2 · b + 2 · h j) (x) (x) (x) = x3

ALGEBRA 1

15

LESSON 2

THE ALGEBRA-GEOMETRY CONNECTION

LESSON 2-A: DISCOVER FORMULAS

In this unit we will discover how formulas are derived as we work through simple measurement projects. We will use measurement instruments: inch tape or yardstick, meter stick, right angle stick, triangles, pencils, erasers, graph paper, plain and colored paper.

Project Activities Perimeter and Area of Rectangles

1. Select one or two rooms of your home. The rooms should be rectangular in shape.

Measure each side in inches and write down that measure. Draw the shape of the room as best you can on plain paper and write the measures on the sides as appropriate. The longer side you have drawn should have the longer measurement, the shorter side you have drawn should have the shorter measurement.

2. Change the measures to feet and then draw the shape of the room again, this time on

graph paper. Use each square to represent feet. Example: If you got 144 inches as the measure on one side of the room, change that to feet. Then use 1 square to represent 1 foot. On your graph paper your room would be 12 squares on one side.

3. Let’s pause and examine what we just did. When we measured the room in inches we had linear (line) measure. This kind of measure is one-dimensional. 144 inches = 1441

or 144 to the first power. 4. It seems that there can be a better way to express the measure of a large object.

There is! Let’s use feet. 5. What did you do to convert inches to feet? Look at your inch tape. Every 12 inches

makes 1 foot. 24 inches makes 2 feet. Dividing 144 ins by 12 = 12 feet. 6. You can build a table that helps to convert inches to feet. We will do that later 7. Find the perimeter of the room. Write what you did. Work out the answer in feet.

Multiply your feet by 12. Did you get the same answer in inches? Example: My room is 10 feet on one side and 12 feet on the other.

P (perimeter) = 10 + 12 + 10 + 12 = 44 feet = 44 x 12 = 528 inches. 8. Now work the answer in inches. 120 + 144 + 120 + 144 = 528 inches. 528 ÷ 12 = 44 feet

ALGEBRA 1

16

9. Now let’s replace the length and width of the room with variables. Length = l and width

= w. Re-write what we did without numbers. Rewrite the perimeter as a formula. P = l + w + l + w. Notice we have 2 sets of l’s and 2 sets of w’s.

In algebra that can be written P = 2l + 2w or 2(l + w). Check the unit 1 review for the formula on measurement. 10. Now measure the other room in inches and change to feet by dividing by 12. Draw on

graph paper using squares. Find the perimeter by adding the number of squares around the room.

11. Use the squares on 2 sides of your diagram to find the area of each room. Can you understand why area of a rectangle is base time height or length times width?

12. In my room the area would be 10 feet times 12 feet or 101 · 121 = 120 feet 2 (The little 2 is the power that represents square measure. It can be read as SQUARE FEET.)

Let’s practice Complete the blanks:

Length Width Perimeter Area 8 12

15 10 6.5 8 11 11 13 12

Suppose your room had the shape of a pentagon, how would you find the perimeter?

1. What does perimeter mean? 2. Framing a picture demonstrates the idea of (a) area (b) perimeter. 3. Washing windows demonstrates the idea of (a) area (b) perimeter. 4. If b = 16cm, and h = 14cm, find the perimeter and area of this figure. 5. Draw this figure on graph paper. Let 1 square equal 2 cm. How many squares would

you have on the longer side? 6. How many squares would define the area on the graph paper? 7. Check your drawing for accuracy.

Problem-solving with area and perimeter

Think Algebra

8. The perimeter of a room is 52 feet. If l = 12 what is w? 9. The area of the same room is 168 ft2. If one side is 12 ft, what is the measure of the

other side? 10. A room measures 18 feet on one side. If its area is 360 square feet, what is its

perimeter?

ALGEBRA 1

17

Project Activities Area of Squares and Triangles

1. Draw a large square on piece of paper. Measure each side in inches and write it down.

Find the perimeter. Find the area. 2. Draw a diagonal across the square so that there are 2 right triangles.

Step 1 Step 2

3. Use graph paper to draw another square, 10 units across by 10 units down. 4. Draw a diagonal across the square.

5. Count the number of squares around = 10 + 10 + 10 + 10 = 40; P = 4s (s = measure of side). 6. Multiply the squares on the height by the number of squares on the base. Area = bh =

100 sq units 7. The diagonal makes two triangles. The area of each triangle is ½ of 100 = 50 sq units

or 0½ (bh) = ½ (10 · 10). You can count the area of the triangle as well. Note that the diagonal line divides each square in half.

8. Area of a square is usually written S2. S = measure of side. The perimeter of a square is usually written P = 4s.

9. Begin to copy all the formulas into a notebook. 10. Note that base and height mean the same as length and width. In the square each

side has the same measure.

Table of Measurement – Units of Length 12 inches = 1 foot 3 feet = 1 yard 36 inches = 1 yard

ALGEBRA 1

18

1760 yards = 1 mile 5280 feet = 1 mile

Project Activities Area of a Trapezoid

Why is the area of a trapezoid the same as area of two triangles? Why is ½ (b1 + b2) h the same as ½ (b1h) + ½ (b2h)? Base 1 10cm Height h = 6 cm Base 2 5 cm A B Area of trapezoid = ½ (b2 + b2)h = ½ (10 + 5) 6 = ½ (15) 6 = 45 cm2 Area of triangles = ½ (b1h) + ½ b2h) = ½ (10 · 6) + ½ (5 · 6) = ½ (60) + ½ (30) = 30 + 15 = 45 cm2

Prove Area of a Trapezoid

1. Draw an isosceles trapezoid on graph paper. Follow the directions: Base 1 Height Base 2 Step 1 Step 2 Base 1 = 6 units (count them) Base 2 = 4 units Height = 4 units 2. Copy both formulas from above or to a piece of paper. 3. Substitute the numbers for the variables and work as in the example. 4. Draw another isosceles trapezoid on graph paper. This time cut it out. 5. Draw a diagonal across so that two triangles are created. One triangle would be acute; name it A. The other would be obtuse; name it B. 6. Write down the measures of each triangle. Make sure you place the height in the appropriate location. 7. Substitute the numbers for the variables and calculate the area of the two triangles as in the example. 8. Practice until you can do this without help.

ALGEBRA 1

19

The Circle Parts of a Circle

Linear measure or the measure of a line refers to the length of the diameter, radius, circle, or any part of the circle (arc). The measure of any part can be used to find the measure of any other part.

Formulas Radius

equals ½ the diameter

Diameter is 2 times the

radius

Circumference is diameter

times π

Area is π times radius times radius

R = ½ d

D = 2r

C = dπ or 2rπ

Area of a

circle = πr2

If R = 7 cm, find D, C, and area of the circle. R = 7cm, therefore D = 14 cm. C = 14 · 3.14 = 43.96cm Area of the circle = 3.14 (7) (7) = 153.86 cm2 Sometimes 22/7 is used as pi. This fraction is a good substitute for 3.14 when the diameter (d) or radius (r) is a multiple of 7. The answer in such situations is a whole number, not a decimal. Look at the example again. Use 22/7 for pi. R = 7cm, therefore d = 14 cm. C = 14 · 22 = 308 ÷ 7 = 44cm 7 Area of the circle = 22 · 7 · 7 = 1078 ÷ 7 = 154 cm2 7 Compare answers when pi = 3.14 or pi = 22/7 Students may be asked to use the best value for pi to make the answer a whole number.

The distance around the circle is the circumference. The center is the spot in the middle, equidistant from all points on the circumference.

The distance across the circle through the center is the diameter. The radius is one half of the diameter or a line that connects the center to any part of the circle. A chord is a line that goes through the circle and touches the circle in two places. The diameter is a chord but a chord may not be the diameter. The measure of the angles inside a circle totals 360º or 4 right angles. Angles are formed at the center of the circle.

ALGEBRA 1

20

Let’s Practice The Trapezoid and the Circle

1. Two triangles were pasted together in a craft project to create a trapezoid. They both

had a height of 8cm. One had a base of 15cm and the other a base of 12 cm. (a) What was the area of each triangle? (b) What was the area of the trapezoid? 2. The height of the right triangle is 4 feet and the base of the

4 ft right triangle is 3 feet. The rectangle is 5 feet wide at the top What is the measure of the longer base? 3. What is the distance around this trapezoid if the slant side is 7 feet? Draw and place

the measures in the right places. 4. Complete each blank space:

Radius Diameter Circumference Area 10cm 7 feet

21 inches 5. If the area of a square was 36 square feet, what was the measure of each side? 6. If the diameter of a circular athletic field was 150 yards, what was the radius?

7. Choose the best value for pi and find the most exact measurement of the

circumference of a circle with a radius of 28 feet.

8. What is the difference between a chord and the diameter of a circle?

9. A dog was on a leash anchored to the base of a tree in the park. If the leash was 7 feet long, what is the largest area in which the dog could play?

10. Draw your answer for #9.

ALGEBRA 1

21

Volume and 3-dimensional measurements

Volume measures capacity or the space inside an object. The shape of the object determines the formula or method. We must multiply 3 dimensions (length, width, and height) to find volume and all answers must have the power of 3 after it (10cm3). Therefore, volume is cubic measure. (Compare this to area, which is square measure.) Recall: Linear measure is 1-dimensional; square measure is 2-dimensional; volume is 3-dimensional.

Activity 1. Get a ream of paper and measure the top. Write down those measures. 2. Measure the height or how tall it is. Write that down. Find the volume of the ream by

multiplying the length by the width by the height. V = l · w · h 3. Sometimes depth may take the place of height but the concept is the same. 4. Open the ream. Take out 1 sheet of paper. Find the length and the width. Find area.

When you multiply the area by height you also find volume. V = A (h) 5. Draw a ream of paper to show the length, width, and height.

Width

Length

Measure the ream Measure 1 sheet

Lesson 2-A wrap up

1. Go back to page 16, # 9. Continue to list formulas discovered in this lesson. 2. Identify those you know well and those that are new. 3. Be able to draw an example for each different formula; practice substituting numbers for

the variables and calculate the answers. There must be a unit of measure for each answer: cm, inches, feet, square feet, square yards, and so on.

4. Short or abbreviated version for: feet = ft; inches = ins; yards = yds; mile = mi; centimeter = cm; meter = m; square feet = sq ft or ft2; cm squared = cm2; square inches = ins2 or sq ins.

5. How would you find length if area and width are given? Would the answer be 2-dimensional or 1-dimensional?

6. How would you find the height if volume and area are given? Would the answer be 1-dimensional, 2-dimensional, or 3-dimensional?

7. How would you find the area if volume and height are given? Would the answer be 1-dimensional, 2-dimensional, or 3-dimensional?

8. Linear measure is (a) 1-dimensional (b) 2-dimensional (c) 3-dimensional 9. Surface measure is (a) 1-dimensional (b) 2-dimensional (c) 3-dimensional 10. Cubic measure is (a) 1-dimensional (b) 2-dimensional (c) 3-dimensional

ALGEBRA 1

22

LESSON 2-B APPLY FORMULAS TO REAL-WORLD PROBLEMS

In this lesson we will convert real-world problems into algebraic equations and solve. Problems of geometric measurement, motion/travel, time/distance, simple interest (money), and profit and loss will be experienced. The variety of real-world issues that are represented in algebra is so large that we could not do them all in this book. It is hoped that our experiences will create enough exposure to enable individual students to make the real-world connections as they go about their daily lives.

Real-world Geometric Measurement

A B C D E A is the shape of a house, B is a child’s play block, C is an empty can, D is the shape of doll’s house, and E is the shape of a playing field.

1. Which of these has area but no volume? 2. Which has volume? 3. How can you find area of those with 3-dimensions? 4. Develop a strategy before getting into the problems.

What is the size of the living space in this house? When you buy a home this is one of the answers you must have in order to compare different homes with their cost and make effective choices.

• Living space = area; area = lw • The house is made up of two sections, a larger and a smaller. • Use graph paper to draw an idea of the house, place your own real life

measures, find the area of each piece and add them together. • Imagine the smaller part of the house to be a family room and make that 14 feet

by 18 feet. On graph paper that could be 7 squares by 9 squares. Each square = 2 feet, so the area of that part of the house would be 14 ft. x 18 ft.= 252 square feet (252 ft2).

• To get real-world measures for the rest of the house, walk around your own home and estimate the number of feet on one side and the number of feet on another side. Use numbers that are easy to work with.

• Go for it!

Child’s Play This is a child’s toy. You are a graphic artist’s assistant and must find the smallest piece of decorative paper to cover several of these blocks.

ALGEBRA 1

23

• What concept are you working with, area or perimeter? • What measures should you find to solve this problem? • Which of the formulas would you use? • Solve.

You are working with area and must find the surface measure of each face. Recall that a cube has 6 faces. Measure across the top on 2 consecutive sides, and then down the height or depth. Complete each set of measurements. 4 ins Measurements of the top = 3 ins Measurements of the side = Measurements of the front = 5 ins Find the area of each face, and then multiply by 2.

1. Write down your calculations. 2. Why multiply by 2? 3. Draw an example of the block as surface measure. You would be able to cut and fold

this example to make the block. 4. What is the volume of this block?

Work at the Canning Factory

20 cm 8 cm On your first day at the cannery you sit at an assembly table and are given a can, a rectangular label, and 2 circles. You must paste the label on to the can. Next, you must place the circular lids on the top and bottom. Later you must calculate the surface area and the volume so that the next group of workers must type in these measures for the last phase of the job. How can you use these measures to find area of the surfaces and volume of this can? Do you have all the measures you need? If not what’s missing?

Clues you can use

1. Find the area of the rectangle using the best formula from your list. 2. Study the formula for area of a circle. Which is missing, r or d? 3. Find the area of 1 circle, and then multiply by 2. 4. Volume may be tricky. Remember area times height = Volume. 5. Which area would you use for volume, area of the rectangle or area of the circle? 6. What would be the height?

ALGEBRA 1

24

The Track Meet

When you run around the track you are actually running around a field that is just like this one. What 2 shapes make up this field? (a) square and circle (b) rectangle and circle (c) circle and rectangle (d) circle and triangle The answer is (b) rectangle and circle. Let’s find perimeter and area of this multi-shaped figure. You will meet many like this in real-life. 100 meters Find the area of the rectangle. Find the area of the Circle (2 half circles make 1 circle). Done! That 7 75 meters was easy. Perimeter is trickier! You want only the circumference plus the measure of 2 sides of the rectangle. C = dπ = 75 (3.14) = 235.5m. The distance of the two sides: + 100m + 100m = 200m Now add the circumference of the circle 235.5m and the two sides 200m to obtain the perimeter of the track. Perimeter is equal to 235.5m + 200m or 435.5m. Area of rectangle = 100m x 75m = 7500m2; Area of the circle = πr2 = 3.14 (37.5)2 = 4415.6m2 . The area of the track is 7500m2 + 4415.6m2 = 11915.6 m2

Motion/Travel or Time/Distance Measurements

60 45 30 15 0 Consider traveling on a number line with 15 minute intervals. Assume that the rate is 60 miles an hour. How many miles would he travel in (a) 30 minutes (b) 45 minutes? This cyclist can travel a mile a minute or 60 miles in 60 minutes. He will therefore ride 30 miles in 30 minutes or 45 miles in 45 minutes. A race car driver can drive at the rate of 120 miles an hour. How far will he get in (a) 3 hours (b) 1 ½ hours (c) 10 minutes. 1 hour = 120 ml; 3 hours = 120 (3) = 360 mi 1 ½ hrs = 120 + 60 = 180 mi 10 min = 10 (120) or fractional part of an hour times 120 60 1200 ÷ 60 = 20 mi Formulas: d = rate · time or d = rt R = d ÷ t t = d ÷ r

ALGEBRA 1

25

Interest on Money

Banks and financial institutions charge money for the use of money. People make money from investments. That money is called Interest. If someone wanted to borrow $100 from a bank, they would be charged a small sum on every dollar for the use of the money borrowed. Interest is calculated yearly. If the current interest rate on consumer loans is 5%, then a person pays 5 cents for every dollar borrowed. A person repays the money borrowed plus the interest. Money borrowed is called the Principal. Money borrowed plus interest is called Amount. We have 3 parts to these problems: Principal, Interest, and Amount. I = P · R · T A = P + I I = A – P P = A – I A man borrowed a small loan of $500 for 1 year. The bank charged him 5%. What is the interest for the use of this money and what will be the total amount repaid? I = P · R · T = 500 · .05 · I = $25; A = P + I = $500 + $25 = $525; A = $525 A bank statement showed the following figures on a new loan: Principal = $10,000; Rate = 4%; Time = 3 years Calculate the interest and amount that would be repaid at the end of the loan. If the borrower wanted to repay in equal installments, what would he have to pay the bank at the end of every month? Round your answer if necessary. I = P · R · T A = P + I I = 10,000 x .04 x 3 = $1200; I = $1200; A = $10,000 + $1200 = $11,200; A = $11,200 Divide $11,200 by 36 months = $311.11111 rounded to $311.11 After 18 months the bank statement read: Amount paid to date = $5,599.98. Calculate the interest and the principal paid up to that point. Interest for 36 months = $1200; interest for 1 month = 1200 ÷ 36 = $33.33; interest for 18 months = $33.33 x 18 = $599.94 P = A – I = $5,599.98 - $599.94 = $5,000.04

Profit and Loss

Many students think of going into business. They should be aware of how their profits and losses are calculated. Do they make a profit? Here is a real case. Carl (not his real name) sold informational cd’s at $5.00 each. He said he paid $50 for copying the cd’s. When asked “how many have you sold”? he said so far about 25. Did he make a profit or a loss with his business? Could this be a profitable business?

ALGEBRA 1

26

What is profit? Profit is the excess between the money you make or earn and the money you spend. You must make more than you spend to make a profit; if not you experience a loss. The formula is P = Earning – Spending. If the answer is positive you make a profit; if the answer is negative, you make a loss. In business terms P = Revenue – Expenditure or P = R - E

The CD Business Revenue Expenditure Profit/Loss 10 cd’s = $50.00 $50.00 $50 − $50 = 0 10 cd’s = $50 00.00 $50 − 0 = $50 5 cd’s = $25 00.00 $25 − 0 = $25 Total Revenue = $125 Total expenditure = $50 Total profit = $125 - $50 = $75 What is the percent profit after 25 cd’s? Would he continue to make a profit forever or is there something that would prevent him from making a profit? Profit % = Total Profit x 100 = 75 x 100 = 7500 = 60% Total revenue 125 125 He made 60% profit on 25 cd’s.

Review of Lesson 2-B

1. Which of these has area but no volume? A B C D E

2. How can you tell if an object has volume?

3. What is the best way to find area of the shape below?

ALGEBRA 1

27

4. Name the different shapes that make up the arrow? What is the area formula for each of

these?

5. List the steps you would take to find the area of a cube.

6. The figure at E has 3 different types of faces. To find the surface area: (a) Multiply length, width, and height (b) Multiply length and width but add height (c) Find area of the 3 different faces, add them and multiply by 2 (d) None of the above

7. Draw a cylinder (can) and place the measures in the appropriate places: height = 10 cm; radius = 4 cm

8. Find the area of this cylinder using the measures in # 7. 9. Volume of the can : (a) V = A · H b) L · W (c) A · V (d) V ÷ H

10. Find the area of this athletic track with a diameter of 70

meters. The rectangle inside the track is 98 meters on the outside.

11. A family wanted to get to another town by 5:00 pm. They had to travel 105 miles. The speed limit was 70 miles an hour. At what time should they leave home? Assume that they would be driving at the speed limit and there would be no traffic jams.

12. A bank lent $15,000 for 3 years at 5%. What was the interest on the loan and how much

would the borrower repay?

13. What would be the monthly installment on the loan?

14. A young family bought a refrigerator for $3500 with a credit card. The credit card company charges 20% per year for use of the card. Identify

(a) The principal and (b) The interest rate. What is the total repayment for this transaction after 1 year?

15. Two young men operated a lemonade stand during 2 months in the summer. Their expenditures were $100 each month and their revenues were $75 one month and $105 the next. Did they make a profit?

16. Calculate their profit or loss in a table like the one in the lesson.

17. Add a column for the percent profit or loss incurred during each month.

ALGEBRA 1

28

LESSON 2-C Mini Project

APPLE CONSTRUCTION COMPANY placed a bid to lay kitchen tiles for a housing project with 25 homes. The bank would provide the loan to you, the owner, as long as you could show that you will make a profit that is greater than 10%. Develop a chart or table to show whether you can earn a profit greater than 10%. Use the information to develop your proposal. Use the format like the one in lesson 2-B. 3 feet 1 15 feet 15 feet

Information

• The small space shaded will have a counter and the rest of the kitchen will have tiles that are 1 ft by 1 ft square.

• There will be 25 homes to tile. • 1 box of tiles contains 200 tiles and costs $40. • Loose tiles are $.60 each. • The entire kitchen is 15 feet by 15 feet • The counter space is 3 feet deep. • There will be no tiles under the counter. • The Housing Project will pay you $3000 for the project. • The bank will lend you $1,000 @ 5% for 6 months if you can show a profit greater than

10%.

Clues you can use

1. Collect all the information you need: Area of kitchen floor, number of tiles needed, and cost of tiles.

2. Compare cost per box of tiles and the cost of single tiles; which is cheaper? 3. Calculate all the expenditures (tiles and loan); calculate all revenues. 4. Profit in $ = Revenue minus Expenditure; profit in percent = Difference/revenue x 100. 5. Compare percents and decide. 6. Draw a sample kitchen to see the result of the completed project.

ALGEBRA 1

29

Assessment

1. Which geometric measurement formulas did you apply? 2. List the math operations you used to determine the number of tiles needed. 3. List any other formulas you used to complete the project. 4. Did the project make a profit? Was the profit greater than 10%? 5. Can this project help you in real life even if you are NOT going to have your own

business? Write any other statements of your own that you observed as you did the project.

Answers to Assessment

1. Area of a rectangle (kitchen floor). 2. Multiply for the area of a rectangle (12 x 15 = 180 sq ft) and area of each tile (1 x 1 = 1 sq

ft). Divide the area of each tile into the area of the kitchen floor to be tiled (180 tiles per home). Multiply by the number of homes (180 x 25 = 4500 tiles). Divide the number of tiles in each box (200) into the number of tiles needed for the entire job (4500) to determine the number of boxes needed (22.5 boxes). You must buy 23 boxes because you cannot get half of a box. This is actually rounding in the real world. Multiply the cost of each box of tiles by the number of boxes $40 x 23 = $920.

3. Other formulas used: Simple interest, profit, profit percent. 4. His expenditures were: Bank repayment ($1,025) + tiles ($920) = $1,945. His revenue

was $3,000; his profit was $3,000 − $1,945 = $1,055. Profit % = 1055/3000 x 100 = 35.16 or 35% (rounded). His profit was greater than 10%.

5. It was cheaper to buy by the box rather than to buy each tile singly. 23 boxes cost $920; buying singly would cost 4500 x .60 = $2,700. The business owner saved a lot of money. He would get the bank loan. This project would help in real life because it provides an opportunity to see how banks make money from loans and how the interest rate affects the way individuals make money. Having your own business is a big responsibility.

LESSON 2-D: CONGRUENCY AND SIMILARITY

Congruency refers to objects that are perfectly matched in size and shape. In geometry we assess whether polygons are congruent and demonstrate ways to prove it. In algebra we will define congruency more broadly to include all objects that are equal in all respects and will look at congruency from a measurement perspective. The sign for congruency is ≡ and it means identical to. Similarity refers to objects that have the same shape but different size. Similar objects are dilations, that is, they are either smaller or larger than each other and can mathematically be expressed as a ratio. The ratio in size remains constant throughout the shape. Example if one triangle is one half the size of another, then each side is half the corresponding side of the other. We meet the idea of the corresponding side that will also be in our unit on Foundations of Geometry. The symbol for similarity is: ~. Similar in mathematics means two figures that are the same shape but not necessarily the same size and one is a scaled down copy of the other.

ALGEBRA 1

30

Examples of congruency: Examples of similarity: These objects are (a) similar (b) congruent (c) neither A B C Answers: A = neither; B = congruent (B is rotated); C = neither We will draw often so let’s prepare by getting the tools we need: Graph paper, ruler or straight edge, eraser, pencil, colored pencils or highlighters. Colored pencils or highlighters help students make a connection and aids recall. Use them often. Sample graph paper is on the next page; make as many copies as you need or purchase graph paper from the local office supply store or discount store that sells office supplies.

ALGEBRA 1

31

GRAPH PAPER

ALGEBRA 1

32

Activities

Use graphs where necessary

1. Draw a square 4 units across and 4 units down. Draw another that is congruent to it. Name your square ABCD as in the diagram. Name the congruent square EFGH. Write ABCDEFGH

A B E F C D G H

2. Draw another square, 6 units by 6 units. Draw another that is half the size. Name square 1 QRST; name the similar one WXYZ. Remember the measure of each side should be half the size of the original square.

3. Draw a right triangle that measures 3 units at the base and 4 units at the height. Create the hypotenuse by connecting the end points of the sides. Name your triangle LMN. Draw another triangle that is twice the size of the first. Name it DOT. What are the measures of each side?

4. OPEN-ENDED: Draw any polygon except a square, rectangle, or a triangle. Measure each side. Draw two more, one that is congruent and one that is similar. Name each using letters of the alphabet. You should have 3 polygons altogether.

5. Write statements that identify one congruent pair and one similar pair.

Identifying Congruency and Similarity

By finding Area and Perimeter

Using the information in the table below: 6. Find the perimeter of the square QRST. Find the area of the same square. 7. Find the perimeter and area of the square that is one half the size (WXYZ). Record

these in the table below.

Figure Measures Perimeter Area Square QRST 6 units by 6 units 24 units 36 square units Square WXYZ 3 units by 3 units 12 units 9 square units Ratios of similar polygons

2 to 1 2 to 1 4 to 1

Congruent polygons have same shape, size, area, and perimeter. Observe the ratio of perimeters of the larger to the smaller is 2 to 1, also written 2:1 Observe the ratio of areas of the larger to the smaller is 4 to 1, also written 4:1. The perimeter of a polygon that is dilated keeps the same ratio as the enlargement or reduction. If a polygon is made 2 times larger, the perimeter will be 2 times the original. If it is made 2 times smaller, the perimeter will be 2 times smaller that the original. Will this work with all polygons?

ALGEBRA 1

33

Let’s work with a triangle. 5 units 4 units 4 units 3 units 3 units A B 8 units C Triangles A and B are congruent; triangle A and C, or B and C are similar. Using this information and the definition of congruent and similar; What would be the measure of the hypotenuse of triangles B and C? Find the perimeter of triangle A. Find the perimeter of triangle B. Find the perimeter of triangle C. Find the area of triangle B, and then determine the area of triangle A without working. Find the area of triangle of C. Compare the area of A and C. Write the ratio of A to C. Write the ratio of triangle C to triangle A. The perimeter of triangle A = 12 units. The perimeter of triangle B = 12 units. They are congruent and their rations are 1:1. The perimeter of triangle C = 24 units or 2 times that of triangle A. Each side is 2 times that of the corresponding side. These two triangles are similar. Ratio of perimeters (A: C) = 1: 2 Area of A = ½ (bh) = ½ (3 · 4) = 6 sq units; Area of C = ½ (bh) = ½ (6 · 8) = 24 sq units Ratio of areas = 1:4 Does it work with other polygons? Yes it does! Complete the sentences:

1. The area of a large polygon is _____________ times the area of a similar smaller polygon.

2. If the measure of one side of a rectangle is 9cm, the measure of a rectangle 3 times smaller is __________________ cm. These rectangles are (a) similar (b) congruent (c) none of the above.

3. A set of 2 similar triangles was drawn in a design. One of the triangles was ¼ of the other. If the measures of the smaller triangle were 5, 7, and 12 cm, the measures of the larger triangle = ________, __________, and _____________

4. What would be the ratio of the area of the smaller to the larger? ___________

ALGEBRA 1

34

Are they similar or congruent? Draw to illustrate if necessary:

1. Triangle A had angles of 60º, 38º, and 82º. Triangle B had angles of 38º and 62º. Are they similar or congruent? What is the measure of the 3rd angle in triangle B?

2. A quadrilateral had two pairs of sides that measured 7 cm and 10 cm. Another quadrilateral had measures twice that size. Are they similar or congruent? Name the quadrilateral.

3. The ratio of the perimeters in a pair of polygons was 1:2. The ratio of the areas in the same pair of polygons was 1:4. Are they similar or congruent?

4. An isosceles triangle (A) has sides of 5cm, 5cm, and 7cm and a height of 3.7cm. Another triangle (B) was dilated 3 times larger. Are these triangles similar or congruent?

5. Find the area and perimeter of both triangles. What is the ratio of the perimeters and areas of triangles A: B?

Find the missing measures. 6. 8m 16m

10m 20m 4m

4cm . 7. 13cm ? 5cm ? 12cm 8. B

A A is twice as big as B. If the area of B is 160m2, what is the area of A?

9.The perimeter of a polygon was 348 inches. The perimeter of another polygon with the same shape was 116 inches. What is the ratio of the smaller to the larger? Are these polygons congruent or similar? What would be the ratio of the area of the smaller to the larger?

10.Use your notes to find the solutions to these activities.

ALGEBRA 1

35

LESSON 2E: STANDARD AND METRIC MEASUREMENTS

Conversion Tables

Linear measurement (Yards, Feet, and Inches)

Volume/Liquid measurement (Gallons, quarts, and pints)

12 inches 3 feet 36 inches 1760 yards 5280 feet

= = = = =

1 foot 1 yard 1 yard 1 mile 1 mile

2 pints 4 quarts 8 pints

= = =

1 quart 1 gallon 1 gallon

Time Linear measures (metric) 60 seconds 60 minutes 7 days 4 weeks 365 days 366 days

= = = = = =

1 minute 1 day 1 week 1 year 1 year 1 leap year

10 millimeters 10 centimeters 10 decimeters 100 centimeters 1000 meters

= = = =

1 centimeter 1 decimeter 1 meter 1 meter 1 kilometer

Weight Mass/solid measurement (metric) 16 ounces 2240 pounds 2000 pounds

= = =

1 pound 1 ton (British) 1 ton (U.S.)

10 milligrams 10 centigrams 100 centigrams 1000 grams

= = = =

1 centigram 1 decigram 1 gram 1 kilogram

Here are a few links that provide conversions of one unit to almost any other unit. Some of these are interactive sites and can be very helpful.

http://www.accelware.com/Unit_Conversion_Tool/ http://convert.springheadmedia.com/volumes.php?select=7

http://www.knowledgedoor.com/

Remember that the internet provides a wealth of information. Access to the internet is available at any public library throughout the United States.

How to use any conversion chart

Use a number line to help

Imagine each piece on the number line is equal to 12 inches. At the end of the number line you have 16 feet (count the number of spaces). Imagine that each 3-foot piece is equal to 1 yard. At the end of the number line you have 5 yards, with 1 foot left over. You have successfully converted 16 feet to yards and feet or 192 inches to yards and feet.

ALGEBRA 1

36

You can use the number line for any conversion. What is important is that you understand the concept so that you always choose the right operation.

Another way of looking at metric conversion

The metric system is based on the decimal system. The prefixes or parts of words at the beginning of each metric measurement have specific meaning. Milli- means 1/1000; centi- means 1/100; deci- means 1/10; kilo- means 1000. It takes 10 millimeters to make 1 centimeter, 10 decimeters to make 1 meter, 100 centimeters to make 1 meter and 1000 meters to make 1 kilometer. Knowing the prefix helps with conversion. Example 1: How many millimeters are there in 2 decimeters? Think 1 dm = 100 mm, so

2 dm = 200 mm. (multiply) Example 2: How many centimeters are there in 3 decimeters? Think 1 dm = 10 cm, so

3 dm = 30 cm. (multiply) Example 3: How many centimeters are there in 2 meters? Think 1m = 100 cm, so

2m = 200 cm. (multiply) Example 4: How many meters are there in 2 kilometers? Think 1km = 1000m, so

2 km = 2000 km. Example 5: How many centimeters are there in 1 decimeter? Think: Deci means 1/10. 1/10

of a dollar is 10 cents. Centi means 1/100. 1/100 of a dollar is ONE cent. Therefore there are 10 centis in one deci.

Example 6: How many meters are there in 1 centimeter? Think: which is smaller, the meter or the centimeter? The centimeter is smaller. The answer will be a fraction. It takes 100 centimeters to make 1 meter, so 1 centimeter can only be .01m (one hundredth) of a meter.

Example 7: How many kilometers will there be in 1 meter? Which is smaller? Answer: the meter. It would take 1,000m to make 1 km; 1 m = .001 km or one thousandth meter.

As with all concepts, thinking things through is better than merely memorizing the formula. If you forget, use logical reasoning to choose an accurate math operation.

ALGEBRA 1

37

Conversion Problems in the Real World

1. A man measured a room to be carpeted in inches (168 inches x 180 inches). Help him convert to feet and then find the area of the room.

2. A mountain climber completed a 4-mile climb. His son, a 7th grader, converted that to inches. How many inches did he get?

3. Create a number line to demonstrate conversion from gallons to quarts and pints. 4. It took 14 hours for me to get from Maryland to Denmark. How many seconds was

that? 5. The speedometer of many cars shows the distance traveled in kilometers. How many

centimeters would a car travel after 17 km? 6. What is the difference between a metric ton and a standard ton? 7. A student converted 3600 pounds to standard tons instead of metric tons. How far off

was his answer? 8. How many pounds and ounces are there in a package that weighs 150 ounces? 9. A yardstick measured 2m. Convert that to cm, dm, and km. 10. 5m = ____________km _______________cm _______________dm

Clues you can use

1. If 12 inches = 1 foot, 168 ins = 168 ÷ 12 (feet) 2. Convert 1 mile to feet, then change 4 miles to feet. 3.

2 pints = 1 quart 4 quarts = 1 gallon and 8 pints = 1 gallon

4. 1 hour = 60 minutes and 1 minute = 60 seconds. 1 hour = 60 x 60 seconds = 3600s; 14 hrs = 14 x 3600 = 50,400 seconds. 5. 1 km = 1000m and 1 m = 100cm; 1 km = 1000 x 100 cm = 100,000 cm 6. The difference between a standard and a metric is 400 lbs. 7. Use the clue at #6 to determine your answer. The answer should be greater than 1000

lbs. 8. Divide the ounces into lbs. The quotient is the pounds; remainder is the ounces. 9. 2m = 200cm, 2000dm, and .002km 10. Use the example at #9 to solve #10.

ALGEBRA 1

38

ALGEBRA 1

39

LESSON 3: FOUNDATIONS OF GEOMETRY

LESSON 3-A: PERPENDICULARITY AND PARALLELISM

Lines have specific relationships when they meet or intersect. Other lines do not meet but have special relationships as well. Lines may be perpendicular, parallel, or neither. Perpendicular lines meet to form right angles. These lines may also form polygons. Squares, rectangles, and some trapezoids have right angles. Lines that are perpendicular perform height or depth functions within geometric figures. The sides of a cylinder are perpendicular to its base. Parallel lines are equidistant, that is they are the same (or equal) distance apart. Parallel lines never meet no matter how far you extend them. The blue lines on notepaper are parallel; train tracks are parallel. Some polygons have parallel lines and right angles. Some polygons have parallel lines but no right angles. State whether each pair of lines (l1 & l2) is perpendicular, parallel, or neither: l1

l1 l1 l1 l1 l2 l1 l1 l2 A B l2 C l2 D E F l l2 l2

G H I J K L Which of these shapes have perpendicular lines, which have parallel lines, and which have neither?

Perpendicular lines Parallel lines

ALGEBRA 1

40

Organize these into three columns: cylinder, parallelogram, regular pentagon, windows of a house, notepaper, television screen, ruler, acute triangle, regular hexagon, feet, eyes, equal sign, train tracks, cube, and circle. Example:

Perpendicular Parallel Neither Cylinder Cylinder Pentagon

Rectangle Square Obtuse scalene triangle

Properties of Perpendicular and Parallel Lines

1. Perpendicular lines meet to form right angles. A small box at the intersection of 2 lines indicates a right angle.

2. Parallel lines create corresponding and alternate angles when intersected by a transversal.

3. Corresponding angles come in pairs and are congruent. 4. Alternate angles come in pairs and are congruent.

Corresponding Angles

1 3 5 9 11 2 7 4 6 10 12 8 transversal

1. Name pairs of corresponding angles: a) 1 and 2 are corresponding and congruent b) 3 and 4 are corresponding and ≡ c) 5 and 6 are corresponding and ≡ d) 9 and 10 are corresponding and ≡ e) ___________________________ f) ___________________________

2. If angle 1 = 93º, then angle 2 = 93º. If angle 3 = 87º, then angle 4 = 87º. 3. Angle 5 = 120º; angle 9 = 90º. State the measure of angles 6 and 10. 4. Study each set of corresponding angles well. Observe that angles are formed on the

straight line. Much information is contained in this diagram. 5. Angles on a straight line add up to 180º. That is worth repeating: Angles on a

straight line add up to180º. 6. If angle 9 = 90º, then angle 11 = 90º. Angle 9 + angle 11 = 180º. Angles 9 and 11 are

supplementary angles. Notice they are also equal to each other (90º). They are called equal supplements.

7. Let’s pause, read, digest, and understand all the information given in this section.

ALGEBRA 1

41

Complete the sentences:

1. Perpendicular lines meet to form ___________________________________ 2. Parallel lines never meet because __________________________________ 3. A transversal is _________________________________________________ 4. Corresponding angles are _________________________________________ 5. Angles formed on a straight line add up to_____________________________ 6. Supplementary angles equal _______________________________________ 7. Two right angles formed on the same line are called ____________________ 8. Parallel lines are __________________________ from one another. 9. Parallel lines form corresponding angles when crossed by a ______________ 10. A transversal crossing parallel lines forms ____________________________

ALGEBRA 1

42

Alternate Angles

Alternate angles form when a transversal crosses a pair of parallel lines. Alternate angles create a Z or zigzag pattern. They come in pairs and are congruent. A 7 8 C 1 3 D 4 2 E 9 10 F B Complete each statement:

1. If angle 1 = 130º, angle 2 = _____________ 2. If angle 1 = 130º, angle 3 = _____________ (supplementary) 3. If angle 7 = 50º, angle 10 = _____________ 4. If angle 7 = 50º, and angle 8 = 130º, then angles 7 and 8 are ________________

(angles on a straight line) 5. If angle 8 = 130º, then angle 9 = _______________ 6. Angles 3 and 4 are ____________________ 7. Angles 8 and 1 are ____________________ 8. Angles 3 and 9 are ____________________ 9. Can angles 3 and 9 be an alternate pair? Y/N ____________ 10. Can angles 9 and 10 be supplementary? Y/N ____________

Vertical angles

Whenever two lines intersect or cross they make vertical angles. Vertical angles are opposite each other (face each other) and are congruent. We have already identified several vertical angles in this section. Compare the locations of angles 1 and 8; 7 and 3, 2 and 9, and 4 and 10. They sit at the intersection of the lines, are opposite and congruent. These are easy to find and can help in solving angle measures when little information is present. Add this one to your math dictionary.

Calculating Angle Measures

We will learn to calculate angle measures from understanding the properties of perpendicular and parallel lines. A picture is really worth a thousand words. Remember the rules you just learned:

• Corresponding angles come in pairs and are congruent. • Alternate angles are come in pairs and are congruent. They form a kind of zig zag

pattern • Vertical angles come in pairs, are opposite and congruent • Corresponding angles are formed when a transversal crosses parallel lines • Lines that are perpendicular meet at right angles • Angles on a straight line are supplementary or equal 180º • Two right angles on a straight line are equal supplements

Pairs of Alternate Angles

1 and 2; 3 and 4 7 and 10; 8 and 9

All pairs are congruent.

ALGEBRA 1

43

Activity

A Q

91º D E 13 14 15 11 12 C

6 1 G 7 8

2 3 F 4 5

9 H 10 B R Line AB is parallel to QR; line CD is parallel to FG. Angle 10 is a right angle. Angle 15 = 40º List 2 pairs of corresponding angles: 1. ____________________ and _________________________ 2. ____________________ and _________________________ List 2 pairs of vertical angles: 3. ____________________ and _________________________ 4. ____________________ and _________________________ List 2 pairs of alternate angles: 5. ____________________ and _________________________ 6. ____________________ and _________________________ Find the measure of angles: 1 = ______________________ 2 = ______________________ 3 = ______________________ 4 = ______________________ 5 = ______________________ 6 = ______________________ 7 = ______________________ 8 = ______________________ 9 = ______________________ 11 = _____________________ 12 = _____________________ 13 = _____________________

ALGEBRA 1

44

IMPORTANT!

This lesson is extremely important. Understanding these rules sets the pace for a full understanding of geometry later. As we work with proofs of congruency in the next lesson, we may need to draw upon the knowledge gained in this lesson. Study the rules well and review often.

LESSON 3B: 2-COLUMN PROOFS 4 Reasons for Congruency:

Reason 1 Triangles are congruent if each of the 3 sides is congruent to a corresponding side of another triangle. We say SSS (side, side, side) as the reason for congruency. In this case corresponding means similar location. These triangles are congruent for the same reason, SSS Reason 2 Corollary: If the 3 sides of a triangle are congruent, then the 3 angles are also congruent. The vertical sides are congruent; there is a right angle in between the vertical sides and the base sides which are also congruent. Reason: Side-Angle between-Side (SAS)

ALGEBRA 1

45

A F E B C D Observe labeling of corresponding sides. Angle A ≅ 30, angle D ≅ 30. Side AB ≅ DE and AC ≅ DF. SAS Corollary: Angles opposite congruent sides are also congruent. Angle C ≅ Angle F. Angle C is opposite side AB and Angle F is opposite side DE Reason 3 A D 25º 25º C F 88º 88º E B The reason for congruency in these two triangles is angle, side in between, angle (ASA). Angle D = Angle A; Side AB = DE; Angle B = Angle E Reason 4

The reason for congruency in these two triangles is angle, angle, side (AAS). AB = DE and angle C = angle F and angle A = angle E

ALGEBRA 1

46

Q L S = 45º N = 45º R S M N Angle R = Angle M (Angle R and Angle M are right angles and all right angles are equal); Side in between RS = MN (Given); Angle S = Angle N (Given) (ASA). Corollary: Sides opposite congruent angles are also congruent. QS (opposite R) = LN (opposite M); QR (opposite S) = LM (opposite N)

Solving Congruency Problems

1. Triangle ABC has side AB = 4cm, BC = 7cm, and AC = 7.5cm. Triangle DEF has side

DE = 7.5cm, EF = 4cm, and DF = 7cm. Are they congruent? Draw and label to illustrate.

2. An isosceles triangle had two sides 4 cm long. The base angles were 50º. Another isosceles had two sides 4 cm long with the angle in between 50º. Draw to show whether they were congruent.

3. These right triangles are congruent. Angle A = 35º and angle D = 35º. AB = DE = 5cm. What reason would you give?

A D 5cm 5cm B C E F

4. Study the triangles and state whether they are congruent. Give reasons for your answer. G Q

75º 75º H R S I GH = QR; Angle H = Angle R; HI = RS . Congruent SAS.

ALGEBRA 1

47

Two-Column Proofs of Congruency We will use the same questions and place the information in a different format in order to compare these two options. Remember there must be 3 pieces of congruent information that match in both triangles for there to be congruency. Look for SSS, SAS, or ASA. Problem #1

Triangle ABC Triangle DEF 1. AB = 7.5cm (given) DE = 7.5 cm (given) 2. BC = 4.0cm (given) FE = 4cm (given) 3. AC = 7.5 (given) DF = 7cm (given) Corresponding sides are congruent. The measures of sides are the same. The triangles are congruent. Problem #2

Triangle 1 Triangle 2 1. Side 1 = 4cm (given) Side 1 = 4cm (given) 2. Base angle 1 = 50º (given) Base angle 2 = 50º (given)

Base angle 1 ≠ 50º (by calculation) Base angle 2 ≠ 50º (by calculation)

3. 3rd angle ≠ 50º (by calculation) 3rd angle = 50º (given) The corresponding sides are congruent but the corresponding angles are not. The triangles are not congruent. Problem #3

Triangle 1 Triangle 2 1. Angle A = 35º (given) Angle D = 35º (given) 2. Side AB = 5cm (given) Side DE = 5cm (given) 3. Angle B = 90º (given; right angles) Angle E = 90º (given; right angles) The triangles are congruent by ASA; the corresponding sides are congruent and the corresponding angles are congruent.

Review of Lesson 3-B

1. List 4 reasons that demonstrate when triangles are congruent. 2. Some statements of congruency have a corollary or logical conclusion following the

reason for congruency. Complete each statement by writing the corollary to each of these reasons for congruency:

a) If the 3 sides of a triangle are congruent, ___________________________ b) If three angles of a triangle are congruent, _________________________ c) Angles opposite congruent sides _________________________________ d) Sides opposite congruent angles _________________________________

3. Write T/F after each statement: a) It does not matter how the triangles are labeled. As long as the angles or

measures are the same the triangles are congruent. b) Corresponding sides mean that each side must be in a similar location on both

triangles. c) One reason for congruency is AAS or angle, angle, side.

ALGEBRA 1

48

d) Side-angle in between-side is one way to prove triangles congruent. e) ASA and SAS mean the same thing. f) Angles opposite congruent sides are also congruent. g) If 2 angles of a triangle match 2 angles of another, there is enough information

to prove the triangles congruent. h) We can use the sum of the angles in a triangle to find the measure of an angle

and prove triangles congruent.

4. L X 6cm 6cm M N Y Z LM = XY; MN and YZ are 2cm less than LM or XY. Both are right triangles. Are these triangles congruent? Use 2-column proofs to explain. 5. Draw a pair of triangles HIJ and KLM with the information: HI = 3cm; Angle I = 60º; IJ = 5cm; KL = 3cm; Angle L = 60º; Angle M = 55º. Prove these triangles congruent using a 2-column chart.

ALGEBRA 1

49

LESSON 3-C: SLOPE, DISTANCE, AND MID-POINT FORMULAS

We use algebra to solve many problems in geometry. Here we are again! We re-introduce the coordinate plane and quickly review the important parts and vocabulary. The rectangle coordinate system is also known as the Cartesian coordinate system after Rene Descartes, who popularized its use in analytic geometry. The rectangle coordinate system is based on a grid, and every point on the plane can be identified by unique x and y coordinates, just as any point on the Earth can be identified by giving its latitude and longitude.

Quadrants are labeled in counter clock-wise direction. Quadrant 1 is Q1, Quadrant 2 is Q2, Quadrant 3 is Q3, and Quadrant 4 is Q4. The X-axis or coordinate is horizontal; the Y-axis is vertical. The origin is the intersection of X and Y. All values to the right of zero on the X-axis are positive, and all values to the left of zero are negative. Values above zero on the Y-axis are positive, and values below zero on the Y-axis are negative. The X-coordinate always comes first followed by the Y-coordinate. Values in Q1 read +x, +y; Q2 read –x, +y; Q3 read –x, -y; and Q4 read +x, -y What is slope and why do we need it? Slope defines the way a line moves upward or downward in a way that creates an angle or incline. A hill slopes and a see-saw slopes.

ALGEBRA 1

50

When toys are made the manufacturer must be sure that the slope of some toys is safe (not too high) for children. They have to calculate it. There is a formula for that! The simplest way to get the idea of slope is to think of the change in movement upward/downward (vertical) divided by the change in movement across (horizontal). Slope = Vertical change (rise) Horizontal change (run)

Vertical change for line AB = 1 (1 block up from A to B) Horizontal change for line AB = (2 blocks across from A to B) Slope =

runrise =

21

Vertical change for line CD = 3 (3 blocks up from C to D) Horizontal change for line CD = 2 (2 blocks across from C to D) Slope =

runrise =

23

If the answer is less than 1, then the slope is gentle or the gradient is low. If the answer is greater than 1, then the slope or gradient is steep. The bigger the fraction is, the steeper the slope gets. These notes help with the idea or concept but we don’t have the algebraic formula yet. It is OK to remember only rise over run but we must be able to recognize the formula and work with it when it is given.

What is the slope of the line AB? What is the slope of the line CD? Find the vertical change by counting the blocks going up from the start of the line to the end of the line. Find the horizontal change by counting the blocks going across from the start of the line to the end of the line.

ALGEBRA 1

51

Quick Practice

Line 1 Line 2 Line 3

1. Find the (x,y) coordinates for the end points of each line. 2. Find slope. Coordinates: Line 1 = (1,1), (4,2); line 2 = (0,0), (3,3); line 3 = (0,1), (3,1) Slope: Line 1 = 1 Line 2 = 3 = 1 Line 3 = 0 = 0 3 3 3 What do these numbers mean?

• When the fraction is positive, the line is ascending or going up; the trend or change is increasing. Lines 1 and 2 show increasing trends or change. If you graphed a line of business profits and it looked like this, the business is doing well.

• When both numerator and denominator are the same, the rate of change is steady. If you simplified the fraction the answer would be 1. The number 1 indicates steady, regular change.

• When the numerator is zero, the line is horizontal, an indication of no change. This is a stagnant business indicator, a bad sign for a business.

Suppose these numbers demonstrated the amount of rain that fell over 3 days, interpret the meaning of these graphs.

• Line 1 shows that more rain fell each day compared to the day before. We can also say how much more rain fell. One third more (33 1/3%) rain fell on day two, and one third more rain fell on day 3.

• Line 2 shows that 100% more rain fell on day 2 and another 100% on day 3. • Line 3 shows that the same amount of rain fell on each day.

Interpreting these graphs is simple but the meanings change depending on the topic. What would the slopes look like when the change is decreasing? Let’s take a look!

ALGEBRA 1

52

Line 1 Line 2 Line 3

All values are the same on the 3 graphs. The data refers to student attendance at 3 different schools over 3 years.

Activity

1. Find the (x,y) coordinates for the end points of each line. 2. Calculate each slope. 3. Interpret the meaning of each graph.

Clues you can use

Remember that the X coordinate is always first. Your information must relate to the fraction that defines the slope. Zero as a numerator or denominator is very important.

The Algebraic Formula for Slope

Slope = Vertical change (rise) of Y values Y2 – Y1 Horizontal change (run) of X values X2 – X1 The small 1 and 2 mean the first value of X or Y and the second value of X or Y. In the last activity the (x,y) coordinates of the end points of line 1 were (0,3) and (4,0). Let’s put this in a table: X Y Study the way the table is set up. This avoids mistakes. 1 0 3 Y2 – Y1 = 0 – 3 = -3 X2 – X1 = 4 – 0 4 2 4 0 Rise over run = -3 4 It is difficult to tell when it is negative unless we recognize that when the vertical change (Y2 – Y1 ) is a drop it is a decrease so the slope is negative. On a coordinate plane all negatives are clearly seen. This is a good reason to learn the algebraic formula.

ALGEBRA 1

53

The (x,y) coordinates of the end points of line 2 are (0,2), (3,1) Set up a table of values: X Y

1 0 2 Y2 – Y1 = 1 − 2 = -1 Rise over run = -1 (down 1 = negative) X2 – X1 3 – 0 3 3 2 3 1 X Y Line 3 coordinates: 1 1 3 Y2 – Y1 = 0 − 3 = -3 X2 – X1 1 – 1 0 2 1 0 Rise over run = -3 (Note that division by zero 0 is undefined.)

Find Slope from the Equation of a Line

What is the equation of a line, and how does this help us find slope? The equation of a line should always start with Y = __________. No matter how we are given the equation, we have to convert it to start with Y =. We say that the form is Y = mx + b. We need that form to get the x-coordinates and be able to plot the points of the line. Example 1: x + y = 4. How do we change that around so that y is on the left side of the equation? Step 1: y = 4 – x (subtract x from each side. –x + x + y = 4 – x; y = 4 – x) Step 2: y = − x + 4 (rewrite the right side so that the equation is in the form y=mx+b) What does this mean and where is the slope? M is the co-efficient of x. That number is the slope of the line. Repeat: The coefficient of x is the slope of the line. In this equation no value is placed before x; that suggests that -1 is the value of m. What does b mean? B is a constant that gives us the location where the line crosses or intersects the Y axis. This is the Y intercept. In our equation b = 4 Example 2: Solve for y (or change to the format y = mx + b); find slope and constant. 2x + y = 3; Solve for y by adding -2x to each side of the equation: -2x + 2x + y = 3 – 2x; y = −2x + 3; slope = -2; b = 3 Example 3: Solve for y, find the slope and the constant. Y + 3x = − 4; Solve for y by adding -3x to each side of the equation; y + 3x – 3x = -4 – 3x; y = -3x-4; slope = −3; the y-intercept = − 4.

ALGEBRA 1

54

A real-world problem

You are a carpenter who must build a ramp for the handicapped in front of this entrance. The slope must not be greater than ¼. You show your drawing to the manager. Will he approve your drawing? The distance from the end of the ramp to the door is 12 feet. The height of the ramp is 2 feet. What is the slope of this ramp?

Solution Use rise over run for slope. Rise = 2, run = 12; slope = 2 = 1 12 6 Is 1/6 greater or less than ¼? Find a common denominator: 12; 1/6 = 2/12; ¼ = 3/12;

2/12 is less than 3/12 or 1/6 is less than ¼. Your ramp will be approved.

Mid-Point Formula

The mid-point formula is used as a short cut. Let’s use a real-world problem to see why the shortcut is sometimes necessary. You work for the National Park System. You are to design a see-saw for the children’s playground. You must use the following information to find the slope of the see-saw and the point at which the fulcrum will be placed. First draw a picture. The see-saw should not go higher than 3 feet. The coordinates of the line on your graph are (0,0) and (10,3). Find the slope of the line and the mid-point (where the fulcrum will be highest). Make a table of values: X Y 1 0 0 2 10 3 Slope formula: Y2 – Y1 = 3 – 0 = 3 X2 – X1 10 – 0 10 Mid-point formula: (X1 + X2), (Y1 + Y2) = (0 + 10), (0 + 3) = 10, 3 = (5, 1.5) 2 2 2 2 2 2 The fulcrum should be placed where the points on the graph are X = 5 and Y = 1.5

Mid-point See-saw Fulcrum

ALGEBRA 1

55

Activity The balance beam at an Olympic event was drawn on graph paper. You have a part-time job to assist in setting up this event. You must place marks at the mid-point of the balance beam on the graph paper. Where will you place the mark?

Use the mid-point formula to demonstrate. Clues: 1. List the coordinates. Write the formula. 2. Substitute the values for x and y. Calculate. 3. This one was easy enough to get the mid-point without calculating. Compare values to see whether the formula worked. 4. Your answer should be (6,2).

The Distance Formula

The distance is another shortcut to find the length of a line and is another way to use the Pythagorean Theorem. Recall that we can use the formula a2 + b2 = C2 to get any of the sides of a right triangle. Fiigure 1 Figure 2

The AB line is the hypotenuse. Consider the AC line as leg b and the BC line as leg a.

Suppose you are given two points A (1,2) and B (4,5) in Fig. 1. Find the distance between the two points Using points A and B, draw the triangle ABC having AB as the hypotenuse as in Fig. 2.

ALGEBRA 1

56

Compare the Pythagorean Theorem with the distance formula. Pythagorean Theorem Distance formula C2 = a2 + b2 D2 = (x2 – x1)2 + (y2 – y1)2 C = √ a2 + b2 D = √(x2 – x1)2 + (y2 – y1)2 (a) (b) Use the above example to find distance of point A to point B: Step 1: Find the coordinates of each point. A = (1, 2); B = (4,5) Step 2: Make a table of values x y 1 1 2 Point A 2 4 5 Point B Step 3: Substitute the values for the variables.

22 )3()3( + = 99+ = 18 or 4.24 Now find the length of BC and AC First AC: D = 22 )22()14( −+− = 22 )0()3( + = 9 = 3 Note: Hypotenuse = 18 or 4.24 Leg AC = 3 Use Pythagorean Theorem to solve for BC or leg a

c2 = a2 + b2 ( 18 )2 = a2 + 32

18 = a2 + 9 18 – 9 = a2

9 = a2 and 3 = a

ALGEBRA 1

57

Unit 3-C Review and Self Check

Use the following websites for personal review and self-check http://regentsprep.org/Regents/math/distance/PracDistance.htm http://www.purplemath.com/modules/slope.htm http://www.stmichaelsacademy.org/faculty/dhaynes/PreCalculus/10/01/01.htm

1. Draw lines that intersect to demonstrate 3 different perpendicular relationships. 2. Draw lines to demonstrate 3 different parallel relationships. 3. Why do parallel lines never intersect? 4. Draw 4 items used everyday that have both perpendicular and parallel relationships

among their parts. 5. What are supplementary angles? Draw a pair of supplementary angles and write their

measures. 6. What is a transversal? Complete the sentence: Transversals form pairs of

a_________________, c__________________, and v________________ angles. These angles are congruent to each other.

7. Identify 2 pairs each of corresponding, alternate, and vertical angles in the diagram: Corresponding angles: _____________________ A B C D _____________________ E F G H Alternate angles: _____________________ _____________________ I J K L Vertical angles: _____________________ M N O P _____________________

8. What are vertical angles? What about their location helps to identify them? 9. Two streets intersected this way. Are there any vertical angles formed? Explain.

Q

10. Triangle QRS is isosceles. Angle Q = 58º X Angle Q = R. Triangle XYZ = QRS. Line QR is parallel to line XY. Use the information to answer the following questions. R S Y Z Angle R = ___________________ Angle Y = _________________________ Angle X = ___________________ Angle S = __________________________ Angle Z = ___________________ Transversal = _______________________

ALGEBRA 1

58

Transversal = ________________ RS = ______________________________ YZ = _______________________ XZ = _____________________________ Triangle QRS ≅ XYZ because of (a) SAS (b) ASA (c) AAA (d) All of the above 11. Use the set of triangles above to name 2 pairs of corresponding angles, 2 pairs of

alternate angles, 2 pairs of vertical angles, and 2 pairs of supplementary angles. Use your own numbers or letters.

12. We can prove triangles congruent by (a) AAS (b) SAS (c) SAA (d) None of the above. 13. F Q ED ≅ RS F = Q = 89º FE ≅ QR E D R S Are these triangles congruent? Use a two-column proof to decide 14. The triangles below are right triangles. One angle in triangle A is congruent to the

corresponding angle in triangle B. One side in between two congruent angles is congruent to the corresponding side in triangle B. What could be the reason for congruency? Draw and highlight what that could look like.

15. If the 3 sides of a triangle are congruent to the 3 corresponding sides in another triangle, then the triangles are congruent by SSS. What is the corollary to this? How can this be true?

16. Answer T/F for each: a) The Y-coordinate is always first b) A decrease in the value of y indicates a downward slope c) An upward slope in revenue is good for business d) A downward slope in sales is good for business e) If the slope of a rainfall graph is 2 over 2, rain has been falling steadily for that period. f) A zero denominator means that the slope is a vertical line. g) A zero numerator means that there is no rise only run. The line is vertical. h) The equation of a line can help us find slope i) The equation of a line should always start with Y=… j) We can change any equation with X and Y to the form Y = mx + b k) M in the equation of a line gives the value of the slope. l) 2x + y = 4 is the same as Y = 2x + 4 m) In the question above, 4 is a constant n) Instead of rise over run you can use Y1 – Y2 X1 – X2 o) The mid-point of (0,10) and (0,3) is (5, 1.5)

ALGEBRA 1

59

18. This is the entry/exit ramp drawn on graph paper. Each space equals 3 feet on the X-axis and 1 foot on the Y-axis. The ramp is 3 feet wide. (6,4) 3 (-3,1) 1 -5 -4 -3 -2 -1 1 2 3 4 5

The coordinates above are (-3,1) and (6,4). • Find the slope. • What is the height of the ramp? • The carpenter needs to build a support beam at the mid-point of the ramp. What

would be the coordinates of the beam? • Find the length of the ramp. (Use Pythagorean Theorem) • How many square feet on the ground would the base of the ramp occupy? (Area

formula) • What area would the ramp platform occupy? (Area formula.) Hint: the ramp

platform’s length is the hypotenuse of the triangle graphed above.

ALGEBRA 1

60

ALGEBRA 1

61

LESSON 4: ALGEBRAIC THINKING

LESSON 4-A: THE LANGUAGE OF ALGEBRA

Basic vocabulary

Key Words or Phrases Definitions Variable A letter used in place of a number Operation +, -, x, ÷ Order of operations The order in which problems are solved when

there are multiple operations in a problem Increase, more than, plus, altogether Means to add Sum The answer to an addition problem Decrease, less, less than, minus, take away Means to subtract Difference The answer to a subtraction problem Times Means to multiply Twice, double Multiply a number by 2 Thrice Multiply a number by 3 Product The result of multiplication Square A number times itself Cube Multiply a number by itself three times Quotient The result of dividing Dividend The number you are dividing Divisor The number you divide by Share Means to divide Among A key word that suggests division Split Can be divide into two or more parts Fraction bar Means to divide Factor The number that divides into another Multiple Many times a number Prime A number that has only 2 factors Exponent The number of times you multiply by the same

factor Term The smallest part of an algebra expression:

Example: a + b; a is a term, b is a term. This example has 2 terms

Expression Two or more terms with an operation between them: a + b is an expression; b2 is an expression that means b times b

Equation Two math expressions with an equal sign between them: a + b = 7; 3b = 12

Substitution Replacing a value for a variable

ALGEBRA 1

62

Algebra Basics Self Assessment 1. a + b means ___________________________________________________ 2. b – a means ___________________________________________________ 3. Simplify a + a + a ______________________________________________ 4. c/d means ____________________________________________________ 5. f2 means _____________________________________________________ 6. 2l + 2b can be written ___________________________________________ 7. If 3a + 2 = 11, a = ______________________________________________ 8. 4b ÷ 2 = 20, b = _______________________________________________ 9. √36 = ________________________________________________________ 10. Simplify (-32) _________________________________________________ 11. Simplify 4 + 7(33) ______________________________________________ 12. Simplify 8(2 + 7) – 20 __________________________________________ 13. Simplify -21 ÷ -3 ______________________________________________ 14. 2c – 6c = 24, c = _______________________________________________ __________________________________________________ = ׀ 5 ׀ + 10- .1516. A young entrepreneur bought some apples at $.25 each and sold them at $.30 each.

Did he make a profit on the sale of 20 apples? __________________ 17. What was his profit percent? _____________________________________ 18. 2l + 2b is an example of what property? ___________________________ 19. 1 + (-1) = 0 is an example of what property? ________________________ 20. The commutative property states that 2 · 3 = 3 · 2 T/F 21. The zero property states that the answer changes if you add zero T/F 22. Two minus three is the same as three minus two. T/F 23. Triangular numbers are created by adding consecutive, counting numbers. T/F 24. Square numbers are created by adding consecutive, odd, counting numbers. T/F 25. A number multiplied by its reciprocal equals 0. T/F 26. Multiplying by a common fraction increases the number. T/F 27. 2 (½) = ____________________________________________________ 28. Adding a fraction increases/decreases the number. Select the answer. 29. Dividing by a common fraction increases/decreases the number. Select the answer. 30. 4 ÷ ¾ = (a) 3 (b) 4¾ (c) 5.3 (d) 3.5 31. The subtrahend is the number you are subtracting. T/F 32. A negative power can be written as a fraction. T/F 33. 23 = (a) 2 · 2 · 2 (b) 2 · 3 (c) 3 · 3 (d) 3 · 3 34. 5-1 = (a) -5 (b) -1 (c) -1/5 (d) 1/5 35. Complete the next number in the series: 4, 9, 16, ________________

ALGEBRA 1

63

LESSON 4-B: INTERPRETING DATA FROM GRAPHS AND CHARTS

Data on attendance for a particular school for 1 academic year (%): A S O N D J F M A M J 98 97 97 92 94 91 96 97 92 97 96

1. Find the mean, median, and mode of the data. 2. Display the data graphically in several ways. List the different types of graphs you

would use and state the reasons for your choice. (There are 9 different types of graphs)

3. Is the mean a good way to represent this data? If not choose another measure of central tendency. Explain your choice.

4. What months does the data cover? 5. What do you think accounts for the lower attendance in November? 6. What might affect attendance in March and May? 7. Which graphs provide the largest amount of data? 8. Which graph looses the largest amount of data? 9. Which graphs allow data comparison? 10. Which graphs use the median as its most important piece of data? 11. Which graph shows a trend clearly? 12. Is there a trend in attendance during the year?

Solutions and Partial Solutions 1. To find the mean, median, and mode first organize the data numerically. 91, 92, 92,

94, 96, 96, 97, 97, 97, 97, 98. Mean = Total ÷ 11 = 1047 ÷ 11 = 95.18 (work to 2 decimal places until the end, then round. Median = 96; mode = 97

2. The nine graphs are: line graph, bar graph, circle graph, pictograph, box and whisker plot, line plot, stem and leaf plot, scatter plot, and histogram. Use the line graph, bar graph, box and whisker plot, line plot, stem and leaf plot, scatter plot, and the histogram. These graphs show possible relationships in the data or trends; the circle graph is not practical because there are too many items to display in a circle; the pictograph is also not practical because there is not a variety of data.

Line Graph

98 * * * *

96 * *

94 *

92 * * *

90 A S O N D J F M A M J

ALGEBRA 1

64

Bar Graph 98 96 94 92 90 A S O N D J F M A M J

Box and Whisker Plot We need 5 elements in order to set up the box and whisker plot: Median, Lower Extreme, Upper Extreme, Lower Quartile, and Upper Quartile. Md = 96; LE = 91; UE = 98; LQ = 92; UQ = 97

M LE LQ UQ UE

90 91 92 93 94 95 96 97 98

Line Plot

* * * * *

* * * * * * 90 91 92 93 94 95 96 97 98

Stem and Leaf Plot Stem Leaf 9 1 2 2 4 6 6 7 7 7 7 8 All the data starts with 9. There are no other stems.

ALGEBRA 1

65

Histogram

98 96 94 92 90 A S O N D J F M A M J

Compare the bar graph with the histogram

Scatter Plot 98 * * * * * 96 * * 94 * 92 * * * 90 A S O N D J F M A M J

Compare the line graph with the scatter plot

3. The mean is a good way to represent this data set. All the measures of central tendency are close; the mean = 95.2, the median = 96, and the mode = 97. When the three measures of central tendency are close any of them represents the data well.

4. There are only 11 months of data because students do not attend school in July. 5. Answers will vary. What happens in November that affects most people’s lifestyle? 6. Think of the school year and what important school events occur during these months.

March indicates end of the 3rd marking period and is close to the end of the school year. High school students will begin to transfer their grades to college and middle school students will begin the registration process to high school. What happens in May?

7. All of the graphs except the box and whisker plot provide all the data. The box and whisker plot gives only partial data.

8. Refer to # 7. 9. The histogram and the bar graph allow data comparison. 10. The box and whisker plot uses the median as the most important piece of data. 11. The line graph clearly shows a trend. 12. There is no clear trend.

ALGEBRA 1

66

What data can we retrieve from these graphs?

Let’s use the questions to find out! Use the line graph to answer these questions:

1. What was the average attendance for the year? Which months were the highest and which were the lowest?

2. What was the range and was there a month when the attendance was much higher or lower than others?

3. Which months demonstrated the greatest fluctuations? Use the bar graph and histogram to answer these questions:

4. What is the difference in the way these graphs are constructed? 5. How are they similar? 6. List the months that provide the same data. Is it easier to locate similarities in these

graphs compared to the others? 7. How easy is it to find the mode? 8. Can you retrieve all the original data from this type of graph?

Use the box and whisker plot to answer these questions and note some basic observations: 9. Can you find the range from a box and whisker plot? 10. Observe that this type of graph demonstrates how the data is scattered or spread;

it shows measures of dispersion. Students generally know measures of dispersion least. Other graphs show measures of central tendency.

11. The median divides the data into 2 halves. The quartiles divide the data into quarters. We can say that 75% of the time attendance is below the median. This is not a good reflection of the attendance at that school. Making a statement like this is better than simply talking about the mean, the median, or the mode.

12. 25% of the time attendance is above the median (from M to UE). 13. It is not necessary to put all the data into this graph to make meaningful analyses

of the data. Compare the stem and leaf plot with the line plot and the box and whisker plot. 14. Is all the data in the graphs? NO, the box and whisker does not have all the data 15. Can you identify or match the months with the percent attendance? NO 16. This is characteristic of these graphs. Why create a graph like this; what is the purpose? The stem and leaf plot and the line plot provide the same data. Amazing how different they seem. Did you find the similarities in the bar graph and the histogram? Did you find the differences? What is the difference between the line graph and the scatter plot? There will be more on graphs in Lesson 3-C of this unit and in LESSON 5 on data analysis.

ALGEBRA 1

67

LESSON 4-C: SIMPLIFY POLYNOMIALS

Characteristics of linear and non-linear graphs

These two apparently different topics are in the same lesson because we will apply our knowledge of polynomials when we solve and graph linear and non-linear equations. Monomial (mono means one) Binomial (bi means two) Trinomial (tri means three) 2a 2a + 4 2a2 + xy - 3 4xy 4xy – 8 x2 + y2 - 5 x3 x3 + x2 x3 + x2 + x

(They are all polynomials)

Add polynomials: Polynomials can be added vertically or horizontally. (Collect like terms) Example 1: (2a + 4) + (7a +3) = 2a + 7a + 4 + 3 = 9a + 7 2a + 4 7a + 3 9a + 7 Example 2: 4x – 1 + -3x + 6 = 4x – 3x – 1 + 6 = x + 5 Example 3: 2b2 – 5 + 3b + 2 + b2 – 6

= 2b2 + b2 + 3b – 5 + 2 – 6 = 3b2 + 3b - 9

Example 4: (- 6y2 + 2y – 8) + (6y2 – 4y + 2) = -6y2 + 6y2 + 2y – 4y – 8 +2

- 2y – 6

ALGEBRA 1

68

Subtract polynomials: It is safest to work vertically; also review subtraction rules. Example 5: 3z + 2 – (z + 1) = 3z + 2 - (z + 1) (add the opposite) 2z + 1 Example 6: -5r2 – 3y + 7 – (-2r2 – 8) = -5r2 – 3y + 7 - (-2r2 - 8) add the opposite - 3r2 - 3y + 15 Example 7: (2ab3 + 7ab – 4a + 9) – (ab3 – 10ab + 2a – 2b) = 2ab3 + 7ab – 4a + 9 -(ab3 -10ab + 2a – 2b) ab3 + 17ab – 6a + 2b + 9

Practice adding and subtracting polynomials

1. 5y + 3z + 2 + (7y + z + 1) 2. (6 + 3a – 2b) + 5a – b – 7 3. 3xy2 – 2x + 4y – 10 – ( 5xy2 + 2x + 7y) 4. -(2p + 7q – 16) + (p – q + 10) 5. 9y – 1 – ( 3y2 + y) 6. 3a2 + 5a + 0 + 4a2 + 8

How did you do?

1. 12y + 4z + 3 2) -1 + 8a -3b 3) -2xy2 – 4x – 3y - 10 4) –p – 8q + 26 5. 8y – 1 – 3y2 6) 7a2 + 5a + 8

Explanations for #4: The negative sign outside the parenthesis affects all the signs inside the parenthesis. – (2p + 7q – 16) = + (-2q – 7q + 16) + (-2p - 7q +16) + p – q + 10 - p - 8q + 26

Multiply Monomials Produce Polynomials

Tips: Multiply numbers as usual; add exponents when multiplying like variables; use the distributive, commutative, and associative properties. Example 1: 2 · a = 2a; a · a = a2; 2a · a = 2a3; (x) y = xy; a(b) = ab; 2(x) · 3x = 6x2

Example 2: xy (x2) y = x3y2; 4a (-2ab) = -8a2b; 5a3 (4a2) = 20a5 6p · p2 · -2p3 = -12p6 (Observe that we add the exponent of like terms)

ALGEBRA 1

69

Example 3: (2x + 1) (x + 2); multiply each term by every other term, and then add. (2x + 1) x = 2x2 + x; (2x + 1) 2 = 4x + 2; add both 2x2 + x + 4x + 2 2x2 + 5x + 2 Example 4: (a + b) (2a – 4) = (a + b) · 2a = 2a2 + 2ab; (a + b) - 4 = - 4a + - 4b; 2a2 + 2ab - 4a + -4b 2a2 + 2ab – 4a + -4b Example 5: We can write multiplication of polynomials vertically. (-3mn + 2m) (5mn + m) -3mn + 2m 5mn + m -15m2n2 + 10m2n - 3m2n + 2m2 -15 m2n2 + 7m2n + 2m2 Example 6: (4ab -3)(a – 2b) 4ab -3 a – 2b_________ 4a2b - 3a - 8ab2 + 6b 4a2b – 3a - 8ab2 + 6b

These could be a little tricky so go slowly as you work.

Practice multiplying monomials

1. (5d + 7e) 7 2) (4r – 2s)(5 + 3r) 3) (a – b) (a – b) 4) (2a – 3b)(ab + 2) 5. 5xy (3x – 4) 6) a2 b · ab2 7) m2n (mn2) 8) (3ab + 4)(2a2 – 5)

How did you do? 1. 35d + 49e 2) 20r – 10s + 12r2 - 6rs 3) a2 – 2ab + b2 4) 2a2b – 3ab2 + 4a – 6b 5) 15x2y – 20xy 6) a3b3 7) m3n3 8) 6a3b + 8a2 - 15ab – 20

Monomials have powers too!

Examples 1: (2a) a = 2a 1+1 = 2a2; x2 (x4) = x 2+4 = x6. When the bases are the same add the exponents. Example 2: (2b2) 3 = 2(b · b) · 2(b · b) · 2(b · b) = 8b6. When raising a number with a power to a power, multiply the exponents. Example 3: Is 4x2 the same as (4x) 2? 4x2 = 4 (x) (x); (4x) 2 = 4x (4x) = 16x2

Note: Remember that you are multiplying!

ALGEBRA 1

70

Squaring a Binomial (Multiplying a binomial by itself)

This is a handy tool that will be useful as we divide or factor polynomials. Pay special attention to the format.

Example 1: (a + 2) (a + 2) = a2 + 2a + 2a + 4 = a2 + 4a + 4 Example 2: (x + 3) (x + 3) = x2 + 3x + 3x + 9 = x2 + 6x +9. There is a pattern or system. Multiply the outside numbers: x times x, and 3 times 3; then multiply the insides 3 by x times 2. We say multiply the extremes, multiply the means (middle numbers), then add the means. Example 3: This is also called the FOIL method. FOIL means first outside inside last. Example 4: (c – 5) (c – 5) = (first) c2; (outside) – 5c; (inside) + − 5c = -10c; (last) + 25. All together: c2 – 10c + 25

FOIL other binomials

Example 5: (y + 2) (y – 4) = y2 -2y -8 Example 6: (d – 6) (-d + 2) = -d2 + 8d -12

Practice using the FOIL Method Work Mentally

1. (y + 3) (y – 3) 2) (a + 2) (a – 6) 3) (x + 6) (x + 4) 4) (a – 9) (a – 9)

How well did you do?

1. y2 – 9 2) a2 -4a -12 3) x2 + 10x + 24 4) a2 -18a + 81 Note: Generalities (a + b)(a + b) = a2 + 2ab + b2 (a – b)(a – b) = a2 – 2ab + b2 (a – b)(a + b) = a2 – b2

ALGEBRA 1

71

Divide Monomials

Example 1: 2a = 1 Example 2: a2 = 1 Example 3: b5 = b3 Example 4: 2x2 = 2x 2a a2 b2 x Explanations for examples 3 and 4: b · b · b · b · b = b3 2 · x · x = 2x b · b x

Subtract exponents when the bases are the same.

Example 5: -10x = -5 Example 6: 12y2 = 6y Example 7: 4x2y = -2x 2x 2y -2xy Example 8: 24 ab3c2 = 8abc Example 9: 15 xy2z3 = -3z2_ Example 10: 4a3b2 = b_ 3b2c -5 x2y3z xy 12a5b 3a2 -3 #9. 15x · y · y · z · z · z = -3z2 #10. 4a · a · a · b · b_ = _b_ -5 x · x · y · y · y · z xy 12 a · a a · a · a · b 3a2

Practice dividing monomials Fractions can be answers too!

1. 36x2y3 2) 5a4b2 3) 18b2c5 4) 7d4e2 3xy a2b3 6b4c2 28d6e4 5. -20abc4 6) x3yz2 7) 5q4rs3 8) – 25a2b3c6 5ab3c5 x4y2z - 35q2s2 − 75ab2c3

Divide Polynomials by Monomials

(Find Common Factors) 2

Example 1: 2l + 2b =2l + 2b = l + b Example 2: 5x – 10y = 5x - 10y = x-2y 2 2 2 5 5 5 Example 3: 18a2 + 3b3 – 6 = 6a2 + b3 - 2 3 Example 4: 4x2yz3 – 2xyz – 8x3y2z = 2xz2 – 1 - 4x2y 2xyz Example 5: 24a3b2 + 4a2b – 12a4b5 = 6a2b + a – 3a3b4 4ab

ALGEBRA 1

72

Long Division of Polynomials

(All the problems are taken from page 69)

Example 1: 35d + 49e we know that 7 is a factor of both 35 and 49; 7 we also know that we multiplied by d and by e __5d_+ 7e___ Work with one term at a time 7) 35d + 49e - 35d_______ Subtract as in a regular division problem 49e 49e Zero remainder Example 2: a3b3 ÷ ab2 Common factors are a and b. Think: how many times does a go into a3? How many times does b go into b2? __a2__b__ ab2) a3 b3 Work with one term at a time - a3______ b3 Subtract as in a regular division problem b3 Zero remainder Example 3: m3n3 ÷ m2n Common factors are m and n. m goes into m3 m2 times. n goes into n3 n2 times. __m__n2_ m2n) m3 n3 m3_____ n3 n3 This is a bit tricky but it is the thought process that helps in factoring. When we factor we separate the common monomials or binomials. Factors may be numbers or variables.

Factor Polynomials 1. 35d + 49e = 7(5d + 7e) 2. 9 + 3y = 3(3 + y) 3. 4x – 10 = 2(2x – 5) 4. 2a2 – 12x = 2(a2 – 6x) 5. 6(x-y) + a(x-y) = (6 + a) (x – y) 6. a(b-c) + x(b-c) = (a + x) (b – c) 7. m3n2 = m2n(mn) 8. 5a3 – 25a2 + 10a = 5a (a2 – 5a + 2)

ALGEBRA 1

73

On Your Own!

1. a2 – 2a 2) 3b2 – 12bc + 9c2 3) 4a2 – 8 + a2b – b 4) 15y2 – 5xy 5. 6y – 15 6) 4a2 + 12ab – 16b 7) 22r + 11s 8) 1 – y2

Check for solutions

1. a (a – 2) 2) 3(b2 – 4bc + 3c2) 3) 4 (a2 – 2) + b (a2 – 1) 4) 5y (3y – x) 5) 3 (2y – 5) 6) (2a + 8b) (2a – 2) 7) 11 (2r + s) 8) (1 + y) (1 – y)

Some Math Facts

1. Any number divided by itself is equal to 1: 2

2

77aa = 1;

10501050 = 1

2. Any number raised to the 0 power is equal to 1: 20 = 1 (4,235,167)0 = 1

3. Any number raised to the power of 1 is that number A1 = a; (27)1 = 27; (475,000)1 = 475,000

4. Order of operations: First do all operations that lie inside parentheses. Next, do any work with exponents or radicals. Working from left to right, do all multiplication and division. Finally, working from left to right, do all addition and subtraction.

5. Subtraction is addition of the negative (opposite) 2x – (7x) = 2x + (-7x) = -5x

6. Division is multiplication of the inverse

61 ÷ 8

3 is 61 × 3

8 = 188 = 9

4

Characteristics of Linear and Non-Linear Graphs Linear and non-linear graphs are created from equations.

We can tell the shape of the graph from the equation. Here we go!

1. Straight line (linear) graphs have a relationship between the independent variable (x) and the dependent variable (y). One is affected by the other. There is only one value for X for every value of Y. This can be read from the table of values. Specific trends are also clear. Sample Equation: The slope of this line is negative. y = -4x

ALGEBRA 1

74

2. Some linear graphs are not perfectly straight lines. They may still exhibit relationships between X and Y but not in a regular pattern. The graph of attendance in lesson 2 demonstrates a relationship between certain months of the year and attendance. Trends may not be as clear. y x 3. Non-linear graphs are curved. Some curved lines demonstrate relationships between the X and Y. Non-linear graphs that are affected by an independent variable may look like this. These shapes are elliptical. Sample Equation: y = x2 – 10x + 25 4. Other non-linear graphs may look like this. This shape is a parabola. Sample Equation: y = x2 – 1 Some graphs are symmetrical; others are not. Some of the non-linear graphs may have more than one value of X for every value of Y. All graphs that have one value of X for every value of Y are functions. Those that have more than one value of X for any value of Y in the same graph are not functions.

Use this part of the lesson as a preparation for the next one

ALGEBRA 1

75

LESSON 4-D: SOLVE PROBLEMS USING TWO FIRST DEGREE EQUATIONS

Example 1: y = 3x; y = -2x + 4. Graph these two equations and find the point of intersection. Step 1: Find ordered pairs for each equation. (Observe that the equations are in the form of y = mx + b) Create a table of values for both equations. y = 3x y = -2x + 4 x y x y 3 9 3 -2 0 0 0 4 -1 -3 -1 6 6 1 -3 -2 -1 0 1 2 3 -2 -4 Step 2: Find appropriate intervals and scales, set up the coordinates, and graph. Step 3: Read the coordinates where both lines intersect. x = 1; y = 2. (1,2) Example 2: Why do we say first degree equations? Every value is written to the first power. No value is x2. Graphs that have a second power curve. Example 3: What does the straight line mean? (Is this graph linear?) This graph is linear; in other words it makes a straight line. That means that there is a relationship between the x other words it makes a straight line. That means that there is a relationship between the x and y axis. The y value changes according to the x. Such situations are called functions. Example 4: Can you test for a function using the graph only? Yes, the vertical line test will determine whether a function exists. Example 5: How do you apply the vertical line test? At any point on the graph a vertical line must cross a horizontal line only once. Let’s say it another way: For every value of x there should be only one value of y. Example 6: Is there another way to use the vertical line test? Yes. In the table of values there should be only one x answer for each y.

ALGEBRA 1

76

Practice

For each set of equations create a table of values, graph each line, and find the point of intersection. Apply the vertical line test to determine whether a function exists. 1. y = 2x + 4; y = 2x 2) y = -3x; y -2x = 8 3) y – x = 5; -3 = -y + 4x The point of intersection is the solution of both graphs expressed as an ordered pair. In problems 4, 5, and 6, find the ordered pair that is a solution of both graphs. 4 6 2 2 -8 -4 0 4 8 -4 -2 0 2 4 -1 -2 -1 1 2 -2 Scales: X = - 8 to 8 X = - 4 to 4 X = -2 to 2 Y = - 1 to 3 Y = - 4 to 4 Y = -2 to 4 Intervals: X = 1 X = 2 X = 1 Y = 4 Y = 2 Y = 2 We can graph inequalities as easily as we can graph equations. Recall that an inequality is a statement that contains an inequality sign. 4 > x is an inequality. We don’t know what x is equal to but we do know that it is more than 4.

Review graphs of Inequalities

y < x + 2 Imagine that y = x + 2 x y Create a table of values for the equation y = x + 2 3 5 1 3 -1 1 Graph these points and then shade all values below the graph y 5 * 3 * -1 0 1 3 x

ALGEBRA 1

77

The solution lies in the shaded area and shows all the points where y < x + 2. This lesson focuses on graphing a system of 2 equalities and locating the intersection of the graphs or the ordered pairs that form the solution.

Activity The shaded area shows where both graphs

intersect. The values assigned to the point of intersection would be solution for both graphs. Your answer should look like this. Find the ordered pairs (table of values) for each system of inequalities, graph them, and find the solution for both graphs. 1. y > x + 1; y > -x + 3 2) y > x + 1; y > x – 3 3) y < 2x - 5; y < x - 3 4. Is there one value of x for each value of y? Y/N 5. Is the graph of an inequality a function? Y/N 6. What will the vertical line test reveal from the graph of an inequality? 7. Change each equation to the form y = mx + b: (a) y + 4 = x (b) 2x – 8 = y (c) 3x + y = 5 (d) - 6 + y = 2x 8. Find the ordered pair that is the solution of both graphs: 2 -6 - 4 -2 2 4 6 2 4

ALGEBRA 1

78

Solutions and Partial Solutions

1. y > x + 1 y > - x + 3 6 If y = x + 1 If y = -x + 3 * x y x y 2 4 5 4 -1 -4 2 4 6 2 3 2 1 0 1 0 3 -2 1 2 1 2 Solution = 1, 2. Prove by substitution: See table The area shaded above the intersection shows all values for y > x + 1 and y > -x + 3 Here is the table of values only for 2 and 3. 2. y > x + 1 y > x – 3 3. y < 2x – 5 y < x - 3 If y = x + 1 If y = x - 3 If y = 2x - 5 x y x y x y x y 4 5 4 1 4 3 4 1 2 3 2 -1 2 -1 2 -1 Solution set 0 1 0 -3 0 -5 0 -3 The graph of Problem 2 will be different from the graph of Problem 3. In Problem 3 the solution set shows up in the table of values. Draw the graphs of Problem 3 for the practice, but in a test if the solution shows up in the table of values it is not necessary to draw the graphs. 6 The graphs do not intersect. There is no solution. In the other 2 problems there were many 2 solutions. -4 0 4 -2 The graph of Problem 2 is unusual and shows that there are no solutions for some graphic problems. Can you tell before you begin to graph? YES, YOU CAN! If the graphs of 2 equations have the same slope, they never will intersect!

ALGEBRA 1

79

4. There are many values of x for each value of y. 5. The graph of an inequality cannot be a function because there may be many values of x for each value of y. 6. See #5. 7. (a) y + 4 = x y = x – 4 (b) 2x – 8 = y - y = -2x + 8 y = 2x – 8 (c) 3x + y = 5 y = - 3x + 5 (d) – 6 + y = 2x y = 2x + 6 8. Solution for the graphs: (2, 2.5)

For more on inequalities go to: http://www.purplemath.com/modules/syslneq.htm

Matrices

Matrices are ways to write numerical data. There are many types of matrices but we will learn just one basic type in this lesson. A matrix is a concise and useful way of uniquely writing data in a rectangular format. We can even write an equation as a matrix. The data is written in rows and columns that are specific. The symbol for writing a matrix is: [ ]. These are called square brackets. Matrices are identified by their sizes. A matrix that has 3 rows and 3 columns is a 3 x 3 (3 by 3) matrix. If there are 3 rows and 2 columns it is a 3 by 2 matrix. Rows come first always. A 3 by 3 is also a square matrix because there is the same number of rows as there are columns. Note: rows are horizontal and columns are vertical. Example 1: is a 2 x 3 matrix Example 2: is a 3 x 4 matrix Example 3: is a 4 x 2 matrix

3 0 1 1 2 3

1 0 4 5 5 2 3 1 6 9 4 3

2 0 1 3 4 2 9 1

ALGEBRA 1

80

Example 4: is a 3 x 3 matrix or square matrix A A matrix is identified by a letter of the alphabet. This letter is always in upper case or capitals. Let’s call the matrix (above) A. Each number in the matrix is called an entry. We can locate an entry by the place it holds in the row and column. Using matrix A above, find entry A1, 1. The answer looks like this: a1,1 = 3; three is the value in the first row of the first column. Notice that we use lower case “a” to record the answer. Find entry A 1, 3. 7 = a 1, 3 Locate entries A 2, 2. Answer: a 2, 2; 5 We can write a system of equations as a matrix. Think of the y = mx + b equation. Now let’s borrow a pair of equations from one of the earlier lessons. Y = x + 1; y = -x + 3. The matrix, C will be a coefficient matrix and we can write only the coefficients: What is the size of this matrix? What is entry C 2, 1. and C 2, 3? 2 x 3 matrix Answers: 1 = c 2,1 3 = c 2, 3 C Create matrices for the following systems of equations: 1. y = 2x + 4; y = x – 3 2. x + y = 4; y + z = 3; z – y = 1 3. What kind of matrices are these? 4. What is the size of each matrix? 5. Create 4 problems that locate entries in each matrix. There should be two problems per matrix.

LESSON 4-E: REVIEW (ALGEBRAIC THINKING)

1. What is the difference between 25 and 52? 2. How does substitution help to prove accuracy in the solution of an algebraic problem?

(One sentence) 3. Complete the next 3 numbers in the series: 35, 33, 31, _______, ______, ______ 4. A negative power is a (a) fraction (b) base (c) the square root (d) the square 5. The measures of central tendency of a data set are: Mean = 85, Median = 78 and

Mode = 70. Which of these is a good representation of the data? (a) The mean (b) The median (c) The mode (d) None of them

6. How is a stem and leaf plot similar to a line plot? (One sentence) 7. The most appropriate graph when comparing data is ______________________

3 8 7 6 5 4 1 2 3

1 1 1 1 -1 3

ALGEBRA 1

81

8. The most appropriate graph indicating a relationship between 2 variables is __________________________________

9. If I want to demonstrate performance of different parts of data I should use a _________________________________________

10. Measures of dispersion are (a) Numbers that show data centered around the mean (b) Numbers that show data scattered or spread away from the mean (c) Numbers that show the mode of the data (d) Numbers that show trends.

11. A box and whisker plot demonstrates measure of central tendency T/F 12. Name 1 graph that provides only partial data. ____________________________ 13. How does the median of a box and whisker plot divide the data? _____________ 14. How do the quartiles divide the data in a box and whisker plot? ______________ 15. The data lists sales performance for a company during the year in order from January

to December: $500,000; $150,000; $100,000; $250,000; $250,000; $100,000; $300,000; $350,000; $250,000; $300,000; $450,000; $700,000. Find the mean, median, and mode of the data.

16. Which 5 elements do you need in order to create a box and whisker plot? 17. Is there a trend evident in the data in #15? 18. What might account for lower sales during February, March, and June? 19. Add or subtract: (a) 4y2 + 2y – 5y2 – 6 – 4y (b) (ab3 + ab2 – 6a + 9) – ( 2ab2 + ab3 + 3a

– 4) 20. Multiply using FOIL: (x + 1)2 21. Factor: 6x2y + 3xy – 15 22. (2x3y2 + 12xy2) ÷ 2xy 23. What is the vertical line test and how is it useful? 24. What is the shape of this equation when graphed: y = a2 + 9? 25. Graph and find the solution set of: y = -x; y = 3x – 4 26. Is there a solution for this system of equations: y = x + 5; y = x – 3? 27. Graph the system of inequalities and shade the area of intersection: y > 3x;

y > -x + 4 28. Arrange the system of equations in #25 as a matrix (A). What is its size? 29. Locate A 1,2 and A 2,1 30. What is a matrix? 31. How can you tell if there is a solution before you graph a system of equations?

ALGEBRA 1

82

ALGEBRA 1

83

LESSON 5

DATA ANALYSIS AND PROBABILITY

LESSON 5-A: MANAGING INFORMATION

We have already learned much about managing information in previous lessons. This lesson will help us to bring it all together in a more meaningful way. There will be a number of questions and answers, and in addition, some real world examples that make it relevant. 1. How do we manage information? Information is usually contained in numbers and we arrange those numbers in groups, individually, in graphs and charts, matrices, and other statistical formats. These different arrangements allow us to make statements, conclusions, or simple observations about data. 2. What are some things we observed as we attempted to manage information? (a) We observed that one number can sometimes be used to represent a large data set (b) Graphs can be helpful in demonstrating information not seen as well in other formats (c) Some graphs do not reproduce all the original data and we may be limited in what we can tell from a graph (d) Graphs have specific purposes (e) Some graphs are more appropriate for large amounts of information and others are more useful with smaller data sets (f) Intervals and scales must be specific and consistent, otherwise information can be misleading (g) Trends or the direction information takes may be better observed with some graphs compared to others. 3. What are some statistical measures used to help us manage data? Measures of central tendency and measures of dispersion are some basic statistical measures used so far. 4. What is the difference between these two? Measures of central tendency or the mean, median, and the mode show data bunched around the mean; measures of dispersion or range, variance, and standard deviation show how data is scattered or spread away from the mean. 5. We know mean, median, and mode well but what about range, variance, and standard deviation? Range will be used in this lesson. Later at college we will learn about variance and standard deviation. 6. Do we use these in everyday life? We do! Some practical examples will introduce ways we use this everyday without even knowing.

Mini Project Imagine that you want to be on a basketball team. There will a selection process and you must come to the selection process with information about your ability to score consistently well. You must determine your probability of success at scoring or you may not be selected on the team. You know you must practice, but you must show what you have done in an organized manner. This is exactly the place to get this accomplished. LET’S GET BUSY!

ALGEBRA 1

84

Step 1: Record your trials; number of successful shots in groups of 10

Day 1 Day 2 Day 3 7 70% 8 80% 7 70% 7 8 8 6 8 9 8 7 9 9 9 8 7 9 9 7 10 8 6 7 8 8 8 7 8 8 7

Mean = 93/10 = 9.3

93% 82/10 = 8.2 82% 80/10 = 8 80%

Day 4 Day 5 Day 6

9 90% 8 80% 8 80% 9 8 8 8 7 7 9 9 9 8 9 9 7 8 9 7 8 8 8 9 9 8 9 8 9 8 8

82/10 = 8.2 82% 83/10 = 8.3 83% 83/10 = 8.3 83% Continue the trials until you have 10 sets of 10. You will be using average data;10 is a small number from a statistical viewpoint. Larger data produces more reliable results. Step 2: Find mean, median, mode, and range of the data. List the average daily percentage score; there should be 10. The next 4 will be given; you should have the real deal: 93, 82, 80, 82, 83, 83, 80, 84, 92, and 90. Numerical order: 80, 80, 82, 82, 83, 83, 84, 90, 92, 93 Mean: 849 ÷ 10 = 84.9; Median = 83; Mode = 80, 82, 83 (Remember there can be more than one mode). This is a tri-modal data set. Range = 93 – 80 = 13 Step 3: Organize your data in graphs and respond to questions about it. 1. What do you want to show in the graphs you choose? What type of graphs would show what you need? 2. How could you use the data to establish your probability of success? This type of probability is called Frequency of Occurrence. 3. Do you demonstrate a trend? Is the trend visible in a graph? Explain.

ALGEBRA 1

85

Step 4: Clues you can use 1. A line graph, scatter plot, line plot, or even a bar graph could be used to display the data. 2. Your probability of success can be determined by looking at the level of consistency. If you score successful shots 80% of the time you can say that your probability of success is 80%. 3. The probability of success for a basketball player means that he/she will do well as long as there are no injuries or the player continues to practice. Human behavior can affect the probability rate. This is one practical way that we express probability on a daily basis. Thoughts and Comments: Players are traded or selected during a draft based on their proven probability of success. Managers and owners offer big bucks based on a player’s proven probability. Injuries or other factors can change the reality and rate of success. We will work more on probability in lesson 2.

Activity Use the data to answer the questions: Data on attendance for a particular school for 2 academic years (%): A S O N D J F M A M J Yr 1: 98 97 97 92 94 91 96 97 92 97 96 Yr 2: 96 98 97 90 95 92 97 98 90 97 96 SEE PAGE 97 FOR ANSWERS. 1. Compare the means of both years. Were they similar? How? 2. What was the range for year 1 and year 2? 3. Can we tell from the data what accounted for the difference in attendance at the start of the school year? 4. Is there a clear trend? Show in a graph of your choice. 5. Many school systems put specific policies in place to increase attendance. In your opinion, do you think these policies have an impact on student attendance?

LESSON 5-B: PROBABILITY Types of Probability 1. Frequency of occurrence: Examples like the shooting rate are based on frequency of occurrence. This type of probability can change if the human factor is involved. We can have frequency of occurrence without the human factor. Probability based on frequency of occurrence without the human factor: What is the chance that the sun will shine tomorrow? 100% What is the chance that night will come? 100%, but it depends on the season and where you are. What is the chance that winter will be cold? It depends on where you live. What is the chance that winter in Maine will be cold? 100% What is the chance that rain will fall everyday in the Amazon? My guess 99%

ALGEBRA 1

86

How can we be so sure that these events will take place? These are scientifically proven facts that these events will occur. We can depend on the sun shining everyday whether we see it or not, or that winter in Maine will be cold, whether we experience it or not. 2. Classical or Simple Probability: In this type we record the favorable outcomes or chances of success as a ratio or fraction of the total outcomes. 2 1 3 3 4 6 6 5 4 2 1 5 0 4 2 4 This is the type you see most on test questions. If you can change 2/6 to a decimal, percent, or ratio, your probability of being accurate on this type is high. Another frequent probability scenario is the deck of cards. There are 52 cards in a deck. Each deck has 13 clubs, 13 diamonds, 13 spades, and 13 hearts. The chances of pulling 1 club are 13 out of 52 or 1 in 4. There are 4 kings, so the chances of getting a king are 4 out of 52 or 1 in 13. To do this one must know the characteristic of a deck of cards. Each question relating to a deck of cards depends on some characteristic of the deck and can affect the probability. It requires some thought before we answer such questions. We write the answers to questions like this: p(4 clubs from a deck of cards) = 4/13

Practice classical probability problems

1. A spinner had 8 equal spaces with 3 twos, 1 five, 2 zeros, 1 four, and 1 three. Draw the spinner and place the numbers.

2. What is the probability of getting 2 on a spin? 3. How many chances do I have to get zero? Express as a fraction and a percent. 4. The probability of obtaining 4 and 3 is 1:4. Is this answer correct? Explain.

A spinner had these numbers on it. What is the probability of getting 1 or 5? The answer would be 2 out of 6. There are 6 chances and there is 2 chances of getting either of those numbers. We may express probability in any of the 4 fractional formats.

Would the chances of getting 4 be the same as getting 1 or 0? NO. Is the probability of getting 5 the same as getting 2? NO. There are more twos than fives; more fours than ones.

ALGEBRA 1

87

5. A deck of cards has 13 each of clubs, hearts, diamonds, and spades; 4 kings (one of each suit); 4 aces (one of each suit); 4 queens (one of each suit), 4 jacks (one of each suit), and 4 of each number from 2 to 10.

a) Find the p (all clubs) b) What is the p (4 aces) c) What is the p (1 queen of hearts and 1 queen of diamonds) d) P (all queens and jacks) e) P (5 of spades)

Dice problems

Each die (a cube) has 6 faces. On each face are small dots: either one dot, two dots, three dots, 4 dots, five dots, or six dots. Two or more of these cubes are called dice. They are picked up, shaken, then rolled onto a surface.

6. What is the probability of rolling an even number on a die? Since a die has 6 numbers

with 1 to 6 on each face, there will be just 3 even numbers on it. The p (even numbers on the roll of a die) = 3/6 or ½ 7. What is p (1 or 4)? 2/6 or .33 or 33% . There are two ways to succeed; a 1 or a 4. 8. The p (5) = 1/6 9. p (a number < 6) = 5/6 10. What are the chances of rolling a number > 3? P = 3/6 or ½

Marbles or M and M’s

11. In a bag of 20 marbles there were 4 red, 3 blue, 5 green, 1 black, 6 orange, and 1

multi-coloreds. What are my chances of taking a green marble out of the bag? 5/20 or ¼ .

12. What is the probability of getting marbles other than orange? 14/20 or 7/10 13. Express as a percent the probability of obtaining only red marbles. 20%

Another type of probability is called permutation. A combination is a variation of a permutation. So whenever we work with permutations we use combinations.

We can play this as a game. Take 3 cards and write the letters I, T and S on each. This arrangement is called a permutation

In how many ways can you arrange the cards differently with no two being alike? 1. tis 2. sit 3. sti 4. tsi 5. ist 6. its

I T S

ALGEBRA 1

88

Suppose we had just 2 cards with A and B on them. How many different ways would we get BA? 2 cards = 1 way that’s different What would we get with 5 cards? How do we figure it out? The 5 cards are: A B C D E Row 1: Keep the arrangement A B C D E Row 2: Keep the first letter the same, change the rest A C D E B Row 3: Keep the first letter, change the rest A D E B C Row 4: Keep the first until we have moved all the others around A E B C D Row 5: Now start with the second letter B C D E A Row 6: Keep the second letter B D E A C Continue… How many did you get? (Should be 120) 5 x 4 x 3 x 2 x 1 = 120 Number of choices for card no. 1 2 3 4 5 Work your own permutation problem: A pizza shop offers pizza with at least 4 toppings: cheese, olives, pepperoni, and mushrooms. In how many ways can you order pizza with 3 of these choices? How many combinations of 4 things can you make using 3 at a time? This problem can be worked by using combination theory C(n, k) = n!___ k!(n-k)!.

C( 4,3) = __4!___ = 4 x 3 x 2 x 1 = 4 3!(4-3)! (3 x 2 x 1)(1) This problem is different from the other one that asks in how many ways can you arrange 3 different objects. Students confuse one with the other. Both are simple and can be solved with accuracy. When should you use the factorial to solve? 1. In how many ways can you arrange 4 objects so that no two arrangements are alike? 2. How many combinations of 4 gifts can you select from 5 choices? Answer: Use combination theory for #2 only. The answer to the first problem is a triangular number. Solution for #2: C(5,4) = ___5!__ = 5 x 4 x 3 x 2 x 1 = 5 4!(5-4)! (4 x 3 x 2 x 1)(1)

Combination Problems 1. How many combinations of 3 pizza toppings can you get from 5 choices?

C (5, 3) = 5! = 5x4x3x2x1 = 10 . 3!(5-3)! (3x2x1)(2x1) 2. How many combinations of 4 students can I get from a 6-person swimming team?

C (6,4) = 6! = 15 4!(6-4)!

Want some neat probability problems? Go to the web at http://www.edhelper.com/math/probability_tg701.htm http://www.edhelper.com/math/probability_tg704.htm http://www.edhelper.com/math/probability_tg703.htm http://www.edhelper.com/math/probability_tg790.htm

ALGEBRA 1

89

LESSON 5-C: LESSON 5 REVIEW AND SELF-CHECK

1. How do we manage information? 2. Name 3 things we may observe when managing information. 3. What is the difference between the measures of central tendency and the measures of

dispersion? 4. Name 2 occasions when we use probability in real life. 5. How will we know when the mean is a good representation of the data? 6. A student entered Spelling Bees every year for the last 5 years. Each year he made it to

the final round. His accuracy rate was 97%. What is the probability that he would be in the final round next year? What kind of probability is that?

7. Name another type of probability and give an example. 8. A spinner at a spring fair had 10 equal sections: 4, 4, 3, 5, 7, 6, 8, 0, 1, 2. Find: p(4). 9. What is the probability of getting a prime number on the roll of a die? 10. How many handshakes will there be among 3 people if each person shook hands with

everyone else once? 11. Jan, Ted, Mary, Bill, and Tom entered a Spelling Bee competition. How many 4-people

combinations can be represented on the team? 12. What is a permutation? 13. What is a factorial?

ALGEBRA 1

90

COMPREHENSIVE REVIEW

1. List formulas from Unit 2 (Pages 15 – 21) 2. Draw an example for each different formula; practice substituting numbers for the variables and calculate the answers. There must be a unit of measure for each answer: cm, inches, feet, square feet, square yards, and so on. 3. Short or abbreviated version for: feet = ft; inches = ins; yards = yds; mile = ml; centimeter = cm; meter = m; square feet = sq ft or ft2; cm squared = cm2; square inches = ins2 or sq ins 4. How would you find length if area and width are given? Would the answer be 2-dimensional or 1-dimensional? 5. Use the graph paper to draw and label the x and y axes, quadrants, and the value of the coordinates in each quadrant.

ALGEBRA 1

91

6.

Figure Measures Perimeter Area Square QRST 5 units by 5 units 20 units 25 square units Square WXYZ 10 units by 10

units 40 units 100 square units

Ratios of similar polygons

2 to 1 2 to 1 4 to 1

Congruent polygons have same shape, size, area, and perimeter. Observe the ratio of perimeters of the larger to the smaller is 2 to 1, also written 2:1 Observe the ratio of areas of the larger to the smaller is 4 to 1, also written 4:1. The perimeter of a polygon that is dilated keeps the same ratio as the enlargement or reduction. If a polygon is made 2 times larger, the perimeter will be 2 times the original. If it is made 2 times smaller, the perimeter will be 2 times smaller that the original. Will this work with all polygons? 7. Complete the sentences:

(a) The area of a large polygon is _____________ times the area of a similar smaller polygon. (b) If the measure of one side of a rectangle is 9cm, the measure of the corresponding side of a rectangle 3 times smaller is __________________ cm. These rectangles are (a) similar (b) congruent (c) none of the above. (c) A set of 2 similar triangles was drawn in a design. One of the triangles was ¼ of the other. If the measures of the smaller triangle were 5, 7, and 12 cm, the measures of the larger triangle = ________, __________, and _____________ (d) What would be the ratio of the area of the smaller to the larger? ___________

8. Study conversion tables on page 35. 9. Parallel lines create corresponding and alternate angles when intersected by a transversal. Corresponding angles come in pairs and are congruent. Alternate angles come in pairs and are congruent. 10. Complete the sentences: Perpendicular lines meet to form ___________________________________ Parallel lines never meet because ___________________________________ A transversal is _________________________________________________ Corresponding angles are _________________________________________ Angles formed on a straight line equal _______________________________ Supplementary angles equal _______________________________________ Two right angles formed on the same line are called ____________________ Parallel lines are __________________________ from one another. Parallel lines form corresponding angles when crossed by a ______________ A transversal crossing parallel lines forms ____________________________

ALGEBRA 1

92

ANSWERS TO PRACTICE QUESTIONS

Let’s Practice, page 20 1. Use the formula for area of a trapezoid: ½ (b1 + b2)h = ½ (15 + 12)8

= ½ (27 · 8) = ½ (216) = 108cm2 Area of triangle A = ½ (15)(8) Area of triangle B = ½ (12)(8)

2. The longer base = 5 feet + 3 feet = 8 feet. 3. Distance around the trapezoid = 5 + 4 + 5 + 3 + 7 = 24 feet.

5 ft 4 ft 7 ft 5 ft 3 ft

4. Radius Diameter Circumference Area 10cm 20 cm 62.8 cm 314 cm2 7 feet 14 feet 44 feet 154 ft2

21 inches 42 inches 132 inches 1386 ins2 Use 22/7 for pi for 2 of the 3 problems.

5. If the area of a square was 36 sq ft, the measure of each side = √36 = 6 ft (What number times itself equals 36?)

6. r = ½ d = 150 ÷ 2 = 75 yards 7. r = 28 feet; c = dπ = 2 · 28 · π = 56(22/7) = 176 feet C = 2rπ same as dπ 8. A chord may not go through the center but a diameter must pass through the center of a circle. 9. Consider the dog’s leash the same as the radius. Area = πr2 = 72 (22/7)

= 154 sq ft 10. The dog’s leash is the radius of the circle.

Lesson 2-B Review, page 26 1. A, B, and D have area but no volume. 2. An object has volume if there are 3 dimensions: Length, width, and height or depth. 3. The best way to find area of this shape is to draw a line to separate the two shapes and find the area of

each shape separately. Add both areas. 4. A rectangle and triangle make up the arrow. Area of the rectangle: LW; area of the triangle: ½ (bh) 5. To find the area of a cube: Multiply l by w of the top; multiply l by w of one side; multiply l by w of another

side; add those areas and then multiply by 2. 6. To find area of a cube: find the area of the 3 different faces, add them and then multiply by 2.

ALGEBRA 1

93

7. radius = 4 cm height = 10 cm 8. Area of cylinder = Area of top + area of bottom + area of rectangular part = r2π + r2π + (πd · h)

= 2(3.14)(16) + 3.14(8 · 10) = 100.48 + 251.2 = 352 cm2 9. V = A · H 10. The track is made up of a rectangle bordered on two sides by two half circles. We must find the area of the

rectangle and add the area of two half circles (one circle). We need radius for the area of the circle (divide the diameter by 2). lw + πr2 = (98 · 70) + (22/7 · 35 ·

35) cm2 = 6860 + 3850 = 10,710 cm2 11. First find out how much time it would take at that speed for that distance, then work backwards to determine

starting time. D =rt; and t = d/r; d = 105 ml; rate = 70 mph; t = 105 ÷ 70 = 1.5 hrs. It will take 1.5 hours to drive that distance. If the family must get there by 5:00 pm, they must leave home by 3:30 pm. 12. Principal = $15,000; Rate = 5%; Time = 3 years; I = P x R x T; A = P + I 15,000 (.05) 3 = $2250; I =

$2250; A = $15,000 + $2250 = $17250 13. The monthly installment on the loan = $17250 ÷ 36 months = $479.17 (rounded) 14. This problem is just like # 12. 15. The young men suffered a loss. Total expenditure, $200, was greater than their revenue $180. 16. Follow the table and insert each item. The numbers in the response at #15 should match the numbers in the

table. Loss: Month 1 = Total revenue – total expenditure = $75 − $100 = −$25 (loss). Loss% = 25/75 x 100 =

33 1/3%; Month 2 = TR – TE = $105 − $100 = $5 (Profit). Profit% = 5/105 x 100 = 4.8%. To find the total profit or loss over both months:

TR (2 months) – TE (2 months) x 100

Complete the Sentences, page 41 1. right angles 2. They are equidistant (same distance apart) 3. A line that crosses a set of parallel lines 4. Congruent 5. 180º 6. 180º 7. Equal supplements 8. Equidistant 9. Transversal 10. Alternate and corresponding angles Complete each statement, page 42 1. 130º 2. 50º 3. 50º 4. Supplementary 5. 130º 6. Congruent 7. Congruent 8. Congruent 9. Yes 10. Yes

Area of top = πr²; area of bottom = πr². Shortcut = work the area of the top and multiply by 2. Rectangular part = height x width; width is same as circumference. Radius is given; multiply r by 2 to get diameter. (d = 2r). Circumference = πd.

ALGEBRA 1

94

Activity, page 43 Corresponding angles: 12 and 5; 12 and 14; 11 and 2; 13 and 7; 14 and 8 Vertical angles: 2 and 5; 1 and 7; 3 and 4; 6 and 8 Alternate angles: 12 and 4; 13 and 1; 91º and 6; 16 and 6 Find the measures: 1 = 91º; 2 = 99º; 3 = 91º; 4 = 91º; 5 = 99º; 6 = 99º; 7 = 91º; 8 = 99º; 9 = 50º; 11 = 99º; 12 = 99º 13 = 91º Solving congruency problems, page 46 C D 1. 4cm B 7cm 7.5cm 7.5cm 7cm E

A F 4 cm These triangles are congruent. C F 2. 50º 4cm 4cm 4cm 4cm 50º 50º A B D E These triangles are not congruent. If the base angles in the first are 50º, then the third angle would be 80º. The third angle in the second triangle is not 80º. 3. A D 35º 35º 5cm 5cm 90º 90º B C E F The reason for congruency is ASA (angle, side, angle). See reason #4 in the lesson. 4. G Q 75º H R 75º I S GH ≅ QR; angle H ≅ R; HI ≅ RS. These triangles are congruent. SAS

NOTE: IN ΔDEF, <FDE = <DEF (angle opposite equal sides in an isocles triangle are equal) <FDE + <DEF = 130 o; (180o – 50o). Then <FDE = <DEF = 65o

ALGEBRA 1

95

Review of lesson 2, page 47 1. 4 reasons to demonstrate when triangles are congruent are SSS, SAS, ASA.

2. Corollaries: (a) If 3 sides of a triangle are congruent, the 3 angles of the same triangle are also

congruent. (b) If the 3 angles of a triangle are congruent, the 3 sides of the same triangle are also congruent. (c) Angles opposite congruent sides are congruent. (d) Sides opposite congruent angles are congruent.

3. (a) F (b) T (c) T (d) T (e) F (f) T (g) F (h) F

4. Triangle 1: LM ≡ XY (given); angle M ≡ angle Y (both are right angles; given); MN ≡ YZ (subtraction; information given). Both triangles are congruent SAS.

H K 3cm 3cm 60º 60º 55º M I J L 5 cm

5. Triangle 1 Triangle 2 HI = 3cm (given) KL = 3cm (given) Angle I = 60 (given) Angle L = 60 (given) IJ = 5cm (given) LM = ? (not given) SAS SA ? (corresponding information not given) There is not enough information to determine whether these triangles are congruent.

Unit 3 Review and Self Check, page 57 1. Refer to page 39 2. Refer to page 39 3. Refer to page 39 4. There may be many different answers. The trash can, soda cans, gift boxes, bookshelves, and similar all

have perpendicular and parallel relationships among them 5.

Any combination of two angles that add up to 180 º Angle A + Angle B = 180 º

A B

6. A transversal is a line that crosses over or intersects parallel lines. Transversals form pairs of alternate, corresponding, and vertical angles.

7. Corresponding angles: A and I, B and J, E and M, F and N, C and K, G and O, D and L, H and P; Alternate angles: E and J, F and I, B and M, A and N, G and L, H and K, C and P, D and O; Vertical angles: A and F, B and E, I and N, J and M, C and H, D and G, K and P, L and O.

8. Page 57. 9. Vertical angles are angles formed in congruent pairs when two lines intersect and cross over each other.

They are located opposite each other and that makes it easy to identify them. 10. Vertical angles are not formed at the intersection of these two streets. They meet but do not cross over and

vertical angles are formed when the lines cross over each other. 11. Angle R = 58º; X = 58º; Y = 58º; S = 64º; Z = 64º; QS, XZ, RS and YZ are all transversals. YZ ≅ RS; XZ

≅ QS. Triangle QRS ≅ XYZ because of (a) SAS. 12. Refer to page 39. 13. We can prove triangles congruent by (b) SAS

ALGEBRA 1

96

14. Triangle DEF Triangle RQS Line ED ≅ RS (given)

Angle F = 89º Angle Q = 89º (given) Line FE ≅ QR (given)

Because of the location of the corresponding parts we get SSA. SSA is not a reason for congruency. These triangles are not congruent.

15. A G

B C H I <A = <G = 90º These triangles are congruent by ASA. AB = GH highlighted side is the side in between.

16. The corollary to SSS: If 3 sides are congruent, the angles are also congruent. Page 43. 17. (a) F (b) T (c) T (d) F (e) F (f) T (g) F (h) T (i) T (j) T (k) T (l) F

(m) T (n) F (o) F 18. The coordinates of the ramp are (-3, 1), (6, 4).

• Rise over run = 3/9 or (slope formula: 4 – 1 = 3 = 1 6 – (-3) 9 3 • Height of the ramp = 3 feet • Mid-point of the ramp: 1.5, 2.5 • Length of the ramp: C = √272 + 2 1/32 = 729 + 9 = √738 = 27.18 feet • The ramp occupies 27 feet on the ground. • The ramp would occupy an area of 34 3/4 square feet

Algebra Basics self Assessment, page 62 1. a plus b 2. b minus a 3. 3a 4. c divided by d 5. t times t 6. 2(l + b) 7. a = 3 8. 10 9. 6 10. 9 11. 193 12. 52 13. 7 14. -6 15. 141 16. Profit 17. 20% 18. Distributive 19. Additive inverse 20. T 21. F 22. F 23. T 24. T 25. F 26. F 27. 1 28. Increases 29. Increases 30. c 31. T 32. T 33. a 34. d 35. 25

Divide polynomials by monomials, page 69 1. 12xy2 2) 5a2 3) 3c3 4) __1__ 5) - 4 6) __z___ b b2 4d2e2 b2c xy 7. – q2rs 8) abc3 7 3 Unit 4 Review, page 80 1. 25 = 32; 52 = 25; difference = 32 – 25 = 7 2. Use the solution in the original equation to make the statement true. 3. 3-1, 3-3, 3-5. 4. A negative power is a fraction 5. (d) none of them 6. The data in both graphs is the same.

ALGEBRA 1

97

7. The bar graph 8. The line graph 9. Box and whisker plot 10. (b) numbers that show data scattered or spread away from the mean 11. F 12. Box and whisker plot 13. Into 2 halves 14. Into quarters 15. Mean = $308,000; Median = $275,000; Mode = $250,000 16. The 5 elements are: LE or lower extreme, LQ or lower quartile, Median, UQ or upper quartile, and the UE or

upper extreme. 17. Yes, sales increase during the end of the year. 18. Sales during February and March could reflect what happens after a holiday. In June it could be the effect of

summer and before the Back to School rush. 19. (a) – y2 – 2y - 6 ;

(b) –ab3 -2ab2 -3a +13 20. (x + 1)2 = (x + 1) (x + 1) = x2 + 2x + 1 21. 3xy (2y + 1) -15 22. x2y + 6y 23. The vertical line test is used to determine when an equation is a function. It is useful because it

demonstrates whether there is one value of x for every value of y. 24. Parabola 25. x y x y *

4 -4 4 8 6 2 -2 2 2 *

0 0 0 - 4 -4 -2 0 2 4 1 -1 1 -1 * -6 y = -x y = 3x – 4; Solution set = 1, -1 26. There is no solution set for this system of equations. 27. y > 3x; y > -x + 4

If y = 3x If y = -x + 4 9 * 3 9 3 1 2 6 2 2 6 1 3 1 3 3 * 1 2 3 Solution set = y > 1, 3 28. y = -x; y = 3x – 4 1 -1 0 This is a 2 by 3 matrix

1 3 -4

29. A 1,2 = -1; A 2,1 = 1 30. A matrix is a rectangular arrangement of data. 31. Lessons about graphing systems: Several things “give away the evidence” as to whether there is a solution

or not: • The value of m in both equations. If m is the same number, the slopes are the same, the graphs will be

parallel and there will be no solution. • If 2 x and y values in the table of values are identical, there is the solution. (2, -3) and (2, -3)

ALGEBRA 1

98

• If you have a graph already, read the rise over run values and determine slope. • The solution to a system of equations may be in the slope (coefficient of x), the table of values, or the

graph itself. Activity, page 85 1. Mean for year 1 = 95.18; mean for year 2 = 95.09. They are similar; in fact the numbers are almost exactly alike. 2. Year 1 range = 7; year 2 range = 8. (3) One cannot tell from the data what accounted for the difference in attendance; we speculate and give opinions in questions like these. 98 * * * Year 1 * * * * * 96 * * * 94 * Year 2 * 92 * * * 90 * * 88 * A S O N D J F M A M J

This is a double line graph

ALGEBRA 1

99

COURSE OBJECTIVES

The purpose of this course is to develop the algebraic concepts and processes that can be used to solve a variety of real-world and mathematical problems. The student will:

• Associate verbal names, written word names and standard numerals with integers, rational

numbers, irrational numbers, real numbers and complex numbers.

• Understand the relative size of integers, rational numbers, irrational numbers and real numbers.

• Understand concrete and symbolic representations of real and complex numbers in real-world

situations.

• Understand that numbers can be represented in a variety of equivalent forms, including

integers, fractions, decimals, percents, scientific notation, exponents, radicals, absolute value

and logarithms.

• Understand and use the real number system.

• Understand and explain the effects of addition, subtraction, multiplication and division on real

numbers, including square roots, exponents and appropriate inverse relationships.

• Select and justify alternative strategies, such as using properties of numbers, including inverse,

identity, distributive, associative and transitive, that allow operational shorcuts for

computational procedures in real-world or mathematical problems.

• Add, subtract, multiply and divide real numbers, including square roots and exponents, using

appropriate methods of computing, such as mental mathematics, paper and pencil and

calculator.

• Use estimation strategies in complex situations to predict results and to check the

reasonableness of results.

• Use concrete and graphic models to derive formulas for finding perimeter, area, surface area,

circumference and volume of two-and three-dimensional shapes, including rectangular solids,

cylinders, cones and pyramids.

• Use concrete and graphic models to derive formulas for finding rate, distance, time, angle

measures and arc lengths.

• Relate the concepts of measurement to similarity and proportionality in real-world situations.

ALGEBRA 1

100

• Select and use direct (measured) and indirect (not measured) methods of measurement as

appropriate.

• Solve real-world problems involving rated measures (miles per hour, feet per second)

• Solve real-world and mathematical problems involving estimates of measurements, including

length, time, weight/mass, temperature, money, perimeter, area and volume and estimate the

effects of measurement errors on calculations.

• Understand geometric concepts such as perpendicularity, parallelism, tangency, congruency,

similarity, reflections, symmetry and transformations including flips, slides, turns,

enlargements, rotations and fractals.

• Represent and apply geometric properties and relationships to solve real-world and

mathematical problems including ratio, proportion, and properties of right triangle

trigonometry.

• Use a rectangular coordinate system (graph), apply and algebraically verify properties of two-

and three-dimensional figures, including distance, midpoint, slope parallelism and

perpendicularity.

• Describe, analyze and generalize relationships, patterns and functions using words, symbols,

variables and tables.

• Determine the impact when changing parameters of given functions.

• Use systems of equations and inequalities to solve real-world problems graphically,

algebraically and with matrices.

• Interpret data that has been collected, organized and displayed in charts, tables and plots.

• Calculate measures of central tendency (mean, median and mode) and dispersion (range,

standard deviation and variance) for complex sets of data and determine the most meaningful

measure to describe the data.

• Analyze real-world data and make predictions of larger populations by applying formulas to

calculate measures of central tendency and dispersion using the sample population data and

using appropriate technology, including calculators and computers.

• Determine probabilities using counting procedures, tables, tree diagrams and formulas for

permutations and combinations.

• Determine the probability for simple and compound events as well as independent and

dependent events.

ALGEBRA 1

101

• Design and perform real-world statistical experiments that involve more than one variable, then

analyze results and report findings.

• Explain the limitations of using statistical technique and data in making inferences and valid

arguments.

Author: Bernice Stephens-AlleyneCopyright 2009

Revision Date:12/2009

Author: Bernice Stephens-AlleyneCopyright 2009

Revision Date:12/2009