MAKING CONNECTIONS: Course 1 - CPM Educational Program

MAKING CONNECTIONS: Course 1 Foundations for Algebra Toolkit Managing Editors / Authors Leslie Dietiker (Both Texts) Michigan State University East Lansing, MI

Evra Baldinger (Course 1)

Barbara Shreve (Course 2)

Phillip and Sala Burton Academic High School San Francisco, CA

San Lorenzo High School San Lorenzo, CA

Technical Assistants: Toolkit Sarah Maile

Aubrie Maize

Andrea Smith

Sacramento, CA

Sebastapool, CA

Placerville, CA

Cover Art Kevin Coffey San Francisco, CA

Program Directors Leslie Dietiker

Brian Hoey

Michigan State University East Lansing, MI

CPM Educational Program Sacramento, CA

Judy Kysh, Ph.D.

Tom Sallee, Ph.D.

Departments of Education and Mathematics San Francisco State University, CA

Department of Mathematics University of California, Davis

Copyright © 2011 by CPM Educational Program. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission should be made in writing to: CPM Educational Program, 1233 Noonan Drive, Sacramento, CA 95822. Email: [email protected]. 1 2 3 4 5 6

16 15 14 13 12 11 10

Printed in the United States of America

ISBN-13: 978-1-60328-049-5

Making Connections: Foundations for Algebra Course 1 Toolkit Chapter 1 Introduction and Representation Learning Log Entries Math Notes 1.1.1 1.1.4 1.2.2 1.2.3 1.3.1 1.3.3

Displays of Data Conjecture and Justify Comparisons Natural Numbers Stem-and-Leaf Plot Measures of Central Tendency

Chapter 2 Length and Integers Learning Log Entries Math Notes 2.1.1 2.1.2 2.1.4 2.2.2 2.2.3 2.2.4 2.3.1 2.4.1

Length Standard Units of Length Units of Length Integers Additive Inverse and Additive Identity Meaning of Multiplication Addition of Integers Order of Operations – Part One

Chapter 3 Arithmetic Strategies and Area

1 2 4 4 4 5 5 5 6

7 8 10 10 10 11 11 11 11 12 12

13

Learning Log Entries

14

Math Notes

19

3.1.1 3.1.2 3.2.1 3.2.2 3.2.3 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5

Place Value Strategies for Mental Addition Base Ten Blocks Area Area, Rectangles, and Square Units Multiplication Using Generic Rectangles Subtracting Integers Multiplication of Integers Connecting Addition and Subtraction of Integers Properties of Addition

19 19 20 20 21 22 22 23 23 24

Chapter 4 Data and Portions

25


26

Math Notes

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4.1.1 4.1.2 4.1.4 4.3.2 4.3.4 4.3.5

Order of Operations – Part Two Graphing Points on an xy-Coordinate Grid Intervals and Scaling 100% Blocks Place Value Percents

Chapter 5 Geometry and Probability Learning Log Entries Math Notes 5.1.1 5.1.3 5.2.2 5.2.4

Definitions of Quadrilaterals Angles and Parallel Lines Probability Vocabulary and Definitions Fraction ! Decimal ! Percent

Chapter 6 Similarity, Multiplying Fractions, and Equivalence Learning Log Entries Math Notes 6.1.4 6.2.2 6.2.4 6.2.6 6.3.1 6.3.3

Ratios Mixed Numbers and Fractions Greater Than One Multiplying Fractions Multiplying Mixed Numbers Multiplying Decimals Multiplicative Identity

Chapter 7 Right Prisms and Adding Portions Learning Log Entries Math Notes 7.1.1 7.2.1 7.2.2 7.2.3 7.3.1 7.3.2

Properties of Right Prisms Volume of Right Prisms Least Common Multiple and Greatest Common Factor Dividing Decimals by Integers Adding and Subtracting Fractions Adding and Subtracting Mixed Numbers

28 28 29 29 30 30

31 32 35 35 36 37 38

39 40 44 44 44 45 45 46 46

47 48 50 50 50 51 51 52 52

Chapter 8 Variables and Dividing Portions

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54

Math Notes

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8.2.2 8.2.3 8.2.4 8.3.1 8.3.2 8.4.1 8.4.2 8.4.3

Describing Growth Patterns Using Variables to Generalize Undefined Fractions Making Graphs of Rules Evaluating Variable Expressions Angle Relationships Multiplicative Inverses and Reciprocals Dividing by Fractions, Part 1

Chapter 9 Percents, Proportions, and Geometry Learning Log Entries Math Notes 9.1.3 9.1.4 9.2.4 9.2.5 9.3.1 9.3.2 9.3.3 9.3.4

Dividing by Fractions, Part 2 Calculating Percents by Composition Proportional Relationship Solving Proportions Circles and Circumference Area of Triangles Area of Circles and ! Prisms, Cylinders, and Their Volumes

Chapter 10 Probability and Survey Design Learning Log Entries Math Notes 10.1.3 Independent and Dependent Events 10.1.5 Calculating Probabilities of Independent Events

Personal Resources

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61 62 66 66 66 67 67 68 69 69 70

71 72 74 74 74

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Dear Math Student, Welcome to your Making Connections: Foundations for Algebra Toolkit! It is designed to help you as you learn math throughout the school year. Inside, you will find all of the Math Notes from your textbook that have useful information about the topics you will study. You will also be able to write in your Toolkit so that you can keep track of what you have learned in your own words and refer back to those notes as you move forward. Many lessons in your math book include a prompt that asks you to think and write about the topic you are learning that day in a Learning Log. There is space in this Toolkit to write your Learning Log entries so that they are all in one place and are easy to use later. It is a good idea to leave some space between your entries so that you can add new ideas to them later, as you learn more. Note that this space has a light grid, which you can use like lined paper, as well as to help you draw diagrams or graphs. At the end of your Toolkit you will find several Personal Reference pages. You can use these to create your own personal dictionary, complete activities, or in other ways suggested by your teacher. Throughout the year, remember to make notes in your Toolkit and add examples if you find them helpful. It is important that the information on these pages—especially the Math Notes—makes sense to you, so be sure to highlight key information, write down important things to remember, and ask questions if something does not make sense. Also remember that the information in your Toolkit can help you solve problems and keep track of important vocabulary words. Keep your Toolkit with you when you are working on math problems, and use it as a source of information as you move through the course. At the end of each chapter in your textbook, there are lists of all of the Learning Log entries and Math Notes for that chapter. There are also lists of important vocabulary words. Take time as you complete each chapter to look through your Toolkit and make sure it is complete. Updating your Toolkit regularly and using it when you are studying are important student habits that will help you to be successful in this and future courses. Have a wonderful year of learning! The CPM Team

Chapter 1: Introduction and Representation

CHAPTER 1

Table of Contents Page No.

Learning Log Entries •

Lesson 1.1.4 – Making Sense of a Challenging Problem

•

Lesson 1.1.5 – Beginning to Think About Proportional Relationships

•

Lesson 1.3.1 – Stem-and-Leaf Plots

•

Lesson 1.3.3 – Finding Mean, Median, Mode, and Range

Math Notes

Toolkit

•

Lesson 1.1.1 – Displays of Data

4

•

Lesson 1.1.4 – Conjecture and Justify

4

•

Lesson 1.2.2 – Comparisons

5

•

Lesson 1.2.3 – Natural Numbers

5

•

Lesson 1.3.1 – Stem-and-Leaf Plot

5

•

Lesson 1.3.3 – Measures of Central Tendency

6

1

Date: Lesson:

2

Learning Log Title:

Making Connections: Course 1


Toolkit

3

Notes:

MATH NOTES DISPLAYS OF DATA Data can be displayed visually in different formats depending on the kind of information collected. In the examples below, Ms. Carpenter collected information about the students in her class.

12 10 Frequency

A histogram is similar to a bar graph except that each bar represents data in an interval of numbers. The intervals for the data are shown on the horizontal axis and the frequency (number of pieces of data in each interval) is represented by the height of a rectangle above the interval.

14

The labels on the horizontal axis represent the lower end of each interval. For example, the histogram at right shows that 10 students take at least 15 minutes but less than 30 minutes to get to school.

8 6 4 2 red blue green purple Favorite color

12 10 Frequency

A bar graph shows the frequency (the number of times something occurs) of the data. It is usually used when separate categories are named. The graph at right shows that green is the favorite color of 14 students.

8 6 4 2 0 15 30 45 60 75 90 Minutes to school

A Venn diagram is two or more overlapping circles used to show relationships between groups of items. The diagram at right shows that seven students have both dogs and cats, 16 students have dogs, 17 students have cats, nine students have only dogs, 10 have only cats, and three students do not have a dog or a cat.

10

7

9 3

cats

dogs

CONJECTURE AND JUSTIFY A conjecture is a statement that appears to be true. It is an educated guess. To justify a conjecture is to give reasons why your conjecture makes sense. Justification may be an observation of a pattern or a logical or algebraic validation. 4



Notes:

COMPARISONS Mathematical symbols are used to compare quantities. The most commonly used symbols are: greater than: > less than: < equal to: =

7>5 3 and >> mean that the sides are parallel (lines that do not ever intersect (cross)). Polygon: A closed, two-dimensional figure that is bounded by straight line segments that do not cross and are connected end to end. Each of these line segments is a side of the polygon. Quadrilateral: A four-sided polygon. Kite (a): A quadrilateral with two pairs of consecutive sides of equal length.

a

Trapezoid (b): A quadrilateral with at least one pair of parallel sides.

b

Parallelogram (c): A quadrilateral with both pairs of opposite sides parallel. It is true that opposite sides are of equal length. Rhombus (d): A quadrilateral with four sides of equal length.

c d e

Rectangle (e): A quadrilateral with four right angles. Square (f): A quadrilateral with four sides of equal length and four right angles.

Toolkit

f

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Notes:

ANGLES AND PARALLEL LINES To understand the meaning of an angle, picture two rays starting at a single point, as shown in the diagram at right. (A ray is a part of a line that starts at a point and goes on without end in one direction.) An angle is formed by two rays (or line segments) that have the same starting point (or endpoint). The measure of an angle is how many degrees you rotate your starting ray to get to the ray on the opposite side of an angle. One way to visualize an angle is as a measure of how “open” the gap is between the two rays.

angle

Angles are named by whether they are less than, greater than, or equal to a right angle. An acute angle measures less than 90° and an obtuse angle measures more than 90° and less than 180°.

acute angle

right angle

obtuse angle

Lines on a flat surface that never meet are called parallel. The >> marks on two lines show that they are parallel. The arrowheads at the end of each line, as well as the ends of the sides of the angles above, indicate that they continue without end.

parallel lines The little box in an angle indicates that it is a right angle (an angle that measures 90°) and also shows that the two lines, rays, or segments are perpendicular, that is, they form a right angle.

perpendicular lines

36


Chapter 5: Geometry and Probability

PROBABILITY VOCABULARY AND DEFINITIONS

Notes:

Outcome: Any possible or actual result or consequence of the action(s) considered, such as rolling a five on a standard number cube or getting tails when flipping a coin. Event: A desired (or successful) outcome or group of outcomes from an experiment, such as rolling an even number on a standard number cube. Probability: A number between zero and one that states the likelihood of an event occurring. Experimental Probability is the probability based on data collected in experiments. Experimental Probability =

number of successful outcomes in the experiment total number of outcomes in the experiment

Theoretical Probability is a calculated probability based on the possible outcomes when they all have the same chance of occurring. Theoretical Probability =

number of successful outcomes (events) total number of possible outcomes

By “successful” we usually mean a desired or specified outcome (event), such as rolling a 3 on a number cube ( 16 ), pulling a king from a deck of cards 4 1 = 13 ( 52 ), or flipping a coin and getting tails ( 12 ). Probabilities are written like this: The probability of rolling a 3 on a number cube is P(3). The probability of pulling a king out of a deck of cards is P(king). The probability of getting tails is P(tails).

Toolkit

37

Notes:

FRACTION  DECIMAL  PERCENT The web diagram at right illustrates that fractions, decimals, and percents are different ways to represent a portion of a whole or a number. The examples below show how to convert from one form to another. To read a description of numbers, see the Math Note about place value.

fraction

words or pictures percent

decimal

Representations of a Portion

Decimal to percent:

Percent to decimal:

Multiply the decimal by 100.

Divide the percent by 100.

Fraction to percent:

Percent to Fraction:

Set up an equivalent fraction using 100 as the denominator. The numerator is the percent.

Use 100 as the denominator. Use the digits in the percent as the numerator. Simplify as needed.

(0.34)(100) = 34%

4 5

80 = 80% = 100

22 = 22% = 100

11 50

Decimal to fraction:

Fraction to decimal:

Use the digits as the numerator. Use the decimal place value as the denominator. Simplify as needed.

Divide the numerator by the denominator.

0.48 =

38

78.6% = 78.6 ÷ 100 = 0.786

48 100

=

12 25

3 8

= 3 ÷ 8 = 0.375


Chapter 6: Similarity, Multiplying Fractions, and Equivalence

CHAPTER 6



Lesson 6.1.2 – Enlarging Figures

•

Lesson 6.1.3 – Ratios

•

Lesson 6.2.3 – Multiplying Fractions

•

Lesson 6.2.5 – Multiplying Mixed Numbers and Fractions Greater than One

•

Lesson 6.3.1 – Multiplication Number Sense

•

Lesson 6.3.2 – The Giant One and Equivalent Fractions

•

Lesson 6.3.3 – Uses of the Giant One

Math Notes

Toolkit

•

Lesson 6.1.4 – Ratios

44

•

Lesson 6.2.2 – Mixed Numbers and Fractions Greater Than One

44

•

Lesson 6.2.4 – Multiplying Fractions

45

•

Lesson 6.2.6 – Multiplying Mixed Numbers

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•

Lesson 6.3.1 – Multiplying Decimals

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•

Lesson 6.3.3 – Multiplicative Identity

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Date: Lesson:

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Learning Log Title:



Toolkit

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42



Toolkit

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Notes:

MATH NOTES RATIOS A ratio is a comparison of two numbers, often written as a quotient; that is, the first number is divided by the second number (but not zero). A ratio can be written in words, in fraction form, or with colon notation. Most often in this class we will write ratios in the form of fractions or state them in words. For example, if there are 38 students in the school band and 16 of them are boys, we can write the ratio of the number of boys to the number of girls as: 16 boys to 22 girls

16 boys 22 girls

16 boys : 22 girls

MIXED NUMBERS AND FRACTIONS GREATER THAN ONE The number 3 41 is called a mixed number because it is composed of a whole number, 3, and a fraction, 41 . The number 13 is called a fraction greater than one because the numerator, 4 which represents the number of equal pieces, is larger than the denominator, which represents the number of pieces in one whole, so its value is greater than one. (Sometimes such fractions are called improper fractions, but this is just an historical term. There is nothing wrong with the fractions themselves.) As you can see in the diagram at right, the fraction 13 can be rewritten as 44 + 44 + 44 + 41 , which 4 shows that it is equal in value to 3 41 . Your choice: Depending on which arithmetic operations you need to perform, you will choose whether to write your number as a mixed number or as a fraction greater than one.

44



MULTIPLYING FRACTIONS

Notes:

You can find the product of two fractions, such as 23 and 43 , by multiplying the numerators (tops) of the fractions together and 6 dividing that by the product of the denominators (bottoms). So 23 ! 43 = 12 , which is equivalent to 12 . Similarly, 47 ! 53 = 12 . If we write this method in 35 a!c algebraic terms, we would say ba ! dc = b!d . The reason that this rule works can be seen using an area model of multiplication, as shown at right, which represents 23 ! 43 . The product of the denominators is the total number of smaller rectangles, while the product of the numerators is the number of the rectangles that are double shaded.

MULTIPLYING MIXED NUMBERS An efficient method for multiplying mixed numbers is to convert them to fractions greater than one, find the product as you would with fractions less than one, and then convert them back to a mixed number, if necessary. (Note that you may also use generic rectangles to find these products.) Here are three examples:

1 23 ! 2

3 4

= 53 ! 11 = 4

55 12

7 = 4 12

2 13 ! 4

Toolkit

1 2

1 53 ! 29 = 85 ! 29 = = 73 ! 92 =

63 6

16 45

= 10 63 = 10 12

45

Notes:

MULTIPLYING DECIMALS There are at least two ways to multiply decimals. One way is to convert the decimals to fractions and use your knowledge of fraction multiplication to compute the answer. The other way is to use the method that you have used to multiply integers; the only difference is that you need to keep track of where the decimal point is as you record each line. Here we show how to compute 1.4(2.35) both ways by using generic rectangles.

2

3 10

5 100

1

2

3 10

5 100

4 10

8 10

12 100

20 1000

2

0.3 0.05

1

2

0.3 0.05

0.4

0.8

0.12 0.020

If you carried out the computation as above, you can calculate the product in either of the two ways shown at right. In the first one, we write down all of the values in the smaller rectangles within the generic rectangle and add the six numbers. In the second example, we combine the values in each row and then add the two rows. We usually write the answer as 3.29 since there are zero thousandths in the product.

2.35 ! 1.4 0.020 0.12 0.8 0.05 0.3 2.0 3.290

2.35 ! 1.4 0.940 2.35 3.29

MULTIPLICATIVE IDENTITY If any number or expression is multiplied by the number one, the result is equal to the original number or expression. The number one is called the multiplicative identity. Formally, the identity is written:

1! x = x !1 = x for all values of x. One way the multiplicative identity is used is to create equivalent fractions using a Giant One. 2 !!!!!! 2 !!!=!! 4 3 2 6

Multiplying any fraction by a Giant One will create a new fraction equivalent to the original fraction.

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Chapter 7: Right Prisms and Adding Portions

CHAPTER 7



Lesson 7.1.2 – Volume of Prisms

•

Lesson 7.2.3 – Converting Fractions to Decimals and Percents

•

Lesson 7.2.5 – Adding and Subtracting Fractions

Math Notes

Toolkit

•

Lesson 7.1.1 – Properties of Right Prisms

•

Lesson 7.2.1 – Volume of Right Prisms

•

Lesson 7.2.2 – Least Common Multiple and Greatest Common Factor

51

•

Lesson 7.2.3 – Dividing Decimals by Integers

51

•

Lesson 7.3.1 – Adding and Subtracting Fractions

52

•

Lesson 7.3.2 – Adding and Subtracting Mixed Numbers

52

50 50-51

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Date: Lesson:

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Learning Log Title:



Toolkit

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Notes:

MATH NOTES PROPERTIES OF RIGHT PRISMS Both of the three-dimensional figures at right are right prisms. Prisms have two identical, parallel, polygonal faces Triangular Prism called bases connected by rectangular faces called lateral faces. They have lateral no holes. The line segments where face Two identical two faces meet are called edges, and and parallel bases the point where three edges meet is edge called a vertex. A prism is named for the shape of its base, as demonstrated vertex by each of the labels at right. The total surface area of a prism is the combined area of all of its faces (bases and lateral faces). Lateral surface area excludes the bases.

Pentagonal Prism

The amount of space enclosed by a three-dimensional figure is called its volume. To measure volume, we use units that measure the three dimensions of space (length, width, and height), which are called cubic units.

lateral face

Two identical and parallel bases

VOLUME OF RIGHT PRISMS The length of a segment is the number of segments of length 1 that are needed to match it exactly. The area of a region is the number of squares of area 1 that are needed to cover it exactly. Similarly, the volume of a solid is the number of cubes of volume 1 that are needed to fill it exactly. The volume of a right rectangular prism (which looks like a standard box) can be computed with the formula: Volume = (Area of base) ! height In this case, the area of the base is the number of squares needed to cover it. If each of those squares is the bottom face of a cube, then the number of these cubes is the same as the area of the base. The height is the number of cubes that from the bottom to the top of the box. The volume of the box is the number of cubes needed to fill the prism, which we may calculate by multiplying the number of layers of cubes (height) times the number of cubes in each layer (area of the base). Math Note continues on next page → 50



Notes:

Math Note continued from previous page. The same formula works for other figures that are right prisms: Volume = (Area of base) ! height The right prism shown in the diagram at right has a height of 6 cm and a triangular base with an area of 48 sq cm. To calculate its volume, multiply the area of the base times the height of the prism, or 48 sq cm ! 6 cm = 288 cu cm .

LEAST COMMON MULTIPLE AND GREATEST COMMON FACTOR The Least Common Multiple of two or more integers is the lowest positive integer that is divisible by both (or all) of the integers. For example, the multiples of 3 and 5 are shown in the table at right. 15 is the Least Common Multiple, because it is the smallest positive integer divisible by both 3 and 5.

3 5

6 10

9 15

12 20

15 25

18 30

The Greatest Common Factor of two or more integers is the greatest positive integer that is a factor of both (or all) of the integers. For example, the factors of 18 are 1, 2, 3, 6, and 18 and the factors of 12 are 1, 2, 3, 4, 6, and 12, so the Greatest Common Factor of 12 and 18 is 6.

DIVIDING DECIMAL BY INTEGERS When using long division to divide one number by another, it is important to be sure that you know the place value of each digit in your result. In the example of dividing 225 by 6 below left, people often say “6 goes into 22 three times,” although “6 goes into 220 thirty-some times” is a better description of what is taking place in this calculation. The 3 of the quotient is written in the tens place to indicate that 6 goes into 225 at least 30 times, but less than 40. The divisor is then multiplied by the 3, and the product, 18, is placed below the 22. Then subtract, getting 4, “bring down” the 5, to get 45, and repeat the same process. In the past, you may have stopped the process at this point, and written that the quotient is 37 with a remainder of 3. The same method works for dividing a decimal. The second example is essentially the same as the first, except that we keep dividing even when we get to the decimal point. Toolkit

37 6 225 – 18 !!!!!45 – !!!!!42 !!!!!!!3

37.5 6 225.0 – 18 !!!!!!45 !!!!!!42 – !! !!! 30 – 30

0 51

Notes:

ADDING AND SUBTRACTING FRACTIONS To add or subtract two fractions that are written with the same denominator, simply add or subtract the numerators. For example, 15 + 25 = 53 . If the fractions have different denominators, rewrite them first as fractions with the same denominator (using the Giant One, for example). Below are examples of adding and subtracting two fractions with different denominators. Addition example:

1 5

+

2 3

3 + 10 = 13 ! 15 !!"!!! 33 !!!+ 23 !!"!!! 55 !!!! 15 15 15

Subtraction example:

5 6

!

1 4

7 " 56 !!#!!! 22 !!!! 14 !!#!!! 33 !!!" 10 ! 3 = 12 12 12

Using algebra to write the general method: a b

+

c d

! ba !!"!!! dd !!!+ dc !!"!!! bb !!!!

a"d b"d

b"c ! + b"d

a"d +b"c b"d

ADDING AND SUBTRACTING MIXED NUMBERS To add or subtract mixed numbers, you can either add or subtract their parts, or you can change the mixed numbers into fractions greater than one. To add or subtract mixed numbers by adding or subtracting their parts, add or subtract the whole-number parts and the fraction parts separately. Adjust if the fraction in the answer would be greater than one or less than zero. For example, the sum of 3 45 + 1 23 is calculated at right. It is also possible to add or subtract mixed numbers by changing them into fractions greater than one and then adding or subtracting as with fractions between zero and one. For example, the sum of 2 16 + 1 45 is calculated at right. 52

!!3 45 = 3 + 45 !!!!!! 33 !!!=!!3 12 15 +1 23 !=!1 + 23 !!!!!! 55 !!!= +1 10 !!!! 15 22 = 5 7 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!4 15 15

2 16 + 1 45 = 13 + 6

9 5

= 13 !!!!!! 55 !!!+ 95 !!!!!! 66 !!! 6 = = =

65 + 54 30 30 119 30 3 29 30


Chapter 8: Variables and Dividing Portions

CHAPTER 8



Lesson 8.2.4 – Graphing Negative Coordinates

•

Lesson 8.3.2 – Angles in a Triangle

•

Lesson 8.4.3 – Using a Super Giant One

Math Notes

Toolkit

•

Lesson 8.2.2 – Describing Growth Patterns

56

•

Lesson 8.2.3 – Using Variables to Generalize

56

•

Lesson 8.2.4 – Undefined Fractions

57

•

Lesson 8.3.1 – Making Graphs of Rules

58

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Lesson 8.3.2 – Evaluating Variable Expressions

58

•

Lesson 8.4.1 – Angle Relationships

59

•

Lesson 8.4.2 – Multiplicative Inverses and Reciprocals

60

•

Lesson 8.4.3 – Dividing by Fractions, Part 1

60

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Date: Lesson:

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Learning Log Title:



Toolkit

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Notes:

MATH NOTES DESCRIBING GROWTH PATTERNS By understanding how a pattern grows, you can then predict how it will continue (assuming it continues to grow the same way). Patterns can usually be described in several different ways. For the dot pattern at right, two different ways of describing the pattern are given below.

Figure 1

Figure 2

Figure 3

Method 1: Figure 1 is one row by two columns, Figure 2 is two rows by three columns, and Figure 3 is three rows by four columns. So, Figure 4 will be four rows by five columns (20 dots) and Figure 5 will be five rows by six columns (30 dots). Method 2: Figure 1 is a single dot plus a single dot to the right, Figure 2 is a two by two square of dots plus a column of two dots. Figure 3 is a three by three square of dots plus a column of three dots. So Figure 4 will have 4 2 + 4 = 20 dots and Figure 5 will have 5 2 + 5 = 30 dots.

USING VARIABLES TO GENERALIZE We use variables to generalize patterns from a few specific numbers to include all possible numbers. For example, if a square is surrounded by smaller square tiles that each measures one centimeter on a side, how many tiles are needed? It helps to look at a specific size square first. The outside square at right has side length 7. One way to see the total number of tiles needed for the frame is that it needs 7 tiles for each of the top and bottom sides and 7 ! 2 = 5 tiles for the left and right sides, as shown in the first diagram at right. The total number of tiles needed for the frame can be counted as 7 + 7 + 5 + 5 = 24 .

7

5

5

7

Math Note continues on next page →

56



Notes:

Math Note continued from previous page. Square frames with different measures will follow the same pattern, so we can generalize by writing an expression for any side length, denoted by x. The second diagram at right shows that the top and bottom each contain x tiles, and the right and left sides each contain x ! 2 tiles, for a total number of x + x + (x ! 2) + (x ! 2) tiles. This expression simplifies to 4 x ! 4 .

x

x–2

x–2

x

Another way to count the number of tiles in the same frame and the variable expression associated with it are shown below. Notice that the expression resulting from this counting method could be written (x ! 1) + (x ! 1) + (x ! 1) + (x ! 1) , or simplified to 4(x ! 1) or 4 x ! 4 . Both methods result in the equivalent expression: 4 x ! 4 .

x–1

6 6

6 6

x–1

x–1 x–1

UNDEFINED FRACTIONS Since it is not possible to divide a number by zero, any fraction written with a denominator of zero is said to be undefined, that is, it has no answer. For example, when finding pairs of numbers that satisfy the rule xy = not possible for y to equal zero, as there is no number x that makes defined.

Toolkit

2 , it is 5x 0

57

Notes:

MAKING GRAPHS OF RULES A graph of a rule (equation) represents all of the points that make that equation true. To make a graph for a rule, it is often useful to make an xy-table by finding pairs of numbers x and y that make the rule true. Then each of the points is plotted on a graph. Whether the points are connected with a line or smooth curve depends on the rule and the context of the problem. Thus, not all graphs are continuous; that is, you have to think about whether or not to connect all of the points. x y For example, to graph the rule xy = 12 , you can start by 2 6 thinking of numbers that will work. For example, x = 2 −1 −12 and y = 6 works, because 2 ! 6 = 12 . This point would go in the table, as shown at right. Another example of a point that works is (!1, !12) , because (!1)(!12) = 12 . When you have several points in your table, draw a pair of xy-axes on graph paper. Decide on a scale that will allow an appropriate number of points to fit on the graph. Then plot each point from the y table. To show that an infinite number of points between the plotted ones also satisfy the rule, connect the points with a smooth curve or line. To show that there are infinite solutions beyond the ones shown on the graph, draw arrowheads x on each end of the curve or line. A graph for the rule xy = 12 is shown at right. Notice that this graph is not continuous, since there is no point on the y-axis corresponding to x = 0 .

EVALUATING VARIABLE EXPRESSIONS A variable expression, also known as an algebraic expression, consists of one or more variables, or a combination of numbers and variables connected by mathematical operations. 4 x , 3(x ! 5) , and 4 x ! 3y + 7 are examples of variable expressions. To evaluate a variable expression for particular values of variables, replace the variables in the expression with their known numerical values (this process is called substitution) and simplify. Two examples are provided below. Evaluate 3(x ! 5) for x = 2 . Replace x with its known value of 2 and simplify.

3(2 ! 5) = 3(!3) = !9

Evaluate 4x ! 3y + 7 for x = 2 and y = !1 . Replace x and y with their known values of 2 and –1, respectively, and simplify.

4 x ! 3y + 7 = 4(2) ! 3(!1) + 7 = 8 + 3+ 7 = 18

58



Notes:

ANGLE RELATIONSHIPS It is common to identify angles using three letters. For example, !ABC means the angle you would find by going from point A to point B to point C in the diagram at right. B is the vertex of the angle (where endpoints of the two sides meet) and !!!" the!!! " BA and BC are the rays that define it. A ray is a part of a line that has an endpoint (starting point) and extends infinitely in one direction.

A

C

B

D

If two angles have measures that add up to 90°, they are called complementary angles. For example, in the diagram above right, !ABC and !CBD are complementary because together they form a right angle. If two angles have measures that add up to 180°, they are called supplementary angles. For example, in the diagram at right, !EFG and !GFH are supplementary because together they form a straight angle.

G E

F

H

Two angles do not have to share a vertex to be complementary or 120° 50° supplementary. The first pair of 60° 40° angles at right are supplementary; the second pair Supplementary Complementary of angles are complementary. Adjacent angles are angles that have a common vertex, share a common side, and have no interior points in common. So c d angles ! c and ! d in the diagram at right are adjacent angles, g f as are ! c and ! f, ! f and ! g, and ! g and ! d. Vertical angles are the two opposite (that is, non-adjacent) angles formed by two intersecting lines, such as angles ! c and ! g in the diagram above right. ! c by itself is not a vertical angle, nor is ! g, although ! c and ! g together are a pair of vertical angles. Vertical angles always have equal measure.

C

The measures of the angles in a triangle add up to 180°. For example, in !ABC at right, m!A + m!B + m!C = 180! . You can verify this statement by carefully drawing a triangle with a ruler, tearing off two of the angles (A and B), and placing them side by side with the third angle (C) on a straight line. The sum of the three angles is the same as the straight angle (line), that is, 180°. Toolkit

80° 60°

A A A

C

40°

B

B B

59

Notes:

MULTIPLICATIVE INVERSES AND RECIPROCALS Two numbers with a product of 1 are called multiplicative inverses. 8 5 ! 5 8

=

40 40

=1

4 3 14 = 13 = 13 !4 = , so 3 14 ! 13 4 4 13

52 52

=1

1 7

!7 =1

In general a ! 1a = 1 and ba ! ba = 1 , where neither a nor b equals zero. We say that 1a is the reciprocal of a and ba is the reciprocal of ba . Note that 0 has no reciprocal.

DIVIDING BY FRACTIONS, PART 1 Method 1: Dividing Fractions Using Diagrams To divide any number by a fraction using a diagram, create a model of the first number using rectangles, a linear model, or some visual representation of it. Then break that model into the fractional parts named by the second fraction. For example, to divide ÷ , you can draw the diagram at right to visualize how many 12 -sized pieces fit into 87 . The diagram shows that one 12 fits, with 83 of a whole left. Since 83 is 43 of 12 , we can see that 1 43 12 -sized pieces fit into 87 , so 87 ÷ 12 = 1 43 . 7 8

For 45

÷

7 8

1 2

3 , you can draw a rectangle, 10

shown at right, divide it into five sections and cut each of them in half. The diagram 3 shows that there are two 10 ths in 45 with 2 2 2 3 ths left. 10 is 3 of 10 , so 45 ÷ 103 = 2 23 . 10

1 2 2 3

of

3 4

of

1 2

3 10

3 10

3 10

Method 2: Dividing Fractions Using Common Denominators To divide a number by a fraction using common denominators, express both numbers as fractions with the same denominator. Then divide the first numerator by the second. An example is at right. Method 3: Dividing Fractions Using a Super Giant One

2 5

4 ÷ 3 ÷ 103 = 10 10

=

4 3

= 1 13

To divide by a fraction using a Super Giant One, write the two numbers as a fraction, make the reciprocal of the super fraction’s denominator the fraction for the Super Giant One, then simplify as shown in the following examples.

6÷

4

3 4

4

3 4

60

÷

2 5

6! 4

= 63 !!!!!! 43 !!!= 13 = 6 ! 43 = 24 =8 3

=

3

3 5 3!5 4 !!!!!! 2 !!!= 4 2 2 5 1 5 2

=

3!5 4 2

= 15 = 1 87 8 Making Connections: Course 1

Chapter 9: Percents, Proportions, and Geometry

CHAPTER 9



Lesson 9.1.4 – Calculating Percents Mentally

•

Lesson 9.2.2 – Proportional Relationships

•

Lesson 9.2.4 – Methods for Finding Missing Information in Proportional Relationships

•

Lesson 9.2.5 – Circumference and Diameter

•

Lesson 9.3.1 – Area of a Triangle

•

Lesson 9.3.2 – π and the Area of a Circle

•

Lesson 9.3.4 – Volume of a Cylinder

Math Notes

Toolkit

•

Lesson 9.1.3 – Dividing by Fractions, Part 2

66

•

Lesson 9.1.4 – Calculating Percents by Composition

66

•

Lesson 9.2.4 – Proportional Relationship

67

•

Lesson 9.2.5 – Solving Proportions

67

•

Lesson 9.3.1 – Circles and Circumference

68

•

Lesson 9.3.2 – Area of Triangles

69

•

Lesson 9.3.3 – Area of Circles and !

69

•

Lesson 9.3.4 – Prisms, Cylinders, and Their Volumes

70

61

Date: Lesson:

62

Learning Log Title:



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63

64



Toolkit

65

Notes:

MATH NOTES DIVIDING BY FRACTIONS, PART 2 Division of Fractions with the Invert and Multiply Method: In Chapter 8 you used a Super Giant One to divide fractions. Now that you have some experience with dividing fractions, you can generalize this process to simplify your work. Read the following example of dividing fractions using the Super Giant One method: 3 4

÷

2 5

=

3 5 3!5 4 !!!!!! 2 !!!= 4 2 2 5 1 5 2

=

3!5 4 2

= 15 8

Notice that the result of multiplying by the Super Giant One in this example, and all of the other examples in Chapter 8 that used a Super Giant One to divide, is that the denominator of the super fraction (also called a complex fraction) is always 1. In addition, the numerator is the product of the first fraction and the reciprocal of the second fraction (divisor). We can generalize division with fractions and name it the invert and multiply method. To use this method, take the first fraction and multiply it by the reciprocal of the second fraction. Some students prefer to say “flip” the second fraction and multiply it by the first fraction. If the first number is an integer, write it as a fraction over 1. Here is the same problem in the example above solved using this method: 3 4

÷

2 5

=

3!5 4 2

= 15 8

CALCULATING PERCENTS BY COMPOSITION Calculating 10% of a number and 1% of a number will help you to calculate other percents by composition.

1 10% = 10 1 1% = 100

To calculate 13% of 25, you can think of 10% of 25 + 3(1% of 25). 1 10% of 25 ! 10 of 25 = 2.5 and 1 1% of 25 ! 100 of 25 = 0.25 so

13% of 25 ! 2.5 + 3(0.25) ! 2.5 + 0.75 = 3.25 To calculate 19% of 4500, you can think of 2(10% of 4500) – 1% of 4500. 1 10% of 4500 ! 10 of 4500 = 450 and 1 1% of 4500 ! 100 of 4500 = 45 so

19% of 4500 ! 2(450) " 45 ! 900 " 45 = 855 66



Notes:

PROPORTIONAL RELATIONSHIP Two quantities are in a (direct) proportional relationship if the value of one of them is always the same multiple of the value of y the other. In algebraic terms, we write y = kx or x = k , where k is the constant of proportionality. For example, if a car drives at a constant speed of 60 miles per hour, the amount of time that the car has been driving (x) and the distance it has traveled (y), are related proportionally. For a 3 hour trip, x = 3 (hrs), k = 60 (miles per hour) and y = 180 (miles). Another example would be if the sales tax rate is 8%, the cost of an item (x) and the amount of tax you need to pay (y) are related proportionally. For a $40 item, x = $40 , k = 8% = 0.08 , and y = $3.20 .

SOLVING PROPORTIONS 5 7

An equation stating that two ratios are equal is called a proportion. Some examples of proportions are:

=

6 mi 2 hr

50 70

=

9 mi 3hr

When two ratios are known to be equal, setting up a proportion is one strategy for solving for an unknown part of one ratio. For example, if the ratios 92 and x x = 9 are equal, setting up the proportion 16 allows you to solve for x. 16 2 Strategy 1: One way to solve this proportion is by using a Giant One to find the equivalent ratio. In this case, since the scale factor between 2 and 16 is 8, we create the Giant One, x 16

= 92 !!!! 88 !! =

9!8 2!8

=

72 so 16

x = 72

Strategy 2: Use Cross Multiplication. This is a solving strategy for proportions that is based on the process of multiplying each side of the equation by the denominators of each ratio and setting the two sides equal.

x 16 x 16

= =

9 2 9 2

2 ! x = 9 !16 2x = 144 x = 72

Toolkit

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Notes:

CIRCLES AND CIRCUMFERENCE A circle is the set of all points that are the same distance from a fixed point, known as its center. The distance from the center to the points on the circle is called the radius (usually denoted r). A line segment drawn through the center of the circle with both endpoints on the circle is called a diameter (denoted d). Note: d = 2r . center

You can think of a circle as the rim of a bicycle wheel. The center of the circle is the hub where the wheel is bolted to the bicycle’s frame. The radius is a spoke of the wheel.

radius diameter circle

The circumference of a circle is the perimeter of the circle, that is, the distance around the circle. To find the circumference of a circle using its diameter, use: C = ! " d . To find the circumference of a circle using its radius, use: C = 2! " r .

Approximations of π Type of Situation

68

Approximation of π to Use

Rough estimation or mental calculation

3

Approximate fraction

3 17 or

Scientific calculation

π (on calculator) or 3.141592654

Decimal to the nearest hundredth

3.14

22 7



Notes:

AREA OF TRIANGLES The area of a triangle can be calculated using the formula:

A=

1 2

bh

where b is the length of the base of the triangle (one side) and h is the height (perpendicular distance from the base or the line through the base to the point where the other two sides of the triangle meet). Note that, unless the triangle is a right triangle, the height is not one of the sides of the triangle. The triangle at right has a base of 10 units and a height of 2 units. The area can be calculated by: A=

1 2

2 10

(10)(2) = 10 square units.

The second triangle at right has a base of 6 units and a height of 7 units. The area can be calculated by:

A=

1 2

6 7


The third triangle at right has a base of 6 units and a height of 5 units. This example illustrates when the height is drawn “to a line through the base” or extended from the base so that it will be perpendicular to that line.

5

The area can be calculated by:

A=

1 2

6


AREA OF CIRCLES AND π For any circle with a diameter, d , and a circumference, C, the number π (pronounced “pie”), one of the most important in mathematics, is defined as ! = Cd " 3.14 where the ratio does not depend on the circle used. If r is the radius of the circle, then the diameter and radius are related by d = 2r .

r

The area, A, of a circle can be calculated using the formula A = ! r 2 .

Toolkit

69

Notes:

PRISMS, CYLINDERS, AND THEIR VOLUMES A right prism is a three-dimensional figure composed of polygonal faces (called sides or lateral sides) and two congruent (same size and shape) parallel faces called bases. No holes are permitted in this solid. The remaining faces are rectangles. A prism is named for the shape of its bases. For example, the solid at right would be called a “square-based prism.”

base

face

height

base

A right circular cylinder is a solid figure much like a prism except its base and top are circles and the lines that join the base and top are perpendicular to both of them. The height of a cylinder is the distance between the base and the top.

base

height

The formula for the volume, V, of either a prism or a cylinder is the same:

V = (area of base) ! (height) For a cylinder, this is the same as V = (! r 2 )h .

70


Chapter 10: Probability and Survey Design

CHAPTER 10



Lesson 10.1.2 – Independent and Dependent Events

•

Lesson 10.1.4 – Finding Probabilities of Compound Events

•

Lesson 10.1.5 – Calculating Compound Probabilities

Math Notes

Toolkit

•

Lesson 10.1.3 – Independent and Dependent Events

74

•

Lesson 10.1.5 – Calculating Probabilities of Independent Events

74

71

Date: Lesson:

72

Learning Log Title:


Chapter 10: Probability and Survey Design

Toolkit

73

Notes:

MATH NOTES INDEPENDENT AND DEPENDENT EVENTS Two events are independent if the outcome of one event does not affect the outcome of the other event. For example, if you draw a card from a standard deck of playing cards but replace it before you draw again, the outcomes of the two draws are independent. Two events are dependent if the outcome of one event affects the outcome of the other event. For example, if you draw a card from a standard deck of playing cards and do not replace it for the next draw, the outcomes of the two draws are dependent.

CALCULATING PROBABILITIES OF INDEPENDENT EVENTS When you know the probability of each of two separate events, the probability of them both occurring is called the compound probability. If the two events are independent, it is easy to find the compound probability of them both occurring: simply multiply the probabilities of the two events together. Using the example of rolling a one followed by a six on a standard number cube:

P(1) =

1 6

and P(6) =

1 6

so P(1 and 6) =

1 6

! 16 =

1 36

Finding the compound probability of dependent events is more complicated and will be developed in future course

74


Personal Resources

Personal Reference Pages These pages are available for you to create reference materials that are useful for you throughout the entire course.

Title and Description

Toolkit

Page No.

75

Personal Resources

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Personal Resources

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Personal Resources

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MAKING CONNECTIONS: Course 1 - CPM Educational Program

MAKING CONNECTIONS: Course 1 - CPM Educational Program

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