Math: Grade 7 Flip Chart

Copyright © NewPath Learning. All rights reserved. www.newpathlearning.com Charts Charts Grade Grade Curriculum Mastery Flip Charts Combine Essential Math Skills with Hands-On Review! ® 33-7001 \|xiBAHBDy01220nzW Sturdy, Free-Standing Design, Perfect for Learning Centers! Reverse Side Features Questions, Math Problems, Vocabulary Review & more!

Phone: 800-507-0966 • Fax: 800-507-0967 www.newpathlearning.com NewPath Learning® products are developed by teachers using research-based principles and are classroom tested. The company’s product line consists of an array of proprietary curriculum review games, workbooks, posters and other print materials. All products are supplemented with web-based activities, assessments and content to provide an engaging means of educating students on key, curriculum-based topics correlated to applicable state and national education standards. Copyright © 2009 NewPath Learning. All Rights Reserved. Printed in the United States of America. Curriculum Mastery® and NewPath Learning® are registered trademarks of NewPath Learning LLC. Math Curriculum Mastery® Flip Charts provide comprehensive coverage of key standards-based curriculum in an illustrated format that is visually appealing, engaging and easy to use. Curriculum Mastery® Flip Charts can be used with the entire classroom, with small groups or by students working independently. Each Math Curriculum Mastery® Flip Chart Set features • 10 double-sided laminated charts covering grade-level specific curriculum content on one side plus write-on/wipe-off charts on reverse side for student use or for small-group instruction. • Built-in sturdy free-standing easel for easy display • Spiral bound for ease of use • Activity Guide with black-line masters of the charts for students to fill-in, key vocabulary terms, corresponding quiz questions for each chart, along with answers Ideal for • Learning centers • In class instruction for interactive presentations and demonstrations • Hands-on student use • Stand alone reference for review of key science concepts • Teaching resource to supplement any program HOW TO USE Classroom Use Each Curriculum Mastery® Flip Chart can be used to graphically introduce or review a topic of interest. Side 1 of each Flip Chart provides graphical representation of key concepts in a concise, grade appropriate reading level for instructing students. The reverse Side 2 of each Flip Chart allows teachers or students to fill in the answers and summarize key concepts. Note: Be sure to use an appropriate dry-erase marker and to test it on a small section of the chart prior to using it. The Activity Guide included provides a black-line master of each Flip Chart which students can use to fill in before, during, or after instruction. On the reverse side of each black-line master are questions corresponding to each Flip Chart topic which can be used as further review or as a means of assessment. While the activities in the guide can be used in conjunction with the Flip Charts, they can also be used individually for review or as a form of assessment or in conjunction with any other related assignment. Learning Centers Each Flip Chart provides students with a quick illustrated view of grade-appropriate curriculum concepts. Students may use these Flip Charts in small group settings along with the corresponding activity pages contained in the guide to learn or review concepts already covered in class. Students may also use these charts as reference while playing the NewPath’s Curriculum Mastery® Games. Independent student use Students can use the hands-on Flip Charts to practice and learn independently by first studying Side 1 of the chart and then using Side 2 of the chart or the corresponding graphical activities contained in the guide to fill in the answers and assess their understanding. Reference/Teaching resource Curriculum Mastery® Charts are a great visual supplement to any curriculum or they can be used in conjunction with NewPath’s Curriculum Mastery® Games. Chart # 1: Chart # 2: Chart # 3: Chart # 4: Chart # 5: Chart # 6: Chart # 7: Chart # 8: Chart # 9: Chart #10: Multistep Equations Inequalities Exponents, Factors & Multiples Numerical & Geometric Pr oportions Finding Volume All About Percents Introduction to Probability The Pythagorean Theorem Slope Nonlinear Functions & Set Theory

Solving Multistep Equations with Fractions Solving Multistep Equations with Like Terms 8y + 2 – 3y = 17 Solve: = 10 Solve: 5x – 5 7 = 10 = 70 5x – 5 7 5x – 5 • Follow the order of operations in reverse when solving equations with more than one operation (multistep equations). • Goals of solving multistep equations: • Place the variables on one side of the equal sign and the numbers on the other side. • Have the number in front of the variable equal to one. • Place the variables on the same side. • The number in front of the variable must be equal to one. • Two inverse operations are needed to solve the equation above – subtraction & division. Solving Two-Step Equations Using Division Solving Equations with Variables on Both Sides Solving Two-Step Equations Using Multiplication 3 x + 8 = 23 3 x + 8 = 15 x = 5 – 8 – 8 4 groups of -3 = -12 5n + 3 = 28 x – 5 = 20 5n = 25 = 5n 5 – 3 – 3 – 2 – 2 – 3x – 3x 4 • • 4 7 • • 7 + 5 + 5 + 5 + 5 0 9 x Subtract 3 from both sides of the equation. Multiply both sides by 4. Multiply both sides by 7. Divide both sides of the equation by 5. Divide both sides by 5. 25 5 1 4 = 25 x 1 4 = n 5n + 3 = 28 5 5x 5 75 5 = = = x 15 y = x 3 6 = x 100 Solve: x – 5 = 20 Solve: 1 4 Add 5 to both sides. 8y + 2 – 3y = 17 5y + 2 = 17 Multiply both sides by 7. Combine like terms (8y – 3y). Combine like terms (8y – 3y). Subtract 2 from both sides. Subtract 2 from both sides. 5y + 2 = 15 Divide both sides by 5. 5y 5 15 5 = 5x – 7 = 3x + 5 5x – 7 = 3x + 5 2x – 7 = 5 Solve: + 7 + 7 2x – 7 = 12 Add 7 to both sides. 2x 2 12 2 = Subtract 3x from both sides. Divide both sides by 2. © Copyright NewPath Learning. All Rights Reserved. 93-4701 www.newpathlearning.com Multistep Equations

\|xiBAHBDy01675lz[ Solving Multistep Equations with Fractions Solving Multistep Equations with Like Terms 8y + 2 – 3y = 17 Solve: = 10 Solve: 5x – 5 7 = 10 = 70 5x – 5 7 5x – 5 7 5x • Follow the order of operations in reverse when solving equations with more than one operation (multistep equations). • Goals of solving multistep equations: • _________________________________________ _________________________________________ • _________________________________________ _________________________________________ • Place the variables on the same side. • The number in front of the variable must be equal to . Solving Two-Step Equations Using Division Solving Equations with Variables on Both Sides Solving Two-Step Equations Using Multiplication 3 x + 8 = 23 3 x + 8 = 15 x = 5 – 8 – 8 5n + 3 = 28 x – 5 = 20 = = – 2 – 2 4 • • 4 7 • • 7 + 5 + 5 Subtract 3 from both sides of the equation. Multiply both sides by 4. Multiply both sides by 7. Divide both sides of the equation by 5. Divide both sides by 5. 1 4 = 25 x 1 4 = n = = x = y x = x 100 5n + 3 = 28 Solve: x – 5 = 20 Solve: 1 4 Add 5 to both sides. 8y + 2 – 3y = 17 5y + 2 = 17 Combine like terms (8y – 3y). Subtract from both sides. 5y + 2 = 15 Divide both sides by 5. 5y = 5x – 7 = 3x + 5 2x – 7 = 5 5x – 7 = 3x + 5 Solve: 2x = Add 7 to both sides. 2x = Subtract 3x from both sides. Divide both sides by 2. Key Vocabulary Terms • equation • inverse operation • multistep equation • operation • order of operations • variable • Two inverse operations are needed to solve the equation above – subtra ction & . . = © Copyright NewPath Learning. All Rights Reserved. 93-4701 www.newpathlearning.com Multistep Equations

• An inequality is a mathematical sentence that does not have an exact solution. Instead, a range of solutions will satisfy the inequality. • All the solutions of an inequality with more than one solution are called the solution set. • Inequalities are used in many real–world situations. An example is a driving speed sign with a number which tells you that your speed must be 65mph. • Graph each inequality separately. • Combine both graphs. • Solve an addition or subtraction inequality the same way as you would solve an equation. • When you multiply or divide both sides of an inequality by a negative integer, reverse the direction of the inequality symbol. Graphing Simple Inequalities Graphing Compound Inequalities Solving Inequalities by Adding or Subtracting Solving Inequalities by Multiplying or Dividing 4 groups of -3 = -12 0 9 x x + 3 5 x + 3 5 x 2 – 3 – 3 Example Meaning Symbol • greater than • more than • above • less than • fewer than • below • greater than or equal to • no less than • at least • less than or equal to • no more than • at most room capacity age – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 1 3 2 1 0 – 4 – 3 – 2 – 1 0 > 50 > > – > – 14 3 5 6 4 > – > > > –> – An inequality uses one of the following symbols, instead of an equal sign: An open circle is used when the variable is or a number. > > A closed circle is used when the variable is or a number. > > – – > x – 4 x 1 or > x – 2 y 1 > – > – x 1 > – x – 2 > x 2 > – > – > – > – – 2x 6 – 2x 6 – 2x 6 – 2 – 2 x – 3 > > > > © Copyright NewPath Learning. All Rights Reserved. 93-4702 www.newpathlearning.com Inequalities

• An inequality is a mathematical sentence that does not have an exact solution. Instead, a range of solutions will satisfy the inequality. • All the solutions of an inequality with more than one solution are called the so . • Inequal ities are used in many real–world situations. An example is a driving speed sign with a number which tells you that your speed must be 65mph. • Graph each inequality separately. • Combine both graphs. • Solve an addition or subtraction inequality the same way as you would solve an equation. • When you multiply or divide both sides of an inequality by a negative integer, revers ethe direction of the inequality . Graphing Simple Inequalities Graphing Compound Inequalities Solving Inequalities by Adding or Subtracting Solving Inequalities by Multiplying or Dividing Example Meaning Symbol • greater than • more than • above • less than • fewer than • below • greater than or equal to • no less than • at least • less than or equal to • no more than • at most room capacity age – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 5 – 4 – 3 – 2 – 1 5 4 3 2 1 0 – 1 3 2 1 0 – 4 – 3 – 2 – 1 0 > – An inequality uses one of the following symbols, instead of an equal sign: An op en circle is used when the variable is or a number. > > A c losed circle is used when the variable is or a number. > > – – > x – 4 x 1 or > x – 2 y 1 > – > – x 1 > – x – 2 > x + 3 5 x + 3 5 x 2 – 3 – 3 x 2 > – > – > – > – – 2x 6 – 2x 6 – 2x 6 x > > > © Copyright NewPath Learning. All Rights Reserved. 93-4702 www.newpathlearning.com Key Vocabulary Terms • closed circle • compound inequality • greater than • greater than or equal to • inequality • less than • less than or equal to • negative integer • open circle • simple inequality • solution • solution set • variable Inequalities \|xiBAHBDy01669kzU

Prime Factorization of 120 Prime Factorization Least Common Factor Greatest Common Factor Exponents Powers of Ten & Scientific Notation • An exponent tells how many times to multiply a number, called the base, by itself. • A number written with a base and an exponent is in exponential form. • Prime factorization is taking a number and breaking it down into its prime factors. • A prime number has exactly two factors – 1 and itself. • A composite number has more than two factors. • The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into each of the numbers. • The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers. • Move the decimal point 6 places to get a number that is greater than or equal to 1 and less than 10. • The exponent is equal to the number of decimal places you moved the point. The Prime Factorization of 120 is Formula Formula 4 = 4 x 4 x 4 = 64 3 10 = 10 x 10 x 10 x 10 = 10,000 = 2.38 x 10 2,380,000 4 6 base exponent Power of ten scientific notation standard form Greatest Common Factor of 36 & 54 Least Common Multiple of 5 & 6 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 54: 1, 2, 3, 6, 9, 18, 27, 54 The GCF is 18 5: 5, 10, 15, 20, 25, 30, 35,... 6: 6, 12, 18, 24, 30, 36, 42,... The LCM is 30 5 • 2 • 3 • 2 • 2 = 5 • 3 • 2 3 5 • 2 • 3 • 2 • 2 5 • 2 • 3 • 4 10 • 12 120 © Copyright NewPath Learning. All Rights Reserved. 93-4703 www.newpathlearning.com Exponents, Factors & Multiples

Prime Factorization of 120 Prime Factorization Least Common Factor Greatest Common Factor Exponents Powers of Ten & Scientific Notation • An exponent tells how many times to multiply a number, called the base, by itself. • A number written with a base and an exponent is in exponential form. • Prime factorization is _________________ _______________________________________ _______________________________________ • A prime number has exactly two factors – and itself. • A composite numb er has more than two factors. • The greatest common factor (GCF) of two or more numbers is ______________ ______________________________________ ______________________________________ • The least common multiple (LCM) of two or more numbers is ____________ ____________________________________ ____________________________________ • Move the decimal point 6 places to get a number that is greater than or equal to 1 and less than 10. • The is equal to the number of decimal places you moved the point. The Prime Factorization of 120 is 4 = 4 x 4 x 4 = 64 3 10 = 10 x 10 x 10 x 10 = 10,000 = 2.38 x 2,380,000 4 Power of ten scientific notation standard form Greatest Common Factor of 36 & 54 Least Common Multiple of 5 & 6 36: , , , , , , , , 54: , , , , , , , The GCF is 18 5: 5 , 5 , 5 , 5 , 5 , 5 ,... 6: 5 , 5 , 5 , 5 , 5 , 5 ,... The LCM is 30 5 • 2 • 3 • 2 • 2 = 5 • 3 • 2 5 • 2 • 3 • 2 • 2 5 • 2 • 3 • 4 10 • 12 120 Key Vocabulary Terms • base • composite number • exponent • exponential form • factor • greatest common factor • least common factor • multiple • power of ten • prime factorization • prime number • scientific notation • standard form © Copyright NewPath Learning. All Rights Reserved. 93-4703 www.newpathlearning.com Exponents, Factors & Multiples \|xiBAHBDy01667qzZ

Numerical Proportions Geometric Proportions corresponding sides corresponding angles A B C D E F • Numerical Proportions compare two numbers. • Ratios are used to compare quantities with the same unit. • Rates are used to compare quantities measured with different units. • Cross products are used to find a missing quantity in a proportion. • Geometric proportions compare two similar figures. • Similar figures have equal corresponding angles and corresponding sides that are in proportion. • Identify the corresponding sides in the two triangles. • Use ratios of the corresponding sides to determine whether the triangles ΔGHI and ΔJKL are similar. • The triangles are similar since their corresponding sides are equivalent. • A cross product is the product of the numerator in one ratio and the denominator in the other ratio. Cross Product Rule • When two ratios are equal, then the cross products, a•d and b•c, are equal. 3 6 1 2 = a b c d = m 5 12 4 = 4m 4 60 4 = 3 5 6 10 = 5 • 6 = 30 3 • 10 = 30 = m • 4 5 • 12 = 4m 60 = m 15 • Cross multiply. • Divide each side by 4 to isolate the variable. Determining whether two triangles are similar G I H 3 cm 6 cm 4 cm K J L 24 cm 12 cm 16 cm 4 16 6 24 = 3 12 = ? ? GH JK IH LK = IG LJ = ? ? 1 4 1 4 = 1 4 = • An equation with two equal ratios is called a proportion. © Copyright NewPath Learning. All Rights Reserved. 93-4704 www.newpathlearning.com GH corresponds to JK IH corresponds to LK IG corresponds to LJ Numerical & Geometric Proportions

Numerical Proportions Geometric Proportions • Numerical Proportions compare two numbers . • Ratios are used to compare quantities with the . • Rate s are used to compare quantities measured with different units. • are used to find a missing quantity in a proportion. • Geometric proportions compare two . • Similar figures have equal corresponding and corresponding that are in . • Identify the corresponding sides in the two triangles. • Use ratios of the corresponding sides to determine whether the triangles ∆GHI and ∆JKL are similar. • The triangles are similar since their corresponding sides are equiva lent. • A cross product is the product of the in in one ratio and the in the other ratio. 3 6 1 2 = Cross Product Rule • When two ratios are equal, then the cross products, a•d and b•c, are equal . a b c d = m 5 12 4 = 3 5 6 10 = 5 • 6 = 5 • 6 = = • • = = m 15 • Cross multiply. • Divide each side by to isolate the variable. A B C D E F Determining whether two triangles are similar • An equation with two equal ratios is called a proportion. G I H 3 cm 6 cm 4 cm K J L 24 cm 12 cm 16 cm = © Copyright NewPath Learning. All Rights Reserved. 93-4704 www.newpathlearning.com Numerical & Geometric Proportions Key Vocabulary Terms • corresponding angle • corresponding side • cross multiply • cross product • denominator • equation • equivalent • geometric proportion • numerator • numerical proportion • proportion • rate • ratio • similar figure • triangle • variable GH corresponds to JK GH corresponds to JK GH corresponds to JK = = ? ? = = \|xiBAHBDy01678mzV

8 cm (d) (h) (r) 9 cm 5 cm 5 cm 14 • 12 3.14 • 16 • 9 3.14 • 4 2 • 9 diameter (d) = radius (r) = radius (r) = r = 4 cm 3 cm 3 cm 3 cm 3 cm 8 cm 3 cm 4 cm 6 cm Area of a rectangular base 3 cm Rectangular Prism Prism (h) ( ) (w) (b) ( ) (h) (r) (h) 8 cm V = x w x h V = r 2 h V V = V = V = V = V = 45 cm3 3.14 168 cm3 452.16 cm3 Base Area Length (A) ( ) b • h, or • 4 • 7 = 14 1 2 1 2 V V d 2 8 cm 2 V = r 3 4 3 V = r 2 h 1 3 V = B • h V = 64 cm3 V = B = B = 1 3 V 75.36 cm3 V 113.04 cm3 V V V V 1 3 1 3 • 3.14 • 32 • 8 4 3 • 3.14 • 33 4 3 • 3.14 • 27 • 3.14 • 9 • 8 1 3 • 24 • 8 4 • 6 = 24 • w A = 8 cm 7cm (h) 4 cm 12 cm • Since the bases are triangles, the area of each triangle is Volume of Cylinders Volume of Prisms Volume of Cones Volume of Spheres Volume of Pyramids (b) © Copyright NewPath Learning. All Rights Reserved. 93-4705 www.newpathlearning.com Finding Volume

\|xiBAHBDy01668nzW 8 cm (d) (h) (r) 9 cm 5 cm 14 • 12 diameter (d) = radius (r) = radius (r) = r = 8 cm 3 cm 4 cm 6 cm • Since the bases are triangles, the area of each triangle is Area of rectangular base 3 cm Volume of Cylinders Volume of Prisms Volume of Cones Volume of Spheres Volume of Pyramids (h) (r) (h) 8 cm V = x w x h V = r 2 h V = V = V = V = V = Base Area Length (A) b•h, or • • = 1 2 1 2 3.14 V 452.16 cm3 3.14 • 16 • 9 V 3.14 • 4 2 • 9 V V = r 3 4 3 V = r 2 h 1 3 V = B • h V = 64 cm3 V = (B) = B = 1 3 V 75.36 cm3 V V V V V 1 3 1 3 • 3.14 • • 4 3 • 3.14 • 33 4 3 • 3.14 • 27 • 3.14 • • 1 3 • • • = • w A = Key Vocabulary Terms • cone • cylinder • diameter • Pi ( π) • prism • pyramid • radius • rectangular prism • sphere • triangle • volume © Copyright NewPath Learning. All Rights Reserved. 93-4705 www.newpathlearning.com Finding Volume 5 cm 3 cm 3 cm Rectangular Prism (h) ( ) (w) Prism (b) ( ) 7cm (h) 4 cm 12 cm (b) ( )

• Percent means “out of 100”. It is a ratio of a number to 100. • The symbol % is added after a number to indicate that it is a percent. • Write the percent as a fraction with a denominator of 100. • Simplify, if needed. Cross multiply. • Write the decimal as a fraction. • Write the decimal as a percent by moving the decimal point two places to the right. • Write the percent as a fraction with a denominator of 100. • Convert the fraction to a decimal. • Write an equivalent fraction with a denominator of 100. • Write the fraction as a percent. Percent Decimal Fraction Percents can be written as decimals or fractions. 0.10 0.25 0.33 0.50 0.75 1.0 10% 25% 50% 75% 100% 33.3% % Divide by 100 to isolate the variable. Cross multiply. Divide by 70 to isolate the variable. Divide by 0.15 to isolate the variable. Write the percent as a decimal. Solving Problems with Percents Percents as Fractions Determining the Percent of a number Using Proportions Using Equations 35 is what percent of 70? 36 is 15% of what number? What is 57% of 80? 57% of 80 is 45.6 Percents as Decimals Decimals as Percents Fractions as Percents 1 10 1 4 1 3 1 2 3 4 4 4 20 20% 9% 60% 0.23 100 = whole part whole part 57 100 m 80 = 4,560 100 100 100m = 57 • 80 100 • m = 45.6 m = 4,560 100m = 1 5 = 9 0.09 100 = = 23 23% 100 60 100 = = 3 • 20 5 • 20 3 5 = = = 35 is 50% of 70 m 100 35 70 = m • 70 100 • 35 = 36 0.15 • m = 36 15% • m = 50 m = 36 is 15% of 240 m 240 = 3,500 70m = 70 70 70m = 3,500 0.15 0.15 36 = 0.15 • m © Copyright NewPath Learning. All Rights Reserved. 93-4706 www.newpathlearning.com All About Percents

• Percent means . It is a ratio of a number to 100. • The symbol is added after a number to indicate that it is a percent. • Write the percent as a fraction with a denominator of 100. • Simplify, if needed. • Write the decimal as a fraction. • Write the decimal as a percent by moving the decimal point two places to the right. • Write the percent as a fraction with a denominator of 100. • Convert the fraction to a decimal. • Write an equivalent fraction with a denominator of 100. • Write the fraction as a percent. Percent Decimal Fraction Percents can be written as decimals or fractions. 0. 0. 0. 0. 0. 10% 25% 50% 75% 100% 33.3% % Divide by to isolate the variable. Divide by to isolate the variable. Cross multiply. Divide by 0.15 to isolate the variable. Write the percent as a decimal. Solving Problems with Percents Percents as Fractions Determining the Percent of a number Using Proportions Using Equations 35 is what percent of 70? 36 is 15% of what number? What is 57% of 80? 57% of 80 is 45.6 Percents as Decimals Decimals as Percents Fractions as Percents 20 20% 9% 0.23 100 = whole part whole part 57 100 m 80 = 45.6 m = 9 0.09 100 = = 23 23% 100 = = • 3 5 = = = 35 is 50% of 70 m 100 35 70 = 36 15% • m = 50 m = 36 is 15% of 240 m 240 = • = Key Vocabulary Terms • decimal • decimal point • denominator • equation • equivalent fraction • fraction • percent • proportion • • = • • = • = = = = = = • © Copyright NewPath Learning. All Rights Reserved. 93-4706 www.newpathlearning.com All About Percents Cross multiply. \|xiBAHBDy01516rzu

What is Probability? Experimental Probability Theoretical Probability Independent Events Dependent Events Combinations Permutations • Probability is the possibility that a certain event will occur. An event that is certain to occur has a probability of 1. An event that cannot occur has a probability of 0. Therefore, the probability of an event occuring is always between 0 and 1. • A cross product is the product of the numberator in one ratio and the denominator in the other ratio. 1 4 1 2 3 4 3 5 6 10 = 5 • 6 = 30 3 • 10 = 30 Impossible Unlikely Likely Certain As likely as not 1 0 1 0.75 100% 75% 0.50 50% 0.25 25% 0% 0 • Experimental probability is the probability that a certain outcome will occur based on an experiment being performed multiple times. • Probability is based on whether events are dependent or independent of each other. An independent event refers to the outcome of one event not affecting the outcome of another event. • A combination is an arrangement of objects or events in which order does not matter. • The number of ways that an event can occur depends on the order. • A permutation is an arrangement of objects or events in which order matters. There are 3 possible letter combinations: DG, OD & GO Example: How many different combinations can be made when picking 2 letters out of the word DOG? Example: Arrange 3 students in 3 chairs. • A dependent event is when the outcome of one event does affect the outcome of the other event. • Theoretical probability is the probability that a certain outcome will occur based on all the possible outcomes. P (a and b) = P (a) • P (b) P (a and b) = P (a) • P (b after a) 3 • 2 • 1 or 3! = 6 number of times an event occurs total number of trials Experimental Probability number of favorable outcomes total number of possible outcomes Theoretical Probability Probability of two independent events Probability of two dependent events probability of both events probability of first event probability of second event probability of both events probability of first event probability of second event Cross out duplicates. DO DG OD OG GD GO D O G 3 2 1 3 • 2 • 1 3 • 1 • 2 2 • 3 • 1 2 • 1 • 3 1 • 2 • 3 1 • 3 • 2 fraction decimal percent • • • • • • • • • • • • © Copyright NewPath Learning. All Rights Reserved. 93-4707 www.newpathlearning.com Introduction to Probability

\|xiBAHBDy01673rzu What is Probability? Experimental Probability Theoretical Probability Independent Events Dependent Events Combinations Permutations • Probability is the possibility that a will occur. An event that is certain to occur has a probability of . An event that cannot occur has a probability of . Therefore, the probability of an event occuring is always between and . • Experimental probability is ____________ _______________________________________ ______________________________________. • Probability is __________________________ _______________________________________ _______________________________________ ______________________________________. • A combination is ______________________ _______________________________________ ______________________________________. • The number of ways that an event can occur depends on the order. • A permutation is ______________________ ______________________________________. There are 3 possible letter combinations: DG, OD & GO Example: How many different combinations can be made when picking 2 letters out of the word DOG? Example: Arrange 3 students in 3 chairs. • A dependent event is __________________ _______________________________________ _______________________________________ ______________________________________. • Theoretical probability is ______________ _______________________________________ ______________________________________. P (a and b) = P (a) • P (b) P (a and b) = P (a) • P (b after a) 3 • 2 • 1 or 3! = 6 number of times an event occurs total number of trials Experimental Probability number of favorable outcomes total number of possible outcomes Theoretical Probability Probability of two independent events Probability of two dependent events probability of both events probability of first event probability of second event probability of both events probability of first event probability of second event Cross out duplicates. D O G 3 2 1 3 • 2 • 1 3 • 2 • 1 3 • 2 • 1 3 • 2 • 1 3 • 2 • 1 3 • 2 • 1 1 4 1 2 3 4 Impossible Unlikely Likely Certain As likely as not 1 0 0% 0 fraction decimal percent © Copyright NewPath Learning. All Rights Reserved. 93-4707 www.newpathlearning.com Introduction to Probability Key Vocabulary Terms • certain • combination • dependent event • event • experimental probability • favorable outcome • impossible • independent event • likely • outcome • permutation • possible outcome • probability • theoretical probability • unlikely

The Pythagorean Theorem • Pythagoras was one of the first mathematicians to recognize the relationship between the sides of a right triangle. This special relationship forms the Pythagorean Theorem. • The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle equals the square of the length of the hypotenuse. • According to the Pythagorean Theorem, the sum of the two green squares, is equal to the area of the blue square. Therefore in algebraic terms, the Pythagorean Theorem is stated as: Find the length of the hypotenuse ( ). Using the Pythagorean Theorem A right triangle is a triangle with an angle of 90º. T he two sides that form the right angle are called legs. The side opposite the right angle is the hypotenuse. Square A a hypotenuse • Area of square A = a2 • Area of square B = b2 • Area of square C = c2 right angle leg leg a2 + b2 = c2 a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 25 = c2 5 = C Square B Square C b c C C 3cm 4cm Substitute for the known variables. Take the square root of both sides. The length of the hypotenuse is 5cm. Pythagoras was a Greek mathematician and philosopher who made important contributions to mathematics and astronomy. The Pythagorean Theorem was Pythagoras’ most famous mathematical contribution. © Copyright NewPath Learning. All Rights Reserved. 93-4708 www.newpathlearning.com The Pythagorean Theorem

\|xiBAHBDy01689sz\ The Pythagorean Theorem • Area of square A = a2 • Area of square B = b2 • Area of square C = c2 • was one of the first mathematicians to recognize the relationship between the sides of a right triangle. This special relationship forms the . • The Pythagorean Theorem states • According to the Pythagorean Theorem, the sum of the two green squares, is equal to the area of the blue square. Therefore in algebraic terms, the Pythagorean Theorem is stated as: Find the length of the hypotenuse ( ). Using the Pythagorean Theorem A right triangle is a triangle with an of . The two sides that form the right angle are called . The side opposite the right angle is the . a2 + b2 = c2 a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 25 = c2 5 = C Square A a Square B Square C b c C C 3cm 4cm Substitute for the known variables. Take the square root of both sides. The length of the hypotenuse is 5cm . Key Vocabulary Terms • angle • hypotenuse • leg • Pythagorean Theorem • right triangle • side Pythagoras was a Greek mathematician and philosopher who made important contributions to mathematics and astronomy. The Pythagorean Theorem was Pythagoras’ most famous mathematical contribution. . © Copyright NewPath Learning. All Rights Reserved. 93-4708 www.newpathlearning.com The Pythagorean Theorem

• Slope is used to describe the steepness or incline of a straight line. A higher slope value indicates a steeper incline. • The slope of a line can be found by dividing the rise by the run. • The y–intercept is the y–value at which a line crosses the y–axis. • The graphs of linear equations are straight lines. Graphing Linear Functions Identifying Slope & y-intercept of a line x y (-5, 3) (-1, -2) -5 -4 -3 -2 -1 1 1 2 3 4 5 2 3 4 5 -1 -2 -3 -4 -5 0 x y (-4, 4) (0, 3) Slope rise run = y–axis x –axis rise run y–intercept • Slope is . The y–intercept is 3. From the y–intercept, move 1 unit up (rise) and 4 units left (run) to find another line coordinate. Draw a line to connect the points. • A horizontal line has a slope of 0. • A vertical line has no slope. • A line with a positive slope resembles a line going uphill. • A line with a negative slope resembles a line going downhill. 1 – 4 The slope–intercept form of a linear equation is: y m x + b = y–intercept slope -4 5 The rise is 5 The run is –4 Slope rise run = = 5 – 4 Graph: y = – x + 3 1 4 y–intercept -5 -4 -3 -2 -1 1 1 2 3 4 5 2 3 4 5 -1 -2 -3 -4 -5 0 © Copyright NewPath Learning. All Rights Reserved. 93-4709 www.newpathlearning.com Slope height (rise) Steepness of the steps = depth (run) rise run

\|xiBAHBDy01684nzW • Slope is used to describe • The slope of a line can be found by dividing the by the . • The y–intercept is the y–value at which a line crosses the . • The graphs of linear equations are straight lines. Graphing Linear Functions Identifying Slope & y-intercept of a line Slope = y–axis x –axis • Slope is . The y–intercept is . From the y–intercept, move unit up (rise) and units left (run) to find another line coordinate. Draw a line to connect the points. • A horizontal line has a slope of 0 . • A vertical line has . • A line with a slope resembles a line going uphill. • A line with a slope resembles a line going downhill. The slope–intercept form of a linear equation is: y m x + b = y–intercept slope The rise is 5 The run is –4 Slope = = 5 – 4 Graph: y = – x + 3 1 4 x y -5 -4 -3 -2 -1 1 1 2 3 4 5 2 3 4 5 -1 -2 -3 -4 -5 0 . x y -5 -4 -3 -2 -1 1 1 2 3 4 5 2 3 4 5 -1 -2 -3 -4 -5 0 ( , ) ( , ) run: rise: © Copyright NewPath Learning. All Rights Reserved. 93-4709 www.newpathlearning.com Slope Key Vocabulary Terms • coordinate • function • incline • linear equation • rise • run • slope • x-axis • y-axis • y-intercept

© Copyright NewPath Learning. All Rights Reserved. 93-4710 www.newpathlearning.com • A function is a rule that is performed on a number, called an input, to produce a result called an output. • There are two types of functions: linear and nonlinear. • Linear functions can be represented by straight lines on a coordinate plane. A linear function has an equation that can be written in the form of y = m x + b. • Equations whose graphs are not straight lines are called nonlinear functions. Nonlinear Functions Set Theory • A set is a collection of individual objects. • Set theory helps us organize things into groups and to understand logic. Subset Definition Notation Example Venn Diagram Set A is a subset of set B if all the elements of set A are in set B. A B A B A B A = {apple, orange} B = {apple, orange, pear, banana} A B All the elements of A are in B. Intersection Definition Notation Example Venn Diagram The intersection of sets A and B consists of only elements which belong to both sets A and B. A B A = {apple, orange, plum} B = {pear, banana, apple} A B = {apple} Apple is the only fruit in both sets. The union of sets A and B consists of all elements which belong to either set A or set B. A ∪ B A = {apple, orange, plum} B = {pear, banana, apple} A ∪ B = {apple, orange, plum, pear, banana} All the elements in both sets are included. As a round balloon gets inflated, its radius changes and its volume increases. Radius (in.) Volume (in 3 ) 1 2 3 4 5 6 4.19 33.52 113.13 268.16 523.75 904.32 0 1 2 3 4 5 6 7 8 9 10 Radius (in.) V olume (in 3 ) 0 100 200 300 400 500 600 700 800 900 1,000 Sample Graphs of Nonlinear Functions A B Union Definition Notation Example Venn Diagram 4 4 3 3 2 2 1 1 -1 -1 -2 -2 -3 -3 -4 -4 0 4 4 3 3 2 2 1 1 -1 -1 -2 -2 -3 -3 -4 -4 0 Nonlinear Functions & Set Theory

© Copyright NewPath Learning. All Rights Reserved. 93-4710 www.newpathlearning.com • A function is • There are two types of functions: and . • Linear functions can be represented by . A linear function has an equation that can be written in the form of . • Equations whose graphs are not straight lines are called . Nonlinear Functions Set Theory • A set is Subset Definition Notation Example Venn Diagram Intersection Definition Notation Example Venn Diagram As a round balloon gets inflated, its radius changes and its volume increases. Radius (in.) Volume (in 3 ) 1 2 3 4 5 6 4.19 33.52 113.13 268.16 523.75 904.32 0 1 2 3 4 5 6 7 8 9 10 Radius (in.) V olume (in 3 ) 0 100 200 300 400 500 600 700 800 900 1,000 Draw a Sample Graph of a Nonlinear Functions Union Definition Notation Example Venn Diagram 4 4 3 3 2 2 1 1 -1 -1 -2 -2 -3 -3 -4 -4 0 Key Vocabulary Terms • coordinate plane • equation • function • input • intersection • linear function • nonlinear function • output • set • set theory • subset • union . . . . . . • Set theory helps us Nonlinear Functions & Set Theory \|xiBAHBDy01676sz\

Math: Grade 7

Mathematics, Grade 7

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