The document is about algebra and graphing. It contains 12 lessons: negative numbers, finding points on a grid, graphing ordered pairs, problem-solving strategies using logical reasoning, functions, and graphing functions. The lessons include examples and practice problems related to these algebra and graphing topics.
This document provides an overview of Chapter 13 from a mathematics textbook on fractions. It includes summaries and examples for 9 lessons:
Lesson 13-1 introduces parts of a whole and identifying fractions for parts of a circle or figure.
Lesson 13-2 covers parts of a set and identifying fractions for parts of groups.
Lesson 13-3 demonstrates using drawings to solve word problems involving fractions.
Lesson 13-4 explains equivalent fractions through multiplication and division.
Lesson 13-5 defines simplest form and writing fractions in their simplest form.
Lesson 13-6 uses a word problem to demonstrate choosing the best problem-solving strategy.
Lesson 13-7 compares and orders fractions using
Let's analyze the pattern to write a rule.
The cost increases by $2 each time.
Rule: Cost = $5 + 2x
To find the cost for 7 people:
Cost = $5 + 2(7) = $5 + 14 = $19
The cost for 8 people is $5 + 2(8) = $5 + 16 = $21
So the amount earned for 7 and 8 people is $19 and $21.
The answer is A.
This document outlines lessons on dividing by one-digit numbers. Lesson 9-1 covers division with and without remainders. Lesson 9-2 discusses dividing multiples of 10, 100, and 1,000. Lesson 9-3 introduces the problem-solving strategy of guess and check. Lesson 9-4 is about estimating quotients. Each lesson provides examples and relates the content to California math standards.
This document contains information about geometry and measurement from a math textbook. It includes 7 lessons: on congruent figures, symmetry, perimeter, solving simpler problems, area, choosing a problem-solving strategy, and finding the area of complex figures. The lessons provide definitions, standards, examples and exercises related to these geometry and measurement topics.
This document contains an overview of Chapter 10 from a geometry textbook. It covers the following topics across 10 lessons: solid figures, plane figures, problem-solving strategies like looking for patterns, lines/segments/rays, angles, and problem-solving investigations. The chapter introduces key concepts, provides examples, and aligns topics to state math standards. It aims to teach students to identify, describe, classify and solve problems involving various geometric shapes and their properties.
This document provides an overview of Chapter 14 from a mathematics textbook. The chapter covers decimals, including tenths, hundredths, relating mixed numbers and decimals, problem-solving strategies involving making models, comparing and ordering decimals, and problem-solving investigations involving choosing the best strategy. It includes learning objectives, standards, examples and explanations for each of the 7 lessons covered in the chapter.
This document provides a summary of Chapter 15 from a mathematics textbook. The chapter covers adding and subtracting decimals through 6 lessons: 1) rounding decimals, 2) estimating decimal sums and differences, 3) using a problem-solving strategy of working backward, 4) adding decimals, 5) choosing a problem-solving strategy, and 6) subtracting decimals. Each lesson includes examples and practice problems to illustrate the concepts and build skills in adding and subtracting decimals.
The document is about algebra and graphing. It contains 7 lessons: negative numbers, finding points on a grid, graphing ordered pairs, problem-solving strategies using logical reasoning, functions, graphing functions, and a problem-solving investigation. Each lesson contains examples and practice problems to teach the concepts and standards covered in that lesson.
This document provides an overview of Chapter 13 from a mathematics textbook on fractions. It includes summaries and examples for 9 lessons:
Lesson 13-1 introduces parts of a whole and identifying fractions for parts of a circle or figure.
Lesson 13-2 covers parts of a set and identifying fractions for parts of groups.
Lesson 13-3 demonstrates using drawings to solve word problems involving fractions.
Lesson 13-4 explains equivalent fractions through multiplication and division.
Lesson 13-5 defines simplest form and writing fractions in their simplest form.
Lesson 13-6 uses a word problem to demonstrate choosing the best problem-solving strategy.
Lesson 13-7 compares and orders fractions using
Let's analyze the pattern to write a rule.
The cost increases by $2 each time.
Rule: Cost = $5 + 2x
To find the cost for 7 people:
Cost = $5 + 2(7) = $5 + 14 = $19
The cost for 8 people is $5 + 2(8) = $5 + 16 = $21
So the amount earned for 7 and 8 people is $19 and $21.
The answer is A.
This document outlines lessons on dividing by one-digit numbers. Lesson 9-1 covers division with and without remainders. Lesson 9-2 discusses dividing multiples of 10, 100, and 1,000. Lesson 9-3 introduces the problem-solving strategy of guess and check. Lesson 9-4 is about estimating quotients. Each lesson provides examples and relates the content to California math standards.
This document contains information about geometry and measurement from a math textbook. It includes 7 lessons: on congruent figures, symmetry, perimeter, solving simpler problems, area, choosing a problem-solving strategy, and finding the area of complex figures. The lessons provide definitions, standards, examples and exercises related to these geometry and measurement topics.
This document contains an overview of Chapter 10 from a geometry textbook. It covers the following topics across 10 lessons: solid figures, plane figures, problem-solving strategies like looking for patterns, lines/segments/rays, angles, and problem-solving investigations. The chapter introduces key concepts, provides examples, and aligns topics to state math standards. It aims to teach students to identify, describe, classify and solve problems involving various geometric shapes and their properties.
This document provides an overview of Chapter 14 from a mathematics textbook. The chapter covers decimals, including tenths, hundredths, relating mixed numbers and decimals, problem-solving strategies involving making models, comparing and ordering decimals, and problem-solving investigations involving choosing the best strategy. It includes learning objectives, standards, examples and explanations for each of the 7 lessons covered in the chapter.
This document provides a summary of Chapter 15 from a mathematics textbook. The chapter covers adding and subtracting decimals through 6 lessons: 1) rounding decimals, 2) estimating decimal sums and differences, 3) using a problem-solving strategy of working backward, 4) adding decimals, 5) choosing a problem-solving strategy, and 6) subtracting decimals. Each lesson includes examples and practice problems to illustrate the concepts and build skills in adding and subtracting decimals.
The document is about algebra and graphing. It contains 7 lessons: negative numbers, finding points on a grid, graphing ordered pairs, problem-solving strategies using logical reasoning, functions, graphing functions, and a problem-solving investigation. Each lesson contains examples and practice problems to teach the concepts and standards covered in that lesson.
This chapter document discusses multiplying by one-digit numbers. It is divided into 7 lessons:
Lesson 7-1 covers multiplying multiples of 10, 100, and 1,000 using patterns.
Lesson 7-2 focuses on determining if answers are reasonable.
Lesson 7-3 introduces estimating products by rounding numbers.
Lesson 7-4 teaches multiplying two-digit numbers by one-digit numbers using different strategies.
Lesson 7-5 has students choose the best strategy to solve problems.
Lessons 7-6 and 7-7 build on these skills to multiply multi-digit numbers and numbers with zeros.
This document outlines lessons from a mathematics textbook on multiplying multi-digit numbers by two-digit numbers. It includes 7 lessons that cover multiplying by tens and hundreds, estimating products, using different problem-solving strategies like acting it out, standard algorithms for multiplying two-digit and three-digit numbers by two-digit numbers, and choosing the best strategy for a given problem. Examples and practice problems are provided for each lesson.
The document is a chapter on addition and subtraction from a math textbook. It contains 7 lessons: 1) addition properties and subtraction rules, 2) estimating sums and differences, 3) problem-solving strategies for estimating or finding exact answers, 4) adding numbers, 5) subtracting numbers, 6) problem-solving investigations for choosing a strategy, and 7) subtracting across zeros. Each lesson provides examples and explanations of the concepts and includes practice problems for students to work through.
This document provides a math review for 6th grade students covering topics like writing numbers in standard, expanded, and word form; exponential form; order of operations; adding, subtracting, multiplying, and dividing decimals; solving equations; analyzing data sets; and making line plots from data tables. It includes examples and practice problems for students to work through related to these various math concepts.
This document discusses solving linear inequalities and compound inequalities. It provides examples of solving various inequalities algebraically and graphing their solution sets. It also gives an example of setting up and solving an inequality to model a real-world situation about payment plans for a house painting job.
1. The document is a mark scheme that provides guidance for examiners marking the Pearson Edexcel International GCSE Mathematics exam.
2. It outlines general marking principles such as marking candidates work positively and awarding all marks that are earned.
3. The mark scheme then provides specific guidance on how to award marks for questions on the exam involving topics like algebra, geometry, statistics, and probability.
This document contains a multi-part math worksheet involving operations with decimals and place value. It includes exercises asking students to:
1) Round decimal numbers to varying places and illustrate on number lines
2) Convert between units like meters and centimeters using exponents
3) Compare and order decimal numbers
3) Express measurements in expanded form using fractions or decimals
The worksheet covers skills like multiplying and dividing by powers of ten, rounding, ordering, converting between units, and expressing decimals in expanded form - all essential skills for understanding decimals and place value.
1. The document contains word problems involving multiplication and division of multi-digit numbers. It provides work for students to practice using standard algorithms and mental math strategies for solving problems.
2. Students are asked to solve problems by drawing area models, rounding factors, and using properties of operations. They also estimate products and solve multi-step word problems.
3. The goal is for students to gain fluency in using various calculation methods and be able to assess the reasonableness of their answers.
Here are the key steps to solve this type of problem:
1) Write the equation relating the quantities: b + 6 = 18
2) Undo the addition by subtracting 6 from both sides: b = 18 - 6
3) Simplify: b = 12
To check the solution, substitute b = 12 back into the original equation:
12 + 6 = 18
18 = 18
The check confirms that b = 12 is indeed the solution.
By learning the process of undoing operations methodically, I can set up and solve two-step equations to find the value of a variable. This helps explain what it means for two quantities to be equal - the operations performed on both sides result in
The document is a table of contents for a mathematics textbook for third grade students in the Philippines. It lists 46 lessons on topics like multiplication, division, properties of operations, and solving word problems involving these operations. The document also provides information about copyright and permissions for using materials in the book. It was developed by the Department of Education of the Republic of the Philippines.
This document provides a mark scheme for the Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 3HR exam from January 2015. It outlines the general marking guidance, including how to award marks for correct working and answers. It also provides specific guidance and mark allocations for each question on the exam. The mark scheme is intended to ensure all candidates receive equal treatment and are rewarded for what they have shown they can do.
This document is a mathematics exam for the International GCSE consisting of 21 multiple-choice questions covering topics like algebra, geometry, trigonometry, and statistics. The exam is 2 hours long and students must show their work. The front page provides instructions for completing the exam, including information about writing implements, how to fill in personal details, and guidance on showing working for partial credit. The back page leaves space for working out solutions to problems.
This document provides the marking scheme for the Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 3H exam from January 2015. It begins with some general marking guidance on how to apply the mark scheme positively and award marks for what students show they can do. It provides details on the types of marks that can be awarded and abbreviations used in the mark scheme. It also provides guidance on aspects like showing working, ignoring subsequent work, and awarding marks for parts of questions. The document then provides the mark scheme for specific questions on the exam.
1. The document is the cover page and instructions for a mathematics exam. It provides information such as the exam date, time allowed, materials permitted, and instructions on how to answer questions and show working.
2. The exam consists of 20 multiple choice and constructed response questions worth a total of 100 marks. Questions cover topics like algebra, geometry, statistics and calculus.
3. Candidates are advised to show all working, use diagrams where appropriate, and check answers if time permits. Calculators are permitted.
This document discusses counting techniques used in probability and statistics. It introduces the fundamental principle of counting and the multiplication rule for determining the total number of possible outcomes of multi-step processes. Specific counting techniques covered include the tree diagram, permutations, and combinations. Examples are provided to demonstrate how to apply these techniques to problems involving determining the number of arrangements of different objects.
This module discusses measures of variability such as range and standard deviation. It provides examples of computing the range of various data sets as the difference between the highest and lowest values. Standard deviation is introduced as a more reliable measure that considers how far all values are from the mean. Students learn to calculate standard deviation by finding the deviation of each value from the mean, squaring the deviations, taking the average of the squared deviations, and extracting the square root. They practice computing and interpreting the range and standard deviation of sample data sets.
The document provides examples of solving two-step inequalities and writing and solving word problems as inequalities. It begins with six examples of solving two-step inequalities by combining like terms and then isolating the variable, including graphing the solution sets on number lines. The next examples involve writing and solving word problems as inequalities, such as writing an inequality to represent the number of magazine subscriptions needed to earn $35. The document concludes by reviewing the process of solving two-step inequalities.
The document is a mark scheme that provides guidance to examiners for marking the Pearson Edexcel International GCSE Mathematics A (4MA0/4HR) Paper 4HR exam. It begins by introducing the Edexcel qualifications and some resources available on their website. It then provides general marking guidance on principles like treating all candidates equally, applying the mark scheme positively, and awarding all marks that are deserved according to the scheme. The rest of the document consists of detailed guidance on marking for each question on the exam.
This document provides the mark scheme for the Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 4HR exam from January 2015. It outlines the general marking guidance, including how to award marks, treat errors, and ignore subsequent working. It then provides detailed mark schemes for 20 multiple part questions on the exam, indicating the maximum marks, working required to earn marks, and acceptable answers.
The document discusses how to convert fractions to decimals and provides examples. It explains that to change a fraction to a decimal, we divide the numerator by the denominator, carrying the division to the desired number of decimal places. Some key points:
- Fractions can be expressed as decimals by dividing the numerator by the denominator
- To change a fraction to a decimal, divide the numerator by the denominator up to the desired number of decimal places
- Examples are provided such as 2/5 = 0.4
This document provides an overview and objectives of a modular workbook on decimal numbers for grade 6 students. It includes lessons on reading, writing, naming, comparing, ordering, rounding decimals, as well as lessons on equivalent fractions and decimals and the four fundamental operations of addition, subtraction, multiplication and division of decimal numbers. The workbook aims to help students understand the language and concepts of decimal numbers through exercises and examples.
This chapter document discusses multiplying by one-digit numbers. It is divided into 7 lessons:
Lesson 7-1 covers multiplying multiples of 10, 100, and 1,000 using patterns.
Lesson 7-2 focuses on determining if answers are reasonable.
Lesson 7-3 introduces estimating products by rounding numbers.
Lesson 7-4 teaches multiplying two-digit numbers by one-digit numbers using different strategies.
Lesson 7-5 has students choose the best strategy to solve problems.
Lessons 7-6 and 7-7 build on these skills to multiply multi-digit numbers and numbers with zeros.
This document outlines lessons from a mathematics textbook on multiplying multi-digit numbers by two-digit numbers. It includes 7 lessons that cover multiplying by tens and hundreds, estimating products, using different problem-solving strategies like acting it out, standard algorithms for multiplying two-digit and three-digit numbers by two-digit numbers, and choosing the best strategy for a given problem. Examples and practice problems are provided for each lesson.
The document is a chapter on addition and subtraction from a math textbook. It contains 7 lessons: 1) addition properties and subtraction rules, 2) estimating sums and differences, 3) problem-solving strategies for estimating or finding exact answers, 4) adding numbers, 5) subtracting numbers, 6) problem-solving investigations for choosing a strategy, and 7) subtracting across zeros. Each lesson provides examples and explanations of the concepts and includes practice problems for students to work through.
This document provides a math review for 6th grade students covering topics like writing numbers in standard, expanded, and word form; exponential form; order of operations; adding, subtracting, multiplying, and dividing decimals; solving equations; analyzing data sets; and making line plots from data tables. It includes examples and practice problems for students to work through related to these various math concepts.
This document discusses solving linear inequalities and compound inequalities. It provides examples of solving various inequalities algebraically and graphing their solution sets. It also gives an example of setting up and solving an inequality to model a real-world situation about payment plans for a house painting job.
1. The document is a mark scheme that provides guidance for examiners marking the Pearson Edexcel International GCSE Mathematics exam.
2. It outlines general marking principles such as marking candidates work positively and awarding all marks that are earned.
3. The mark scheme then provides specific guidance on how to award marks for questions on the exam involving topics like algebra, geometry, statistics, and probability.
This document contains a multi-part math worksheet involving operations with decimals and place value. It includes exercises asking students to:
1) Round decimal numbers to varying places and illustrate on number lines
2) Convert between units like meters and centimeters using exponents
3) Compare and order decimal numbers
3) Express measurements in expanded form using fractions or decimals
The worksheet covers skills like multiplying and dividing by powers of ten, rounding, ordering, converting between units, and expressing decimals in expanded form - all essential skills for understanding decimals and place value.
1. The document contains word problems involving multiplication and division of multi-digit numbers. It provides work for students to practice using standard algorithms and mental math strategies for solving problems.
2. Students are asked to solve problems by drawing area models, rounding factors, and using properties of operations. They also estimate products and solve multi-step word problems.
3. The goal is for students to gain fluency in using various calculation methods and be able to assess the reasonableness of their answers.
Here are the key steps to solve this type of problem:
1) Write the equation relating the quantities: b + 6 = 18
2) Undo the addition by subtracting 6 from both sides: b = 18 - 6
3) Simplify: b = 12
To check the solution, substitute b = 12 back into the original equation:
12 + 6 = 18
18 = 18
The check confirms that b = 12 is indeed the solution.
By learning the process of undoing operations methodically, I can set up and solve two-step equations to find the value of a variable. This helps explain what it means for two quantities to be equal - the operations performed on both sides result in
The document is a table of contents for a mathematics textbook for third grade students in the Philippines. It lists 46 lessons on topics like multiplication, division, properties of operations, and solving word problems involving these operations. The document also provides information about copyright and permissions for using materials in the book. It was developed by the Department of Education of the Republic of the Philippines.
This document provides a mark scheme for the Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 3HR exam from January 2015. It outlines the general marking guidance, including how to award marks for correct working and answers. It also provides specific guidance and mark allocations for each question on the exam. The mark scheme is intended to ensure all candidates receive equal treatment and are rewarded for what they have shown they can do.
This document is a mathematics exam for the International GCSE consisting of 21 multiple-choice questions covering topics like algebra, geometry, trigonometry, and statistics. The exam is 2 hours long and students must show their work. The front page provides instructions for completing the exam, including information about writing implements, how to fill in personal details, and guidance on showing working for partial credit. The back page leaves space for working out solutions to problems.
This document provides the marking scheme for the Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 3H exam from January 2015. It begins with some general marking guidance on how to apply the mark scheme positively and award marks for what students show they can do. It provides details on the types of marks that can be awarded and abbreviations used in the mark scheme. It also provides guidance on aspects like showing working, ignoring subsequent work, and awarding marks for parts of questions. The document then provides the mark scheme for specific questions on the exam.
1. The document is the cover page and instructions for a mathematics exam. It provides information such as the exam date, time allowed, materials permitted, and instructions on how to answer questions and show working.
2. The exam consists of 20 multiple choice and constructed response questions worth a total of 100 marks. Questions cover topics like algebra, geometry, statistics and calculus.
3. Candidates are advised to show all working, use diagrams where appropriate, and check answers if time permits. Calculators are permitted.
This document discusses counting techniques used in probability and statistics. It introduces the fundamental principle of counting and the multiplication rule for determining the total number of possible outcomes of multi-step processes. Specific counting techniques covered include the tree diagram, permutations, and combinations. Examples are provided to demonstrate how to apply these techniques to problems involving determining the number of arrangements of different objects.
This module discusses measures of variability such as range and standard deviation. It provides examples of computing the range of various data sets as the difference between the highest and lowest values. Standard deviation is introduced as a more reliable measure that considers how far all values are from the mean. Students learn to calculate standard deviation by finding the deviation of each value from the mean, squaring the deviations, taking the average of the squared deviations, and extracting the square root. They practice computing and interpreting the range and standard deviation of sample data sets.
The document provides examples of solving two-step inequalities and writing and solving word problems as inequalities. It begins with six examples of solving two-step inequalities by combining like terms and then isolating the variable, including graphing the solution sets on number lines. The next examples involve writing and solving word problems as inequalities, such as writing an inequality to represent the number of magazine subscriptions needed to earn $35. The document concludes by reviewing the process of solving two-step inequalities.
The document is a mark scheme that provides guidance to examiners for marking the Pearson Edexcel International GCSE Mathematics A (4MA0/4HR) Paper 4HR exam. It begins by introducing the Edexcel qualifications and some resources available on their website. It then provides general marking guidance on principles like treating all candidates equally, applying the mark scheme positively, and awarding all marks that are deserved according to the scheme. The rest of the document consists of detailed guidance on marking for each question on the exam.
This document provides the mark scheme for the Pearson Edexcel International GCSE Mathematics A (4MA0) Paper 4HR exam from January 2015. It outlines the general marking guidance, including how to award marks, treat errors, and ignore subsequent working. It then provides detailed mark schemes for 20 multiple part questions on the exam, indicating the maximum marks, working required to earn marks, and acceptable answers.
The document discusses how to convert fractions to decimals and provides examples. It explains that to change a fraction to a decimal, we divide the numerator by the denominator, carrying the division to the desired number of decimal places. Some key points:
- Fractions can be expressed as decimals by dividing the numerator by the denominator
- To change a fraction to a decimal, divide the numerator by the denominator up to the desired number of decimal places
- Examples are provided such as 2/5 = 0.4
This document provides an overview and objectives of a modular workbook on decimal numbers for grade 6 students. It includes lessons on reading, writing, naming, comparing, ordering, rounding decimals, as well as lessons on equivalent fractions and decimals and the four fundamental operations of addition, subtraction, multiplication and division of decimal numbers. The workbook aims to help students understand the language and concepts of decimal numbers through exercises and examples.
Here are the steps to make a double bar graph from the given data:
1. Draw two sets of bars side by side on the graph. Label one set "Weekday" and the other "Weekend".
2. For each activity (sleeping, eating, etc.), draw the appropriate length bar for the weekday amounts underneath the "Weekday" label.
3. Do the same for the weekend amounts, drawing the bars underneath the "Weekend" label.
4. Be sure to label the axes and provide a title for the double bar graph.
Let me know if any part needs more explanation! Making graphs from data takes some practice but gets easier with experience.
The document discusses the Dewey Decimal System, which was invented by Melvil Dewey to categorize books into 10 main subject groups represented by 3-digit numbers. It explains the general categories including 000s for general works, 100s for philosophy, 200s for religion, and so on up to 900s for history and geography. Nonfiction books are organized on shelves first by their Dewey Decimal number, which helps readers find books on the same subject near each other.
The document discusses simple interest calculations. It defines key terms like principal, rate, and time used to calculate simple interest using the formula I=PRT. It provides examples of simple interest problems, such as calculating interest earned on a $500 savings account with an annual interest rate of 2.5% over 18 months. The document also discusses using simple interest to calculate the total cost of a $7,000 car loan with 9% annual interest over 4 years.
This document provides an overview and objectives of a modular workbook on decimal numbers. It introduces decimals and their place value, explaining how to read, write, name, compare, order, and round decimal numbers. Exercises are included to help learners evaluate their understanding of decimals.
1. The document discusses multiplying decimal numbers, including multiplying decimals by whole numbers, decimals by decimals, and decimals by 10, 100, and 1,000.
2. Key rules covered are counting decimal places to determine the product's decimal placement and moving the decimal over when multiplying by powers of 10.
3. Examples provide step-by-step workings of multiplying decimals using partial products and placing the decimal point correctly in the final product.
This document provides an overview and objectives of a modular workbook on learning decimal numbers for 6th grade students. It covers reading, writing, naming, comparing, ordering, and rounding decimal numbers. It also includes lessons on equivalent fractions and decimals, and the four arithmetic operations of addition, subtraction, multiplication and division of decimal numbers. The workbook aims to help students understand and work with decimal numbers in a fun and engaging way through various exercises and activities.
The document provides a lesson on evaluating numerical expressions using the order of operations. It includes examples of simplifying expressions with addition, subtraction, multiplication, division and exponents. It also has examples of word problems involving amounts of money spent on items. The lesson concludes with a quiz to assess understanding of simplifying expressions and applying the order of operations.
This chapter discusses using algebra to represent and solve problems involving addition, subtraction, and finding patterns and rules. It includes the following key points:
- Lesson 3-1 covers writing and evaluating expressions with variables and addition/subtraction.
- Lesson 3-2 explains how to solve addition and subtraction equations mentally without using models.
- Lesson 3-3 introduces identifying extra and missing information in word problems in order to write and solve the correct equations.
- Lesson 3-4 teaches finding patterns in tables and writing rules as equations that can be used to determine future terms in the pattern.
Pre-Calculus Quarter 4 Exam
1
Name: _________________________
Score: ______ / ______
1. Find the indicated sum. Show your work.
2. Locate the foci of the ellipse. Show your work.
𝑥2
36
+
𝑦2
11
= 1
Pre-Calculus Quarter 4 Exam
2
3. Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
4. Graph the function. Then use your graph to find the indicated limit. You do not have to
provide the graph
f(x) = 5x - 3, f(x)
5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
Pre-Calculus Quarter 4 Exam
3
6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
Short Answer Questions: Type your answer below each question. Show your work.
8. A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true. Show your work.
Sn: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
𝑛(6𝑛2−3𝑛−1)
2
Pre-Calculus Quarter 4 Exam
4
9. A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely. Show your work.
Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10. Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of p
This chapter discusses multiplication and division facts. It includes 10 lessons: relating multiplication and division; algebra properties; facts through 5; problem solving skills; facts through 10; multiplying with 11 and 12; problem solving investigations; multiplying three numbers; factors and multiples; and prime and composite numbers. The lessons provide examples and practice with multiplication and division concepts and skills.
The document outlines the daily lesson plan for a math class which includes warm-up problems on solving equations, working through examples of adding, subtracting, multiplying and dividing equations, and explaining how to solve equations involving division. Students are provided practice problems to work through involving various equation types and instructed on the key steps to solving different kinds of equations.
This document provides examples and explanations for multiplying and dividing integers. It discusses how the sign of the product is determined by whether the signs of the integers being multiplied are the same or different. It also uses examples involving golf scores to demonstrate adding and multiplying integers to calculate a total score.
The document provides instructions and examples for solving integer equations and working with absolute values. It discusses key concepts like:
- The rules for adding, subtracting, multiplying and dividing integers
- Classifying numbers as natural, whole, integer, rational or real
- Understanding that the absolute value of a number represents its distance from zero
- Solving problems involving absolute values and performing operations inside and outside of absolute value signs
The document provides an introduction and overview of inequalities for a math class. It includes:
1) A discussion of the key differences between equations and inequalities, noting that inequalities can have a range of solutions rather than a single value.
2) Examples of how to write inequalities using appropriate symbols (<, >, ≤, ≥) and an explanation of open vs. closed circles on a number line.
3) Steps for solving linear inequalities, with the reminder that the inequality sign must be flipped when multiplying or dividing both sides by a negative number.
4) Practice problems for students to solve and graph inequalities on a number line.
The document provides instructions for several math concepts:
1. Multiplying integers with the same or different signs. When signs are the same, the product is positive, and when signs differ, the product is negative.
2. Exponents - When multiplying a negative number with an exponent, multiply the base by itself the number of times the exponent indicates and then apply the negative sign.
3. The distributive property - Multiplying numbers both inside and outside parentheses according to the property.
The document provides instructions for several math concepts:
1. Multiplying integers with the same or different signs. When signs are the same, the product is positive, and when signs differ, the product is negative.
2. Exponents - When multiplying a negative number with an exponent, multiply the base by itself the number of times the exponent indicates and then apply the negative sign.
3. The distributive property - Multiplying numbers both inside and outside parentheses according to the property.
The document provides instructions for multiplying integers, exponents, the distributive property, and adding/subtracting integers. It includes examples of:
- Multiplying integers with the same or different signs
- Working with negative exponents
- Using the distributive property to simplify expressions
- Combining like terms by adding/subtracting variables
- Rules for adding/subtracting integers based on sign
The document provides instructions for several math concepts:
1. Multiplying integers with the same or different signs. When signs are the same, the product is positive, and when signs differ, the product is negative.
2. Exponents - When multiplying a negative number with an exponent, multiply the base by itself the number of times the exponent indicates and then apply the negative sign.
3. The distributive property - Multiplying numbers both inside and outside parentheses according to the property.
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The document is a math lesson on dividing integers. It provides examples of dividing integers with the same sign or different signs. It explains that if the signs are the same, the quotient is positive, and if the signs are different, the quotient is negative. It also notes that you cannot divide by zero. The lesson includes practice problems evaluating integer expressions and word problems involving integer division.
The document provides information about absolute values and the real number system. It includes:
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1) When multiplying integers with the same sign, the product is positive, but with different signs, the product is negative.
2) For exponents, if the base is in parentheses, you raise that number to the power. If the base is negative without parentheses, you raise the absolute value to the power and then make the answer negative.
3) The distributive property distributes the number being multiplied over terms in parentheses by multiplying each term individually and then combining like terms.
This document contains a lesson on solving equations using the division and multiplication properties of equality. It includes examples of solving equations by dividing or multiplying both sides of the equation by the same number. It also contains examples of writing and solving rate problems using the formula distance = rate x time. The lesson emphasizes understanding what it means for two quantities to be equal and using properties of equality to solve equations.
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1. Chapter 12
Algebra and Graphing
Click the mouse or press the space bar to continue.
2. Algebra and Graphing
12
Lesson 12-1 Negative Numbers
Lesson 12-2 Find Points on a Grid
Lesson 12-3 Graph Ordered Pairs
Lesson 12-4 Problem-Solving Strategy: Use
Logical Reasoning
Lesson 12-5 Functions
Lesson 12-6 Graph Functions
Lesson 12-7 Problem-Solving Investigation:
Choose a Strategy
3. 12-1 Negative Numbers
Five-Minute Check (over Chapter 11)
Main Idea and Vocabulary
California Standards
Example 1
Example 2
Example 3
4. 12-1 Negative Numbers
• I will understand and use negative numbers.
• positive number
• negative number
5. 12-1 Negative Numbers
Standard 4NS1.8 Use concepts of
negative numbers (e.g., on a number line, in
counting, in temperature, and in “owing”).
6. 12-1 Negative Numbers
Write the number that represents the situation.
Then show the number on a number line. Sara
owes her mom $5.
When you owe money, it is a decrease.
The number is –5.
7. 12-1 Negative Numbers
Write the number that represents the situation.
The temperature is 7 below zero.
A. –7
B. 7
C. 0
D. –8
8. 12-1 Negative Numbers
Write the number that represents the situation.
Then show the number on a number line. Grant
earned $7 for shoveling Mr. Lincoln’s driveway.
When you earn money, it is an increase.
The number is 7 or +7.
9. 12-1 Negative Numbers
Write the number that represents the situation.
Julia deposits $25 into her bank account.
A. 0
B. 25
C. –25
D. 30
10. 12-1 Negative Numbers
Write the number of each letter on the number line.
A is between –4 and –2. So, A is –3.
B is between –2 and 0. So, B is –1.
C is the same distance from zero as +4.
Answer: So, C is +4.
11. 12-1 Negative Numbers
Write the number of each letter on the number line.
A. –5; –4; –3
B. –3; –1; 4
C. –3; –2; –1
D. –3; 1; 4
12.
13. 12-2 Find Points on a Grid
Five-Minute Check (over Lesson 12-1)
Main Idea and Vocabulary
California Standards
Example 1
Example 2
14. 12-2 Find Points on a Grid
• I will use ordered pairs to find and name points on
a grid.
• coordinate plane • y-axis
• origin • ordered pair
• x-axis • coordinates
15. 12-2 Find Points on a Grid
Preparation for Standard 4MG2.1 Draw the
points corresponding to linear relationships on
graph paper (e.g., draw 10 points on the graph of
the equation y = 3x and connect them by using a
straight line).
16. 12-2 Find Points on a Grid
What is
located at
point (2, 5)?
To find (2, 5), start at
(0, 0). Move right 2
units. Then, move up
5 units.
Answer: The ordered pair (2, 5) locates the post office.
17. 12-2 Find Points on a Grid
What is located at
point (1, 3)?
A. red house
B. blue house
C. tan house
18. 12-2 Find Points on a Grid
What letter is located
at (–3, 5)?
To find (–3, 5), start at
(0, 0). The –3 tells you
to move 3 units to the
left. The 5 tells you to
move 5 units up.
Answer: The ordered pair (–3, 5) locates the letter C.
19. 12-2 Find Points on a Grid
What letter is located
at (–4, 2)?
A. Q
B. R
C. S
20.
21. 12-3 Graph Ordered Pairs
Five-Minute Check (over Lesson 12-2)
Main Idea and Vocabulary
California Standards
Key Concept: Length of Line Segments
Example 1
Example 2
22. 12-3 Graph Ordered Pairs
• I will graph ordered pairs and find the lengths
of line segments on a coordinate grid.
• graph
23. 12-3 Graph Ordered Pairs
Standard 4MG2.2 Understand that the
length of a horizontal line segment equals
the difference of the x-coordinates.
Standard 4MG2.3 Understand that the
length of a vertical line segment equals the
difference of the y-coordinates.
25. 12-3 Graph Ordered Pairs
P
Graph point P at (–2, 4).
Step 1 Start at (0, 0).
Step 2 The x-coordinate
is –2. So, move 2
units to the left.
Step 3 The y-coordinate
is 4. So, move 4
units up.
Step 4 Graph a point at (–2, 4). Label it P.
26. 12-3 Graph Ordered Pairs
Choose which graph shows point A at (1, 4).
A. B.
27. 12-3 Graph Ordered Pairs
Choose which graph shows point A at (1, 4).
C. D.
28. 12-3 Graph Ordered Pairs
Choose which graph shows point A at (1, 4).
C.
29. 12-3 Graph Ordered Pairs
Find the distance
between (2, 2) and
(2, 6).
The line segment is
vertical. Subtract the y-
coordinates.
length of segment
=6–2
=4
Answer: The length is 4 units.
30. 12-3 Graph Ordered Pairs
Find the distance between (1, 5) and (1, 2).
A. 2 units
B. 3 units
C. 4 units
D. 5 units
31.
32. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Five-Minute Check (over Lesson 12-3)
Main Idea
California Standards
Example 1: Problem-Solving Strategy
33. 12-4 Problem-Solving Strategy: Use Logical Reasoning
• I will solve problems using logical reasoning.
34. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Standard 4MR1.1 Analyze problems by
identifying relationships, distinguishing
relevant from irrelevant
information, sequencing and prioritizing
information, and observing patterns.
35. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Standard 4SDAP1.1 Formulate survey questions;
systematically collect and represent data on a
number line; and coordinate graphs, tables, and
charts.
36. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Bella, Devan, Carl, and Jill live on
Ash, Pine, Maple, and Oak Streets. Bella lives on
Ash. Devan does not live on Pine. Carl lives on
Maple. What street does Jill live on?
37. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Understand
What facts do you know?
• The four students live on
Ash, Pine, Maple, and Oak Streets.
• Bella lives on Ash Street.
• Devan does not live on Pine Street.
• Carl lives on Maple Street.
What do you need to find?
• Find what street Jill lives on.
38. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Plan
You can use logical reasoning and a table to solve
the problem.
39. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Solve
• Bella lives on Ash and Carl lives on Maple. So Jill
cannot live on either of these streets.
• Devan does not live on Pine Street. He must live
on Oak Street.
Answer: So, Jill must live on Pine Street.
40. 12-4 Problem-Solving Strategy: Use Logical Reasoning
Check
Look back at the problem. The answer makes sense
for the facts given in the problem.
So, the answer is correct.
41.
42. 12-5 Functions
Five-Minute Check (over Lesson 12-4)
Main Idea and Vocabulary
California Standards
Example 1
Example 2
Example 3
43. 12-5 Functions
• I will complete function tables.
• function
44. 12-5 Functions
Standard 4AF1.5 Understand that an
equation, such as y = 3x + 5, is a prescription
for determining a second number when a first
number is given.
45. 12-5 Functions
Jake makes a $2 profit for each magazine
subscription he sells. Complete the function
table to show the profits earned for the number
of subscriptions sold.
Make a table to find how
much money Jake will
make for the number of
subscriptions sold. 4 2 8
Multiply each input by 2 6 2 12
to find each output.
46. 12-5 Functions
Anita makes $5 for every yard she rakes leaves for.
Complete the function table to show the profits
earned for the number of yards raked.
A. 4 × 2 = 8; 6 × 2 = 12
B. 5 × 4 = 25; 5 × 6 = 35
C. 5 × 4 = 20; 5 × 6 = 30
D. 4 × 4 = 16; 6 × 6 = 36
47. 12-5 Functions
Use the rule y = 4x to complete a function table
where the input is 4, 6, 8, 10.
The rule y = 4x
means to multiply x
by 4 to get y.
48. 12-5 Functions
Use the rule y = 5x to complete a function table
where the input is 2, 4, 6, 8. What is the output for
each input?
A. 10, 20, 30, 40
B. 5, 10, 15, 20
C. 7, 9, 11, 13
D. 2, 4, 6, 8
49. 12-5 Functions
Use the rule y = 2x + 5 to complete a function table
where the input is 3, 5, 7.
First, multiply x by 2.
Then, add 5 to the
product to get y.
50. 12-5 Functions
Use the rule y = 4x + 1 to complete a function table
where the input is 2, 4, 6. What is the output for
each input?
A. 8, 16, 24
B. 9, 17, 25
C. 2, 4, 6
D. 3, 5, 7
51.
52. 12-6 Graph Functions
Five-Minute Check (over Lesson 12-5)
Main Idea
California Standards
Example 1
Example 2
Graph Functions
53. 12-6 Graph Functions
• I will graph functions.
54. 12-6 Graph Functions
Standard 4MG2.1 Draw the points
corresponding to linear relationships on graph
paper (e.g., draw 10 points on the graph of the
equation y = 3x and connect them by using a
straight line).
55. 12-6 Graph Functions
Each smoothie at the bakery costs $3. The function
table shows how much it will cost if you buy
1, 2, 3, or 4 smoothies. Write the ordered pairs and
graph the function y = 3x. Then use the graph to
find how much 6 smoothies will cost.
Step 1 Write the
ordered pairs.
Then graph.
56. 12-6 Graph Functions
Step 2 Extend the
pattern in the
graph by drawing
a straight line.
The straight line
will help you see
the pattern.
Answer: So, 6 smoothies
will cost $18.
57. 12-6 Graph Functions
Noah gets $11 a week for allowance. Find ordered
pairs and graph y = 11x to find how much money
Noah will have after 8 weeks.
A. $70
B. $75
C. $80
D. $88
58. 12-6 Graph Functions
Graph 10 points on the graph of the function
y = 3x – 1.
Complete a table to
find the ordered pairs.
59. 12-6 Graph Functions
Then graph the ordered
pairs on a coordinate
plane. Connect the
points with a straight line.
60. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
A.
61. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
B.
62. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
C.
63. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
D.
64. 12-6 Graph Functions
Choose the correct chart of ordered pairs and graph
that go with the equation y = 2x + 2.
D.
65.
66. 12-7 Problem-Solving Investigation: Choose a Strategy
Five-Minute Check (over Lesson 12-6)
Main Idea
California Standards
Example 1: Problem-Solving Investigation
67. 12-7 Problem-Solving Investigation: Choose a Strategy
• I will solve problems by choosing the best
strategy.
68. 12-7 Problem-Solving Investigation: Choose a Strategy
Standard 4MR1.1 Analyze problems by
identifying relationships, distinguishing
relevant from irrelevant
information, sequencing and prioritizing
information, and observing patterns.
69. 12-7 Problem-Solving Investigation: Choose a Strategy
Standard 4AF1.5 Understand that an
equation such as y = 3x + 5 is a prescription
for determining a second number when a
first number is given.
70. 12-7 Problem-Solving Investigation: Choose a Strategy
AIDEN: I just got a new video game
system. Games cost $20. Felice has
the older version of the video game
system. Her games cost $15. How
many video games can we each buy if
we each have $60?
YOUR MISSION: Find out how many
games each person can buy.
71. 12-7 Problem-Solving Investigation: Choose a Strategy
Understand
What facts do you know?
• Games for Aiden’s game system cost $20.
• Games for Felice’s game system cost $15.
• Each has $60 to spend on video games.
What do you need to find?
• Find how many games each person can buy.
72. 12-7 Problem-Solving Investigation: Choose a Strategy
Plan
To find the answer, organize the data to show the
number of games and the total amount of money
spent.
73. 12-7 Problem-Solving Investigation: Choose a Strategy
Solve
Answer: Since Aiden’s games cost more, he can
only buy 3, whereas Felice can buy 4.
74. 12-7 Problem-Solving Investigation: Choose a Strategy
Check
Look back at the problem. Since 20 × 3 = 60 and
15 × 4 = 60, you know that the answer is correct.
78. Algebra and Graphing
12
(over Chapter 11)
Find the area of the figure.
4
A. 60 6
6
B. 36 4
C. 56
D. 48
79. Algebra and Graphing
12
(over Lesson 12-1)
Write the number of letter A on the number line.
A. –2
B. 3
C. –3
D. –1
80. Algebra and Graphing
12
(over Lesson 12-1)
Write the number of letter B on the number line.
A. 3
B. 1
C. –2
D. 2
81. Algebra and Graphing
12
(over Lesson 12-1)
Write the number of letter C on the number line.
A. –4
B. 4
C. 2
D. 3
82. Algebra and Graphing
12
(over Lesson 12-2)
Write the ordered pair that names point A.
A. (2, –2)
B. (–1, 2)
C. (–2, 2)
D. (–2, –2)
83. Algebra and Graphing
12
(over Lesson 12-2)
Write the ordered pair that names point B.
A. (–5, –1)
B. (5, 1)
C. (–5, 2)
D. (–5, 1)
84. Algebra and Graphing
12
(over Lesson 12-2)
Write the ordered pair that names point C.
A. (–4, 3)
B. (4, –3)
C. (–4, –3)
D. (4, 3)
85. Algebra and Graphing
12
(over Lesson 12-3)
Find the length of the horizontal or vertical line
segment formed by the following set of ordered
pairs: (5, 4), (1, 4).
A. 6
B. 9
C. 4
D. 5
86. Algebra and Graphing
12
(over Lesson 12-3)
Find the length of the horizontal or vertical line
segment formed by the following set of ordered
pairs: (–3, 6), (–3, 0).
A. 3
B. 6
C. 9
D. 0
87. Algebra and Graphing
12
(over Lesson 12-4)
Hugo has horses and ducks on his farm. He has 3
times as many horses as he does ducks. Together,
the animals have 14 legs. How many horses and
ducks does Hugo have?
A. 1 horse, 3 ducks
B. 2 horses, 3 ducks
C. 1 horse, 5 ducks
D. 3 horses, 1 duck
88. Algebra and Graphing
12
(over Lesson 12-5)
Use the rule to complete the function table.
A. 5, 6, 7
B. 9, 18, 27
C. 6, 9, 12
D. 5, 8, 11
89. Algebra and Graphing
12
(over Lesson 12-6)
Choose the set of points that belong on the graph of
the function y = 2x.
A. (1, 2), (3, 6), (5, 8)
B. (1, 3), (2, 4), (3, 5)
C. (2, 1), (4, 2), (6, 3)
D. (1, 2), (2, 4), (3, 6)
90. Algebra and Graphing
12
(over Lesson 12-6)
Choose the set of points that belong on the graph of
the function 2x = y.
A. (20, 10), (18, 9), (16, 8)
B. (10, 12), (9, 11), (8, 10)
C. (10, 20), (9, 18), (8, 16)
D. (10, 20), (18, 9), (8, 16)