(7) Lesson 1.8 - Slope
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This document provides instruction on identifying proportional relationships and calculating slope. It includes examples of finding the constant rate of change from tables and graphs, as well as calculating slope from points on a line. The document emphasizes that slope represents the rate of change between any two points and can be used to show proportional relationships between quantities in real-world scenarios.
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(8) Lesson 3.3
1. The document provides examples of finding slopes of lines from given points and determining if points form a parallelogram. It also discusses direct variation relationships and using equations, tables, and graphs to represent and compare them.
2. A constant of variation in a direct variation relationship is the constant that relates the ratio of the output quantity to the input quantity. It represents the unit rate or slope in the direct variation equation and graph.
3. Graphs are helpful for comparing different proportional relationships represented in different ways and finding the unit rate between quantities in a relationship from its graphical representation.
(7) Lesson 1.7 - Constant Rate of Change
Here is a possible table with a constant rate of change:
Hours Worked | Money Earned
1 | $10
2 | $20
3 | $30
4 | $40
The constant rate of change is $10 per hour. The money earned increases by $10 for every 1 hour worked.
(8) Lesson 5.2 - Geometric Proof
The document discusses paragraph proofs and two-column proofs in geometry. A paragraph proof states the given information, what is to be proven, then uses a logical chain of statements and reasons to justify the conclusion. A two-column proof separates the statements from the reasons into two columns, with the given information and conclusion stated. Both proof styles use definitions, properties, and theorems to logically justify each step. The document provides examples of each type of proof to illustrate the process.
(7) Lesson 3.3 - Subtract Integers
The document is a lesson on adding and subtracting integers:
1. It provides examples of adding and subtracting integers and finding the sums and differences.
2. It explains that to subtract an integer, you add its additive inverse, such as 4 - 9 = 4 + (-9) = -5.
3. It contains practice problems evaluating integer expressions and solving real-world problems involving adding and subtracting integers.
(8) Lesson 3.4
This document contains a mathematics lesson on graphing linear functions and interpreting slope and y-intercept. It includes examples of writing equations from graphs, finding slope and y-intercept, graphing linear equations, and interpreting slope and y-intercept in real world contexts. Key concepts covered are slope-intercept form, slope, y-intercept, rate of change, and using graphs to model proportional relationships.
(8) Lesson 2.1 - Solve Equations with Rational Coefficients
This document provides instruction on solving linear equations with one variable. It begins by defining key terms like multiplicative inverse and coefficient. It then provides examples of solving various equations by multiplying or dividing both sides of the equation by the same term to isolate the variable. These examples include equations with fractions, decimals, and percentages. The document emphasizes using the inverse operation property to solve equations and checking solutions. It aims to help students solve equations that may require expanding and combining like terms.
(8) Lesson 3.7
1. The document provides examples for writing and solving systems of linear equations by graphing. It explains how to write equations in slope-intercept form, graph the lines on the same coordinate plane, and determine if the system has no solution, one solution, or an infinite number of solutions based on where the lines intersect.
2. Key steps shown include writing equations in y=mx+b form, finding the slope and y-intercept, graphing the lines and analyzing where they intersect to determine the number of solutions.
3. Graphing systems of equations allows you to visually see where lines intersect, satisfying both equations simultaneously and showing the solution to the system.
(8) Lesson 5.3 - Angles of Triangles
1. The document discusses applying algebraic concepts to solve geometric problems involving triangles. It provides examples of writing and solving equations using angle sums, exterior angles, and ratios of interior angles to find missing angle measures.
2. Algebraic concepts like writing equations, using properties of equality, and solving for unknowns allow for finding missing angles in triangles given information about other angles or ratios of angles.
3. Examples show setting up and solving equations from statements about angle sums equaling 180 degrees, exterior angles equaling the sum of remote interior angles, and ratios of interior angles. This allows for determining missing angle measures.
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1. The document provides examples of finding slopes of lines from given points and determining if points form a parallelogram. It also discusses direct variation relationships and using equations, tables, and graphs to represent and compare them.
2. A constant of variation in a direct variation relationship is the constant that relates the ratio of the output quantity to the input quantity. It represents the unit rate or slope in the direct variation equation and graph.
3. Graphs are helpful for comparing different proportional relationships represented in different ways and finding the unit rate between quantities in a relationship from its graphical representation.
(7) Lesson 1.7 - Constant Rate of Change
Here is a possible table with a constant rate of change:
Hours Worked | Money Earned
1 | $10
2 | $20
3 | $30
4 | $40
The constant rate of change is $10 per hour. The money earned increases by $10 for every 1 hour worked.
(8) Lesson 5.2 - Geometric Proof
The document discusses paragraph proofs and two-column proofs in geometry. A paragraph proof states the given information, what is to be proven, then uses a logical chain of statements and reasons to justify the conclusion. A two-column proof separates the statements from the reasons into two columns, with the given information and conclusion stated. Both proof styles use definitions, properties, and theorems to logically justify each step. The document provides examples of each type of proof to illustrate the process.
(7) Lesson 3.3 - Subtract Integers
The document is a lesson on adding and subtracting integers:
1. It provides examples of adding and subtracting integers and finding the sums and differences.
2. It explains that to subtract an integer, you add its additive inverse, such as 4 - 9 = 4 + (-9) = -5.
3. It contains practice problems evaluating integer expressions and solving real-world problems involving adding and subtracting integers.
(8) Lesson 3.4
This document contains a mathematics lesson on graphing linear functions and interpreting slope and y-intercept. It includes examples of writing equations from graphs, finding slope and y-intercept, graphing linear equations, and interpreting slope and y-intercept in real world contexts. Key concepts covered are slope-intercept form, slope, y-intercept, rate of change, and using graphs to model proportional relationships.
(8) Lesson 2.1 - Solve Equations with Rational Coefficients
This document provides instruction on solving linear equations with one variable. It begins by defining key terms like multiplicative inverse and coefficient. It then provides examples of solving various equations by multiplying or dividing both sides of the equation by the same term to isolate the variable. These examples include equations with fractions, decimals, and percentages. The document emphasizes using the inverse operation property to solve equations and checking solutions. It aims to help students solve equations that may require expanding and combining like terms.
(8) Lesson 3.7
1. The document provides examples for writing and solving systems of linear equations by graphing. It explains how to write equations in slope-intercept form, graph the lines on the same coordinate plane, and determine if the system has no solution, one solution, or an infinite number of solutions based on where the lines intersect.
2. Key steps shown include writing equations in y=mx+b form, finding the slope and y-intercept, graphing the lines and analyzing where they intersect to determine the number of solutions.
3. Graphing systems of equations allows you to visually see where lines intersect, satisfying both equations simultaneously and showing the solution to the system.
(8) Lesson 5.3 - Angles of Triangles
1. The document discusses applying algebraic concepts to solve geometric problems involving triangles. It provides examples of writing and solving equations using angle sums, exterior angles, and ratios of interior angles to find missing angle measures.
2. Algebraic concepts like writing equations, using properties of equality, and solving for unknowns allow for finding missing angles in triangles given information about other angles or ratios of angles.
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This document provides examples and step-by-step explanations of how to add, subtract, multiply and divide integers according to their signs. It begins with writing integers for word problems, then evaluates expressions. The main content explains that to add integers with the same sign, add their absolute values and the sum is positive if both are positive or negative if both are negative. To add integers with different signs, subtract the absolute values and the sum is positive if the positive value is greater or negative if the negative value is greater. Several worked examples are provided to illustrate the rules.
(8) Lesson 3.8
1. The document presents three systems of equations related to word problems involving Justin and Karin's allowance, costs for recreation programs, and ages of Dewan and Adrianne.
2. Graphs are helpful for solving systems of equations because they allow you to visually check solutions obtained through algebraic methods. Graphing the equations shows their point of intersection, which indicates the solution.
3. Learning to solve systems of equations algebraically and using graphs helps answer word problems leading to two linear equations in two variables, as required by the Common Core standards.
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- Mauldin Middle School wants to make $4,740 from yearbook sales. Print yearbooks cost $60 each and digital yearbooks cost $15 each.
- This relationship can be represented by the equation 60x + 15y = 4,740.
- Setting y = 0 in the equation yields an x-intercept of 79, meaning 79 print yearbooks would earn the school $4,740.
- Setting x = 0 yields a y-intercept of 316, meaning 316 digital yearbooks would earn the school $4,740.
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This document provides information and examples for determining congruence and similarity using slopes of lines and triangles. It explains that the ratio of the rise to the run of two slope triangles formed by a line is equal to the slope of the line. It also gives examples of finding the slope of a roof, stairs, and verifying that the slope is the same between points on the line. Finally, it discusses that right triangles that have their hypotenuses on the same line in a coordinate plane are called slope triangles, and if two triangles are slope triangles then they are similar.
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This document discusses determining the perimeters and areas of similar figures:
1) If figure B is similar to figure A by a scale factor, the perimeter of B is equal to the perimeter of A times the scale factor. The area of B is equal to the area of A times the square of the scale factor.
2) An example problem demonstrates finding the perimeter of a rectangle similar to one with a given perimeter and length, using the scale factor relationship.
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This document contains information about ratios, proportions, and using proportional relationships to solve problems:
1. It provides examples of writing and solving proportions to determine unknown values. Proportions can be written as fraction or ratio equations and solved using cross products.
2. It discusses how to identify if a relationship is proportional by determining if the graph is a straight line that passes through the origin, and how to write an equation to represent a proportional relationship using a unit rate.
3. It contains lessons on recognizing and representing proportional relationships between quantities, identifying constants of proportionality, and using proportional relationships to solve multi-step ratio and percent problems.
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(8) Lesson 7.4 - Properties of Similar Polygons
This document provides instruction on determining if two figures are similar using transformations. It includes examples of checking if figures have corresponding congruent angles and proportional sides. The document also covers the definitions of similar polygons, scale factors, and using ratios and proportions to find missing measures in similar figures.
Lesson 1.9 grade 8
This document provides examples for estimating square roots and cube roots of numbers that are not perfect squares or cubes. It shows how to estimate the value by finding the largest perfect square or cube below the number and the smallest perfect square or cube above it. The square or cube roots of these numbers are then used to determine an integer estimate between them. Several step-by-step examples are provided to demonstrate this process. Common Core State Standards addressed in the lesson are also listed.
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This document contains a lesson on ratios, proportional relationships, and direct variation. It includes:
1) Examples of identifying the constant of proportionality from graphs and equations and explaining what it represents. The constant of proportionality represents the rate of change and can be interpreted as dollars per hour or words per minute.
2) Explanations of how to recognize a direct variation by checking if quantities have equivalent ratios or if a graph is a straight line through the origin.
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Proportionality and percentage
1. The document discusses key concepts in proportions and percentages including ratios, proportions, direct and inverse proportionality, and simple interest.
2. Ratios compare two quantities as a fraction and proportions are equalities between two ratios. Direct proportionality means two variables increase or decrease together, while inverse proportionality means one increases as the other decreases.
3. Percentages express a number as a fraction of 100. Simple interest depends on principal amount, interest rate, and time and uses the formula Interest = Principal x Rate x Time. Worked examples demonstrate calculating proportions, percentages, and simple interest.
(8) Lesson 3.1
Here are some real-world examples of relationships that have a constant rate of change:
- The speed of a car traveling at a constant velocity - As time passes, the car's position changes at a steady rate.
- Interest accumulating in a bank account - The interest earned each time period is a constant percentage of the current balance.
- Population growth over time - If the birth and death rates remain steady, the population will increase at a constant percentage each year.
- Depreciation of an asset - Most assets lose value at a steady rate each year due to age and wear.
- Temperature change from heating or cooling an object - The temperature will change at a steady rate until the object reaches equilibrium with its
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This document provides examples and step-by-step explanations of how to add, subtract, multiply and divide integers according to their signs. It begins with writing integers for word problems, then evaluates expressions. The main content explains that to add integers with the same sign, add their absolute values and the sum is positive if both are positive or negative if both are negative. To add integers with different signs, subtract the absolute values and the sum is positive if the positive value is greater or negative if the negative value is greater. Several worked examples are provided to illustrate the rules.
(8) Lesson 3.8
1. The document presents three systems of equations related to word problems involving Justin and Karin's allowance, costs for recreation programs, and ages of Dewan and Adrianne.
2. Graphs are helpful for solving systems of equations because they allow you to visually check solutions obtained through algebraic methods. Graphing the equations shows their point of intersection, which indicates the solution.
3. Learning to solve systems of equations algebraically and using graphs helps answer word problems leading to two linear equations in two variables, as required by the Common Core standards.
(7) Lesson 5.4 - The Distributive Property
The document discusses the distributive property in mathematics. It provides examples of using the distributive property to evaluate numerical expressions and rewrite algebraic expressions. It also gives a step-by-step example of using the distributive property to find the total cost of baseball helmets when the price per helmet is given. The distributive property allows numbers and symbols to represent mathematical ideas by distributing a number across terms within parentheses during multiplication.
Chapter 3
- Mauldin Middle School wants to make $4,740 from yearbook sales. Print yearbooks cost $60 each and digital yearbooks cost $15 each.
- This relationship can be represented by the equation 60x + 15y = 4,740.
- Setting y = 0 in the equation yields an x-intercept of 79, meaning 79 print yearbooks would earn the school $4,740.
- Setting x = 0 yields a y-intercept of 316, meaning 316 digital yearbooks would earn the school $4,740.
(8) Lesson 7.6 - Slope and Similar Triangles
This document provides information and examples for determining congruence and similarity using slopes of lines and triangles. It explains that the ratio of the rise to the run of two slope triangles formed by a line is equal to the slope of the line. It also gives examples of finding the slope of a roof, stairs, and verifying that the slope is the same between points on the line. Finally, it discusses that right triangles that have their hypotenuses on the same line in a coordinate plane are called slope triangles, and if two triangles are slope triangles then they are similar.
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This document discusses determining the perimeters and areas of similar figures:
1) If figure B is similar to figure A by a scale factor, the perimeter of B is equal to the perimeter of A times the scale factor. The area of B is equal to the area of A times the square of the scale factor.
2) An example problem demonstrates finding the perimeter of a rectangle similar to one with a given perimeter and length, using the scale factor relationship.
3) Another example finds the area of a porch similar to one with given dimensions, where the similar porch has double the dimensions, using the scale factor squared relationship.
(8) Lesson 5.1 - Lines
This document discusses classifying angles formed when parallel lines are cut by a transversal. It defines key terms like interior angles, exterior angles, alternate interior angles, alternate exterior angles, and corresponding angles. Examples are provided to illustrate each type of angle relationship. The document also provides step-by-step examples of classifying angle pairs and solving for missing angle measures using properties of parallel lines cut by a transversal.
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This document contains a lesson on determining congruence between figures using transformations. It provides examples of identifying congruent triangles and determining the transformations needed to map one onto the other. It also gives examples of using congruence to find missing angle measures. The lesson explains that two figures are congruent if their corresponding sides are congruent and corresponding angles are congruent.
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This document provides examples for applying algebraic concepts to geometry problems. It demonstrates using the Pythagorean theorem and distance formula to calculate distances between points on a coordinate plane. Sample problems are worked through step-by-step and answers are provided. Common Core standards addressed include applying the Pythagorean theorem to find distances between points and using square root and cube root symbols to represent solutions to equations.
Chapter 5
This document provides examples and explanations for using algebraic concepts in geometry, specifically regarding angles formed when lines are cut by a transversal. It defines key terms like parallel lines, transversal, interior angles, exterior angles, alternate interior angles, alternate exterior angles, and corresponding angles. Examples are given to classify pairs of angles and find missing angle measures. Step-by-step examples walk through using properties of parallel lines cut by a transversal to find missing angles. The document also explains how to prove conjectures using paragraph and two-column proofs.
(7) Lesson 1.6 - Solve Proportional Relationships
This document contains information about ratios, proportions, and using proportional relationships to solve problems:
1. It provides examples of writing and solving proportions to determine unknown values. Proportions can be written as fraction or ratio equations and solved using cross products.
2. It discusses how to identify if a relationship is proportional by determining if the graph is a straight line that passes through the origin, and how to write an equation to represent a proportional relationship using a unit rate.
3. It contains lessons on recognizing and representing proportional relationships between quantities, identifying constants of proportionality, and using proportional relationships to solve multi-step ratio and percent problems.
Chapter 7
The document provides examples and step-by-step explanations for solving geometry problems involving similar and congruent shapes using transformations and properties of similarity and congruence. In one example, it is determined that two triangles are similar because corresponding angles are congruent. In another example, transformations of a rotation and translation are used to show that two figures are congruent.
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This document discusses equivalence and solving equations. It begins with defining equivalence as the product of a number and its multiplicative inverse being 1. It then provides examples of solving various equations using properties of equality like addition, subtraction, multiplication and division. These include one-step, two-step and multi-step equations with or without variables on both sides. It also discusses equations having no solutions, one solution, or infinitely many solutions.
(8) Lesson 7.3 - Similarity and Transformations
This document provides instruction and examples for determining congruence and similarity using transformations. It explains that two figures are similar if one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations. Examples show using transformations and writing side ratios to determine if figures are similar. The last example finds the dimensions of an image that is enlarged twice by different scale factors, showing the results are similar.
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Chapter 1 Review
- To write fractions as decimals and decimals as fractions, it is helpful to write numbers in different ways. This allows for easier comparisons and calculations.
- Rational numbers can be written as ratios of integers, repeating decimals can be written as fractions, and terminating decimals can be written as fractions.
- Writing numbers in different forms, such as fractions and decimals, allows for easier evaluation of expressions and calculations.
(8) Lesson 7.4 - Properties of Similar Polygons
This document provides instruction on determining if two figures are similar using transformations. It includes examples of checking if figures have corresponding congruent angles and proportional sides. The document also covers the definitions of similar polygons, scale factors, and using ratios and proportions to find missing measures in similar figures.
Lesson 1.9 grade 8
This document provides examples for estimating square roots and cube roots of numbers that are not perfect squares or cubes. It shows how to estimate the value by finding the largest perfect square or cube below the number and the smallest perfect square or cube above it. The square or cube roots of these numbers are then used to determine an integer estimate between them. Several step-by-step examples are provided to demonstrate this process. Common Core State Standards addressed in the lesson are also listed.
Sequences, Series, and the Binomial Theorem
An infinite sequence is a function whose domain is the set of natural numbers, while a finite sequence has a domain of natural numbers up to some limit. A sequence can be described by its general term, which gives a rule for calculating each term based on its position in the sequence. The sum of the terms of a sequence is called a series, which is finite if it includes a finite number of terms and infinite if it includes all terms.
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(8) Lesson 7.7 - Area and Perimeter of Similar Figures
(8) Lesson 7.7 - Area and Perimeter of Similar Figures
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(7) Lesson 1.9 - Direct Variation
This document contains a lesson on ratios, proportional relationships, and direct variation. It includes:
1) Examples of identifying the constant of proportionality from graphs and equations and explaining what it represents. The constant of proportionality represents the rate of change and can be interpreted as dollars per hour or words per minute.
2) Explanations of how to recognize a direct variation by checking if quantities have equivalent ratios or if a graph is a straight line through the origin.
3) Clarification that a situation is not a direct variation if it does not have a constant ratio between quantities or if its graph does not pass through the origin.
Proportionality and percentage
1. The document discusses key concepts in proportions and percentages including ratios, proportions, direct and inverse proportionality, and simple interest.
2. Ratios compare two quantities as a fraction and proportions are equalities between two ratios. Direct proportionality means two variables increase or decrease together, while inverse proportionality means one increases as the other decreases.
3. Percentages express a number as a fraction of 100. Simple interest depends on principal amount, interest rate, and time and uses the formula Interest = Principal x Rate x Time. Worked examples demonstrate calculating proportions, percentages, and simple interest.
(8) Lesson 3.1
Here are some real-world examples of relationships that have a constant rate of change:
- The speed of a car traveling at a constant velocity - As time passes, the car's position changes at a steady rate.
- Interest accumulating in a bank account - The interest earned each time period is a constant percentage of the current balance.
- Population growth over time - If the birth and death rates remain steady, the population will increase at a constant percentage each year.
- Depreciation of an asset - Most assets lose value at a steady rate each year due to age and wear.
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Chapter 03 scatterplots and correlation
This chapter discusses exploring relationships between two quantitative variables using scatterplots and measuring the strength of linear relationships using correlation coefficients. Scatterplots show the joint distribution of two variables by plotting one on the x-axis and the other on the y-axis. Interpreting scatterplots involves examining the overall pattern, direction, strength, and outliers of the relationship. Correlation coefficients measure the direction and strength of linear relationships between -1 and 1. The chapter also discusses adding categorical variables to scatterplots and facts about correlation such as its sensitivity to outliers.
1. A small accounting firm pays each of its five clerks $35,000, t.docx
1. A small accounting firm pays each of its five clerks $35,000, two junior accountants $80,000 each, and the firm's owner $350,000. What is the mean salary paid at this firm? (Round your answer to the nearest whole number.)
$
How many of the employees earn less than the mean?
employees
What is the median salary?
$
2. A small accounting firm pays each of its five clerks $35000, two junior accountants $90000 each, and the firm's owner $256000.
What is the mean salary paid at this firm?
How many of the employees earn less than the mean?
What is the median salary?
If this firm gives no raises to the clerks and junior accountants, but the owner now has a salary of $435000.
How does this change affect the mean?
The mean increases by $ .
How does it affect the median?
The median increases by $ .
3. A study of diet and weight gain deliberately overfed 16 volunteers for eight weeks. The mean increase in fat was x = 2.63 kilograms and the standard deviation was s = 1.21 kilograms. What are x and s in pounds? (A kilogram is 2.2 pounds.)
x
=
s
=
4.Every few years, the National Assessment of Educational Progress asks a national sample of eighth-graders to perform the same math tasks. The goal is to get an honest picture of progress in math. Suppose these are the last few national mean scores, on a scale of 0 to 500.
Year
1990
1992
1996
2000
2003
2005
2008
Score
265
266
270
271
278
279
281
(a) Make a time plot of the mean scores, by hand. This is just a scatterplot of score against year. There is a slow linear increasing trend. (Do this on your own.)
(b) Find the regression line of mean score on time step-by-step. First calculate the mean and standard deviation of each variable and their correlation (use a calculator with these functions). Then find the equation of the least-squares line from these. (Round your answers to two decimal places.)
= + x
Draw the line on your scatterplot. What percent of the year-to-year variation in scores is explained by the linear trend? (Round your answer to one decimal place.)
%
(c) Now use software or the regression function on your calculator to verify your regression line. (Do this on your own.
5. A student wonders if tall women tend to date taller men than do short women. She measures herself, her dormitory roommate, and the women in the adjoining rooms; then she measures the next man each woman dates. The data (heights in inches) are listed below.
Women (x)
65
63
63
64
69
64
Men (y)
72
67
69
69
69
68
(a) Make a scatterplot of these data. (Do this on paper. Your instructor may ask you to turn this in.) Based on the scatterplot, do you expect the correlation to be positive or negative? Near ± 1 or not?
The correlation should be positive. It should be near 1.The correlation should be negative. It should be near -1. The correlation should be positive. It should not be near 1.The correlation should be negative. It should not be near -1.
(b) Find the correlation r between the heigh ...
(7) Lesson 10.4
Here is an example of how measures of center and variation can be used to compare two populations:
The double box plot shows the test scores on the last math exam for two different classes - Algebra 1 and Geometry. To compare the centers of the populations, we look at the medians. The median score for Algebra 1 is 80, while the median for Geometry is 85. This indicates that on average, the Geometry students scored higher on the exam.
To compare the variations between the two populations, we look at the interquartile ranges. The IQR for Algebra 1 is 15, while the IQR for Geometry is 10. Although the Geometry class had a higher median score, the Algebra 1 scores were more consistent
Proportionality and percentage
The document discusses various mathematical concepts related to ratios, proportions, percentages, and simple interest. It provides examples and formulas for calculating ratios, proportions, percentage increase and decrease, and simple interest. Key concepts covered include defining ratios and proportions, the properties of proportions, direct and inverse proportionality, calculating percentages, and the formulas for simple interest based on principal, rate, time, and interest amount.
(8) Lesson 4.7
The document is a lesson on identifying linear and nonlinear functions from tables of values or graphs. It includes examples of determining whether functions are linear or nonlinear based on whether the rate of change is constant as the input variable changes. It also provides examples of writing equations in y=mx+b form from linear functions and describing qualitative features of functions from graphs or descriptions.
(8) Lesson 9.2
This document provides step-by-step examples for determining a line of best fit from a scatter plot and using the line of best fit to make predictions. It explains how to construct a scatter plot, draw a line that best represents the data, write the equation in slope-intercept form, and use the equation to predict values. The examples illustrate how to find the slope and y-intercept, write the line of best fit equation, and make conjectures for data points not explicitly in the original data set.
(8) Lesson 3.2
This document provides instruction on determining the slope of a line from graphs, tables, and point pairs. It includes examples of finding the slope from a graph showing patients seen per day based on hygienists working, a table showing pages read over time, and using the slope formula with point pairs. Key steps shown are finding the rise over run and using the slope formula m=(y2-y1)/(x2-x1).
(7) Lesson 1.4 - Proportional and Nonproportional Relationships
This document provides examples and explanations for identifying proportional and nonproportional relationships between quantities. It begins with examples of converting between different measurement units. The next sections explain how to identify if two quantities have a proportional relationship by checking if their ratios are equivalent. Examples are provided of proportional relationships where the ratios are constant, as well as nonproportional relationships where the ratios differ. The document emphasizes using tables and graphs to determine if ratios between quantities simplify to the same value, indicating a proportional relationship.
Correlation: Bivariate Data and Scatter Plot
This document discusses bivariate data and scatter plots. Bivariate data involves collecting values of two variables from the same population. A scatter plot can show the relationship between two quantitative variables by plotting their ordered pairs. The direction and strength of the pattern in a scatter plot indicates whether the variables have a positive, negative, or no correlation. The correlation coefficient (r) measures the strength of the linear relationship between variables on a scale from -1 to 1.
Unit 7 jcs md
This document summarizes key concepts about proportional reasoning. It defines proportional reasoning as a mathematical relationship between two quantities that involves a constant multiplicative relationship. It discusses proportional reasoning as developing between concrete and formal operations. It also provides examples of using proportional relationships to solve problems and discusses research on how to best teach proportional reasoning concepts to students.
Understanding graphs and analyzing graphs
Scientific graphs are usually line graphs with best-fit straight or curved lines that get close to but don't necessarily touch all data points. The lines are not connect-the-dots. Graphs have two axes corresponding to the variables being related and show the relationship between them. Points are plotted on the graph and lines are drawn to best fit the data points.
Math-9_Q2_Mod3.pdf
1. The tailor in Gulliver's Travels used direct variation to determine Gulliver's measurements based on his thumb size. He said twice around the thumb equals the wrist, twice the wrist equals the neck, and twice the neck equals the waist.
2. Tables show the direct variations between thumb and wrist, neck and waist, and thumb and neck. Graphs of these relationships show straight lines passing through the origin, indicating direct variation.
3. When given that x is 6 and y is 2, an inverse variation equation relating y and x is found to be xy = 12, with 12 as the constant of variation.
Linear vs exponential growth (Simplified)
Students will construct, compare, and interpret linear and exponential function models and solve problems involving each model. Key differences are:
- Linear functions change at a constant rate per unit interval while exponential functions change by a constant factor over equal intervals.
- Linear functions are of the form y=mx+b while exponential functions are of the form y=ax.
- Students will determine whether examples describe a linear or exponential pattern of change by analyzing whether the rate of change is constant or changing by a common factor over time.
(8) Lesson 4.9
1. The document provides instruction on graphing functions and qualitative graphs. It includes examples of sketching and describing qualitative graphs based on verbal descriptions of relationships between quantities over time.
2. It provides examples of sketching qualitative graphs to represent a bathtub filling with water, the number of people at a restaurant, a bouncing tennis ball, and a swinging child.
3. Qualitative graphs can be used to model relationships between quantities without specific numerical values by representing the essential features of the situation through the shape of the graph over time.
(8) Lesson 8.1
Here are some key ways to tell the relationship shown in a scatter plot:
- Positive relationship: As one variable increases, the other variable also increases. The data points will fall generally above or to the right of the origin.
- Negative relationship: As one variable increases, the other decreases. The data points will fall generally below or to the left of the origin.
- No relationship: There is no consistent pattern to how the variables change together. The data points will be randomly scattered with no discernible upward or downward trend.
Some other clues:
- For positive or negative relationships, the data points tend to fall near a straight line. For no relationship, they are more randomly arranged.
-
Statistics For Management 3 October
The document provides information about statistics and economics tutorials being offered after school, including regression analysis, correlation, and the normal distribution. It gives examples of calculating rank correlation, finding regression equations, and using the standard normal distribution table. It also explains key aspects of the normal distribution like the 68-95-99.7 rule and how to calculate probabilities using the normal distribution function in Excel.
(8) Lesson 4.1
1. The document provides examples for writing equations to model linear relationships between quantities using words, tables, graphs, and ordered pairs.
2. Equations can be used to find output values for a relationship given any input value. Examples show writing equations and using them to calculate values.
3. Modeling relationships allows quantities and their relationships to be represented and understood in multiple ways including visually with tables and graphs.
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(7) Lesson 1.4 - Proportional and Nonproportional Relationships
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Chapter 2 inquiry_lab_presentation_course_3 (1)
A bar diagram helps solve a two-step equation problem by visually representing the quantities and their relationships. The bar diagram shows the total number of postcards Miranda bought and the total cost. It can be used to write an equation relating the unknown cost of a large postcard to the total quantity and cost. Solving this equation reveals that the cost of one large postcard is $2.50.
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The document provides examples for solving linear equations using the work backward strategy. It includes 4 example equations with the step-by-step work shown. The answers are then provided. Additional material is included on equivalence, solving equations with variables on both sides, and using properties of equality.
Lesson 5.1
This document discusses classifying angles formed when parallel lines are cut by a transversal. It defines key terms like alternate interior angles, alternate exterior angles, and corresponding angles. It then provides examples of classifying angle pairs based on their relationship. Sample problems are worked through, applying the concepts to find missing angle measures by justifying answers using angle properties of parallel lines cut by a transversal.
Lesson 1.8 grade 8
1. The document provides examples for evaluating expressions with scientific notation, square roots, and cube roots. It includes step-by-step workings for finding square roots of perfect squares and cube roots of perfect cubes.
2. Key concepts explained are square roots, perfect squares, radical signs, cube roots, and perfect cubes. Examples are provided to find square roots of numbers less than 100 and cube roots of small numbers.
3. Knowing how to find the exact square roots of perfect squares helps estimate the square roots of nearby non-perfect squares. Being able to write the perfect square root without the radical provides understanding and allows for computations.
Lesson 7.4
This document provides instruction on determining if two figures are similar using transformations. It includes examples of determining if figures are similar by checking if corresponding angles are congruent and if corresponding sides are proportional. It also covers finding missing measures in similar figures using scale factors and proportions. Determining similarity involves rotations, reflections, translations, and dilations, while congruence only involves rigid motions.
Lesson 7.3 (8)
This document provides instruction and examples for determining congruence and similarity using transformations. It explains that two figures are similar if one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations. Examples show using transformations and writing side ratios to determine if figures are similar. The last example finds the dimensions of an image that is enlarged twice by different scale factors, showing the results are similar.
Lesson 7.2
This document contains a lesson on determining congruence between figures using transformations. It provides examples of identifying congruent triangles and determining the transformations needed to map one onto the other. It also gives examples of using congruence to find missing angle measures. The lesson explains that two figures are congruent if their corresponding sides are congruent and corresponding angles are congruent.
Pat05 ppt 0201
This document provides an overview of functions and concepts covered in Chapter 2, including:
- Graphing functions to identify intervals where they are increasing, decreasing, or constant.
- Modeling real-world applications with appropriate functions.
- Graphing functions defined piecewise using different formulas for different parts of the domain.
- Identifying relative maximum and minimum values of functions.
- Defining functions as increasing, decreasing, or constant on intervals.
- Describing the greatest integer function.
Chapter 6
This document provides examples and explanations for describing geometric transformations on a coordinate plane, including translations, reflections, rotations, and dilations. It defines key terms like preimage, image, and scale factor. Examples show how to perform and describe each type of transformation by:
1) Graphing a figure and identifying its vertices with coordinates
2) Performing the transformation by applying the rules for that specific transformation
3) Graphing the transformed image and identifying the new coordinates of its vertices
The document ensures understanding through multiple examples of each transformation type and encourages practicing the skills on additional example problems. It emphasizes that transformations preserve properties like distance and angle measures.
Pat05 ppt 0106
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.
Pat05 ppt 0105
This document discusses linear equations and functions. It provides examples of solving linear equations, including special cases where equations have no solution or infinitely many solutions. Applications involving linear models for distance, rate, and time as well as simple interest are presented. The key concepts of zeros of linear functions and finding the zero of a given linear function are also covered.
Lesson 1.10 grade 8
This document provides examples and explanations of concepts related to classifying and ordering different types of numbers:
1) It defines rational and irrational numbers, and provides examples of each. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
2) Exercises are included for classifying numbers as rational or irrational, comparing irrational numbers on a number line, and ordering sets of numbers from least to greatest.
3) Answering "Why is it helpful to write numbers in different ways?" the document explains that not all numbers can be written the same way, like irrational numbers that cannot be expressed as integers. Being able to classify numbers facilitates understanding and operations involving different types of numbers.
Lesson 1.7 grade 8
1. The document provides examples of writing numbers in standard form and scientific notation. It also gives examples of performing arithmetic operations such as addition, subtraction, multiplication, and division on numbers expressed in scientific notation.
2. The examples calculate population sizes, planet diameters, and perform other calculations to illustrate how scientific notation makes working with very large and small numbers easier.
3. Writing numbers in scientific notation allows for easier operations like multiplication and division of very large and small quantities, as well as comparing the relative sizes of numbers like population and distances in the solar system.
Lesson 1.7 grade 8
1) The document provides examples of writing numbers in standard form and scientific notation. It also gives examples of performing arithmetic operations such as addition, subtraction, multiplication, and division on numbers expressed in scientific notation.
2) The examples illustrate how writing very large or very small numbers in scientific notation makes it easier to perform operations with them. It also allows for estimating quantities that would otherwise be difficult to comprehend in standard form.
3) Understanding scientific notation helps to compare quantities that differ vastly in magnitude, such as populations and diameters of astronomical objects. It enables estimating ratios and relative sizes in a straightforward manner.
Lesson 1.6 (8)
Here are the steps to write a number between 0 and 1 in scientific notation with an example:
1. Write the number in decimal form. For example, 0.000375.
2. Count the number of decimal places in the number. In this example there are 4 decimal places.
3. Move the decimal point to the left by that number of places. This gives 3.75.
4. Write the number as a coefficient and exponent of 10. Since we moved the decimal point 4 places to the left, the exponent is -4.
So, 0.000375 written in scientific notation is 3.75 × 10-4.
Lesson 1.5 (8)
This document provides examples for writing numbers using negative exponents. It begins with examples of writing expressions like 6-3 and 4-4 using positive exponents. It then provides step-by-step examples of writing fractions with negative exponents other than -1, like writing 5-2. Further examples show simplifying expressions using the properties of exponents, like 53 • 5-5 = 53 + (-5). In the examples, writing numbers with negative exponents allows writing very small or large numbers without fractions and makes multiplying or dividing numbers with the same base easier.
Lesson 1.4 (8)
1. This document provides examples and explanations for simplifying expressions using exponents. It discusses the laws for taking the power of a power and the power of a product.
2. For the power of a power, you multiply the exponents. For the power of a product, you find the power of each factor and multiply them.
3. Several multi-step examples are worked through applying these laws to simplify expressions with variables and integers. The document also provides explanations of how rewriting numbers with exponents can make calculations easier.
Lesson 1.3 (8)
1) The document provides examples of writing expressions using exponents and evaluating those expressions. It gives the expressions 2c8, 85, and m5n3p3 as examples and evaluates the expressions 512, 216, and m5n3p3.
2) It then discusses rules for multiplying and dividing powers with the same base: to multiply powers with the same base, add the exponents, and to divide powers with the same base, subtract the exponents.
3) Several step-by-step examples demonstrate simplifying expressions using these rules to multiply and divide exponents.
Lesson 1.2 (8)
1. This document provides examples and explanations for writing expressions using exponents. It defines key terms like base, exponent, and power.
2. Step-by-step examples are provided for writing expressions with exponents, evaluating expressions, and applying properties of exponents. Examples include writing expressions for multiplication involving repeated factors, evaluating expressions given values for variables, and using the distributive property.
3. Exponents provide a concise way to write repeated multiplication and to evaluate large expressions. Working through examples helps understand how to use exponents to write expressions compactly and evaluate them efficiently.
Lesson 1.1 (8)
This document contains a mathematics lesson on writing numbers in different forms, including:
1. Converting fractions to decimals and decimals to fractions
2. Examples of solving proportions and writing decimals as fractions
3. The lesson discusses why it is helpful to write numbers in different ways, such as for measurements or when dealing with money.
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(7) Lesson 1.8 - Slope
- 1. Course 2, Lesson 1-8 1. The table shows the number of miles Greg walked during a walk-a-thon. Find the constant rate of change for the table. 2. The graph represents the number of patients that can be seen in a doctor’s office based on the number of nurses on duty. Find the constant rate of change for the graph. 3. Use the information in the table to find the rate of change. Time (h) 1 2 3 Distance (mi) 4 8 12
- 2. Course 2, Lesson 1-8 ANSWERS 1. 4 miles per hour 2. 5 patients per nurse 3. 3 slices per student
- 3. HOW can you show that two objects are proportional? Ratios and Proportional Relationships Course 2, Lesson 1-8
- 4. Ratios and Proportional Relationships Course 2, Lesson 1-8 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. • 7.RP.2 Recognize and represent proportional relationships between quantities. • 7.RP.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics.
- 5. • To find the slope of a line • To interpret what the slope of a line represents Ratios and Proportional Relationships Course 2, Lesson 1-8
- 6. Ratios and Proportional Relationships Course 2, Lesson 1-8 slope = rise run
- 7. Ratios and Proportional Relationships Course 2, Lesson 1-8 Slope is the rate of change between any two points on a line. slope = vertical change horizontal change = or 2 c c h ha an nge ge in in y x 2 1
- 8. 1 Need Another Example? 2 Definition of slope 3 4 Use (3, 15) and (5, 25). seconds 5 6 Simplify. Step-by-Step Example 1. The table below shows the relationship between the number of seconds y it takes to hear thunder after a lightning strike and the miles x you are from the lightning. Graph the data and find the slope. Explain what the slope represents. miles So, for every 5 seconds between a lightning flash and the sound of thunder, there is 1 mile between you and the lightning strike.
- 9. Answer Need Another Example? The table below shows the relationship between the number of hours Carlos works and the amount of money that he earns. Graph the data. Then find the slope of the line. Explain what the slope represents. The slope of the graph is 15 and represents the amount earned per hour.
- 10. 1 Need Another Example? Step-by-Step Example 2. Renaldo opened a savings account. Each week he deposits $300. Draw a graph of the account balance versus time. Find the numerical value of the slope and interpret it in words. The slope of the line is the rate at which the account balance rises, or .
- 11. Answer Need Another Example? Nina drains a 20-gallon aquarium at a rate of 0.5 gallon per minute. Draw a graph of the water in the aquarium over time. Find the numerical value of the slope and interpret it in words. ; the slope is the rate at which the water in the tank drains.
- 12. How did what you learned today help you answer the HOW can you show that two objects are proportional? Course 2, Lesson 1-8 Ratios and Proportional Relationships
- 13. How did what you learned today help you answer the HOW can you show that two objects are proportional? Course 2, Lesson 1-8 Ratios and Proportional Relationships Sample answers: • By using slope to find the rate of change between two points on a line • By interpret the numerical value of a slope in the context of a real-world situation
- 14. Give an example of how being able to find the constant rate of change helped you with today’s lesson on finding slope. Ratios and Proportional RelationshipsRatios and Proportional Relationships Course 2, Lesson 1-8