(8) Lesson 3.1
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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 discusses determining the perimeters and areas of similar figures:
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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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- 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.
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1 | $10
2 | $20
3 | $30
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This document provides a daily lesson log for a Grade 8 mathematics class taught by Luisa F. Garcillan. The lesson covers linear inequalities in two variables over four class sessions. The objectives are to illustrate, differentiate, and graph linear inequalities in two variables. References and learning resources include textbooks, guides, and online materials. The procedures involve reviewing concepts, presenting examples, discussing new concepts, and practicing skills through activities, graphing, and word problems. The goal is for students to understand and be able to solve problems involving linear inequalities in two variables.
Ratio and proportion
This document contains materials for a mathematics lesson on ratios and proportions. It includes examples of writing ratios using fractions and colons, forming proportions, and finding missing terms in proportions. Activities guide students to form ratios, write proportions, solve word problems involving ratios, and evaluate their understanding through questions and applications using visual representations. Cooperative learning strategies and using various tools like charts and presentations are suggested for instruction.
RATIOS_PROPORTIONS_AND_SIMILAR_FIGURES.ppt
This document discusses ratios, proportions, and similar figures. It provides examples of using ratios to compare quantities and solve problems involving scale drawings. A ratio compares two quantities using division, and ratios that make the same comparison are equivalent ratios. Proportions state that two ratios are equal, and can be used to solve for unknown values. Figures are similar if corresponding angles are congruent and corresponding sides are proportional, forming equivalent ratios. Examples are given of setting up and solving proportions to find unknown side lengths of similar figures.
PD Slide Show
This document outlines a professional development program to help teachers better understand proportional reasoning needed for ratios, proportions, and rates. It will:
1) Examine common ways of working with ratios and identify student misconceptions with ratio problems.
2) Connect ratios to topics in elementary, algebra, geometry, and more using examples and frameworks.
3) Highlight the multiplicative nature of ratios and proportions, and how equivalent ratios can be created through partitioning and iterating units.
G6 m1-b-lesson 10-t
The document describes a lesson on ratio tables that have both additive and multiplicative structures. Students create three ratio tables showing amounts of blueberries and strawberries needed for fruit salads serving different numbers of people. The tables all use a 1:3 ratio of blueberries to strawberries. Students analyze patterns in the tables, such as entries increasing by addition or multiplication. They also describe how the blueberry and strawberry columns relate both to each other and within themselves.
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.
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This document provides an overview of functions and concepts covered in Chapter 2, including:
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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.
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- 1. Course 3, Lesson 3-1 Solve each equation. Check your solution. 1. 3(a + 3) = 12 2. 4(2y – 3) = 20 3. –5(m – 5) = 4(m + 1) 4. 5. Alexis has 6 more collector’s cards than Danny. Together, the two friends have 24 cards. How many cards does Alexis have? 1 2 ( 25) ( 30) 5 5 z z
- 2. Course 3, Lesson 3-1 ANSWERS 1. 1 2. 4 3. 4. –85 5. 15 7 3
- 3. WHY are graphs helpful? Expressions and Equations Course 3, Lesson 3-1
- 4. • Preparation for 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 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 Use appropriate tools strategically. Course 3, Lesson 3-1 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Expressions and Equations
- 5. To • determine if a relationship is linear, • find the constant rate of change in a linear relationship, • determine if a relationship is proportional Course 3, Lesson 3-1 Expressions and Equations
- 6. • linear relationship • constant rate of change Course 3, Lesson 3-1 Expressions and Equations
- 7. Course 3, Lesson 3-1 Expressions and Equations Words Two quantities a and b have a proportional linear relationship if they have a constant ratio and a constant rate of change. Symbols is constant and is constant. a b change in change in b a
- 8. 1 Need Another Example? 2 Step-by-Step Example 1. The balance in an account after several transactions is shown. Is the relationship between the balance and number of transactions linear? If so, find the constant rate of change. If not, explain your reasoning. As the number of transactions increases by 3, the balance in the account decreases by $30. Since the rate of change is constant, this is a linear relationship. The constant rate of change is or –$10 per transaction. This means that each transaction involved a $10 withdrawal.
- 9. Answer Need Another Example? The amount a babysitter charges is shown. Is the relationship between the number of hours and the amount charged linear? If so, find the constant rate of change. If not, explain your reasoning. Yes; the constant rate of change is , or $8 per hour.
- 10. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 2. Use the table to determine if there is a proportional linear relationship between a temperature in degrees Fahrenheit and a temperature in degrees Celsius. Explain your reasoning. Since the rate of change is constant, this is a linear relationship. 7 Constant Rate of Change To determine if the two scales are proportional, express the relationship between the degrees for several columns as a ratio. = 8.2 = 5 ≈ 3.9 Since the ratios are not the same, the relationship between degrees Fahrenheit degrees Celsius is not proportional. Check: Graph the points on the coordinate plane. Then connect them with a line. The points appear to fall in a straight line so the relationship is linear. The line connecting the points does not pass through the origin so the relationship is not proportional.
- 11. Answer Need Another Example? Use the table to determine if there is a proportional linear relationship between the speed (meters per second) and the time since a ball has been dropped. Explain your reasoning. Yes; the ratio of speed to time is a constant 9.8, so the relationship is proportional.
- 12. How did what you learned today help you answer the WHY are graphs helpful? Course 3, Lesson 3-1 Expressions and Equations
- 13. How did what you learned today help you answer the WHY are graphs helpful? Course 3, Lesson 3-1 Expressions and Equations Sample answers: • You can determine if a relationship is linear by looking at the graph. • You can use the graph to determine if a relationship is proportional.
- 14. Give a real-world example of a relationship that has a constant rate of change. Course 3, Lesson 3-1 Ratios and Proportional RelationshipsExpressions and Equations