(7) Inquiry lab - Multiply Integers
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The document discusses when the product of two integers is positive or negative. It provides examples using counters to model multiplication of integers. The product of two integers is positive when both integers are positive or both are negative. The product is negative when the integers have opposite signs, with one being positive and the other negative.
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- Setting x = 0 yields a y-intercept of 316, meaning 316 digital yearbooks would earn the school $4,740.
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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.
Chapter 2 Review
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.
Chapter 2 lesson_4_presentation_course_3
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.
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.
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.
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.
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.
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.
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 1.9 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.
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.
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(7) Inquiry lab - Multiply Integers
- 1. WHEN is the product of two integers a positive number? WHEN is the product a negative number? The Number System Course 2, Inquiry Lab before Lesson 3-4
- 2. โข 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. โข 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. 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. The Number System Course 2, Inquiry Lab before Lesson 3-4 Common Core State Standards ยฉ Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
- 3. Course 2, Inquiry Lab before Lesson 3-4 Activity 1 Continued ๏ The Number System The number of students who bring their lunch to Phoenix Middle School has been decreasing at a rate of 4 students each month. What integer represents the total change in the number of students bringing their lunch after three months? The integer represents a decrease of 4 students each month. After three months, the total change will be 3 x (โ4). Add 3 sets of 4 negative counters to the mat.
- 4. Course 2, Inquiry Lab before Lesson 3-4 The Number System The number of students who bring their lunch to Phoenix Middle School has been decreasing at a rate of 4 students each month. What integer represents the total change in the number of students bringing their lunch after three months? Count the number of negative counters. There are negative counters. So, 3 x (โ4) = . After three months, the total change in the number of students bringing their lunch will be . The model shows that adding sets of negative counters results in negative counters on the mat.
- 5. Course 2, Inquiry Lab before Lesson 3-4 Activity 2 Continued ๏ The Number System Use counters to find โ2 x 3. If the first factor is negative, you need to remove counters from the mat. There are no counters on the mat, so add two sets of three zero pairs on the mat. The value on the mat is zero.
- 6. Course 2, Inquiry Lab before Lesson 3-4 The Number System Use counters to find โ2 x 3. If the first factor is negative, you need to remove counters from the mat. Remove two sets of three positive counters from the mat. There are negative counters remaining. So, โ2 x 3 = . The model shows that removing sets of positive counters results in sets of negative counters that are left.
- 7. Course 2, Inquiry Lab before Lesson 3-4 Activity 3 Continued ๏ The Number System Use counters to find โ2 x (โ4). Both factors are negative. Remove sets of negative counters from the mat. There are no counters on the mat, so draw two sets of four zero pairs on the mat.
- 8. Course 2, Inquiry Lab before Lesson 3-4 The Number System Use counters to find โ2 x (โ4). Cross out two sets of four negative counters from the mat. There are positive counters remaining. So, โ2 x (โ4) = . The model shows that removing sets of negative counters results in sets of positive counters that are left.
- 9. Course 2, Inquiry Lab before Lesson 3-4 WHEN is the product of two integers a positive number? WHEN is the product a negative number? The Number System