The document describes a numeric pattern where each row contains consecutive odd integers centered around 1. It asks students to conjecture the pattern and sum of terms in each row. It also provides homework questions on conditional statements, deductive reasoning, and analyzing the truth value of related conditional statements.
The document summarizes key concepts in geometry including conditional statements, counter-examples, definitions, bi-conditionals, deductive reasoning, laws of logic, algebraic proofs, segment and angle properties, two-column proofs, the linear pair postulate, congruent complement and supplement theorems, the vertical angles theorem, and the common segments theorem. Examples are provided for each concept.
The document provides instructions for students to complete an opener assignment. It instructs students to draw two identical acute triangles, each filling half of the provided box. Students are then asked to sketch the perpendicular bisector of each side and the angle bisector of each angle of the triangles. The document asks students to notice anything interesting about their drawings. It then provides a radicals worksheet for students to check over and mark any incorrect problems on.
The document provides instructions for students to complete an opener worksheet. It notes that the teacher will check that students have started working when the bell rings and that students won't receive opener points for the day if they haven't started. It also wishes students a happy new semester.
The document provides answers to homework problems related to geometry concepts like inductive reasoning, conditional statements, congruence, and proofs. It includes the answers to problems about complementary angles, eligibility criteria, conditional statements, properties of equality, and definitions used in proofs. The final pages discuss proof strategies, using statements and reasons, and provide examples of setting up geometry proofs.
The document contains information about triathlon-style teams competition with three legs:
1. The first two legs have 15 possible points each but are graded out of 12, with extra credit for speed and accuracy.
2. The third leg is a 25-minute group mini quiz.
3. Teams and team members are listed, and homework assignments are given.
This document provides homework questions on solving equations and graphing equations. It includes:
1) Solving equations with one variable like 3x + 2 = 5x + 3 and determining the number of solutions.
2) Solving a two variable equation 2x + 3y = 12 and finding points that satisfy the equation.
3) Graphing equations like y = x^2 - 3x + 2 on a graphing calculator and determining features of the graph like where it crosses the x-axis.
4) Applying a transformation like (x,y) -> (x + 5, y) to the points on a graph.
5) Identifying points that do or do not
The document summarizes key concepts in geometry including conditional statements, counter-examples, definitions, bi-conditionals, deductive reasoning, laws of logic, algebraic proofs, segment and angle properties, two-column proofs, the linear pair postulate, congruent complement and supplement theorems, the vertical angles theorem, and the common segments theorem. Examples are provided for each concept.
The document provides instructions for students to complete an opener assignment. It instructs students to draw two identical acute triangles, each filling half of the provided box. Students are then asked to sketch the perpendicular bisector of each side and the angle bisector of each angle of the triangles. The document asks students to notice anything interesting about their drawings. It then provides a radicals worksheet for students to check over and mark any incorrect problems on.
The document provides instructions for students to complete an opener worksheet. It notes that the teacher will check that students have started working when the bell rings and that students won't receive opener points for the day if they haven't started. It also wishes students a happy new semester.
The document provides answers to homework problems related to geometry concepts like inductive reasoning, conditional statements, congruence, and proofs. It includes the answers to problems about complementary angles, eligibility criteria, conditional statements, properties of equality, and definitions used in proofs. The final pages discuss proof strategies, using statements and reasons, and provide examples of setting up geometry proofs.
The document contains information about triathlon-style teams competition with three legs:
1. The first two legs have 15 possible points each but are graded out of 12, with extra credit for speed and accuracy.
2. The third leg is a 25-minute group mini quiz.
3. Teams and team members are listed, and homework assignments are given.
This document provides homework questions on solving equations and graphing equations. It includes:
1) Solving equations with one variable like 3x + 2 = 5x + 3 and determining the number of solutions.
2) Solving a two variable equation 2x + 3y = 12 and finding points that satisfy the equation.
3) Graphing equations like y = x^2 - 3x + 2 on a graphing calculator and determining features of the graph like where it crosses the x-axis.
4) Applying a transformation like (x,y) -> (x + 5, y) to the points on a graph.
5) Identifying points that do or do not
This document contains instructions for students to complete various math exercises on their TI-Nspire calculators. It asks students to match expressions to steps, write an expression for the area of a rectangle, simplify an algebraic expression, evaluate expressions for different variable values, and complete an activity on their calculators worth daily work points. Students are told to work with partners but can ask other group members for help if needed and to raise their hand once finished.
1. The document presents problems involving solving equations for x and factoring expressions to find values of x that make the expression equal to zero. It discusses using the zero product property (ZPP), where the product of two numbers is zero if one or both numbers is zero. Students are asked to factor expressions using the greatest common factor (GCF) and using factoring, then apply ZPP to find solutions for x. Graphing an expression is used to visualize where it crosses the x-axis, relating to solutions found earlier.
2. Students are asked to factor equations using GCF or factoring techniques, then apply ZPP to find the solutions to each equation.
3. The document provides practice with
The document provides instruction and examples for graphing related quantities qualitatively. It discusses using graphs to represent how a bouncing basketball or Curious George's skydiving experience might change over time. Students are asked to come up with stories about money growing or shrinking and graph them, including an important feature from the lesson. The homework assigned is to complete practice problems and a writing assignment.
Day 2 Opener:
- Review problem solving linear equations from the previous day
- Introduce different forms of writing linear equations: point-slope form, slope-intercept form, standard form
- Students work with a partner to practice identifying properties of lines from their equations
Section 4.7 Jiffy Graphs:
- Explain how to quickly write an equation of a line given certain information using the different forms
- Demonstrate graphing lines using point-slope and slope-intercept form by plotting points
- Ask students to consider how to graph a line in standard form for the exit slip
The document provides examples and explanations of key concepts related to rates, expressions, and order of operations in mathematics:
1) It gives an example of calculating rate using the equation r = d/t, showing how to find the rate of a person running 100 meters in 40 seconds.
2) It lists two "number tricks" involving multiple steps of operations on a chosen number, but notes that only one trick allows determining the original number.
3) It discusses combining like terms in expressions and the rules of arithmetic for expressions.
4) Finally, it reviews the proper order of operations to simplify expressions.
The document provides instructions and examples for graphing related quantities qualitatively. It discusses using graphs to represent a bouncing basketball with varying speeds and prices of Nike shoes over time. Students are asked to identify independent and dependent variables, construct their own qualitative graphs representing scenarios, and interpret other qualitative graphs. The homework assignment is to complete practice problems and a writing assignment by a due date.
This document discusses how to calculate the lateral area and surface area of pyramids and cones. It provides the formulas for lateral area of a pyramid and cone and surface area of a pyramid and cone. It then gives examples of applying the formulas to find the lateral and surface areas of specific pyramids and cones where the dimensions are provided.
The document provides a quick review for a quiz on writing equations of lines. It includes 4 problems: writing the equation of a line with a given slope and point, through two given points, a horizontal line through a given point, and a vertical line through a given point. It then transitions to discussing solving systems of equations using the elimination method, providing 3 examples of solving systems by eliminating a variable through adding or subtracting the equations.
This document contains a homework assignment on rates of change and graphs involving distance and time. It includes questions about finding the average speed between points on a distance-time graph, identifying which graph represents steady movement versus speeding up or slowing down, examples of other rate of change problems, and determining if three points are collinear based on their coordinates and slopes. The document provides context, definitions, and problems for students to practice skills relating to rates of change and interpreting distance-time and speed-time graphs.
1. The geometric mean between 2 and 10 is 5.
2. A right triangle problem is presented with missing side lengths for QS, QT, and RT given side lengths of 8, 32, and an unknown side R.
3. A homework problem asks to write trigonometric ratios as fractions and decimals and use special right triangles to write ratios in terms of degrees. It also asks which ratio would be used to find missing side lengths in sample right triangles.
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. It includes examples of using the midpoint formula to find the midpoint of a segment and using given midpoints and endpoints to find missing endpoints.
3. There are also examples of using the Pythagorean theorem and distance formula to find the length of a segment between two points.
The document discusses a 4 step process but provides no details on the actual steps or content of the process. It references numbered sections but provides no information within those sections. Overall, the document does not contain any substantive information that could be summarized due to the lack of details provided within the numbered sections.
This document discusses deductive reasoning and how to evaluate arguments using deductive logic. It introduces the law of detachment and law of syllogism as valid forms of deductive reasoning. Examples are provided to illustrate identifying the hypothesis and conclusion of conditionals, and determining whether conjectures follow validly from given information using the laws of detachment and syllogism. Practice problems are included for the reader to apply these concepts.
This document covers conditional statements and their components. It defines key terms like hypothesis, conclusion, converse, inverse, and contrapositive. Examples are provided to demonstrate how to identify the hypothesis and conclusion of conditional statements, write conditionals from sentences, and determine truth values. The last part discusses logical equivalence and how a conditional is equivalent to its contrapositive based on the Law of Contrapositive.
The student is able to (I can):
Use inductive reasoning to identify patterns and make conjectures
Find counterexamples to disprove conjectures
Identify, write, and analyze the truth value of conditional statements.
Write the inverse, converse, and contrapositive of a conditional statement.
The document defines conditional statements, converse, inverse, and contrapositive statements and provides examples of identifying each. It also discusses identifying the hypothesis and conclusion of conditional statements and determining the truth value of conditionals and their converses. The document provides guidance on writing indirect proofs using contradictions.
This document discusses biconditional statements and definitions. It begins by defining a biconditional statement as one that can be written "p if and only if q," meaning "if p then q" and "if q then p." Examples are provided of writing the conditionals within biconditionals. The document also discusses analyzing the truth value of biconditionals and providing counterexamples if false. Finally, it notes that definitions can be written as true biconditionals, and examples are given of writing geometric definitions as biconditionals.
Here are the converses, inverses, and contrapositives of the given statements:
1. Conditional: If two angles form a linear pair, then they are supplementary.
Converse: If two angles are supplementary, then they form a linear pair.
Inverse: If two angles do not form a linear pair, then they are not supplementary.
Contrapositive: If two angles are not supplementary, then they do not form a linear pair.
2. Conditional: If a parallelogram has a right angle, then it is a rectangle.
Converse: If a parallelogram is a rectangle, then it has a right angle.
Inverse: If a parallelogram does not have
This document discusses conditional statements and their components. It defines a conditional statement as having a hypothesis and conclusion connected by "if...then". It provides examples of writing the converse, inverse, and contrapositive of conditional statements. The key points are:
- A conditional statement has two parts: a hypothesis and conclusion connected by "if...then"
- The converse switches the hypothesis and conclusion
- The inverse negates both the hypothesis and conclusion
- A statement is equivalent to its contrapositive
This document contains a geometry drill with various geometry concepts and problems. It also contains a logic drill with conditional statements and vocabulary about conditionals. It discusses writing conditionals, their converses, inverses, and contrapositives. It provides examples of evaluating the truth value of these and using Venn diagrams to represent conditional statements.
This document contains instructions for students to complete various math exercises on their TI-Nspire calculators. It asks students to match expressions to steps, write an expression for the area of a rectangle, simplify an algebraic expression, evaluate expressions for different variable values, and complete an activity on their calculators worth daily work points. Students are told to work with partners but can ask other group members for help if needed and to raise their hand once finished.
1. The document presents problems involving solving equations for x and factoring expressions to find values of x that make the expression equal to zero. It discusses using the zero product property (ZPP), where the product of two numbers is zero if one or both numbers is zero. Students are asked to factor expressions using the greatest common factor (GCF) and using factoring, then apply ZPP to find solutions for x. Graphing an expression is used to visualize where it crosses the x-axis, relating to solutions found earlier.
2. Students are asked to factor equations using GCF or factoring techniques, then apply ZPP to find the solutions to each equation.
3. The document provides practice with
The document provides instruction and examples for graphing related quantities qualitatively. It discusses using graphs to represent how a bouncing basketball or Curious George's skydiving experience might change over time. Students are asked to come up with stories about money growing or shrinking and graph them, including an important feature from the lesson. The homework assigned is to complete practice problems and a writing assignment.
Day 2 Opener:
- Review problem solving linear equations from the previous day
- Introduce different forms of writing linear equations: point-slope form, slope-intercept form, standard form
- Students work with a partner to practice identifying properties of lines from their equations
Section 4.7 Jiffy Graphs:
- Explain how to quickly write an equation of a line given certain information using the different forms
- Demonstrate graphing lines using point-slope and slope-intercept form by plotting points
- Ask students to consider how to graph a line in standard form for the exit slip
The document provides examples and explanations of key concepts related to rates, expressions, and order of operations in mathematics:
1) It gives an example of calculating rate using the equation r = d/t, showing how to find the rate of a person running 100 meters in 40 seconds.
2) It lists two "number tricks" involving multiple steps of operations on a chosen number, but notes that only one trick allows determining the original number.
3) It discusses combining like terms in expressions and the rules of arithmetic for expressions.
4) Finally, it reviews the proper order of operations to simplify expressions.
The document provides instructions and examples for graphing related quantities qualitatively. It discusses using graphs to represent a bouncing basketball with varying speeds and prices of Nike shoes over time. Students are asked to identify independent and dependent variables, construct their own qualitative graphs representing scenarios, and interpret other qualitative graphs. The homework assignment is to complete practice problems and a writing assignment by a due date.
This document discusses how to calculate the lateral area and surface area of pyramids and cones. It provides the formulas for lateral area of a pyramid and cone and surface area of a pyramid and cone. It then gives examples of applying the formulas to find the lateral and surface areas of specific pyramids and cones where the dimensions are provided.
The document provides a quick review for a quiz on writing equations of lines. It includes 4 problems: writing the equation of a line with a given slope and point, through two given points, a horizontal line through a given point, and a vertical line through a given point. It then transitions to discussing solving systems of equations using the elimination method, providing 3 examples of solving systems by eliminating a variable through adding or subtracting the equations.
This document contains a homework assignment on rates of change and graphs involving distance and time. It includes questions about finding the average speed between points on a distance-time graph, identifying which graph represents steady movement versus speeding up or slowing down, examples of other rate of change problems, and determining if three points are collinear based on their coordinates and slopes. The document provides context, definitions, and problems for students to practice skills relating to rates of change and interpreting distance-time and speed-time graphs.
1. The geometric mean between 2 and 10 is 5.
2. A right triangle problem is presented with missing side lengths for QS, QT, and RT given side lengths of 8, 32, and an unknown side R.
3. A homework problem asks to write trigonometric ratios as fractions and decimals and use special right triangles to write ratios in terms of degrees. It also asks which ratio would be used to find missing side lengths in sample right triangles.
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. It includes examples of using the midpoint formula to find the midpoint of a segment and using given midpoints and endpoints to find missing endpoints.
3. There are also examples of using the Pythagorean theorem and distance formula to find the length of a segment between two points.
The document discusses a 4 step process but provides no details on the actual steps or content of the process. It references numbered sections but provides no information within those sections. Overall, the document does not contain any substantive information that could be summarized due to the lack of details provided within the numbered sections.
This document discusses deductive reasoning and how to evaluate arguments using deductive logic. It introduces the law of detachment and law of syllogism as valid forms of deductive reasoning. Examples are provided to illustrate identifying the hypothesis and conclusion of conditionals, and determining whether conjectures follow validly from given information using the laws of detachment and syllogism. Practice problems are included for the reader to apply these concepts.
This document covers conditional statements and their components. It defines key terms like hypothesis, conclusion, converse, inverse, and contrapositive. Examples are provided to demonstrate how to identify the hypothesis and conclusion of conditional statements, write conditionals from sentences, and determine truth values. The last part discusses logical equivalence and how a conditional is equivalent to its contrapositive based on the Law of Contrapositive.
The student is able to (I can):
Use inductive reasoning to identify patterns and make conjectures
Find counterexamples to disprove conjectures
Identify, write, and analyze the truth value of conditional statements.
Write the inverse, converse, and contrapositive of a conditional statement.
The document defines conditional statements, converse, inverse, and contrapositive statements and provides examples of identifying each. It also discusses identifying the hypothesis and conclusion of conditional statements and determining the truth value of conditionals and their converses. The document provides guidance on writing indirect proofs using contradictions.
This document discusses biconditional statements and definitions. It begins by defining a biconditional statement as one that can be written "p if and only if q," meaning "if p then q" and "if q then p." Examples are provided of writing the conditionals within biconditionals. The document also discusses analyzing the truth value of biconditionals and providing counterexamples if false. Finally, it notes that definitions can be written as true biconditionals, and examples are given of writing geometric definitions as biconditionals.
Here are the converses, inverses, and contrapositives of the given statements:
1. Conditional: If two angles form a linear pair, then they are supplementary.
Converse: If two angles are supplementary, then they form a linear pair.
Inverse: If two angles do not form a linear pair, then they are not supplementary.
Contrapositive: If two angles are not supplementary, then they do not form a linear pair.
2. Conditional: If a parallelogram has a right angle, then it is a rectangle.
Converse: If a parallelogram is a rectangle, then it has a right angle.
Inverse: If a parallelogram does not have
This document discusses conditional statements and their components. It defines a conditional statement as having a hypothesis and conclusion connected by "if...then". It provides examples of writing the converse, inverse, and contrapositive of conditional statements. The key points are:
- A conditional statement has two parts: a hypothesis and conclusion connected by "if...then"
- The converse switches the hypothesis and conclusion
- The inverse negates both the hypothesis and conclusion
- A statement is equivalent to its contrapositive
This document contains a geometry drill with various geometry concepts and problems. It also contains a logic drill with conditional statements and vocabulary about conditionals. It discusses writing conditionals, their converses, inverses, and contrapositives. It provides examples of evaluating the truth value of these and using Venn diagrams to represent conditional statements.
The document discusses inductive and deductive reasoning. Inductive reasoning involves forming general conclusions from specific observations, while deductive reasoning draws specific conclusions from general statements. Examples are given of inductive arguments building from specific cases to a general rule, and deductive arguments applying a general premise to specific cases. The key features of deductive reasoning, including conditional statements and the five types of if-then logical structures (conditional, converse, counter example, inverse, and contrapositive), are also explained through examples.
Inductive and deductive reasoning are two important types of logical reasoning.
[1] Inductive reasoning involves observing specific examples and patterns to derive a general conclusion. [2] Deductive reasoning uses logical rules and known statements to derive a conclusion. [3] Venn diagrams can help determine the validity of deductive arguments by visually representing logical relationships between categories.
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
This document provides information on deductive reasoning and the laws of detachment and syllogism. It defines deductive reasoning as using facts and rules to reach a logical conclusion. The law of detachment states that if p implies q and p is true, then q must be true. The law of syllogism states that if p implies q and q implies r, then p implies r. The document includes examples applying these concepts and determining whether conclusions are valid deductive reasoning or not.
Identify, write, and analyze conditional statements.
Write the converse, inverse, and contrapositive of a conditional statement.
Write biconditional statements.
This document discusses syllogisms and their use in logical reasoning. It provides examples of valid and invalid syllogisms using the law of syllogism, which states that if p → q and q → r are true statements, then p → r is also a true statement. It also shows how syllogisms can be used to set up chains of conditionals and how proofs expand on this idea by providing justifications for each step.
Conditional statements can have a converse, negation, inverse, or contrapositive formed from them. Deductive reasoning uses stated rules or principles to draw conclusions, while inductive reasoning draws conclusions based on patterns or samples without being conclusive. The key difference is that deductive reasoning's conclusions are certain if the premises are true, while inductive reasoning's conclusions are probable but not certain.
This document discusses conditional statements, their converses, inverses, and contrapositives. It begins by defining biconditional statements as combining a conditional statement with its converse. It provides examples of determining if a conditional statement, its converse, inverse, and contrapositive are true or false. It explains that a conditional statement is logically equivalent to its contrapositive, and the converse is logically equivalent to the inverse. The document also discusses using biconditional statements when a conditional and its converse are both true.
This document discusses inductive reasoning and how it can be used to identify patterns and make predictions. It provides examples of using inductive reasoning to find subsequent terms in numeric sequences and to make conjectures based on patterns observed in geometric figures like rectangles. Key terms discussed include inductive reasoning, conjecture, and counterexample. Readers are guided through activities to practice applying inductive reasoning to sequences and geometric patterns.
CONDITIONAL STATEMENTS AND TRUTH VALUE.pptxJasminAndAngie
This document discusses conditional statements and if-then statements. It provides examples of identifying the hypothesis and conclusion of conditional statements and transforming statements into equivalent if-then forms. The document also contains practice problems for learners to identify the hypothesis and conclusion of conditional statements and to write conditional statements in if-then form given a hypothesis and conclusion.
1) The document provides 4 math word problems and equations to solve using techniques like backtracking and defining variables.
2) It asks the reader to simplify the expression 2(x+6) - 4(3x - 1) and evaluate it for x = -2.
3) Another problem defines a variable to represent the number of giraffes at Brookfield Zoo and writes an expression relating it to the number of giraffes at Lincoln Park Zoo.
1) The document provides 4 math word problems and equations to solve using techniques like backtracking and defining variables.
2) It asks the reader to simplify the expression 2(x+6) - 4(3x - 1) and evaluate it for x = -2.
3) Another problem defines a variable to represent the number of giraffes at Brookfield Zoo and writes an expression relating it to the number of giraffes at Lincoln Park Zoo.
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of segments. It also defines different types of transformations, such as translations, reflections, and rotations. Students are given homework problems applying these concepts, and examples of identifying transformations and describing them with arrow notation.
1. The document discusses solving equations by backtracking. It begins by defining an equation and identifying true and false equations. It then discusses the concept of solutions - the values that make the equation true.
2. Examples are provided to illustrate finding solutions to equations by reversing the steps of number tricks. Backtracking involves working backwards through the operations to determine the starting value.
3. Readers are instructed to solve sample equations by listing the number trick steps and then reversing the steps through backtracking to find the solution values.
This document provides homework questions and a review packet for a chapter. It lists 7 numbered sections that appear to be questions or tasks related to reviewing material from the first chapter. The document aims to help students review and reinforce their understanding of the concepts covered in the initial chapter through completing the homework and review activities.
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of segments. It also defines different types of transformations, such as translations, reflections, and rotations. Students are given homework problems applying these concepts, and examples of identifying transformations and describing them with arrow notation.
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of line segments. It also discusses the different types of transformations (translations, reflections, rotations, and dilations) and provides examples of identifying each type using arrow notation. The homework assignments are to complete practice problems from sections 1.6 and 1.7 in the textbook, as listed.
This document provides examples and explanations of the midpoint and distance formulas, as well as transformations in the coordinate plane. It includes 4 examples of using the midpoint and distance formulas to find midpoints and lengths of line segments. It also discusses the different types of transformations - translations, reflections, rotations, and dilations - and provides examples of identifying each type of transformation using arrow notation. The homework assignments are to complete practice problems from sections 1.6 and 1.7 in the textbook, as listed.
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. Students are asked to work on their vocabulary packet silently after finishing the quiz. Then the class will review geometry formulas.
3. The homework assignments are to complete problems on page 38 from section 1.5 and page 47 from section 1.6 in the textbook. These cover midpoints, distances, and the midpoint and distance formulas.
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6.
2. Students are asked to work on their vocabulary packet silently after finishing the quiz. Then the class will review geometry formulas.
3. The homework assignments are to complete problems on page 38 from section 1.5 and page 47 from section 1.6 in the textbook. These cover midpoints, distances, and the midpoint and distance formulas.
1. The document provides instructions for a quiz and homework assignments on geometry chapters 1.5 and 1.6. Students are asked to work on vocabulary and postulates after the quiz.
2. Examples are given for finding midpoints and distances between points on a coordinate plane using formulas like the midpoint formula, Pythagorean theorem, and distance formula.
3. Homework assignments include problems from the textbook on the topics of formulas in geometry, midpoints, and distance.
1. The document provides instructions and tasks for students to complete mathematical expressions, homework questions, and a lesson on reversible and non-reversible operations.
2. Students are asked to simplify expressions, complete homework problems, and determine whether example operations are reversible by considering if the starting number can be determined.
3. The document demonstrates how to "backtrack" through a multi-step operation to find the original starting number using reversible operations.
This document contains notes and instructions for a math lesson that includes:
1) Solving expressions and evaluating them for given values.
2) Completing an in-class activity with partners to review basic arithmetic rules.
3) Practicing the basic rules of arithmetic through examples of simplifying expressions using properties like commutative, associative, and distributive properties.
1. Students were asked to put math problems on the board from previous homework. The document then provides examples of expressions and teaches how to simplify them using order of operations and properties like the distributive property. Students are asked to simplify sample expressions involving variables.
2. The document reviews that expressions need to have "like terms" to be simplified, such as terms with the same variables. Students practice simplifying expressions with multiple variables and terms by combining like terms.
3. To conclude, students are instructed to write their name and ID number on raffle tickets and provide just the simplified answer, practicing the skills of defining a variable, writing an expression, simplifying it, and evaluating it.
1. When entering class each day, students should say hi, have their homework out, write any questions on the board, and start the opener problem.
2. The document then provides examples of algebra problems involving variables to represent unknown quantities and expressions combining variables, operators, and constants.
3. Students are instructed to complete a set of practice problems from page 94 in their workbook and have their work checked by the teacher.
This document contains notes from multiple sections on solving systems of equations by graphing. It includes questions on graphing systems like y = 3x - 4 and y = -2x + 5. It also asks how to solve an absolute value equation like |x+4| = x^2 - 3 and discusses the pros and cons of the graphing method, listing advantages and disadvantages. Finally, it addresses solving an equation like x^3 + 5x = 7x^2 - 5 by graphing and reflects on graphing techniques from prior lessons.
This document contains notes from multiple sections on solving systems of equations by graphing. It includes questions about graphing systems like y = 3x - 4 and y = -2x + 5. It also asks how to solve an absolute value equation like |x+4| = x^2 - 3 and discusses the pros and cons of the graphing method, listing advantages and disadvantages.
The document discusses a 4 step process but provides no details on the actual steps or content of the process. It references numbered sections but provides no information within those sections. Overall, the document does not contain any substantive information that could be summarized due to the lack of details provided within the numbered sections.
This document contains notes from multiple sections on solving systems of equations by graphing. It includes questions on graphing systems like y = 3x - 4 and y = -2x + 5. It also asks how to solve an absolute value equation like |x+4| = x^2 - 3 and discusses the pros and cons of the graphing method, listing advantages and disadvantages.
1. Opener:
Study the following pattern. Row 1: 1
Row 2: 1 1
1. If the pattern is extended, find Row 3: 1 2 1
the terms in row 7. Row 4: 1 3 3 1
Row 5: 1 4 6 4 1
Row 6: 1 5 10 10 5 1
2. Make a conjecture for the pattern.
3. Make a conjecture for the sum of the
terms in each row.
1
3. 2‐2: Conditional Statements
What are they?
Hypothesis Conclusion
Conditional Statements
Notation Venn Diagram
Examples:
1) If you are 16, then you are a teenager.
2) You live in Illinois, if you live in Northbrook.
Venn Diagrams to represent conditionals:
1) If you are 16, then you are a teenager.
2) You live in Illinois, if you live in Northbrook.
3
4. Writing Conditional Statements:
1) Tarantulas are spiders.
2) A 25 degree angle is acute.
Analyzing truth values:
1) If this is September, then the next month is October.
2) If 2 angles are obtuse, then they are congruent.
3) If an even number greater than 2 is prime, then 5 + 4 = 8.
Important conclusion:
If p is true, then ___________________________________
If p is false, then __________________________________
4
5. Negation Converse
Related
Conditionals
Inverse Contrapositive
Examples:
1) FACT: Glenbrook North students are Spartans.
Conditional:
Converse:
Inverse:
Contrapositive:
2) Conditional:If m A = 95 degrees, then A is obtuse.
TRUE or FALSE
Converse: TRUE or FALSE
Inverse: TRUE or FALSE
Contrapositive: TRUE or FALSE
Look for the pattern (what kind of reasoning?):
Which statements always have the same truth value?
5
6. 2‐3: Deductive Reasoning
A process of using logic to draw conclusions from given facts, definitions, and
properties
How is this different from inductive reasoning?
Inductive or Deductive?
1)
2) There is a myth that birds will abandon their young if you touch them.
6
7. Types of Deductive Reasoning
Law of Detachment Law of Syllogism
If p q is a true If p q and q r, are
statement,and p is true, true statements, then
then q is also true. p r is a true statement.
Example: If I oversleep,
Example: If I get over
then I will miss the bus.
a 90%, then I get an A.
If I miss the bus, then I
I got a 96%.
will have to walk to school.
Conjecture: I got an A.
Conjecture: If I oversleep
then I will have to walk to
school.
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8. With your partner, decide whether each of the following examples is valid using
Deductive Reasoning. Then decide which law it is demonstrating. Hold up the
appropriate response cards.
A) Given: If side lengths of a triangle are 5, 12, and 13, then the area of the
triangle is 30.
The area of PQR is 30.
Conjecture: The side lengths of the triangle are 5, 12, and 13.
B) Given: In the World Series, if a team wins 4 games, the team wins the
Series.
In 2004, the Red Sox won 4 games in the World Series.
Conjecture: The Red Sox won the World Series.
C) Given: If a figure is a kite, then it is a quadrilateral. If a figure is a
quadrilateral, then it is a polygon.
Conjecture: If a figure is a kite, then it is a polygon.
D) Given:If a number is divisible by 2, then it is even. If a number is even,
then it is an integer.
Conjecture: If a number is an integer, then it is divisible by 2.
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9. Draw a conclusion using Deductive Reasoning:
A) Given: If 2y = 4, then z = ‐1. If x + 3 = 12, then 2y = 4.
x + 3 = 12.
Conclusion:
B) Given: If the sum of two angles is 180 degrees, then the angles are
supplementary. If 2 angles are supplementary, then they are not the
angles in a triangle.
m<A = 135 degrees and m<B = 45 degrees
Conclusion:
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11. Conditional: If an animal is a brown bear,
then the animal sleeps for 3 months each year.
Related Statement:
If an animal does not sleep for 3 months each year,
then the animal is not a brown bear.
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13. Yard
Enclosed by a fence
Has a swimming pool
Conditional: If a yard has a swimming pool, then it is enclosed by a fence.
Related Statement: If a yard does not have a swimming pool, then it is not
enclosed by a fence.
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