The document provides instruction on the Pythagorean theorem. It defines key vocabulary like hypotenuse and leg. It explains how to use the theorem to find the length of an unknown side of a right triangle if two sides are known. Examples show setting up and solving equations using the theorem. The document also discusses using the theorem to determine if a triangle is right based on the lengths of its sides.
This module covers similarity and the Pythagorean theorem as they relate to right triangles. It discusses how the altitude to the hypotenuse of a right triangle divides it into two smaller right triangles that are similar to each other and the original triangle. It also explains how the altitude is the geometric mean of the hypotenuse segments. Special right triangles like 45-45-90 and 30-60-90 triangles are examined, relating side lengths through their properties. The Pythagorean theorem is derived and used to solve for missing sides of right triangles. Students work through examples and multi-step problems applying these concepts.
This module introduces ratio, proportion, and the Basic Proportionality Theorem. Students will learn about ratios, proportions, and how to use the fundamental law of proportions to solve problems involving triangles. The module is designed to teach students to apply the definition of proportion of segments to find unknown lengths and illustrate and verify the Basic Proportionality Theorem and its Converse. Examples are provided to demonstrate how to express ratios in simplest form, find missing values in proportions, determine if ratios form proportions, and solve problems involving angles and segments in triangles using ratios and proportions.
This document provides information about Module 17 on similar triangles. The key points covered are:
1. The module discusses the definition of similar triangles, similarity theorems, and how to determine if two triangles are similar or find missing lengths using properties of similar triangles.
2. Students are expected to learn how to apply the definition of similar triangles, verify the AAA, SAS, and SSS similarity theorems, and use proportionality theorems to calculate lengths of line segments.
3. Several examples and exercises are provided to help students practice determining if triangles are similar, citing the appropriate similarity theorem, finding missing lengths, and applying properties of similar triangles.
This document is a lesson on the Pythagorean theorem from a Holt Geometry textbook. It begins with examples of using the Pythagorean theorem to solve for missing sides of right triangles. It then discusses how to identify if a set of three sides forms a Pythagorean triple. The lesson continues with examples of using the converse of the Pythagorean theorem and Pythagorean inequalities to classify triangles as right, acute, or obtuse based on their side lengths. It concludes with a two-part lesson quiz reviewing the material.
This document contains a mathematics lesson on solving word problems involving parallelograms, trapezoids, and kites. It begins with an introduction to the key properties of parallelograms, trapezoids, and kites. It then provides 4 examples of word problems involving these shapes and shows the step-by-step work and reasoning to solve each problem using the relevant geometric properties. The lesson aims to teach students how to illustrate, set up, and solve word problems involving parallelograms, trapezoids, and kites.
The document provides examples and explanations of the Fundamental Theorem of Variation. It begins with warm-up problems about how increasing the side of a square patio or the edge of a cube by 50% or tripling the length affects the area or volume. It then states the Fundamental Theorem - if x is multiplied by a constant c, a direct variation y is multiplied by c^n and an inverse variation y is divided by c^n. Examples are given of how doubling the distance from a light source affects intensity, which varies inversely with the square of the distance. It concludes by stating how doubling x would affect y in different variation problems.
Who is Pythagoras? What is The Pythagoras Theorem?Nuni Yustini
This document discusses the Pythagorean theorem. It provides a brief history of Pythagoras and his contributions to mathematics. It then presents two proofs of the Pythagorean theorem - using the area of squares and the area of triangles. Finally, it discusses applications of the theorem in mathematics and physics, such as determining if a triangle is right, acute, or obtuse and calculating impedance in electrical systems.
This geometry exam review covers topics that will be on the final exam. It includes true/false questions, multiple choice, matching, and free response problems involving geometry concepts like triangles, circles, polygons, and three-dimensional shapes. Calculators may be used but the exam may have non-calculator sections, so students should prepare with and without calculators. The review is due before the scheduled final exam date.
This module covers similarity and the Pythagorean theorem as they relate to right triangles. It discusses how the altitude to the hypotenuse of a right triangle divides it into two smaller right triangles that are similar to each other and the original triangle. It also explains how the altitude is the geometric mean of the hypotenuse segments. Special right triangles like 45-45-90 and 30-60-90 triangles are examined, relating side lengths through their properties. The Pythagorean theorem is derived and used to solve for missing sides of right triangles. Students work through examples and multi-step problems applying these concepts.
This module introduces ratio, proportion, and the Basic Proportionality Theorem. Students will learn about ratios, proportions, and how to use the fundamental law of proportions to solve problems involving triangles. The module is designed to teach students to apply the definition of proportion of segments to find unknown lengths and illustrate and verify the Basic Proportionality Theorem and its Converse. Examples are provided to demonstrate how to express ratios in simplest form, find missing values in proportions, determine if ratios form proportions, and solve problems involving angles and segments in triangles using ratios and proportions.
This document provides information about Module 17 on similar triangles. The key points covered are:
1. The module discusses the definition of similar triangles, similarity theorems, and how to determine if two triangles are similar or find missing lengths using properties of similar triangles.
2. Students are expected to learn how to apply the definition of similar triangles, verify the AAA, SAS, and SSS similarity theorems, and use proportionality theorems to calculate lengths of line segments.
3. Several examples and exercises are provided to help students practice determining if triangles are similar, citing the appropriate similarity theorem, finding missing lengths, and applying properties of similar triangles.
This document is a lesson on the Pythagorean theorem from a Holt Geometry textbook. It begins with examples of using the Pythagorean theorem to solve for missing sides of right triangles. It then discusses how to identify if a set of three sides forms a Pythagorean triple. The lesson continues with examples of using the converse of the Pythagorean theorem and Pythagorean inequalities to classify triangles as right, acute, or obtuse based on their side lengths. It concludes with a two-part lesson quiz reviewing the material.
This document contains a mathematics lesson on solving word problems involving parallelograms, trapezoids, and kites. It begins with an introduction to the key properties of parallelograms, trapezoids, and kites. It then provides 4 examples of word problems involving these shapes and shows the step-by-step work and reasoning to solve each problem using the relevant geometric properties. The lesson aims to teach students how to illustrate, set up, and solve word problems involving parallelograms, trapezoids, and kites.
The document provides examples and explanations of the Fundamental Theorem of Variation. It begins with warm-up problems about how increasing the side of a square patio or the edge of a cube by 50% or tripling the length affects the area or volume. It then states the Fundamental Theorem - if x is multiplied by a constant c, a direct variation y is multiplied by c^n and an inverse variation y is divided by c^n. Examples are given of how doubling the distance from a light source affects intensity, which varies inversely with the square of the distance. It concludes by stating how doubling x would affect y in different variation problems.
Who is Pythagoras? What is The Pythagoras Theorem?Nuni Yustini
This document discusses the Pythagorean theorem. It provides a brief history of Pythagoras and his contributions to mathematics. It then presents two proofs of the Pythagorean theorem - using the area of squares and the area of triangles. Finally, it discusses applications of the theorem in mathematics and physics, such as determining if a triangle is right, acute, or obtuse and calculating impedance in electrical systems.
This geometry exam review covers topics that will be on the final exam. It includes true/false questions, multiple choice, matching, and free response problems involving geometry concepts like triangles, circles, polygons, and three-dimensional shapes. Calculators may be used but the exam may have non-calculator sections, so students should prepare with and without calculators. The review is due before the scheduled final exam date.
This document contains notes from a math class lesson on the Pythagorean theorem. It includes the classroom jobs, warm-up problems, objectives to define and apply the theorem, explanations of Pythagoras and the theorem, examples of using the theorem to determine if triangles are right triangles and to find missing hypotenuses, and directions for a poster assignment on the theorem.
The document discusses the Pythagorean theorem and square roots. It begins by defining a right triangle and its components. It then states the Pythagorean theorem, which relates the lengths of the sides of a right triangle. An example problem demonstrates using the theorem to find the height of a wall. The document concludes by defining the square root, explaining that the square root of a number is its positive root and how to calculate and approximate square roots.
Pythagoras was an ancient Greek mathematician born around 500 BC outside of Athens. The document discusses the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides examples of using the theorem to find missing side lengths of right triangles.
This document discusses square numbers and how to identify them. It explains that a number is square if you can form a square using the same length for all four sides. Students investigate which area values represent square numbers by making rectangles with unit blocks. They determine that the areas 4, 9, 16, 25, and others represent perfect squares where the number is multiplied by itself. The document concludes by asking students to show that 49 is a square number and calculate the perimeter of a square with an area of 144 cm2.
The document discusses eight laws of exponents:
1) The exponent indicates how many times the base is multiplied by itself.
2) If the bases are the same and the operation is multiplication, the exponent is the sum of the individual exponents.
3) If the bases are the same and the operation is division, the exponent is the difference of the individual exponents.
4) If an exponential form is raised to another exponent, the result is the base raised to the product of the exponents.
This document discusses various methods for estimating and calculating square roots. It begins by defining irrational numbers as non-repeating decimals that can result from taking the square root of a non-perfect square. It then discusses estimating square roots by knowing perfect square numbers and using guess and check methods. Later sections cover using calculators, applying square roots in formulas, exponent rules including multiplication, division, powers, and scientific notation for writing extremely large numbers.
Estimating Square Roots (Number Line Method)Nicole Gough
The document describes using a number line method to estimate square roots. It reviews perfect squares from 1 to 144. To estimate the square root of 40, it identifies the nearest perfect squares of 36 and 49 and draws a number line between them. It determines that the square root of 40 is closer to 6 than 7. Similarly, to estimate the square root of 58, it identifies the nearest perfect squares of 49 and 64 and draws a number line between them, determining the square root is closer to 8 than 7. It concludes by having the reader practice estimating square roots using the number line method.
The document discusses square numbers, square roots, and estimating square roots. It defines a square number as a number that is the product of a whole number multiplied by itself. Square roots are defined as numbers that when multiplied by themselves produce another given number. The document provides examples of calculating square roots of perfect squares by factoring them into smaller perfect square factors. It also describes a method for estimating square roots of non-perfect squares by placing them on a number line between the adjacent perfect squares and interpolating to the nearest tenth.
This document discusses grid references and how they are used to locate features on maps. It explains that topographic maps use a grid system of eastings and northings to specify locations, and that area references use a four-digit code to indicate a specific grid square. It also introduces six-figure grid references which can pinpoint an exact location within a grid square down to the tenth of the easting and northing. Examples are provided to demonstrate how to read and use both area references and more precise grid references.
The document summarizes the LE550 laser sensor from Banner Engineering. It can measure distances from 100-1000 mm and has an intuitive two-line display. It has accuracy and repeatability for various targets. It is ideal for applications like loop control, thickness measurement, and positioning. It has analog and discrete output options and is easy to adjust for different applications.
El documento proporciona información sobre The Brock Group, un proveedor líder de servicios industriales especializados. Brock ofrece una variedad de servicios como andamios, aislamiento, pintura, instrumentación eléctrica y más. Genera más de $1 mil millones en ingresos anuales y tiene una cartera diversificada de más de 500 clientes en múltiples industrias como refinería, química y energía. Brock se destaca por su sólido historial de seguridad y relaciones a largo plazo con clientes de la lista Fortune 500.
El diablo alienta tres estrategias propuestas por demonios para alejar a las personas de Dios: 1) decir que Dios no existe, pero Satanás dice que los humanos tienen un sentido innato de Dios; 2) decir que no hay infierno, pero Satanás dice que la vida sin Dios es un infierno; 3) decir que no hay prisa para aceptar a Dios, ya que siempre hay un mañana, pero Satanás advierte que nadie está seguro de tener un mañana.
Lawrence embarked on his first contract from June 2012 to November 2012, gaining experience across multiple departments on board two cruise ships, the Quest for Adventure and Saga Sapphire. Over this period, he worked a total of 403.5 hours, spending time in areas like the galley, dining room, provisions, and reception. Lawrence determined that he was most interested in pursuing a career in food and beverage.
This document discusses the history of feminism and women's rights movements from the 1st wave to present. It outlines key events like women gaining the right to vote in the early 20th century and the 2nd wave focusing on equal rights in the workplace in the 1960s. It also summarizes international agreements like CEDAW that aim to eliminate discrimination against women. Finally, it provides overviews of relevant Philippine laws regarding sexual harassment, rape, and anti-trafficking.
The document explains the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides examples of using the theorem to calculate the length of the hypotenuse or legs of a right triangle when the other two sides are known. These include using the theorem to calculate the distance between two points if the distances travelled in each direction are known.
The document provides instruction on using the Pythagorean theorem to solve for missing sides of right triangles. It defines right triangles as having one 90 degree angle, with the two sides forming the right angle called the legs and the side opposite the hypotenuse. Examples are given of using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. Students are asked to solve problems finding missing sides of right triangles.
This document contains notes from a math class lesson on the Pythagorean theorem. It includes the classroom jobs, warm-up problems, objectives to define and apply the theorem, explanations of Pythagoras and the theorem, examples of using the theorem to determine if triangles are right triangles and to find missing hypotenuses, and directions for a poster assignment on the theorem.
The document discusses the Pythagorean theorem and square roots. It begins by defining a right triangle and its components. It then states the Pythagorean theorem, which relates the lengths of the sides of a right triangle. An example problem demonstrates using the theorem to find the height of a wall. The document concludes by defining the square root, explaining that the square root of a number is its positive root and how to calculate and approximate square roots.
Pythagoras was an ancient Greek mathematician born around 500 BC outside of Athens. The document discusses the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides examples of using the theorem to find missing side lengths of right triangles.
This document discusses square numbers and how to identify them. It explains that a number is square if you can form a square using the same length for all four sides. Students investigate which area values represent square numbers by making rectangles with unit blocks. They determine that the areas 4, 9, 16, 25, and others represent perfect squares where the number is multiplied by itself. The document concludes by asking students to show that 49 is a square number and calculate the perimeter of a square with an area of 144 cm2.
The document discusses eight laws of exponents:
1) The exponent indicates how many times the base is multiplied by itself.
2) If the bases are the same and the operation is multiplication, the exponent is the sum of the individual exponents.
3) If the bases are the same and the operation is division, the exponent is the difference of the individual exponents.
4) If an exponential form is raised to another exponent, the result is the base raised to the product of the exponents.
This document discusses various methods for estimating and calculating square roots. It begins by defining irrational numbers as non-repeating decimals that can result from taking the square root of a non-perfect square. It then discusses estimating square roots by knowing perfect square numbers and using guess and check methods. Later sections cover using calculators, applying square roots in formulas, exponent rules including multiplication, division, powers, and scientific notation for writing extremely large numbers.
Estimating Square Roots (Number Line Method)Nicole Gough
The document describes using a number line method to estimate square roots. It reviews perfect squares from 1 to 144. To estimate the square root of 40, it identifies the nearest perfect squares of 36 and 49 and draws a number line between them. It determines that the square root of 40 is closer to 6 than 7. Similarly, to estimate the square root of 58, it identifies the nearest perfect squares of 49 and 64 and draws a number line between them, determining the square root is closer to 8 than 7. It concludes by having the reader practice estimating square roots using the number line method.
The document discusses square numbers, square roots, and estimating square roots. It defines a square number as a number that is the product of a whole number multiplied by itself. Square roots are defined as numbers that when multiplied by themselves produce another given number. The document provides examples of calculating square roots of perfect squares by factoring them into smaller perfect square factors. It also describes a method for estimating square roots of non-perfect squares by placing them on a number line between the adjacent perfect squares and interpolating to the nearest tenth.
This document discusses grid references and how they are used to locate features on maps. It explains that topographic maps use a grid system of eastings and northings to specify locations, and that area references use a four-digit code to indicate a specific grid square. It also introduces six-figure grid references which can pinpoint an exact location within a grid square down to the tenth of the easting and northing. Examples are provided to demonstrate how to read and use both area references and more precise grid references.
The document summarizes the LE550 laser sensor from Banner Engineering. It can measure distances from 100-1000 mm and has an intuitive two-line display. It has accuracy and repeatability for various targets. It is ideal for applications like loop control, thickness measurement, and positioning. It has analog and discrete output options and is easy to adjust for different applications.
El documento proporciona información sobre The Brock Group, un proveedor líder de servicios industriales especializados. Brock ofrece una variedad de servicios como andamios, aislamiento, pintura, instrumentación eléctrica y más. Genera más de $1 mil millones en ingresos anuales y tiene una cartera diversificada de más de 500 clientes en múltiples industrias como refinería, química y energía. Brock se destaca por su sólido historial de seguridad y relaciones a largo plazo con clientes de la lista Fortune 500.
El diablo alienta tres estrategias propuestas por demonios para alejar a las personas de Dios: 1) decir que Dios no existe, pero Satanás dice que los humanos tienen un sentido innato de Dios; 2) decir que no hay infierno, pero Satanás dice que la vida sin Dios es un infierno; 3) decir que no hay prisa para aceptar a Dios, ya que siempre hay un mañana, pero Satanás advierte que nadie está seguro de tener un mañana.
Lawrence embarked on his first contract from June 2012 to November 2012, gaining experience across multiple departments on board two cruise ships, the Quest for Adventure and Saga Sapphire. Over this period, he worked a total of 403.5 hours, spending time in areas like the galley, dining room, provisions, and reception. Lawrence determined that he was most interested in pursuing a career in food and beverage.
This document discusses the history of feminism and women's rights movements from the 1st wave to present. It outlines key events like women gaining the right to vote in the early 20th century and the 2nd wave focusing on equal rights in the workplace in the 1960s. It also summarizes international agreements like CEDAW that aim to eliminate discrimination against women. Finally, it provides overviews of relevant Philippine laws regarding sexual harassment, rape, and anti-trafficking.
The document explains the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. It provides examples of using the theorem to calculate the length of the hypotenuse or legs of a right triangle when the other two sides are known. These include using the theorem to calculate the distance between two points if the distances travelled in each direction are known.
The document provides instruction on using the Pythagorean theorem to solve for missing sides of right triangles. It defines right triangles as having one 90 degree angle, with the two sides forming the right angle called the legs and the side opposite the hypotenuse. Examples are given of using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. Students are asked to solve problems finding missing sides of right triangles.
Pythagoras' Theorem allows you to calculate the lengths of sides in a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be used to find missing side lengths or to determine if a triangle is right-angled. It is commonly applied to problems involving distances, lengths, and geometrical shapes containing right triangles.
The document summarizes the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides the background of Pythagoras and how he developed this important theorem around 570 BC. Examples are given to demonstrate how to use the theorem to calculate unknown side lengths of right triangles.
Day 2 - Unit 3 - Work Period - Pythagorean Theorem PowerPoint Accomodations.pptJamaicaLalaguna
The document discusses the Pythagorean theorem and how to use it to solve for missing sides of right triangles. It defines key terms like hypotenuse, leg, and base. It then explains the two scenarios for using the theorem - when finding the hypotenuse given two legs, and when finding a leg given the hypotenuse and other leg. The steps are to square all sides, add the leg squares in one case and subtract in the other, then take the square root of the result to find the missing side. Several examples are worked through to demonstrate the process.
The document provides an explanation of the Pythagorean theorem using examples of right triangles found in baseball diamonds and ladders. It begins by defining a right triangle and the key terms of hypotenuse, legs, and the Pythagorean theorem formula a2 + b2 = c2. It then works through two word problems, solving for missing side lengths by setting up the appropriate equations and calculations based on the given information.
The document provides an explanation of the Pythagorean theorem using examples of right triangles found in baseball diamonds and ladders. It begins by defining a right triangle and its components - the hypotenuse and two legs. It then states the Pythagorean theorem formula that the sum of the squares of the two legs equals the square of the hypotenuse. Several word problems are worked through step-by-step using the theorem to calculate missing side lengths of right triangles.
Here are the steps to solve problems 1-15 and 35 from page 740 of the textbook using the Pythagorean theorem:
1. Identify the formula: a^2 + b^2 = c^2
2. Substitute the given values for a, b, and c into the formula
3. Simplify by combining like terms
4. Solve for the unknown side length by taking the square root of each side
5. Round your answer to the nearest tenth if instructed to do so
Be sure to show the setup and work for each problem. Label answers clearly. Bring any questions to class tomorrow.
This document explains the Pythagorean theorem and how to use it to solve problems involving right triangles. It begins by defining a right triangle as one with a 90 degree angle. It then explains that Pythagoras discovered that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Several examples are provided to demonstrate how to use the theorem to calculate missing side lengths of right triangles in various contexts like finding the diagonal of a rectangle or calculating distance traveled.
This document provides examples for applying the Pythagorean theorem and algebraic concepts to solve geometric problems involving right triangles. It includes:
1. Five multi-step examples that write and solve equations using the Pythagorean theorem (a^2 + b^2 = c^2) to find missing side lengths of right triangles in real-world contexts.
2. Information on the Common Core State Standards for applying the Pythagorean theorem to determine unknown side lengths in two and three dimensional figures.
3. A question about how using the Pythagorean theorem in this lesson connects to finding the distance between two points on a coordinate plane in the next lesson.
This document provides information about Pythagoras' theorem, including:
- The theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- It gives examples of using the theorem to find missing sides of right triangles.
- The converse of the theorem is also stated - if the square of one side equals the sum of the squares of the other two sides, then the triangle is right.
This document provides instruction on using the Pythagorean theorem to solve for unknown sides of right triangles. It defines key terms like hypotenuse and legs, shows the Pythagorean theorem formula, and provides examples of using it to find the length of unknown sides. Historical context is given about Pythagoras and that other ancient cultures also understood this relationship between triangle sides. Word problems are worked through as practice applications of the theorem.
This document provides a tutorial on using the Pythagorean theorem to solve for missing sides of right triangles. It explains the theorem and its formula, how to identify the sides of a right triangle, and how to solve for both missing hypotenuses and legs. Examples are worked through step-by-step and opportunities for practice are provided throughout the tutorial. The goal is for students to demonstrate mastery in solving for missing sides of right triangles.
These slides contain the pathagorean theorem and right trinagles. How to prove the oathagorean theorem and how to vind the area of triangles by the pathagorean theorem. There are some slides that explains that how the pathagorean theorem was discovrers. Some slides explain the pathagorean triple theorem and c^2=a^2 + b^2.
The document provides instruction on using the Pythagorean theorem to find the length of sides in right triangles. It begins with examples of estimating square roots, then presents the key vocabulary and formula. Several examples are worked through step-by-step to demonstrate applying the theorem to find missing side lengths. Practice problems are provided for students to check their understanding.
This document provides examples and explanations of using the Pythagorean theorem to solve problems involving right triangles. It begins with examples of calculating missing side lengths of right triangles and identifying whether three given side lengths form a right triangle. It then presents an example of applying the Pythagorean theorem to solve a real-world problem about finding the distance between corners of a square field. Additional examples follow the same format of presenting the problem, setting up the Pythagorean theorem equation, solving, and checking the reasonableness of the answer. The document concludes with a lesson quiz involving similar right triangle problems.
today we reviewed the Pythagorean Theorem and there was one sheet handed out to the B and D class and two handed out to the C class.
B and D make sure both sides are done, C make sure the front of each is done.
The document discusses Pythagoras' theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of applying the theorem to calculate missing side lengths. Pythagoras discovered this relationship around 2500 years ago. The theorem can be used to solve problems involving right triangles, such as calculating the total distance traveled by someone who walks two legs of a right triangle.
The document provides a lesson on the Pythagorean theorem. It includes examples of using the theorem to find the length of hypotenuses and legs of right triangles. It also gives practice problems and their step-by-step solutions. The document aims to teach students how to apply the Pythagorean theorem to solve problems involving right triangles.
This document provides examples and explanations for using algebra to solve for unknown angle measures involving complementary and supplementary angles. It begins with warm up problems labeling angle pairs as complementary, supplementary, or neither. Examples are then worked through step-by-step to find missing angles given information about one angle. Students are assigned complementary, supplementary, or 3 times larger problems to work through independently and explain to the class. Homework and an upcoming test are assigned before ending.
1.5 Complementary and Supplementary Angles Dee Black
Some slides lifted from: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CEsQFjAD&url=http%3A%2F%2Fdionmath.wikispaces.com%2Ffile%2Fview%2F2.3%2BComplementary%2Band%2BSuppl.%2BAngles.ppt&ei=_wVFUbzHCa-o4AP9ooGwBQ&usg=AFQjCNF-KDyDx_yiVaUuMJMdM6yOJqHASQ&sig2=wH2TZ9xGxsHgtc4cCnn2QQ&bvm=bv.43828540,d.dmg&cad=rja
The document discusses angles and how to measure them using a protractor. It provides examples of angles in everyday life and notes that the ancient Babylonians were the first to divide a circle into 360 equal parts called degrees. It explains how to properly use a protractor to measure angles, including making sure the protractor is lined up with the vertex of the angle and reading the measurement. The document also classifies different types of angles such as acute, right, obtuse, and straight angles.
The document discusses three geometric postulates:
1) The ruler postulate establishes a one-to-one correspondence between points on a line and real numbers on the number line, where the distance between points equals the absolute value of the difference of their corresponding numbers.
2) The ruler placement postulate allows choosing a number line such that two given points correspond to 0 and a positive number.
3) The segment addition postulate states that if one point is between two others, the sum of the distances to the end points equals the distance between the outer points.
The document discusses properties of real numbers including commutative, associative, identity, zero, and multiplication properties of addition and multiplication. These properties allow expressions to be rewritten and compared, and are useful rules for solving problems using mental math. Examples are provided to demonstrate applying properties like commutative and associative to solve problems.
The document discusses classifying, graphing, and comparing real numbers, including finding and estimating square roots. It defines real numbers as numbers that can be located on the number line, including integers, rational numbers, and irrational numbers. The document also discusses the differences between finding the square root of a perfect square versus a non-perfect square.
This document provides instructions on evaluating expressions by substituting values for variables and simplifying. It includes examples of evaluating algebraic expressions by substituting values for variables a, b, and c. The document also defines key vocabulary like "evaluate" and provides practice problems and assignments for students to complete. It concludes by emphasizing that evaluating expressions involves substituting values before simplifying.
Algebra Foundations Series- 1.1 Variables and ExpressionsDee Black
This document appears to be from a math class and includes the following elements:
1. An outline of the daily agenda which includes picking a seat, getting supplies, writing a verb to describe summer, and using class time well.
2. Sections on vocabulary words like quantity, variable, algebraic expression, and numerical expression with definitions and examples.
3. Word phrases used in algebraic expressions like addition, subtraction, multiplication, and division terms.
4. Practice writing algebraic expressions using variables and numbers.
5. A focus question asking how algebraic expressions differ from numerical expressions.
6. An assignment listing specific math problems to complete for homework.
7. Instructions to write in a math journal about the difficulty of
4. BIG Ideas
The lengths of the sides of a right triangle have a special
relationship.
5. BIG Ideas
The lengths of the sides of a right triangle have a special
relationship.
If the lengths of any two sides of a right triangle are known, the
length of the third side can be found.
6. BIG Ideas
The lengths of the sides of a right triangle have a special
relationship.
If the lengths of any two sides of a right triangle are known, the
length of the third side can be found.
If the length of each side of a triangle is known, then whether the
triangle is a right triangle can be determined.
10. Vocabulary to Know
Pythagorean Theorem
In any right triangle, the sum of the squares of the lengths of the legs is
equal to the square of the length of the hypotenuse.
a2 + b2 = c2
11. Finding the Hypotenuse and Leg
Lengths
You can use the Pythagorean Theorem to find the length of a right
triangle’s hypotenuse given the lengths of its legs.
Using the Pythagorean Theorem to solve for a side length involves
finding a principal square root because side lengths are always
positive.
12. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the
hypotenuse of the right triangle shown?
6 in.
6 in.
13. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the
hypotenuse of the right triangle shown?
Use the Pythagorean Theorem
a2 + b2 = c2
6 in.
6 in.
14. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the
hypotenuse of the right triangle shown?
Use the Pythagorean Theorem
a2 + b2 = c2
Substitute 6 for a and b.
6 in.
62 + 62 = c2
6 in.
15. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the
hypotenuse of the right triangle shown?
Use the Pythagorean Theorem
a2 + b2 = c2
Substitute 6 for a and b.
6 in.
62 + 62 = c2
Simplify
36 + 36 = c2
6 in.
16. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the
hypotenuse of the right triangle shown?
Use the Pythagorean Theorem
a2 + b2 = c2
Substitute 6 for a and b.
6 in.
62 + 62 = c2
Simplify
36 + 36 = c2
6 in.
Simplify
72 = c2
17. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the
hypotenuse of the right triangle shown?
Use the Pythagorean Theorem
a2 + b2 = c2
Substitute 6 for a and b.
62 + 62 = c2
Simplify
6 in.
36 + 36 = c2
Simplify
72 = c2
6 in.
Find the principal square root.
=c
18. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the
hypotenuse of the right triangle shown?
Use the Pythagorean Theorem
a2 + b2 = c2
Substitute 6 for a and b.
62 + 62 = c2
Simplify
36 + 36 = c2
6 in.
Simplify
72 = c2
Find the principal square root.
=c 6 in.
Use a calculator
8.5 ≈ c
Convert to a fraction!!
19. Finding the Hypotenuse
The tiles are squares with 6-in. sides. What is the length of the hypotenuse of the
right triangle shown?
Use the Pythagorean Theorem
a2 + b2 = c2
Substitute 6 for a and b.
62 + 62 = c2
Simplify
36 + 36 = c2
Simplify
6 in.
72 = c2
Find the principal square root.
=c
Use a calculator 6 in.
8.5 ≈ c
Convert to a fraction!!
Write answer sentence.
The hypotenuse is 8 ½ in.
20. Finding the hypotenuse – Dry erase
practice
What is the length of the hypotenuse of a right triangle with leg
lengths 9 cm and 12 cm?
21. Finding the Length of a Leg
What is the side length b in the triangle?
5 cm 13 cm
b
22. Finding the Length of a Leg
What is the side length b in the triangle?
Use the Pythagorean Theorem.
a2 + b2 = c2
5 cm 13 cm
b
23. Finding the Length of a Leg
What is the side length b in the triangle?
Use the Pythagorean Theorem.
a2 + b2 = c2
Substitute 5 for a and 13 for c.
52 + b2 = 132
5 cm 13 cm
b
24. Finding the Length of a Leg
What is the side length b in the triangle?
Use the Pythagorean Theorem.
a2 + b2 = c2
Substitute 5 for a and 13 for c.
52 + b2 = 132
Simplify. 5 cm 13 cm
25 + b2 = 169
b
25. Finding the Length of a Leg
What is the side length b in the triangle?
Use the Pythagorean Theorem.
a2 + b2 = c2
Substitute 5 for a and 13 for c.
52 + b2 = 132
Simplify. 5 cm 13 cm
25 + b2 = 169
Remember how we solve equations. We need to isolate the
variable by using the inverse operation.
b
26. Finding the Length of a Leg
What is the side length b in the triangle?
Use the Pythagorean Theorem.
a2 + b2 = c2
Substitute 5 for a and 13 for c.
52 + b2 = 132
Simplify. 5 cm 13 cm
25 + b2 = 169
Remember how we solve equations. We need to isolate the
variable by using the inverse operation.
Subtract 25 from each side.
b2 = 144
b
27. Finding the Length of a Leg
What is the side length b in the triangle?
Use the Pythagorean Theorem.
a2 + b2 = c2
Substitute 5 for a and 13 for c.
52 + b2 = 132
Simplify. 5 cm 13 cm
25 + b2 = 169
Remember how we solve equations. We need to isolate the
variable by using the inverse operation.
Subtract 25 from each side.
b2 = 144
b
Find the principal square root of each side.
b = 72
28. Find the Length of a Leg – Dry Erase
Practice
What is the side length a in the triangle?
12
a 15
29. Vocabulary to Know
Conditional
An if-then statement such as “If an animal is a horse, then it has four
legs.”
30. Vocabulary to Know
Hypothesis
The “if” part of the conditional.
Conclusion
The “then” part of the conditional.
Converse
Switches the hypothesis and conclusion.
31. Conditional
You can write the Pythagorean Theorem as a conditional:
“If a triangle is a right triangle with legs of lengths a and b and
hypotenuse of length c, then a2 + b2 = c2.
The converse of the Pythagorean Theorem is always true.
32. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
6 in., 24 in., 25 in.
4 m, 8 m, 10 m
10 in., 24 in., 26 in.
8 ft., 15 ft., 16 ft.
33. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
6 in., 24 in., 25 in.
62 + 242 ___ 252
34. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
6 in., 24 in., 25 in.
62 + 242 ___ 252
36 + 576 ___ 625
35. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
6 in., 24 in., 25 in.
62 + 242 ___ 252
36 + 576 ___ 625
612 ≠ 625
NO
36. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
4 m, 8 m, 10 m
42 + 82 = 102
37. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
4 m, 8 m, 10 m
42 + 82 ___ 102
16 + 64 ___ 100
38. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
4 m, 8 m, 10 m
42 + 82 ___ 102
16 + 64 ___ 100
80 ≠ 100
NO
39. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
10 in., 24 in., 26 in.
102 + 242 ___ 262
100 + 576 ___ 676
40. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
10 in., 24 in., 26 in.
102 + 242 ___ 262
100 + 576 ___ 676
676 = 676
YES
41. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
8 ft., 15 ft., 16 ft.
82 + 152 ___ 162
42. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
8 ft., 15 ft., 16 ft.
82 + 152 ___ 162
64 + 225 ___ 256
43. Identifying Right Triangles
Which set of lengths could be the side lengths of a right triangle?
8 ft., 15 ft., 16 ft.
82 + 152 ___ 162
64 + 225 ___ 256
289 ≠ 256
44. Identifying Right Triangles
Dry Erase Practice
Could the lengths 20 mm, 47 mm, and 52 mm be the side lengths of
a right triangle? Explain.
46. BIG Ideas
The lengths of the sides of a right triangle have a special
relationship.
If the lengths of any two sides of a right triangle are known, the
length of the third side can be found.
If the length of each side of a triangle is known, then whether the
triangle is a right triangle can be determined.
This assignment is for Algebra I Foundation by Pearson. It was the recommended assignment for a block schedule. I usually end up altering assignments since I teach Special Education.