Alg2 lesson 9-2
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This document provides examples for operations using the least common denominator (LCD), including adding and subtracting fractions with different denominators, simplifying expressions with fractions, and dividing fractions. Examples 2-3a through 2-5a demonstrate finding the LCD and performing the specified operations on fractions and fractional expressions.
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Alg2 lesson 9-2
The document describes performing operations on like terms with LCD. It shows adding a term and subtracting a term with a common denominator, then simplifying the expression.
1150 day 11
Ratios, proportions, and percents can be used to compare quantities and solve problems. A ratio compares two quantities, like the number of squares to circles. Equal ratios form a proportion, which can be used to solve for unknown values. Percents represent a number out of 100. Common percent calculations include finding a percent of a number and converting between fractions, decimals, and percents. The percent proportion states that the percent equals the part divided by the whole quantity.
1150 day 10
This document covers topics related to repeating decimals, irrational numbers, and square roots. It includes examples of writing repeating decimals as fractions, ordering decimals, finding square roots, and using the Pythagorean theorem. Various math problems are presented along with step-by-step solutions to illustrate these concepts.
1150 day 9
1) Decimals represent numbers using place value with decimal points. To write decimals in expanded form, they are broken into place value terms using positive and negative exponents.
2) Fractions can be converted to decimals using division or by writing the fraction as a ratio of two numbers and setting up a proportion to solve for the decimal.
3) Scientific notation is used to write very large or small numbers in a standard form, such as 3.603 x 107. Operations can be done on decimals by lining up the decimal points or moving them with multiplication/division.
1150 day 8
This document provides an overview of rational numbers including:
- Integers and fractions written in the form a/b where a and b are integers and b ≠ 0
- Equivalent fractions represented by the same number
- Ordering and comparing rational numbers
- Converting between improper fractions and mixed numbers
- Basic operations of addition, subtraction, multiplication, and division of rational numbers
- Word problems involving rational numbers
1150 day 7
This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.
1150 day 6
Integers include positive and negative whole numbers. Absolute value is the distance from zero. Addition and subtraction on a number line involve moving left or right. Multiplication follows patterns based on sign. Division is the inverse of multiplication regarding sign. A number is divisible by another if it can be written as a product of an integer multiple. Divisibility tests identify patterns in a number's digits.
1150 day 5
Algebraic thinking involves recognizing patterns, modeling situations with symbols, and analyzing change. It relies on understanding variables to represent unknown quantities. The document traces the evolution of algebraic thinking from simple equations to more complex concepts like functions, composite functions, and properties of equations. It provides examples of how algebraic reasoning and symbols can be used to represent and solve real-world problems.
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Alg2 lesson 9-2
The document describes performing operations on like terms with LCD. It shows adding a term and subtracting a term with a common denominator, then simplifying the expression.
1150 day 11
Ratios, proportions, and percents can be used to compare quantities and solve problems. A ratio compares two quantities, like the number of squares to circles. Equal ratios form a proportion, which can be used to solve for unknown values. Percents represent a number out of 100. Common percent calculations include finding a percent of a number and converting between fractions, decimals, and percents. The percent proportion states that the percent equals the part divided by the whole quantity.
1150 day 10
This document covers topics related to repeating decimals, irrational numbers, and square roots. It includes examples of writing repeating decimals as fractions, ordering decimals, finding square roots, and using the Pythagorean theorem. Various math problems are presented along with step-by-step solutions to illustrate these concepts.
1150 day 9
1) Decimals represent numbers using place value with decimal points. To write decimals in expanded form, they are broken into place value terms using positive and negative exponents.
2) Fractions can be converted to decimals using division or by writing the fraction as a ratio of two numbers and setting up a proportion to solve for the decimal.
3) Scientific notation is used to write very large or small numbers in a standard form, such as 3.603 x 107. Operations can be done on decimals by lining up the decimal points or moving them with multiplication/division.
1150 day 8
This document provides an overview of rational numbers including:
- Integers and fractions written in the form a/b where a and b are integers and b ≠ 0
- Equivalent fractions represented by the same number
- Ordering and comparing rational numbers
- Converting between improper fractions and mixed numbers
- Basic operations of addition, subtraction, multiplication, and division of rational numbers
- Word problems involving rational numbers
1150 day 7
This document discusses prime and composite numbers, greatest common divisors (GCD), and least common multiples (LCM). It provides examples of finding the GCD and LCM of various numbers using different methods like the intersection of sets method, prime factorization method, and Euclidean algorithm. Key definitions include: a prime number has exactly two distinct positive divisors, a composite number has factors other than itself and 1, the GCD is the largest integer that divides numbers, and the LCM is the smallest number that is a multiple of the given numbers.
1150 day 6
Integers include positive and negative whole numbers. Absolute value is the distance from zero. Addition and subtraction on a number line involve moving left or right. Multiplication follows patterns based on sign. Division is the inverse of multiplication regarding sign. A number is divisible by another if it can be written as a product of an integer multiple. Divisibility tests identify patterns in a number's digits.
1150 day 5
Algebraic thinking involves recognizing patterns, modeling situations with symbols, and analyzing change. It relies on understanding variables to represent unknown quantities. The document traces the evolution of algebraic thinking from simple equations to more complex concepts like functions, composite functions, and properties of equations. It provides examples of how algebraic reasoning and symbols can be used to represent and solve real-world problems.
1150 day 4
The document discusses various aspects of whole number operations including:
1) Place value systems and how numbers are represented in bases other than 10 such as base 5 and base 12.
2) Algorithms for addition, subtraction, multiplication, and division using various models and representations.
3) Properties of operations like the commutative, associative, identity, and zero properties of multiplication.
1150 day 3
The document discusses different number systems used throughout history including Hindu-Arabic, Egyptian, Babylonian, Mayan, and Roman systems. It also covers basic concepts of whole numbers such as addition, subtraction, and their properties. Different models are presented to demonstrate whole number operations including set, number line, take-away, comparison, and missing addend models.
1150 day 2
The document discusses sets and set operations including defining sets, elements, cardinal and ordinal numbers, equal and equivalent sets, subsets, Venn diagrams, and set operations like union, intersection, and complement. Examples are provided to illustrate concepts like finite and infinite sets, subsets, Venn diagrams representing multiple sets and operations, and using Venn diagrams to solve problems involving sets and their relationships.
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Alg2 lesson 13-4
The document discusses solving for unknown sides and angles of triangles using trigonometric functions like the law of sines. Several multi-step example problems are worked through that involve determining if a triangle is possible based on given information, finding missing side lengths, and calculating angles. Diagrams are included to illustrate each step of the example problems.
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
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6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-2
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Alg2 lesson 13-1
This document discusses trigonometric functions and their ratios, including sine, cosine, tangent, cosecant, secant, and cotangent. It provides examples of using trigonometric functions to solve values in right triangles, including finding missing side lengths and angle measures. Special right triangles with ratios of 1/2, 1/√3, and 1/√2 are also covered.
Alg2 lesson 11-2 and 11-4
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Alg2 lesson 10-4 and 10-5
A common logarithm has a base of 10. The log key on a calculator will find common logs. Solving log equations involves changing bases using log properties. Natural exponential functions involve the constant e, which is approximately 2.71828. The inverse of an exponential function is the natural log function.
Alg2 lesson 10-3
1) This document contains examples of evaluating logarithmic expressions and solving logarithmic equations.
2) Logarithmic expressions are broken down using properties such as loga(bc) = loga(b) + loga(c) and their values are calculated.
3) Logarithmic equations are set up and solved by isolating the logarithmic term and using inverse logarithm properties.
Alg2 lesson 10-2
This document provides an introduction to converting between exponential and logarithmic forms. It includes examples of writing expressions in exponential form given the logarithmic form, and vice versa. It also gives examples of evaluating logarithmic expressions and solving simple logarithmic equations.
Alg2 lesson 10-1
The document discusses exponential functions of the form y=ab^x. Exponential growth occurs when b>1 and exponential decay occurs when 0<b<1. It provides examples of graphs and equations that represent exponential growth and decay, and asks the reader to identify which is which and simplify and solve example equations.
Alg2 lesson 8-7
The system of equations is solved as follows:
1) x + 2y = 5 and y = 3x - 1 are substituted into each other and simplified
2) This results in 7x = 7, so x = 1
3) Substituting x = 1 into y = 3x - 1 gives y = 2
4) Therefore, the solution is (1, 2).
Alg2 lesson 8-6
The document contains 5 equations that represent different types of curves:
1) A parabola with vertex (1,4) and focus (3,4)
2) A circle with center (0,-3) and radius 6
3) An ellipse with center (1,0), vertices of (1±2/√8,0), and foci of (3,0) and (-1,0)
4) A hyperbola with center (-1,2), vertices of (-1,3), (-1,1), and foci of (-1,2±√20)
5) Key properties are summarized for each curve.
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1150 day 4
The document discusses various aspects of whole number operations including:
1) Place value systems and how numbers are represented in bases other than 10 such as base 5 and base 12.
2) Algorithms for addition, subtraction, multiplication, and division using various models and representations.
3) Properties of operations like the commutative, associative, identity, and zero properties of multiplication.
1150 day 3
The document discusses different number systems used throughout history including Hindu-Arabic, Egyptian, Babylonian, Mayan, and Roman systems. It also covers basic concepts of whole numbers such as addition, subtraction, and their properties. Different models are presented to demonstrate whole number operations including set, number line, take-away, comparison, and missing addend models.
1150 day 2
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Alg2 lesson 13-5
This document discusses solving an ambiguous triangle where it is not a right triangle and the measures of angles and sides are not fully known. It determines that trigonometric functions and the Law of Sines cannot be used. It identifies that the Law of Cosines is the appropriate theorem to use to begin solving since it requires only knowing the measures of three sides.
Alg2 lesson 13-4
The document discusses solving for unknown sides and angles of triangles using trigonometric functions like the law of sines. Several multi-step example problems are worked through that involve determining if a triangle is possible based on given information, finding missing side lengths, and calculating angles. Diagrams are included to illustrate each step of the example problems.
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-3
The document contains instructions for finding trigonometric function values given specific angle measurements or terminal side locations. It includes:
1) Finding the six trig functions of an angle with a terminal side at (8, -15)
2) Finding the six trig functions of an angle with a terminal side at (-3, 4)
3) Finding the reference angle of 330°
4) Finding the reference angle of an angle in Quadrant III
5) Finding the value of sin(135°)
6) Finding the remaining five trig functions of an angle in Quadrant III with a terminal side of (-4, -3)
Alg2 lesson 13-2
The document describes how to draw angles in standard position on a unit circle and convert between degrees and radians. It provides examples of drawing angles such as 210°, -45°, and 540° in standard position and rewriting angles such as 30°, 45°, and an unspecified angle in radians and degrees. It also gives examples of finding coterminal angles for 210° with one positive and one negative measure.
Alg2 lesson 13-1
This document discusses trigonometric functions and their ratios, including sine, cosine, tangent, cosecant, secant, and cotangent. It provides examples of using trigonometric functions to solve values in right triangles, including finding missing side lengths and angle measures. Special right triangles with ratios of 1/2, 1/√3, and 1/√2 are also covered.
Alg2 lesson 11-2 and 11-4
The document provides information about arithmetic and geometric series. It defines arithmetic and geometric series, provides examples of finding sums of arithmetic series using formulas, and defines the key terms (first term, common ratio, number of terms, last term) used in the formula to calculate the sum of a geometric series.
Alg2 lesson 11-3
The document discusses geometric sequences, which are patterns of numbers where each term is found by multiplying the previous term by a constant called the common ratio. It provides examples of geometric sequences with different common ratios and shows how to write equations to find the nth term in a sequence. It also explains how to find specific terms in a geometric sequence and calculates the geometric means between two nonconsecutive terms.
Alg2 lesson 11-1
The document discusses arithmetic sequences, which are patterns of numbers where each term is found by adding a constant value to the previous term. It provides examples of arithmetic sequences with different common differences and shows how to write equations to determine the nth term. It also explains how to find terms within an arithmetic sequence and determine arithmetic means between two non-consecutive terms.
Alg2 lesson 10-6
The document discusses exponential growth and decay models, including that exponential decay models how much of a substance is left over time. It provides an example of calculating the decay rate (k) of Sodium-22 given its half-life of 2.6 years, and uses this rate to determine that a meteorite containing only 10% as much Sodium-22 as when it reached Earth must have arrived about 9 years ago.
Alg2 lesson 10-4 and 10-5
A common logarithm has a base of 10. The log key on a calculator will find common logs. Solving log equations involves changing bases using log properties. Natural exponential functions involve the constant e, which is approximately 2.71828. The inverse of an exponential function is the natural log function.
Alg2 lesson 10-3
1) This document contains examples of evaluating logarithmic expressions and solving logarithmic equations.
2) Logarithmic expressions are broken down using properties such as loga(bc) = loga(b) + loga(c) and their values are calculated.
3) Logarithmic equations are set up and solved by isolating the logarithmic term and using inverse logarithm properties.
Alg2 lesson 10-2
This document provides an introduction to converting between exponential and logarithmic forms. It includes examples of writing expressions in exponential form given the logarithmic form, and vice versa. It also gives examples of evaluating logarithmic expressions and solving simple logarithmic equations.
Alg2 lesson 10-1
The document discusses exponential functions of the form y=ab^x. Exponential growth occurs when b>1 and exponential decay occurs when 0<b<1. It provides examples of graphs and equations that represent exponential growth and decay, and asks the reader to identify which is which and simplify and solve example equations.
Alg2 lesson 8-7
The system of equations is solved as follows:
1) x + 2y = 5 and y = 3x - 1 are substituted into each other and simplified
2) This results in 7x = 7, so x = 1
3) Substituting x = 1 into y = 3x - 1 gives y = 2
4) Therefore, the solution is (1, 2).
Alg2 lesson 8-6
The document contains 5 equations that represent different types of curves:
1) A parabola with vertex (1,4) and focus (3,4)
2) A circle with center (0,-3) and radius 6
3) An ellipse with center (1,0), vertices of (1±2/√8,0), and foci of (3,0) and (-1,0)
4) A hyperbola with center (-1,2), vertices of (-1,3), (-1,1), and foci of (-1,2±√20)
5) Key properties are summarized for each curve.
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Alg2 lesson 9-2
- 2. LCD: 42a2b2 Add Example 2-3a
- 3. Subtract LCD 6(x– 5) Example 2-4a
- 5. a b Simplify b a Example 2-5a