This document outlines instructional strategies for teaching multiplication and division of whole numbers, decimals, and fractions using the concrete-representational-abstract (CRA) approach. It provides examples of using physical objects, drawings, and standard algorithms to develop conceptual understanding at each stage. The CRA approach is demonstrated for topics like multiplying large whole numbers, dividing with decimals, and solving word problems involving fractions.
The document provides examples and instructions for adding and subtracting fractions with unlike denominators using two different methods:
1) Find a common denominator by multiplying the denominators or finding the least common denominator. Then add or subtract the numerators and keep the common denominator.
2) Write the prime factorization of each denominator, circle the common factors, and use those factors to find the lowest common denominator. Then multiply fractions to equivalent fractions with the common denominator before adding or subtracting.
3) An example problem walks through subtracting amounts of ribbon from a total length to find the amount left over.
Three students were asked to draw rectangles with an area of 24 square inches. The first student drew a rectangle with dimensions of 2 inches by 12 inches. The second drew a rectangle of 3 inches by 8 inches. The third drew a rectangle of 1 inch by 24 inches. Since all three rectangles have an area of 24 square inches, this shows that multiplying different factors can result in the same product.
This document provides instruction on multiplying fractions. It includes examples of multiplying fractions using models, the cancellation method, and multiplying a fraction by a whole number. It also covers converting between mixed numbers and improper fractions. Students are asked to complete practice exercises assessing their understanding of multiplying fractions.
GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLEJohdener14
1) Greatest common factors (GCF) and least common multiples (LCM) are important concepts in mathematics. The GCF is the largest integer that divides two or more numbers, while the LCM is the smallest integer that is a multiple of two or more numbers.
2) There are several methods to find the GCF and LCM, including listing factors, using a factor tree, and continuous division. These methods involve breaking numbers down into their prime factors to identify common factors or multiples.
3) For the example of finding the GCF and LCM of 36 and 48, the GCF is 12 and the LCM is 144 using either the factor tree or continuous division methods.
This document introduces some basic concepts in number theory, including primes, least common multiples, greatest common divisors, and modular arithmetic. It then discusses different number systems such as decimal, binary, octal, and hexadecimal. Methods are provided for converting between these different bases, including dividing or grouping bits and multiplying by the place value. Prime numbers are defined as integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every integer can be written as a unique product of prime factors.
This document provides instructions for multiplying two-digit numbers. It explains that the place values must be lined up and then each digit of the top number is multiplied by each digit of the bottom number. When multiplying, you carry numbers to the next place value if the result is greater than 9. Several examples of multiplying two-digit numbers are shown step-by-step to demonstrate the process.
This document provides instructions and examples for multiplying two-digit numbers. It explains that the process involves two steps: 1) multiplying the ones digits and regrouping the products of ten or more, and 2) multiplying the tens digits and adding any regrouped digits from step one. Four examples of multiplying two-digit numbers are shown step-by-step to demonstrate this process.
This document outlines instructional strategies for teaching multiplication and division of whole numbers, decimals, and fractions using the concrete-representational-abstract (CRA) approach. It provides examples of using physical objects, drawings, and standard algorithms to develop conceptual understanding at each stage. The CRA approach is demonstrated for topics like multiplying large whole numbers, dividing with decimals, and solving word problems involving fractions.
The document provides examples and instructions for adding and subtracting fractions with unlike denominators using two different methods:
1) Find a common denominator by multiplying the denominators or finding the least common denominator. Then add or subtract the numerators and keep the common denominator.
2) Write the prime factorization of each denominator, circle the common factors, and use those factors to find the lowest common denominator. Then multiply fractions to equivalent fractions with the common denominator before adding or subtracting.
3) An example problem walks through subtracting amounts of ribbon from a total length to find the amount left over.
Three students were asked to draw rectangles with an area of 24 square inches. The first student drew a rectangle with dimensions of 2 inches by 12 inches. The second drew a rectangle of 3 inches by 8 inches. The third drew a rectangle of 1 inch by 24 inches. Since all three rectangles have an area of 24 square inches, this shows that multiplying different factors can result in the same product.
This document provides instruction on multiplying fractions. It includes examples of multiplying fractions using models, the cancellation method, and multiplying a fraction by a whole number. It also covers converting between mixed numbers and improper fractions. Students are asked to complete practice exercises assessing their understanding of multiplying fractions.
GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLEJohdener14
1) Greatest common factors (GCF) and least common multiples (LCM) are important concepts in mathematics. The GCF is the largest integer that divides two or more numbers, while the LCM is the smallest integer that is a multiple of two or more numbers.
2) There are several methods to find the GCF and LCM, including listing factors, using a factor tree, and continuous division. These methods involve breaking numbers down into their prime factors to identify common factors or multiples.
3) For the example of finding the GCF and LCM of 36 and 48, the GCF is 12 and the LCM is 144 using either the factor tree or continuous division methods.
This document introduces some basic concepts in number theory, including primes, least common multiples, greatest common divisors, and modular arithmetic. It then discusses different number systems such as decimal, binary, octal, and hexadecimal. Methods are provided for converting between these different bases, including dividing or grouping bits and multiplying by the place value. Prime numbers are defined as integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every integer can be written as a unique product of prime factors.
This document provides instructions for multiplying two-digit numbers. It explains that the place values must be lined up and then each digit of the top number is multiplied by each digit of the bottom number. When multiplying, you carry numbers to the next place value if the result is greater than 9. Several examples of multiplying two-digit numbers are shown step-by-step to demonstrate the process.
This document provides instructions and examples for multiplying two-digit numbers. It explains that the process involves two steps: 1) multiplying the ones digits and regrouping the products of ten or more, and 2) multiplying the tens digits and adding any regrouped digits from step one. Four examples of multiplying two-digit numbers are shown step-by-step to demonstrate this process.
Lesson plan multiple and factors.ppt v 3Kavita Grover
This lesson plan outlines 10 lessons to teach students about multiples, factors, prime and composite numbers, divisibility rules, factorization, exponents, least common multiples (LCM), and highest common factors (HCF). Each lesson includes the topic, time, location, content overview, and learning objectives. Methods for finding LCM, HCF, prime factorization, and factorization are discussed. Practice problems are provided for students.
The document provides instructions and examples for multiplying 2-digit numbers. It explains that the process involves two steps: 1) multiplying the ones and regrouping the product, and 2) multiplying the tens, adding any regrouped amounts, and regrouping further if needed. Several examples are shown step-by-step to illustrate multiplying numbers such as 35 x 9, 42 x 7, 23 x 5, and 58 x 6.
The document provides information about various calculation techniques in mathematics, including addition, subtraction, multiplication, and division. It discusses place value, rounding numbers, comparing numbers, and using finger multiplication for times tables up to 9. Examples are provided to illustrate different methods like using a lattice method for multi-digit multiplication problems or breaking down multi-digit multiplication into repeated addition.
Addition is the process of combining numbers to find a total sum. The key principles of addition are that the order and grouping of numbers does not change the sum. Subtraction is the inverse of addition, where the minuend is the starting number and the subtrahend is subtracted to find the difference. Multiplication provides a shortcut to repeated addition by multiplying the multiplicand by the multiplier to get the product. The principles of multiplication are that the order or grouping of factors does not change the product.
Addition is the process of combining numbers to find a total sum. The key principles of addition are that the order and grouping of numbers does not change the sum. Subtraction is the inverse of addition, where the minuend is the starting number and the subtrahend is subtracted to find the difference. Multiplication provides a shortcut to repeated addition by multiplying the multiplicand by the multiplier to find the product. The principles of multiplication state that the order or grouping of factors does not change the product, and multiplying a number by 1 or 0 does not change the number or results in 0, respectively.
This document appears to be an assignment submission for a financial engineering course. It includes a plagiarism declaration signed by the student, Andrew Hair. The assignment contains 11 questions addressing interest rate derivatives and modeling using the Vasicek model. Code is provided in MATLAB to generate simulations and analyze interest rate data based on the questions.
This document discusses solving trigonometric equations graphically. It provides examples of using the intersection method and x-intercept method to find approximate solutions to trig equations by graphing them. Key steps outlined are writing the equation in the form f(x)=0, determining the period, graphing over the interval, and finding solutions graphically. It notes that trig equation solutions can also be expressed in degrees. Examples are provided for solving equations like sin(x)=a and tan(x)=b graphically in both radians and degrees.
The document discusses the order of operations in mathematics. It explains that the order of operations (PEMDAS) - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction - provides rules for which operations to perform first in a mathematical expression without changing the result. It provides examples of evaluating expressions using the proper order of operations and also provides links to online games for practicing order of operations skills.
The document discusses divisibility rules and algorithms for integers. It provides divisibility rules for integers 2 through 20, describing how to determine if a number is divisible by each factor. For example, a number is divisible by 2 if the last digit is even. It also describes algorithms for determining the greatest common divisor of two numbers, such as the Euclidean algorithm which uses successive division. The Goldschmidt division algorithm iteratively multiplies factors to converge the divisor to 1. Diophantine equations allow only integer solutions.
This document discusses using the Scratch coding platform to teach mathematical concepts aligned with the Next Generation Science Standards. It provides an example of using Scratch to calculate the sum of numbers from 1 to 100 by creating variables, loops, and conditional statements. The document outlines the steps taken, discusses issues with initial solutions, and describes refining the code to arrive at the correct solution. It also provides additional examples and resources for using Scratch in math education.
1) The document discusses factors, prime numbers, and composite numbers and how to use them to reduce fractions and find equivalent fractions.
2) It explains how to find all factors of a number using the rainbow method and defines prime and composite numbers.
3) It also covers prime factorization, using factor trees to break numbers down into their prime factors, and how to reduce fractions to lowest terms.
This document discusses place value, rounding numbers, and calculations with whole numbers. It provides examples of rounding 437 to the nearest ten and 432 to the nearest ten. It also discusses order of operations for calculations. The document then provides 10 questions about whole numbers, including calculating values, rounding, and evaluating expressions. It also discusses number patterns and sequences, including odd/even numbers, prime numbers, least common multiples, and highest common factors. It provides 10 additional questions involving number patterns and sequences.
Ways to construct Triangles in an mxn Array of Dotsguest1f2d6d
1. The document describes research to find the relationship between the number of triangles that can be constructed in an m x n array of dots.
2. The researchers used a counting stage and formularizing stage. In the counting stage, they drew arrays and counted the triangles. In the formularizing stage, they broke the arrays into rows to derive formulas relating m, n, and the number of triangles.
3. Formulas were derived for 3xn, 4xn, and 5xn arrays. The formulas found relationships between the size of the array and the number of triangles possible.
The document appears to be notes from a calculus workshop or class covering several topics:
- Conic sections and putting equations into standard form.
- Quadric surfaces and their applications in physics for cooling towers.
- Limits, derivatives, integrals, and their definitions and applications in physics for concepts like velocity and acceleration.
- Substitution techniques for integrals and trigonometric substitutions.
- Riemann sums and using them to define the definite integral.
- References are provided for further reading on topics like quadric surfaces, conic sections, derivatives and integrals, and their uses in physics.
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Lesson plan multiple and factors.ppt v 3Kavita Grover
This lesson plan outlines 10 lessons to teach students about multiples, factors, prime and composite numbers, divisibility rules, factorization, exponents, least common multiples (LCM), and highest common factors (HCF). Each lesson includes the topic, time, location, content overview, and learning objectives. Methods for finding LCM, HCF, prime factorization, and factorization are discussed. Practice problems are provided for students.
The document provides instructions and examples for multiplying 2-digit numbers. It explains that the process involves two steps: 1) multiplying the ones and regrouping the product, and 2) multiplying the tens, adding any regrouped amounts, and regrouping further if needed. Several examples are shown step-by-step to illustrate multiplying numbers such as 35 x 9, 42 x 7, 23 x 5, and 58 x 6.
The document provides information about various calculation techniques in mathematics, including addition, subtraction, multiplication, and division. It discusses place value, rounding numbers, comparing numbers, and using finger multiplication for times tables up to 9. Examples are provided to illustrate different methods like using a lattice method for multi-digit multiplication problems or breaking down multi-digit multiplication into repeated addition.
Addition is the process of combining numbers to find a total sum. The key principles of addition are that the order and grouping of numbers does not change the sum. Subtraction is the inverse of addition, where the minuend is the starting number and the subtrahend is subtracted to find the difference. Multiplication provides a shortcut to repeated addition by multiplying the multiplicand by the multiplier to get the product. The principles of multiplication are that the order or grouping of factors does not change the product.
Addition is the process of combining numbers to find a total sum. The key principles of addition are that the order and grouping of numbers does not change the sum. Subtraction is the inverse of addition, where the minuend is the starting number and the subtrahend is subtracted to find the difference. Multiplication provides a shortcut to repeated addition by multiplying the multiplicand by the multiplier to find the product. The principles of multiplication state that the order or grouping of factors does not change the product, and multiplying a number by 1 or 0 does not change the number or results in 0, respectively.
This document appears to be an assignment submission for a financial engineering course. It includes a plagiarism declaration signed by the student, Andrew Hair. The assignment contains 11 questions addressing interest rate derivatives and modeling using the Vasicek model. Code is provided in MATLAB to generate simulations and analyze interest rate data based on the questions.
This document discusses solving trigonometric equations graphically. It provides examples of using the intersection method and x-intercept method to find approximate solutions to trig equations by graphing them. Key steps outlined are writing the equation in the form f(x)=0, determining the period, graphing over the interval, and finding solutions graphically. It notes that trig equation solutions can also be expressed in degrees. Examples are provided for solving equations like sin(x)=a and tan(x)=b graphically in both radians and degrees.
The document discusses the order of operations in mathematics. It explains that the order of operations (PEMDAS) - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction - provides rules for which operations to perform first in a mathematical expression without changing the result. It provides examples of evaluating expressions using the proper order of operations and also provides links to online games for practicing order of operations skills.
The document discusses divisibility rules and algorithms for integers. It provides divisibility rules for integers 2 through 20, describing how to determine if a number is divisible by each factor. For example, a number is divisible by 2 if the last digit is even. It also describes algorithms for determining the greatest common divisor of two numbers, such as the Euclidean algorithm which uses successive division. The Goldschmidt division algorithm iteratively multiplies factors to converge the divisor to 1. Diophantine equations allow only integer solutions.
This document discusses using the Scratch coding platform to teach mathematical concepts aligned with the Next Generation Science Standards. It provides an example of using Scratch to calculate the sum of numbers from 1 to 100 by creating variables, loops, and conditional statements. The document outlines the steps taken, discusses issues with initial solutions, and describes refining the code to arrive at the correct solution. It also provides additional examples and resources for using Scratch in math education.
1) The document discusses factors, prime numbers, and composite numbers and how to use them to reduce fractions and find equivalent fractions.
2) It explains how to find all factors of a number using the rainbow method and defines prime and composite numbers.
3) It also covers prime factorization, using factor trees to break numbers down into their prime factors, and how to reduce fractions to lowest terms.
This document discusses place value, rounding numbers, and calculations with whole numbers. It provides examples of rounding 437 to the nearest ten and 432 to the nearest ten. It also discusses order of operations for calculations. The document then provides 10 questions about whole numbers, including calculating values, rounding, and evaluating expressions. It also discusses number patterns and sequences, including odd/even numbers, prime numbers, least common multiples, and highest common factors. It provides 10 additional questions involving number patterns and sequences.
Ways to construct Triangles in an mxn Array of Dotsguest1f2d6d
1. The document describes research to find the relationship between the number of triangles that can be constructed in an m x n array of dots.
2. The researchers used a counting stage and formularizing stage. In the counting stage, they drew arrays and counted the triangles. In the formularizing stage, they broke the arrays into rows to derive formulas relating m, n, and the number of triangles.
3. Formulas were derived for 3xn, 4xn, and 5xn arrays. The formulas found relationships between the size of the array and the number of triangles possible.
The document appears to be notes from a calculus workshop or class covering several topics:
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- Quadric surfaces and their applications in physics for cooling towers.
- Limits, derivatives, integrals, and their definitions and applications in physics for concepts like velocity and acceleration.
- Substitution techniques for integrals and trigonometric substitutions.
- Riemann sums and using them to define the definite integral.
- References are provided for further reading on topics like quadric surfaces, conic sections, derivatives and integrals, and their uses in physics.
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