This document provides an overview of the 3rd grade Common Core standards for operations and algebraic thinking, and number and operations in base ten. It lists each standard and indicates which units in the 3rd grade Investigations curriculum cover each standard. The standards address representing and solving problems involving multiplication and division, understanding properties of multiplication and the relationship between multiplication and division, multiplying and dividing within 100, solving two-step word problems using the four operations, identifying patterns in arithmetic, using place value to perform multi-digit arithmetic, and multiplying one-digit numbers by multiples of 10 from 10 to 90. The curriculum units are mapped to the standards to show where each standard is addressed.
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
This document discusses methods for inferring multiple graph structures from high-dimensional genomic data across multiple experimental conditions that have a low sample size. It proposes handling the scarcity of data by pooling information across conditions using multi-task learning. Specifically, it considers inferring a common graph structure by maximizing a joint penalized likelihood objective rather than optimizing each experimental condition independently. It outlines statistical graphical models, neighborhood selection versus penalized likelihood approaches, and algorithms for multi-task learning of graphical structures from multiple related genomic datasets.
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
Absolute and Relative Clustering
4th MultiClust Workshop on Multiple Clusterings, Multi-view Data, and Multi-source Knowledge-driven Clustering (Multiclust 2013)
Aug. 11, 2013 @ Chicago, U.S.A, in conjunction with KDD2013
Article @ Official Site: http://dx.doi.org/10.1145/2501006.2501013
Article @ Personal Site: http://www.kamishima.net/archive/2013-ws-kdd-print.pdf
Handnote: http://www.kamishima.net/archive/2013-ws-kdd-HN.pdf
Workshop Homepage: http://cs.au.dk/research/research-areas/data-intensive-systems/projects/multiclust2013/
Abstract:
Research into (semi-)supervised clustering has been increasing. Supervised clustering aims to group similar data that are partially guided by the user's supervision. In this supervised clustering, there are many choices for formalization. For example, as a type of supervision, one can adopt labels of data points, must/cannot links, and so on. Given a real clustering task, such as grouping documents or image segmentation, users must confront the question ``How should we mathematically formalize our task?''To help answer this question, we propose the classification of real clusterings into absolute and relative clusterings, which are defined based on the relationship between the resultant partition and the data set to be clustered. This categorization can be exploited to choose a type of task formalization.
In this paper, the notion a -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The
homomorphic images (pre images) ofa -anti fuzzy new-ideal under homomorphism of a PU-algebras has
been obtained. Some related result have been derived.
International Journal of Computational Engineering Research(IJCER)ijceronline
The document presents some fixed point theorems for expansion mappings in complete metric spaces. It begins with definitions of terms like metric spaces, complete metric spaces, Cauchy sequences, and expansion mappings. It then summarizes several existing fixed point theorems for expansion mappings established by other mathematicians. The main result proved in this document is Theorem 3.1, which establishes a new fixed point theorem for expansion mappings under certain conditions on the metric space and mapping. It shows that if the mapping satisfies the given inequality, then it has a fixed point. The proof of this theorem constructs a sequence to show that it converges to a fixed point.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
This document discusses methods for inferring multiple graph structures from high-dimensional genomic data across multiple experimental conditions that have a low sample size. It proposes handling the scarcity of data by pooling information across conditions using multi-task learning. Specifically, it considers inferring a common graph structure by maximizing a joint penalized likelihood objective rather than optimizing each experimental condition independently. It outlines statistical graphical models, neighborhood selection versus penalized likelihood approaches, and algorithms for multi-task learning of graphical structures from multiple related genomic datasets.
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
Absolute and Relative Clustering
4th MultiClust Workshop on Multiple Clusterings, Multi-view Data, and Multi-source Knowledge-driven Clustering (Multiclust 2013)
Aug. 11, 2013 @ Chicago, U.S.A, in conjunction with KDD2013
Article @ Official Site: http://dx.doi.org/10.1145/2501006.2501013
Article @ Personal Site: http://www.kamishima.net/archive/2013-ws-kdd-print.pdf
Handnote: http://www.kamishima.net/archive/2013-ws-kdd-HN.pdf
Workshop Homepage: http://cs.au.dk/research/research-areas/data-intensive-systems/projects/multiclust2013/
Abstract:
Research into (semi-)supervised clustering has been increasing. Supervised clustering aims to group similar data that are partially guided by the user's supervision. In this supervised clustering, there are many choices for formalization. For example, as a type of supervision, one can adopt labels of data points, must/cannot links, and so on. Given a real clustering task, such as grouping documents or image segmentation, users must confront the question ``How should we mathematically formalize our task?''To help answer this question, we propose the classification of real clusterings into absolute and relative clusterings, which are defined based on the relationship between the resultant partition and the data set to be clustered. This categorization can be exploited to choose a type of task formalization.
In this paper, the notion a -anti fuzzy new-ideal of a PU-algebra are defined and discussed. The
homomorphic images (pre images) ofa -anti fuzzy new-ideal under homomorphism of a PU-algebras has
been obtained. Some related result have been derived.
International Journal of Computational Engineering Research(IJCER)ijceronline
The document presents some fixed point theorems for expansion mappings in complete metric spaces. It begins with definitions of terms like metric spaces, complete metric spaces, Cauchy sequences, and expansion mappings. It then summarizes several existing fixed point theorems for expansion mappings established by other mathematicians. The main result proved in this document is Theorem 3.1, which establishes a new fixed point theorem for expansion mappings under certain conditions on the metric space and mapping. It shows that if the mapping satisfies the given inequality, then it has a fixed point. The proof of this theorem constructs a sequence to show that it converges to a fixed point.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This document provides an overview of functions and function notation that will be used in Calculus. It defines a function as an equation where each input yields a single output. Examples demonstrate determining if equations are functions and evaluating functions using function notation. The key concepts of domain and range of a function are explained. The document concludes by finding the domains of various functions involving fractions, radicals, and inequalities.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This document discusses operations on rational expressions, including:
1) Reducing rational expressions to lowest terms by dividing the numerator and denominator by their greatest common factor.
2) Multiplying rational expressions follows the same rules as multiplying fractions - the numerators are multiplied and the denominators are multiplied.
3) Dividing rational expressions follows the same rules as dividing fractions - the expression is written as the first rational expression over the second.
This document provides an overview of mathematical functions and relations through a series of lessons:
1. It defines key concepts like domains, ranges, and intervals used to describe functions and relations. Functions are defined as relations where no two ordered pairs have the same first element.
2. One-to-one functions are introduced, which satisfy both vertical and horizontal line tests. Only one-to-one functions can have inverse functions.
3. The process for finding the inverse of a function is described. The inverse is formed by swapping the inputs and outputs of the original function and solving for the new output. The domain of the original becomes the range of the inverse, and vice versa.
4
Integrals with inverse trigonometric functionsindu thakur
The document discusses techniques for integrating trigonometric functions. It begins by reviewing definitions of trig functions like sine, cosine, tangent, and cotangent. It then provides examples of trig integrals using trig identities and u-substitution. Examples include integrals of sine, cosine, tangent, and secant functions. The document concludes by stating that practicing these types of integrals will help students perform well on exams involving calculus.
Presentation of Birnbaum's Likelihood Principle foundational paper at the Reading Statistical Classics seminar, Jan. 20, 2013, Université Paris-Dauphine
The document defines key concepts related to functions and relations, including:
- Sets, set notation, and operations like intersection and union
- Different types of number sets like natural, integer, rational, and real numbers
- Ordered pairs and how they are used to define relations and functions
- The domain and range of relations and functions
- What defines a function versus a relation
- Examples of functions, their graphs, and evaluating functions for given inputs
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
This document discusses techniques for setting linear algebra problems in a way that ensures relatively easy arithmetic. Some key techniques discussed include:
1. Using Pythagorean triples and sums of squares to generate vectors with integer norms in R2 and R3.
2. Using the PLU decomposition theorem to generate matrices with a given determinant, such as ±1, to avoid fractions.
3. Extending a basis for the kernel of a matrix to generate matrices with a given kernel.
4. Ensuring the coefficients for a Leontieff input-output model are nonnegative to generate a productive consumption matrix. Examples and Maple routines are provided.
FABIA: Large Data Biclustering in Drug DesignMartin Heusel
Biclustering groups features and samples simultaneously. It is an emerging tool for analyzing large data sets like transcriptomics or chemoinformatics data.
In this work we apply FABIA biclustering to the ChEMBL database where compounds are described by the substructures they possess (the fingerprints). Chemical substructures are assumed to cause or inhibit biological activity and serve, thereby, as building blocks in drug design. ChEMBL biclusters group compounds together which have the same substructure. If a bicluster can be related to a biological activity then the biological effect of a substructure is identified. For example, FABIA found a large ChEMBL bicluster where the compounds have a common substructure which could be related to a bioactivity.
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
This document introduces the concept of order of an element modulo n and uses it to prove theorems about when an integer n satisfies n^2 + 1 or more generally satisfies a cyclotomic polynomial modulo a prime p. It begins by stating and proving the n^2 + 1 lemma, which says a prime p satisfies p | n^2 + 1 if and only if p ≡ 1 (mod 4). It introduces the concepts of order, primitive roots, and cyclotomic polynomials to generalize this result. It concludes by stating and proving a theorem about when a cyclotomic polynomial of an integer a is divisible by a prime p.
This document contains a mathematics test for 7th grade students on the topic of sets. The test has 10 questions and provides 3 different types of questions for each number - Type A questions are worth 80 points, Type B questions are worth 90 points, and Type C questions are worth 100 points. Students must choose one type of question for each number and show their work. The questions cover topics like examples of sets in daily life, set notation, Venn diagrams, subsets, and relationships between sets. Students are given 60 minutes to complete the test and must sign the answer sheet along with their teacher and parent.
This document discusses Joseph-Louis Lagrange and interpolation. It provides:
1) A brief biography of Joseph-Louis Lagrange, an Italian mathematician who made significant contributions to calculus and probability.
2) A definition of interpolation as producing a function that matches given data points exactly and can be used to approximate values between points.
3) An explanation of Lagrange's interpolation formula for finding a polynomial that fits a set of data points, including an example of applying the formula.
Universal Approximation Theorem
Here, we prove that the perceptron multi-layer can approximate all continuous functions in the hypercube [0,1]. For this, we used the Cybenko proof... I tried to include the basic in topology and mathematical analysis to make the slides more understandable. However, they still need some work to be done. In addition, I am a little bit rusty in my mathematical analysis, so I am still not so convinced with my linear functional I defined for the proof...!!! Back to the Rudin and Apostol!!! So expect changes in the future.
This curriculum map outlines the essential concepts, skills, activities, and assessments for 3rd grade mathematics over the school year. From September to October, students will learn addition and subtraction of 3-digit numbers through strategies like rounding, estimating, and regrouping. From October to December, the focus is on multiplication and division, including interpreting situations, properties, and solving word problems. Students will also learn to measure area from December to January by finding the area of rectangles and composing figures. Fractions will be covered from January to March, where students will describe, compare, and represent fractions. Finally, measurement and data will be addressed in two parts, with graphing and data displayed covered from March to April, and geometry and
These are the unpacking documents to better help you understand the expectations for Third gradestudents under the Common Core State Standards for Math. The examples should be very helpful.
The document provides an overview of a third grade mathematics unit on fractions and decimals that is 16 sessions long. It includes the big ideas, essential questions, unit vocabulary, and Arizona math standards covered. It also provides explanations and examples for key concepts like representing fractions as parts of a whole, using models to demonstrate equivalent fractions, comparing fractions, and representing fractions on a number line.
The document provides an overview of Unit 6 of the third grade mathematics curriculum for the Isaac School District. The unit focuses on patterns, functions, and change through stories, tables, and graphs over 16 sessions. It covers key ideas such as recognizing and extending patterns, variables representing numbers, and functions showing relationships. The unit vocabulary and Arizona math standards are outlined. Explanations and examples are provided for core concepts like properties of multiplication and division, and fluently multiplying and dividing within 100.
This document provides an overview of functions and function notation that will be used in Calculus. It defines a function as an equation where each input yields a single output. Examples demonstrate determining if equations are functions and evaluating functions using function notation. The key concepts of domain and range of a function are explained. The document concludes by finding the domains of various functions involving fractions, radicals, and inequalities.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This document discusses operations on rational expressions, including:
1) Reducing rational expressions to lowest terms by dividing the numerator and denominator by their greatest common factor.
2) Multiplying rational expressions follows the same rules as multiplying fractions - the numerators are multiplied and the denominators are multiplied.
3) Dividing rational expressions follows the same rules as dividing fractions - the expression is written as the first rational expression over the second.
This document provides an overview of mathematical functions and relations through a series of lessons:
1. It defines key concepts like domains, ranges, and intervals used to describe functions and relations. Functions are defined as relations where no two ordered pairs have the same first element.
2. One-to-one functions are introduced, which satisfy both vertical and horizontal line tests. Only one-to-one functions can have inverse functions.
3. The process for finding the inverse of a function is described. The inverse is formed by swapping the inputs and outputs of the original function and solving for the new output. The domain of the original becomes the range of the inverse, and vice versa.
4
Integrals with inverse trigonometric functionsindu thakur
The document discusses techniques for integrating trigonometric functions. It begins by reviewing definitions of trig functions like sine, cosine, tangent, and cotangent. It then provides examples of trig integrals using trig identities and u-substitution. Examples include integrals of sine, cosine, tangent, and secant functions. The document concludes by stating that practicing these types of integrals will help students perform well on exams involving calculus.
Presentation of Birnbaum's Likelihood Principle foundational paper at the Reading Statistical Classics seminar, Jan. 20, 2013, Université Paris-Dauphine
The document defines key concepts related to functions and relations, including:
- Sets, set notation, and operations like intersection and union
- Different types of number sets like natural, integer, rational, and real numbers
- Ordered pairs and how they are used to define relations and functions
- The domain and range of relations and functions
- What defines a function versus a relation
- Examples of functions, their graphs, and evaluating functions for given inputs
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
This document discusses techniques for setting linear algebra problems in a way that ensures relatively easy arithmetic. Some key techniques discussed include:
1. Using Pythagorean triples and sums of squares to generate vectors with integer norms in R2 and R3.
2. Using the PLU decomposition theorem to generate matrices with a given determinant, such as ±1, to avoid fractions.
3. Extending a basis for the kernel of a matrix to generate matrices with a given kernel.
4. Ensuring the coefficients for a Leontieff input-output model are nonnegative to generate a productive consumption matrix. Examples and Maple routines are provided.
FABIA: Large Data Biclustering in Drug DesignMartin Heusel
Biclustering groups features and samples simultaneously. It is an emerging tool for analyzing large data sets like transcriptomics or chemoinformatics data.
In this work we apply FABIA biclustering to the ChEMBL database where compounds are described by the substructures they possess (the fingerprints). Chemical substructures are assumed to cause or inhibit biological activity and serve, thereby, as building blocks in drug design. ChEMBL biclusters group compounds together which have the same substructure. If a bicluster can be related to a biological activity then the biological effect of a substructure is identified. For example, FABIA found a large ChEMBL bicluster where the compounds have a common substructure which could be related to a bioactivity.
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
This document introduces the concept of order of an element modulo n and uses it to prove theorems about when an integer n satisfies n^2 + 1 or more generally satisfies a cyclotomic polynomial modulo a prime p. It begins by stating and proving the n^2 + 1 lemma, which says a prime p satisfies p | n^2 + 1 if and only if p ≡ 1 (mod 4). It introduces the concepts of order, primitive roots, and cyclotomic polynomials to generalize this result. It concludes by stating and proving a theorem about when a cyclotomic polynomial of an integer a is divisible by a prime p.
This document contains a mathematics test for 7th grade students on the topic of sets. The test has 10 questions and provides 3 different types of questions for each number - Type A questions are worth 80 points, Type B questions are worth 90 points, and Type C questions are worth 100 points. Students must choose one type of question for each number and show their work. The questions cover topics like examples of sets in daily life, set notation, Venn diagrams, subsets, and relationships between sets. Students are given 60 minutes to complete the test and must sign the answer sheet along with their teacher and parent.
This document discusses Joseph-Louis Lagrange and interpolation. It provides:
1) A brief biography of Joseph-Louis Lagrange, an Italian mathematician who made significant contributions to calculus and probability.
2) A definition of interpolation as producing a function that matches given data points exactly and can be used to approximate values between points.
3) An explanation of Lagrange's interpolation formula for finding a polynomial that fits a set of data points, including an example of applying the formula.
Universal Approximation Theorem
Here, we prove that the perceptron multi-layer can approximate all continuous functions in the hypercube [0,1]. For this, we used the Cybenko proof... I tried to include the basic in topology and mathematical analysis to make the slides more understandable. However, they still need some work to be done. In addition, I am a little bit rusty in my mathematical analysis, so I am still not so convinced with my linear functional I defined for the proof...!!! Back to the Rudin and Apostol!!! So expect changes in the future.
This curriculum map outlines the essential concepts, skills, activities, and assessments for 3rd grade mathematics over the school year. From September to October, students will learn addition and subtraction of 3-digit numbers through strategies like rounding, estimating, and regrouping. From October to December, the focus is on multiplication and division, including interpreting situations, properties, and solving word problems. Students will also learn to measure area from December to January by finding the area of rectangles and composing figures. Fractions will be covered from January to March, where students will describe, compare, and represent fractions. Finally, measurement and data will be addressed in two parts, with graphing and data displayed covered from March to April, and geometry and
These are the unpacking documents to better help you understand the expectations for Third gradestudents under the Common Core State Standards for Math. The examples should be very helpful.
The document provides an overview of a third grade mathematics unit on fractions and decimals that is 16 sessions long. It includes the big ideas, essential questions, unit vocabulary, and Arizona math standards covered. It also provides explanations and examples for key concepts like representing fractions as parts of a whole, using models to demonstrate equivalent fractions, comparing fractions, and representing fractions on a number line.
The document provides an overview of Unit 6 of the third grade mathematics curriculum for the Isaac School District. The unit focuses on patterns, functions, and change through stories, tables, and graphs over 16 sessions. It covers key ideas such as recognizing and extending patterns, variables representing numbers, and functions showing relationships. The unit vocabulary and Arizona math standards are outlined. Explanations and examples are provided for core concepts like properties of multiplication and division, and fluently multiplying and dividing within 100.
This document provides an introduction to functions and their representations. It defines what a function is and gives examples of functions expressed through formulas, numerically, graphically and verbally. It also covers properties of functions including monotonicity, symmetry, even and odd functions. Key examples are used throughout to illustrate concepts.
The document explains the Rational Root Theorem, which can be used to find rational zeros of a polynomial equation. The theorem states that if a polynomial has a rational zero of the form p/q, then p must be a factor of the constant term and q must be a factor of the leading coefficient. Several examples demonstrate using the Rational Root Theorem to list all possible rational zeros and test them to find the actual zeros. The document also discusses the multiplicity of zeros, noting whether a zero is crossed or touched based on whether its multiplicity is odd or even.
This document provides a reference guide for the 4th grade Common Core standards addressed by the Investigations curriculum. It lists the domains, clusters, and individual standards covered in each of the 9 units. The domains include Operations and Algebraic Thinking, Number and Operations in Base Ten, Number and Operations - Fractions, Measurement and Data, and Geometry. The guide also includes descriptions of the Mathematical Practices for 4th grade. In 3 sentences or less: This reference guide maps the 4th grade Common Core standards to the Investigations curriculum units, showing which standards are addressed in each unit for the domains of number, operations, measurement, data, geometry and the mathematical practices.
I am learning advanced proportional and multiplicative reasoning skills such as working with fractions, decimals, percentages, ratios, and proportions. Some of the key areas I am focusing on include finding least common factors and highest common multiples, converting between fractions, decimals and percentages, ordering fractions with different denominators, and solving problems involving combining different proportions. I am also practicing solving multiplication and division problems with fractions and decimals using standard place value and compensating methods.
Kungfu math p4 slide4 (equivalent fraction)pdfkungfumath
This document discusses equivalent fractions and provides examples for listing equivalent fractions and determining numerators of fractions. It explains that equivalent fractions have the same value even though they may look different. It then provides examples of listing the first 8 equivalent fractions of 1/5 by multiplying the numerator and denominator by successive integers from 2 to 8. The document emphasizes understanding the concept of multiplying corresponding parts of a fraction to obtain equivalent fractions.
This document discusses logarithmic functions. It defines logarithms as the inverse of exponents and explains how to convert between logarithmic and exponential forms. Both common (base 10) and natural (base e) logarithms are covered. Examples show how to evaluate logarithms, rewrite expressions between logarithmic and exponential forms, and solve logarithmic equations for the variable. Practice problems are assigned from the textbook to reinforce the concepts taught.
This document provides a curriculum guide for a third grade mathematics unit on equal groups involving multiplication and division. The unit focuses on helping students understand multiplication and division as the grouping and ungrouping of objects to solve real-world problems. It includes essential questions, vocabulary, and standards aligned to specific mathematical practices. Students are expected to interpret and represent multiplication and division problems using various models like arrays, pictures, and equations. They also solve word problems involving equal groups and measurement quantities using multiplication and division within 100.
Rational expressions are fractions with polynomials in the numerator and denominator. The domain of a rational expression excludes values that would make the denominator equal to zero, as division by zero is undefined. To simplify rational expressions, factor the numerator and denominator and reduce where possible. When combining rational expressions through addition, subtraction, multiplication or division, the key steps are to factor denominators, find the common domain, and reduce. Rationalizing techniques include using the conjugate or rationalizing the numerator.
This document discusses polynomials and partial fractions. It covers arithmetic operations on polynomials, identities involving polynomials, and dividing polynomials. Specifically, it teaches how to:
1) Perform addition, subtraction, and multiplication on polynomial expressions.
2) Identify polynomial equations and identities based on their properties.
3) Use the long division process to divide one polynomial by another.
To solve this rational equation, we first find the LCD of the fractions, which is x^2(x-5)(x-6). Then we multiply both sides of the equation by the LCD:
x^2(x-5)(x-6) = x^2(x-5)(x-6)
Canceling the common factors, we obtain:
x^2 = x^2
Since this holds for all real values of x, the solution is all real numbers. Therefore, the solution is x ∈ R.
Section 3.1 graphs of polynomials, from college algebra corrAKHIL969626
This document provides an overview of polynomials and their graphs from a college algebra textbook. It begins with definitions of polynomial functions as functions that can be written as the sum of terms involving integer powers of x, with the highest power determining the degree. Examples are provided to illustrate these concepts. The document then discusses determining if functions are polynomials based on their forms, and defines key properties of polynomials like degree, leading term, and coefficients. Overall, the document introduces the basic concepts and terminology used to define and analyze polynomial functions and their graphs.
The document discusses how Japanese textbooks teach multiplication and division of fractions through problems and representations. It explores using students' prior understanding of multiplication and division of whole numbers to help them understand operations with fractions. It outlines how Japanese textbooks represent fraction operations concretely using area models and number lines to emphasize the meanings of factors and relate new concepts to previous learning.
1) The document discusses graphing and properties of exponential and logarithmic functions, including: graphing exponential functions by substituting values of the variable into the equation, graphing logarithmic functions using the change of base formula, and properties like the product, quotient, and power properties of logarithms.
2) Examples are provided of solving exponential and logarithmic equations using properties like changing bases to the same value, multiplying or dividing arguments using the product and quotient properties, and applying exponents using the power property.
3) Steps shown include using properties to isolate the variable, set arguments or exponents equal to each other, and solve the resulting equation.
This document outlines math lessons on multiplication and division using units of 4. It includes exercises for students to practice multiplying by 4, counting by numbers, and reading tape diagrams. Students work on application problems such as representing a seating chart as an array. The main lesson teaches students to decompose larger multiplication facts into the sum of smaller facts using a pattern of 5 groups and additional groups, such as writing 8 x 4 as (5 x 4) + (3 x 4). Students represent the decompositions using arrays, number bonds and equations.
Similar to 3rd Grade cc Reference Guide for Investigations (20)
This document provides an 8th grade math curriculum map that outlines the units, clusters, standards, and resources to be covered over the school year. The map introduces the organization of the curriculum into units and clusters with essential questions, big ideas, standards, and resources listed for each cluster. It explains that all units and clusters must be taught in the specified sequence before the 2013 AIMS assessment. The curriculum map is intended to guide teachers in delivering the mandated 2010 Arizona Mathematical Standards for 8th grade in a logical progression.
The document provides a curriculum map for 7th grade math standards organized into units and clusters. It includes essential questions, big ideas, standards, mathematical practices, vocabulary, and resources for each cluster. The purpose is to provide a logical progression of content and ensure all teachers follow the same sequence of instruction. It explains how the standards and practices are paired and priorities are designated for certain standards. Assessments and possible projects are included at the end of each cluster.
This document provides a 6th grade math curriculum map that outlines the standards, units, and clusters to be taught over the school year. It includes essential questions, big ideas, vocabulary, and resources for each cluster. The curriculum is organized into units on ratios and proportional relationships, the number system including fractions, decimals, percents, and integers, and decimal operations. Standards are prioritized and paired with mathematical practices. Suggested formative assessments, projects, and resources are included to support instruction.
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This 6th grade math curriculum map outlines the units, clusters, standards, and resources for teaching math over the course of the year. It includes 5 units: Ratios and Proportional Relationships; The Number System; Decimal Operations; Absolute Value and Integers; and Expressions and Equations. Each unit contains multiple clusters that break down the standards and mathematical practices. The map provides essential questions, key vocabulary, web resources, and suggested assessments for teachers to use in their planning and instruction.
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The document provides an overview of the 8th grade science curriculum map for Arizona. It outlines the organization of the curriculum into units, clusters, and standards. The map provides essential questions, big ideas, common misconceptions, standards, and resources for each cluster to guide instruction. The purpose is to ensure all required content is taught in a logical progression. Suggested projects are also included to encourage connections between topics and higher-order thinking. Formative assessments are to be given for each cluster to check student understanding of the standards.
This 7th grade science curriculum map outlines the units and clusters to be taught over the school year. It begins with an introduction that describes the organization of the state standards into units and clusters, with performance objectives sequenced in a logical progression. The map then provides information about approximate times, essential questions, big ideas, common misconceptions, standards, vocabulary, and resources for each cluster. Key details include that the Earth has unique layers with distinct compositions and properties, and that the Earth's surface is constantly changing due to processes like erosion, deposition, and plate tectonics. Teachers are expected to follow the sequence of units and clusters and assess student learning regularly using formative assessments.
This document provides an overview of the 6th grade science curriculum map for an elementary school district. It outlines the units, clusters, and standards that will be covered over the school year. The curriculum is organized by units representing major scientific domains, with clusters representing related concepts within each domain. For each cluster, the document provides essential questions, big ideas, common misconceptions, priority standards, vocabulary, and suggested resources and assessments. The goal is to logically sequence the content standards while integrating skill and process standards to facilitate conceptual understanding and connections across clusters and units.
This document provides an overview of the 5th grade science curriculum map for an elementary school district. It outlines the organization of the curriculum into units and clusters, and provides details about essential questions, big ideas, common misconceptions, standards, vocabulary, and resources for each cluster. The sample cluster summarized focuses on how humans and the environment impact each other, with standards addressing how human behavior impacts the environment through global warming and how the environment can impact humans through events like inclement weather or limited natural resources.
This document provides an overview of the 4th grade science curriculum map for the Isaac Elementary School District. It outlines the organization of the curriculum including units, clusters, essential questions, big ideas, standards, and resources. The curriculum is organized into units on life science, earth science, and physical science. Each unit contains multiple clusters which focus on specific concepts and are taught over approximately 2 weeks. The document provides guidance for teachers on teaching the standards and assessing student learning.
This document outlines a fifth grade mathematics curriculum guide from the Isaac School District. Unit 9 focuses on data analysis and probability. Students will understand probability, how it can be represented, and how mathematical methods can maximize efficiency. They will learn to fluently multiply multi-digit numbers, find quotients of whole numbers with up to four-digit dividends and two-digit divisors, and make and interpret line plots displaying fractional data. Examples are provided to illustrate different strategies and models for solving problems within these domains.
This document outlines a 5th grade mathematics unit on growth patterns from the Isaac School District curriculum guide. The unit focuses on calculating area and perimeter, analyzing patterns that change over time, and generating and comparing numerical patterns based on given rules. Key concepts include rate of change, area, perimeter, writing numerical expressions, and generating ordered pairs from patterns to graph. Students will work on skills like comparing decimals to thousandths, writing expressions to represent word problems, and justifying relationships between patterns.
The document is a mathematics curriculum guide for 5th grade from the Isaac School District. It outlines several units of study around operations and algebraic thinking, number and operations in base ten, and number and operations with fractions. The guide provides standards, essential questions, unit vocabulary, explanations and examples for key concepts like understanding the place value system, performing operations with multi-digit numbers and decimals, writing and interpreting numerical expressions, and using equivalent fractions to add and subtract fractions. Teachers are provided guidance on teaching strategies like using visual models and making connections to other math practices.
The document provides information about a 5th grade mathematics curriculum guide from the Isaac School District. It includes:
1) Details on Unit 6 which covers decimals on grids and number lines, including essential questions, vocabulary, and math standards.
2) Explanations and examples for each math standard, such as how to use parentheses and exponents, place value of decimals, comparing and rounding decimals, and algorithms for multiplication and long division.
3) The document is intended to outline the key concepts and skills students should master related to decimals and fractions for 5th grade mathematics.
This document outlines a 14-session unit on measuring polygons for a 5th grade mathematics curriculum. The unit focuses on classifying and measuring two-dimensional figures. Students will learn about attributes of 2D figures like angles and sides, and how these relate to calculating figures' areas and perimeters. They will classify figures into hierarchies based on their geometric properties and understand how categories are related. Examples include learning all rectangles have four right angles, so squares do as well. The unit aligns with Arizona math standards and incorporates mathematical practices like reasoning quantitatively.
The document provides information about a 5th grade mathematics unit on fractions and percents from the Isaac School District. The unit covers adding, subtracting, and multiplying fractions, as well as solving word problems involving fractions. It lists the essential questions, vocabulary, and Arizona state math standards covered. Examples are provided to demonstrate how to add and subtract fractions with unlike denominators, solve word problems, and multiply fractions. Estimation strategies for fraction calculations are also discussed.
The document is a mathematics curriculum guide for 5th grade from the Isaac School District. It describes Unit 3 which focuses on addition, subtraction, and the number system. The unit has 13 sessions and aims to teach students how algebra can be used to solve real-world problems and that numbers can be represented in multiple ways. Key vocabulary includes terms like million, billion, and algorithm. The unit covers understanding place value in multi-digit numbers up to the hundredths place and performing operations like multiplication and long division with multi-digit whole numbers and decimals. Students are expected to use standard algorithms and strategies like area models to solve problems.
This document outlines the fifth grade mathematics curriculum for Unit 2 on prisms and pyramids. The unit focuses on determining the volume of 3D objects and relating volume to multiplication and addition. Students will learn about rectangular prisms, pyramids, cylinders and cones. They will measure volume using cubic units and apply formulas to find the volume of right rectangular prisms. Students will also explore how volume is additive by finding the total volume of objects made of two combined prisms. The unit aims to develop students' geometric reasoning and quantitative skills through 13 sessions of instruction and activities.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
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Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
1. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
Domain Cluster Standard Unit Unit Unit Unit Unit Unit Unit Unit Unit
1 2 3 4 5 6 7 8 9
Represent and solve 3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as X
the total number of objects in 5 groups of 7 objects each. For example,
problems involving describe a context in which a total number of objects can be expressed
multiplication and division. as 5 × 7.
3.OA.2. Interpret whole-number quotients of whole numbers, e.g.,
interpret 56 ÷ 8 as the number of objects in each share when 56 X
objects are partitioned equally into 8 shares, or as a number of shares
when 56 objects are partitioned into equal shares of 8 objects each.
For example, describe a context in which a number of shares or a
Operations and Algebraic Thinking (OA)
number of groups can be expressed as 56 ÷ 8.
3.OA.3. Use multiplication and division within 100 to solve word
problems in situations involving equal groups, arrays, and X X X
measurement quantities, e.g., by using drawings and equations with a
symbol for the unknown number to represent the problem.
3.OA.4. Determine the unknown whole number in a multiplication or
division equation relating three whole numbers. For example, X
determine the unknown number that makes the equation true in each of
the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?.
Understand properties of 3.OA.5. Apply properties of operations as strategies to multiply and X X X
divide. (Students need not use formal terms for these properties.)
multiplication and the Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.
relationship between (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 ×
5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30.
multiplication and division. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 ×
2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 =
56. (Distributive property.)
3.OA.6. Understand division as an unknown-factor problem. For
example, find 32 ÷ 8 by finding the number that makes 32 when X X
multiplied by 8.
Multiply and divide within 3.OA.7. Fluently multiply and divide within 100, using strategies such X X
as the relationship between multiplication and division (e.g., knowing
100. that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By
the end of Grade 3, know from memory all products of two one-digit
numbers.
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -1-
2. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
Solve problems involving 3.OA.8. Solve two-step word problems using the four operations. X X X
Represent these problems using equations with a letter standing for the
Algebraic Thinking
the four operations, and unknown quantity. Assess the reasonableness of answers using mental
Operations and
identify and explain computation and estimation strategies including rounding. (This
standard is limited to problems posed with whole numbers and having
patterns in arithmetic. whole-number answers; students should know how to perform
(OA)
operations in the conventional order when there are no parentheses to
specify a particular order (Order of Operations).
3.OA.9. Identify arithmetic patterns (including patterns in the addition
table or multiplication table), and explain them using properties of X X X
operations. For example, observe that 4 times a number is always
even, and explain why 4 times a number can be decomposed into two
equal addends.
Use place value 3.NBT.1. Use place value understanding to round whole numbers to X X X X
Number and
the nearest 10 or 100.
in Base Ten
understanding and
Operations
3.NBT.2. Fluently add and subtract within 1000 using strategies and
properties of operations to algorithms based on place value, properties of operations, and/or the X X X X X X
relationship between addition and subtraction.
perform multi-digit
(NBT)
3.NBT.3. Multiply one-digit whole numbers by multiples of 10 in the
arithmetic. range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place
value and properties of operations.
X X
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -2-
3. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
Develop understanding of 3.NF.1. Understand a fraction 1/b as the quantity formed by 1 part X X X
when a whole is partitioned into b equal parts; understand a fraction a/b
fractions as numbers. as the quantity formed by a parts of size 1/b.
3.NF.2. Understand a fraction as a number on the number line;
represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram X X X
by defining the interval from 0 to 1 as the whole and
Number and Operations—Fractions (NF)
partitioning it into b equal parts. Recognize that each
part has size 1/b and that the endpoint of the part
based at 0 locates the number 1/b on the number
line.
b. Represent a fraction a/b on a number line diagram X X
by marking off a lengths 1/b from 0. Recognize that
the resulting interval has size a/b and that its
endpoint locates the number a/b on the number line.
3.NF.3. Explain equivalence of fractions in special cases, and compare
fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if X
they are the same size, or the same point on a
number line.
b. Recognize and generate simple equivalent fractions, X
e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions
are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize
fractions that are equivalent to whole numbers.
Examples: Express 3 in the form 3 = 3/1; recognize X
that 6/1 = 6; locate 4/4 and 1 at the same point of a
number line diagram.
d. Compare two fractions with the same numerator or
the same denominator by reasoning about their size. X
Recognize that comparisons are valid only when the
two fractions refer to the same whole. Record the
results of comparisons with the symbols >, =, or <,
and justify the conclusions, e.g., by using a visual
fraction model.
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -3-
4. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
Solve problems involving 3.MD.1. Tell and write time to the nearest minute and measure time X X
intervals in minutes. Solve word problems involving addition and
measurement and subtraction of time intervals in minutes, e.g., by representing the
estimation of intervals of problem on a number line diagram.
Measurement and Data (MD)
3.MD.2. Measure and estimate liquid volumes and masses of objects
time, liquid volumes, and using standard units of grams (g), kilograms (kg), and liters (l). X
masses of objects. (Excludes compound units such as cm3 and finding the geometric
volume of a container.) Add, subtract, multiply, or divide to solve one-
step word problems involving masses or volumes that are given in the
same units, e.g., by using drawings (such as a beaker with a
measurement scale) to represent the problem. Excludes multiplicative
comparison problems.
Represent and interpret 3.MD.3. Draw a scaled picture graph and a scaled bar graph to X X
represent a data set with several categories. Solve one- and two-step
data. “how many more” and “how many less” problems using information
presented in scaled bar graphs. For example, draw a bar graph in
which each square in the bar graph might represent 5 pets.
3.MD.4. Generate measurement data by measuring lengths using X X
rulers marked with halves and fourths of an inch. Show the data by
making a line plot, where the horizontal scale is marked off in
appropriate units— whole numbers, halves, or quarters.
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -4-
5. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
Geometric measurement: 3.MD.5. Recognize area as an attribute of plane figures and X
understand concepts of area measurement.
understand concepts of a. A square with side length 1 unit, called “a unit square,” is
area and relate area to said to have “one square unit” of area, and can be used to X X
measure area.
multiplication and to b. A plane figure which can be covered without gaps or X X
addition. overlaps by n unit squares is said to have an area of n square units.
3.MD.6. Measure areas by counting unit squares (square cm, square
X X
X
Measurement and Data (MD)
m, square in, square ft, and improvised units).
3.MD.7. Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side
lengths by tiling it, and show that the area is the same as X X
would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole-
number side lengths in the context of solving real world X X
and mathematical problems, and represent whole-number
products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a X
rectangle with whole-number side lengths a and b + c is
the sum of a × b and a × c. Use area models to represent
the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear X X
figures by decomposing them into non-overlapping
rectangles and adding the areas of the non-overlapping
parts, applying this technique to solve real world problems.
Geometric measurement: 3.MD.8. Solve real world and mathematical problems involving X X
perimeters of polygons, including finding the perimeter given the side
recognize perimeter as an lengths, finding an unknown side length, and exhibiting rectangles with
attribute of plane figures the same perimeter and different areas or with the same area and
different perimeters.
and distinguish between
linear and area measures.
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -5-
6. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
Reason with shapes and 3.G.1. Understand that shapes in different categories (e.g., rhombuses, X X
rectangles, and others) may share attributes (e.g., having four sides),
their attributes.
Geometry (G)
and that the shared attributes can define a larger category (e.g.,
quadrilaterals). Recognize rhombuses, rectangles, and squares as
examples of quadrilaterals, and draw examples of quadrilaterals that do
not belong to any of these subcategories.
3.G.2. Partition shapes into parts with equal areas. Express the area of X
each part as a unit fraction of the whole. For example, partition a shape
into 4 parts with equal area, and describe the area of each part as 1/4
of the area of the shape.
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -6-
7. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
3.MP.1. Make sense of problems and persevere in solving them.
In third grade, students know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem
and look for ways to solve it. Third graders may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking
themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.
3.MP.2. Reason abstractly and quantitatively.
Third graders should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at
hand, considering both the appropriate units involved and the meaning of quantities.
3.MP.3. Construct viable arguments and critique the reasoning of others.
In third grade, students may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they
participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’
thinking.
3.MP.4. Model with mathematics.
Students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making
a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of
these representations as needed. Third graders should evaluate their results in the context of the situation and reflect on whether the results make sense.
3.MP.5. Use appropriate tools strategically.
Third graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use
graph paper to find all the possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table, and determine whether they have all the
possible rectangles.
3.MP.6. Attend to precision.
As third graders develop their mathematical communication skills, they try to use clear and precise language in their discussions with others and in their own reasoning. They are
careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the area of a rectangle they record their answers in
square units.
3.MP.7. Look for and make use of structure.
In third grade, students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to multiply and divide (commutative and
distributive properties).
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -7-
8. 2010 Arizona Standards
Common Core Reference Guide for
Investigations
3rd Grade
3.MP.8. Look for and express regularity in repeated reasoning.
Students in third grade should notice repetitive actions in computation and look for more shortcut methods. For example, students may use the distributive property as a strategy
for using products they know to solve products that they don’t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then
multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually evaluate their work by asking themselves, “Does this make sense?”
3rd Grade Reference Guide
Created on 2/4/2012 8:51 PM -8-