1. The document provides information about teaching quantity, magnitude, and numeration to students. It includes definitions of key concepts, early indicators of difficulties with these topics, and assessments to diagnose students' skills.
2. The research section discusses concrete, representational, and abstract approaches to teaching concepts like subitizing. It recommends hands-on materials and strategies to develop number sense.
3. The classroom application section gives examples of linking physical quantities to numerical representations using tools like number lines and algebra blocks.
Unit 6 presentation base ten equality form of a number with trainer notes 7.9.08jcsmathfoundations
The document discusses concepts related to base ten, equality, and forms of numbers. It defines these concepts, examines how students develop an understanding through research on cognitive development, and provides classroom applications and strategies for teaching these concepts effectively. Diagnostic questions are presented to assess student understanding, and examples show how to respond to common student errors or misconceptions in working with numbers.
This document provides instructions and materials for administering a mathematics assessment to 8th grade students. It details the preparation, administration, and after testing procedures. The assessment consists of multiple choice, short answer, and extended performance tasks questions about pre-algebra concepts. It is designed to be administered over several class periods. The document provides sample questions and directions for completing each part of the test. Student responses will be scored and results reported according to district policy.
1. The lesson plan aims to teach students how to find the prime factors of numbers.
2. Students will practice finding prime factors using factor trees, continuous division by primes, and listing the prime factors.
3. The lesson concludes with students practicing finding the prime factors of various numbers independently using any method taught.
This document provides a curriculum guide for a 5th grade mathematics unit on multiplication and division. It includes big ideas, essential questions, vocabulary, standards, and examples for content covering writing and interpreting numerical expressions, understanding place value systems, and performing operations with multi-digit whole numbers and decimals. The unit aims to help students understand algebraic problem solving and number representations, as well as master skills with operations and place value.
This document provides a teaching guide for a module on special products and factors. It includes:
1. Learning outcomes and standards that students should understand key concepts of special products and factors and be able to solve real-life problems accurately.
2. An assessment map that lists different assessment types (pre, formative, summative) and what knowledge, skills, and understanding they assess.
3. A planning section that introduces the module and lessons, which cover identifying and finding special products and factors, as well as factoring polynomials. It suggests questions to engage students and emphasizes goals.
This document provides a lesson on finding missing numbers in mathematical equations. It discusses different properties of operations like the commutative, associative, distributive, and identity properties. Examples of equations are provided and students are asked to determine the missing numbers by identifying which property is being used. Practice problems are included for students to solve. The goal is for students to understand how to use properties of operations to determine missing numbers in equations.
This lesson introduces students to the relationship between addition and subtraction using tape diagrams and algebraic expressions. Students represent addition and subtraction expressions using tape diagrams with squares and explore how adding and then subtracting the same number results in the original amount. They realize this is true regardless of the specific numbers used. Students then write number sentences with variables to represent these identities, like w + x - x = w. The lesson aims to build students' understanding that identities will be true for any numbers substituted into the variables.
This document provides materials for a 4th grade mathematics unit. It includes lessons, activities, practice problems and games related to fractions, data analysis, geometry, measurement, number sense, and problem solving. Some key lessons include dividing objects into halves or other fractions, collecting and graphing data, solving word problems, and playing numerical games like a dice game called "Corn Shucks." The document offers guidance for teachers on discussing concepts and assessing student understanding.
Unit 6 presentation base ten equality form of a number with trainer notes 7.9.08jcsmathfoundations
The document discusses concepts related to base ten, equality, and forms of numbers. It defines these concepts, examines how students develop an understanding through research on cognitive development, and provides classroom applications and strategies for teaching these concepts effectively. Diagnostic questions are presented to assess student understanding, and examples show how to respond to common student errors or misconceptions in working with numbers.
This document provides instructions and materials for administering a mathematics assessment to 8th grade students. It details the preparation, administration, and after testing procedures. The assessment consists of multiple choice, short answer, and extended performance tasks questions about pre-algebra concepts. It is designed to be administered over several class periods. The document provides sample questions and directions for completing each part of the test. Student responses will be scored and results reported according to district policy.
1. The lesson plan aims to teach students how to find the prime factors of numbers.
2. Students will practice finding prime factors using factor trees, continuous division by primes, and listing the prime factors.
3. The lesson concludes with students practicing finding the prime factors of various numbers independently using any method taught.
This document provides a curriculum guide for a 5th grade mathematics unit on multiplication and division. It includes big ideas, essential questions, vocabulary, standards, and examples for content covering writing and interpreting numerical expressions, understanding place value systems, and performing operations with multi-digit whole numbers and decimals. The unit aims to help students understand algebraic problem solving and number representations, as well as master skills with operations and place value.
This document provides a teaching guide for a module on special products and factors. It includes:
1. Learning outcomes and standards that students should understand key concepts of special products and factors and be able to solve real-life problems accurately.
2. An assessment map that lists different assessment types (pre, formative, summative) and what knowledge, skills, and understanding they assess.
3. A planning section that introduces the module and lessons, which cover identifying and finding special products and factors, as well as factoring polynomials. It suggests questions to engage students and emphasizes goals.
This document provides a lesson on finding missing numbers in mathematical equations. It discusses different properties of operations like the commutative, associative, distributive, and identity properties. Examples of equations are provided and students are asked to determine the missing numbers by identifying which property is being used. Practice problems are included for students to solve. The goal is for students to understand how to use properties of operations to determine missing numbers in equations.
This lesson introduces students to the relationship between addition and subtraction using tape diagrams and algebraic expressions. Students represent addition and subtraction expressions using tape diagrams with squares and explore how adding and then subtracting the same number results in the original amount. They realize this is true regardless of the specific numbers used. Students then write number sentences with variables to represent these identities, like w + x - x = w. The lesson aims to build students' understanding that identities will be true for any numbers substituted into the variables.
This document provides materials for a 4th grade mathematics unit. It includes lessons, activities, practice problems and games related to fractions, data analysis, geometry, measurement, number sense, and problem solving. Some key lessons include dividing objects into halves or other fractions, collecting and graphing data, solving word problems, and playing numerical games like a dice game called "Corn Shucks." The document offers guidance for teachers on discussing concepts and assessing student understanding.
This document outlines lessons on dividing by one-digit numbers. Lesson 9-1 covers division with and without remainders. Lesson 9-2 discusses dividing multiples of 10, 100, and 1,000. Lesson 9-3 introduces the problem-solving strategy of guess and check. Lesson 9-4 is about estimating quotients. Each lesson provides examples and relates the content to California math standards.
This document introduces special products and factors of polynomials. It discusses how patterns can be used to simplify algebraic expressions and solve geometric problems. Students will learn to identify special products through pattern recognition, find special products of polynomials, and apply these concepts to real-world problems. The goals are to demonstrate understanding of key concepts and solve practice problems accurately using different strategies.
This document provides a curriculum guide for a third grade mathematics unit on solids and boxes. The unit focuses on understanding two-dimensional and three-dimensional figures and their attributes. Students will learn about prisms, pyramids, edges, faces, and vertices. They will represent and solve word problems involving multiplication and division up to 10x10. Students will also measure and estimate liquid volumes and masses using standard units like grams, kilograms, and liters. They will recognize area as an attribute of plane figures and understand concepts of area measurement using unit squares.
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 contains a collection of math word problems and exercises for students. It includes problems involving geometry, algebra, time, money, fractions, probability and other topics. After each problem section, it provides the answers and an explanation of the problem solving strategies and concepts involved. The purpose is to challenge students with complex multi-step problems and help them improve their problem solving skills.
The document provides an overview of the fourth grade mathematics curriculum for Unit 8 on multiplication and division. It includes 3 key ideas: that there are multiple strategies for multiplying and dividing whole numbers, that multiplication and division are related, and that learning these skills has value. The unit covers multiplying up to 4-digit numbers by 1-digit numbers and dividing up to 4-digit dividends by 1-digit divisors. Students will represent and solve multi-step word problems involving all four operations. They will also generate and analyze number patterns that follow given rules.
This document introduces continued fraction expansions by discussing rational approximations of real numbers using the mediant, or Farey sum, of two fractions. It describes an activity where students explore continued fractions by drawing and analyzing paths on a Farey diagram, which represents rational numbers as endpoints. The activity aims to reinforce fraction addition and introduce patterns in continued fractions that relate to operations on the path endpoints.
3rd Grade Math Activity: Metric Mango Tree (measurement; number sense)Mango Math Group
A sample math lesson from Mango Math's 3rd grade math curriculum.
Mango Math provides grade level math games and activities that reinforce core math concepts. Our activities are designed to enhance and compliment existing curriculum and are aligned with NCTM standards. Our innovative and fun math curriculum products are designed to assist teachers, resource room instructors, home school organizations, and parents build positive attitudes towards math while reinforcing key math skills.
for more information visit www.mangomathgroup.com
The document is a curriculum guide for 4th grade mathematics that outlines the key concepts and standards for Unit 1 on factors, multiples, and arrays involving multiplication and division. The unit focuses on helping students understand relationships between multiplication and division and strategies for solving word problems using the four operations. It provides examples of how students can find factor pairs, determine if numbers are multiples, and identify prime and composite numbers between 1-100. The unit aims to build students' abstract reasoning skills and ability to model mathematical concepts.
This challenge asks students to construct numbers from 1 to 50 using exactly four 4s and no other numerals. Students can use non-numeric symbols to combine the 4s. The goals are to increase number sense and develop understanding of order of operations. Hints include using exponents, square roots, decimals and other concepts. Discussing order of operations is encouraged when students get different answers using different calculators or methods.
K to 12 - Grade 8 Math Learners Module Quarter 2Nico Granada
Here are the completed statements based on the conclusions:
1. n(A × B) = n(B × A).
2. A × B ≠ B × A.
The key conclusions are:
1. The cardinalities of the Cartesian products A × B and B × A are equal, since n(A × B) = n(B × A).
2. However, the sets A × B and B × A are not equal, since the ordered pairs will be arranged differently, so A × B ≠ B × A.
This document provides examples and explanations of concepts related to classifying and ordering different types of numbers:
1) It defines rational and irrational numbers, and provides examples of each. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
2) Exercises are included for classifying numbers as rational or irrational, comparing irrational numbers on a number line, and ordering sets of numbers from least to greatest.
3) Answering "Why is it helpful to write numbers in different ways?" the document explains that not all numbers can be written the same way, like irrational numbers that cannot be expressed as integers. Being able to classify numbers facilitates understanding and operations involving different types of numbers.
This curriculum map outlines the mathematics curriculum for 10th grade students. It details the content, standards, learning competencies, assessments, activities and resources that will be covered during the first grading period, with a focus on patterns and algebra. Key topics include sequences, polynomials, arithmetic and geometric sequences, and their related concepts. A variety of assessment types are listed, like problem solving, oral recitations, and group presentations. The goals are for students to understand key concepts and solve problems related to sequences and polynomials.
This is one of three workshops conducted by the specialist team from MCI and SIF. This is part of MCI's community involvement programme. This is the workshop on secondary mathematics.
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.
Presentation math workshop#may 25th newUmber Tariq
It was prepared for the staff of our school , in order to guide that how to make, teaching and leaning for Maths, interesting and fun .
To reduce boredom for kids and to relate the concepts with the nature and universe.
1. The document is about a book titled "Primary 6 Mathematics Ace The Exams with My 24/7 Personal Tutor" which contains 10 mock exam papers and detailed explanations for Primary 6 math questions on a CD-ROM.
2. The book is meant to help pupils prepare for Primary 6 math exams by including exam-style questions that highlight common misconceptions.
3. The accompanying CD-ROM contains video lessons explaining how to solve each question in the mock exams.
The document provides instruction on absolute value including:
1) Absolute value refers to the distance of a number from zero on the number line. It is never negative.
2) Examples are provided to show that the absolute values of -15 and 15 are both 15 units from zero.
3) Students are asked to determine the absolute value of various integers and match pairs of absolute values.
The document provides an agenda for an October 28 math class. It includes a do now, homework check on integers and absolute value using a Frayer model vocabulary template, and homework assignments from a pre-algebra textbook. Expressions are categorized as variable expressions, numerical expressions, or variables. Ways to write multiplication and division are demonstrated, along with examples of evaluating variable expressions by substituting values for variables. Vocabulary on evaluating, integers, opposite numbers, and absolute value are defined using Frayer models.
This document outlines lessons on dividing by one-digit numbers. Lesson 9-1 covers division with and without remainders. Lesson 9-2 discusses dividing multiples of 10, 100, and 1,000. Lesson 9-3 introduces the problem-solving strategy of guess and check. Lesson 9-4 is about estimating quotients. Each lesson provides examples and relates the content to California math standards.
This document introduces special products and factors of polynomials. It discusses how patterns can be used to simplify algebraic expressions and solve geometric problems. Students will learn to identify special products through pattern recognition, find special products of polynomials, and apply these concepts to real-world problems. The goals are to demonstrate understanding of key concepts and solve practice problems accurately using different strategies.
This document provides a curriculum guide for a third grade mathematics unit on solids and boxes. The unit focuses on understanding two-dimensional and three-dimensional figures and their attributes. Students will learn about prisms, pyramids, edges, faces, and vertices. They will represent and solve word problems involving multiplication and division up to 10x10. Students will also measure and estimate liquid volumes and masses using standard units like grams, kilograms, and liters. They will recognize area as an attribute of plane figures and understand concepts of area measurement using unit squares.
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 contains a collection of math word problems and exercises for students. It includes problems involving geometry, algebra, time, money, fractions, probability and other topics. After each problem section, it provides the answers and an explanation of the problem solving strategies and concepts involved. The purpose is to challenge students with complex multi-step problems and help them improve their problem solving skills.
The document provides an overview of the fourth grade mathematics curriculum for Unit 8 on multiplication and division. It includes 3 key ideas: that there are multiple strategies for multiplying and dividing whole numbers, that multiplication and division are related, and that learning these skills has value. The unit covers multiplying up to 4-digit numbers by 1-digit numbers and dividing up to 4-digit dividends by 1-digit divisors. Students will represent and solve multi-step word problems involving all four operations. They will also generate and analyze number patterns that follow given rules.
This document introduces continued fraction expansions by discussing rational approximations of real numbers using the mediant, or Farey sum, of two fractions. It describes an activity where students explore continued fractions by drawing and analyzing paths on a Farey diagram, which represents rational numbers as endpoints. The activity aims to reinforce fraction addition and introduce patterns in continued fractions that relate to operations on the path endpoints.
3rd Grade Math Activity: Metric Mango Tree (measurement; number sense)Mango Math Group
A sample math lesson from Mango Math's 3rd grade math curriculum.
Mango Math provides grade level math games and activities that reinforce core math concepts. Our activities are designed to enhance and compliment existing curriculum and are aligned with NCTM standards. Our innovative and fun math curriculum products are designed to assist teachers, resource room instructors, home school organizations, and parents build positive attitudes towards math while reinforcing key math skills.
for more information visit www.mangomathgroup.com
The document is a curriculum guide for 4th grade mathematics that outlines the key concepts and standards for Unit 1 on factors, multiples, and arrays involving multiplication and division. The unit focuses on helping students understand relationships between multiplication and division and strategies for solving word problems using the four operations. It provides examples of how students can find factor pairs, determine if numbers are multiples, and identify prime and composite numbers between 1-100. The unit aims to build students' abstract reasoning skills and ability to model mathematical concepts.
This challenge asks students to construct numbers from 1 to 50 using exactly four 4s and no other numerals. Students can use non-numeric symbols to combine the 4s. The goals are to increase number sense and develop understanding of order of operations. Hints include using exponents, square roots, decimals and other concepts. Discussing order of operations is encouraged when students get different answers using different calculators or methods.
K to 12 - Grade 8 Math Learners Module Quarter 2Nico Granada
Here are the completed statements based on the conclusions:
1. n(A × B) = n(B × A).
2. A × B ≠ B × A.
The key conclusions are:
1. The cardinalities of the Cartesian products A × B and B × A are equal, since n(A × B) = n(B × A).
2. However, the sets A × B and B × A are not equal, since the ordered pairs will be arranged differently, so A × B ≠ B × A.
This document provides examples and explanations of concepts related to classifying and ordering different types of numbers:
1) It defines rational and irrational numbers, and provides examples of each. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
2) Exercises are included for classifying numbers as rational or irrational, comparing irrational numbers on a number line, and ordering sets of numbers from least to greatest.
3) Answering "Why is it helpful to write numbers in different ways?" the document explains that not all numbers can be written the same way, like irrational numbers that cannot be expressed as integers. Being able to classify numbers facilitates understanding and operations involving different types of numbers.
This curriculum map outlines the mathematics curriculum for 10th grade students. It details the content, standards, learning competencies, assessments, activities and resources that will be covered during the first grading period, with a focus on patterns and algebra. Key topics include sequences, polynomials, arithmetic and geometric sequences, and their related concepts. A variety of assessment types are listed, like problem solving, oral recitations, and group presentations. The goals are for students to understand key concepts and solve problems related to sequences and polynomials.
This is one of three workshops conducted by the specialist team from MCI and SIF. This is part of MCI's community involvement programme. This is the workshop on secondary mathematics.
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.
Presentation math workshop#may 25th newUmber Tariq
It was prepared for the staff of our school , in order to guide that how to make, teaching and leaning for Maths, interesting and fun .
To reduce boredom for kids and to relate the concepts with the nature and universe.
1. The document is about a book titled "Primary 6 Mathematics Ace The Exams with My 24/7 Personal Tutor" which contains 10 mock exam papers and detailed explanations for Primary 6 math questions on a CD-ROM.
2. The book is meant to help pupils prepare for Primary 6 math exams by including exam-style questions that highlight common misconceptions.
3. The accompanying CD-ROM contains video lessons explaining how to solve each question in the mock exams.
The document provides instruction on absolute value including:
1) Absolute value refers to the distance of a number from zero on the number line. It is never negative.
2) Examples are provided to show that the absolute values of -15 and 15 are both 15 units from zero.
3) Students are asked to determine the absolute value of various integers and match pairs of absolute values.
The document provides an agenda for an October 28 math class. It includes a do now, homework check on integers and absolute value using a Frayer model vocabulary template, and homework assignments from a pre-algebra textbook. Expressions are categorized as variable expressions, numerical expressions, or variables. Ways to write multiplication and division are demonstrated, along with examples of evaluating variable expressions by substituting values for variables. Vocabulary on evaluating, integers, opposite numbers, and absolute value are defined using Frayer models.
This document provides instructions for 8 mathematics laboratory activities for grades 3-5. Activity 1 involves drawing shapes like a hut, joker and flower on dot paper. Activity 2 uses stickers to represent and compare numbers on a number line. Later activities involve measuring body parts, estimating room dimensions, counting edges and corners of objects, making shapes with paper folding, and representing multiplication tables visually using lines and dots. The goal is to help students learn concepts through hands-on experimentation and observation.
This document provides guidance for conducting mathematics laboratory activities for classes 3-8 in CBSE schools in India. It begins with an introduction explaining the importance of making learning joyful and relating school subjects to real-life. It then discusses how CBSE has introduced mathematics laboratories to overcome mathematical phobias by taking a hands-on approach. The document provides a detailed process for teachers to conduct various activities listed in an earlier CBSE circular. It aims to illustrate how students can construct mathematical knowledge themselves through experimentation.
Risk assessment principles and guidelinesHaris Tahir
Risk assessment principles and guidelines is a presentation slides was created and presented at Mission Critical Workshop. This slides is part of Business Continuity Management (BCM) presentation which intended for professional who is responsible for BCM or Risk Assessment Program.
Ib grade 11 physics lesson 1 measurements and uncertainitiesMESUT MIZRAK
This document discusses measurement and uncertainties in physics. It covers:
1) Expressing quantities using orders of magnitude to easily understand scale.
2) The ranges of distances, masses, and times that occur in the universe from smallest to greatest.
3) Calculating ratios as differences in orders of magnitude to compare quantities.
This document provides a daily lesson log for a 7th grade mathematics class covering operations on integers. The lesson covers addition, subtraction, multiplication, and division of integers over four sessions. Each session includes objectives, content, learning resources, procedures, and an evaluation. The procedures describe activities to motivate students, present examples, discuss concepts, and apply the skills to word problems. The goal is for students to understand and be able to perform the four fundamental operations on integers.
NCTM 2015 Filling Knowledge Gaps with Critical Singapore Math® Approach (Gr. ...Cassandra Turner
Exhibitor Workshop for Singapore Math, Inc. April 17, 2015
Description:
During Singapore Math® implementation, gaps in student knowledge can be a challenge. Upper elementary students frequently lack the foundation provided by the 1-3rd grade Singapore Math® Curriculum. In this session, you’ll learn the classroom-tested critical lessons and concepts students must master before jumping into their grade-level content.
The document provides a math review for kindergarten students covering key concepts such as patterns, addition, subtraction, comparing quantities, and number sequences. It includes review questions to test students' understanding of counting numbers to 20, completing number patterns and sequences, comparing quantities to identify who has more or less, and performing simple addition and subtraction problems up to 10. The review is designed to reinforce foundational math skills outlined in the state frameworks for kindergarten.
Presentation Math Workshop#May 25th New Help our teachers understa...guest80c0981
This is presented by a Math teacher,in Army Burn Hall College For Girls ,Abbottabad.
The target group was the teachers of school section. There were certain activities also performed an demonstrated in order to introduce new teaching methodologies and to prepare our teachers to meet the need of the day.
Umber
This chapter discusses multiplication and division facts. It includes 10 lessons: relating multiplication and division; algebra properties; facts through 5; problem solving skills; facts through 10; multiplying with 11 and 12; problem solving investigations; multiplying three numbers; factors and multiples; and prime and composite numbers. The lessons provide examples and practice with multiplication and division concepts and skills.
This document discusses numeracy development stages and strategies for addition and subtraction. It outlines 8 stages of development from emergent to advanced proportional reasoning. For each stage, it provides examples of the skills, knowledge, and strategies students at that stage typically use. It also lists and describes common addition and subtraction strategies like counting on, making 10s, partitioning, compensation, and the standard algorithm. Finally, it provides websites for additional math resources.
This document discusses numeracy development stages and strategies for addition and subtraction. It outlines 8 stages of development from emergent to advanced proportional reasoning. For each stage, it provides examples of the skills, knowledge, and strategies students at that stage typically use. It also lists and describes common addition and subtraction strategies like counting on, making 10s, partitioning, compensation, and the standard algorithm. Finally, it provides websites for additional math resources.
This document provides examples and step-by-step explanations of how to add, subtract, multiply and divide integers according to their signs. It begins with writing integers for word problems, then evaluates expressions. The main content explains that to add integers with the same sign, add their absolute values and the sum is positive if both are positive or negative if both are negative. To add integers with different signs, subtract the absolute values and the sum is positive if the positive value is greater or negative if the negative value is greater. Several worked examples are provided to illustrate the rules.
This document provides information from a Maths Information Evening for parents. It discusses what progress in maths entails, how maths is taught in key stages 1 and 2, and different maths concepts covered, including place value, addition, subtraction, multiplication, division, and problem solving. Parents are advised to praise their children's efforts, play maths games at home, and focus on building confidence rather than stressing workbooks or written methods.
1) Students will collect data on favorite fast foods in their class and grade to create bar graphs. They will write a report sharing their findings.
2) Fraction concepts are explored through examples of parts of wholes, such as one-third and two-sixths being equivalent.
3) A problem solving activity involves arranging digits 1-9 in groups so the sum is the same in each group, with discussion of multiple solutions.
This document is an instructional material for mathematics grade 3 published by the Department of Education of the Republic of the Philippines. It was collaboratively developed by educators and contains lessons on multiplication and division of whole numbers up to 3 digits long, including properties of multiplication and division. The material is freely available for public use but prior approval is needed for commercial exploitation.
This document provides information about math instruction at the school. It discusses what constitutes progress in math, how math is taught in key stages 1 and 2, and key concepts taught like number sense, addition, subtraction, multiplication, division, word problems, and problem solving. It emphasizes building understanding of place value and using various methods like number lines, arrays, and partitioning. It suggests ways for parents to help children at home, such as playing math games and giving praise, rather than workbooks or stressing written algorithms.
This document outlines the first grade math curriculum which focuses on developing students' understanding of key math concepts like basic algebra, geometry, problem solving and reasoning. The curriculum covers topics like addition, subtraction, data analysis, measurement, patterns and shapes over 9 units. It provides examples of math problems and activities students will experience to build their skills in a hands-on way. The document also notes that students will receive materials like math bookmarks and handbooks to use at home for homework.
This document outlines the first grade math curriculum which focuses on developing students' understanding of key math concepts like basic algebra, geometry, problem solving and reasoning. The curriculum covers topics like addition, subtraction, data analysis, measurement, patterns and shapes over 9 units. It aims for students to become proficient in computation and able to apply math concepts. The summary also notes that students will receive homework help materials to work on math skills at home with their families.
This document provides information about differentiating arithmetic and geometric sequences. It begins with an introduction explaining the learning objectives are to identify sequences as arithmetic or geometric, differentiate between the two types of sequences, and provide examples of each. It then provides examples of arithmetic and geometric sequences with their common differences or ratios. The document features group and individual practice problems identifying sequences and their properties. It concludes with a two-column chart comparing the key differences between arithmetic and geometric sequences.
This document contains a math practice worksheet for a 3rd period exam at Colegio San Patricio. The worksheet includes problems on the binary system, number lines, equations, word problems, simplifying expressions, and writing algebraic expressions. It provides practice with various math concepts to help students prepare for their upcoming exam.
This chapter introduces integers and their operations. Students will learn to use negative numbers, draw integers on a number line, compare integers, and order integers in sequences. Key terms include integers, positive integers, negative integers, and number line. The chapter discusses representing temperatures below zero as negative numbers, finding opposites on the number line, and using properties like commutativity and associativity to simplify integer calculations mentally.
This document discusses problem solving and reasoning in mathematics. It outlines various problem solving strategies and techniques including understanding the problem, devising a plan, carrying out the plan, and checking answers. Examples are provided to illustrate applying these steps to word problems involving ages, ratios, and logic puzzles. Different problem solving approaches are described such as looking for patterns, making organized lists, guessing and checking, using tables, and working backwards. The document also discusses inductive and deductive reasoning as well as recreational math problems.
Connect with Maths ~Maths leadership series- Session 3- the right knowledgeRenee Hoareau
Connect with Maths ~Maths leadership series- Session 3- the right knowledge presented by Rob Proffitt-White
The right knowledge – A clear valuing and understanding of mathematical content, the connections and a working knowledge of the proficiency strands underpins successful teaching
This workshop targets teachers and school leaders and aims to upskill their assessment literacy by:
• Creating cognitive activation tasks that promote critical thinking in all students
• Ensuring a consistent and shared responsibility for numeracy transfer
• Differentiating tasks through a focus on the proficiency strands
• Classifying the different problem solving types.
Connect with Maths ~ supporting the teaching of mathematics ONLINE
Engaging All Students community ~ http://connectwith.engaging.aamt.edu.au
Similar to Presentation 5 quantity magnitude and numeration january 2 (20)
This document summarizes key concepts about proportional reasoning. It defines proportional reasoning as a mathematical relationship between two quantities that involves a constant multiplicative relationship. It discusses proportional reasoning as developing between concrete and formal operations. It also provides examples of using proportional relationships to solve problems and discusses research on how to best teach proportional reasoning concepts to students.
The document discusses the importance of using the "Components of Number Sense" to connect math lessons and build students' understanding. It lists the components as Quantity and Magnitude, Numeration, Equality, Base Ten, Form of a Number, Proportional Reasoning, Algebraic and Geometric Thinking. Examples are given showing how story problems can be used to teach the components. The document emphasizes using language to develop mathematical understanding and representing problems visually and symbolically.
1. The document discusses using the "Components of Number Sense" framework to connect math lessons and build students' understanding of mathematical concepts.
2. It explains how the Components of Number Sense align with the National Council of Teachers of Mathematics standards and help teachers teach the standards more effectively.
3. The Components of Number Sense include quantity and magnitude, numeration, equality, base ten, form of a number, proportional reasoning, and algebraic and geometric thinking.
1. The document discusses using the "Components of Number Sense" framework to connect math lessons and build students' understanding of mathematical concepts.
2. It explains how the Components of Number Sense align with the National Council of Teachers of Mathematics standards and help teachers effectively teach the standard domains like number and operations.
3. The Components of Number Sense include concepts like quantity, numeration, equality, proportional reasoning, and others that can be incorporated into math lessons.
The document discusses differences between how American and Chinese teachers approach teaching elementary math concepts. It analyzes teacher responses to sample math problems involving subtraction, multiplication, and division of fractions. American teachers tended to focus on procedures and memorization, while Chinese teachers emphasized mathematical structure and developing conceptual understanding. The document advocates for teaching math in a way that builds students' number sense and understanding of mathematical concepts and properties.
- The document discusses research on mathematics education in the United States, finding that only about a third of students are proficient in math based on national assessments. It also discusses research showing US students performing poorly compared to other nations.
- The research emphasizes the need for a well-designed curriculum, quality teacher preparation, and explicitly teaching concepts and making connections to help students succeed in algebra and beyond. It discusses characteristics of students with learning difficulties in math.
- The document provides an overview of effective teaching practices informed by research, including concrete-representational-abstract instruction, explicit teaching, sequencing skills appropriately, and providing cumulative practice and review.
This document provides information about the North Carolina State Improvement Project's (NCSIP) math instruction foundation training course. The course aims to help teachers improve math instruction for students with disabilities through understanding research-based teaching strategies. It covers topics like number sense, assessment, and connections. Requirements include participating in workshops, readings, and assessments. Research-based math programs discussed include Transitional Mathematics, which has shown growth in student math proficiency at Statesville Middle School. Factors in sustained implementation of programs include assessment data, support systems, and leadership commitment.
12. Name__________________________ Score _____ Date________ Preliminary Let’s see if you can count from 1 to 10. Go ahead. Number Knowledge Test Level 0 (4-year-old level) ____ 1. Can you count these Counters and tell me how many there are? (Place 3 Counters in a row in front of the child.) ____ 2a. (Show stacks of counters, 5 vs. 2, same color) Which pile has more? ____ 2b. (Show stacks of counters, 3 vs. 7, same color) Which pile has more? ____ 3a. This time, I’m going to ask you which pile has less. (Show stacks of counters, 2 vs. 6, same color.) Which pile has less? ____ 3b. (Show stacks of Counters, 8 vs. 3, same color.) Which pile has less? ____ 4. I’m going to show you some Counters. (Show a line of 3 blue and 4 white Counters in a row, as follows: BWBWBWW.) Count just the white Counters and tell me how many there are. ____ 5. (Pick up all the Counters from the previous question. Show a mixed array—not a row—of 7 white and 8 blue Counters) Here are some more Counters. Count just the blue counters and tell me how many there are. *Go to Level 1 if 3 or more correct In order to receive credit an item that has an a and b, both a and b must be correct to be counted in the score.
13. Name__________________________ Score _____ Date________ Number Knowledge Test Level 1(6-year-old level) __ 1. If you had 4 chocolates and someone gave you 3 more, how many chocolates would you have altogether? __ 2.What number comes right after 7? __ 3.What number comes two numbers after 7? __ 4a Which is bigger: 5 or 4 ? 4bWhich is bigger: 7 or 9 ? __ 5a This time, I’m going to ask you about smaller numbers Which is smaller: 8 or 6? 5b Which is smaller: 5 or 7? __ 6a Which number is closer to 5: 6 or 2? (show visual array after asking) 6b Which number is closer to 7: 4 or 9? (show visual array after asking) __ 7. How much is 2 + 4 ? (okay to use fingers) ___ 8. How much is 8 take away 6 ? (okay to use fingers) ___ 9a When you are counting, which of these numbers do you say first? (show visual array 8526-ask child to point and say each numeral) 9b When you are counting, which of these numbers do you say last? *Go to level 2 if 5 or more correct In order to receive credit an item that has an a and b, both a and b must be correct to be counted in the score. .
14. Name__________________________ Score _____ Date________ Number Knowledge Test Level 2 (8-year-old level) _____ 1.What number comes 5 numbers after 49? _____ 2.What number comes 4 numbers before 60? _____ 3a.Which is bigger: 69 or 71 ? 3b.Which is bigger: 32 or 28 ? _____ 4a.This time I’m going to ask you about smaller numbers. Which is smaller: 27 or 32 ? 4b Which is smaller: 51 or 39 ? ____ 5a Which number is closer to 21: 25 or 18? (Show visual array after asking the question) 5b Which number is closer to 28: 31 or 24? (Show visual array after asking the question) ____ 6. How many numbers are there in between 2 and 6? (Accept either 3 or 4) ____ 7. How many numbers are there in between 7 and 9? (Accept 1 or 2) ____8. (Show visual array 12 54.) How much is 12+54? ____9. (Show visual array 47 21.) How much is 47-21? (use term take away) *Go to Level 3 if 5 or more correct In order to have an Item that has a and b, both a and be must be correct to be counted in the score.
15. Name__________________________ Score_____ Date________ Number Knowledge Test Level 3 (10-year-old level ____ 1. What number comes 10 numbers after 99? _____ 2. What number comes 9 numbers after 99? _____ 3a. Which difference is bigger, the difference between 9 and 6 or the difference between 8 and 3? _____ 3b. Which difference is bigger, the difference between 6 and 2 or the difference between 8 and 5? _____ 4a. Which difference is smaller, the difference between 99 and 92 or the difference between 25 and 11? _____ 4b. Which difference is smaller, the difference between 48 and 36 or the difference between 84 and 73? _____ 5. How much is 13 + 39? (Show visual array of 13 and 39.) _____ 6. How much is 36 – 18? (Show visual array of 36 and 18.) _____ 7. How much is 301 take away 7? In order to receive credit an item that has an a and b, both a and b must be correct to be counted in the score.
16. Developmental Level Comparison Chart Raw Score Developmental Level Score C.A. Equivalents 1-3 -0.5 2-3 years 4-6 0.0 3-4 years 7-8 0.5 4-5 years 9-14 1.0 5-6 years 15-19 1.5 6-7 years 20-25 2.0 7-8 years 26-28 2.5 8-9 years 29-30 3.0 9-10 years
17. Where Research Meets the Road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
18. Different Forms of a Number-- Linking to Magnitude to Number Lines Number Worlds Griffin
23. Different Forms of a Number-- Linking to Magnitude to Number Lines Number Worlds Griffin
24. Example: Percentages Predictions, Box scores, % vs. # 70/100 = 70 % 7/10 = .7 = 70% 35/50 = .7 = 70% 1 2 Recording the information Sense-making Touchable or visual
25. Even Algebra can fit this mold! How does Speed affect $; What is the constant? 1 2 4(s-65) + 10 Faulkner adapting Leinwald, Griffin Quantity: Concrete display of concept Verbal: Mathematical Structure Symbols: Simply record keeping!
26. Example: Algebra—Solving Equations 1 2 Sense-making 8 Equations Maintaining Balance X+X = 8 2X = 8 X = 4 Adapted from Borenson: Hands on Equations Touchable or visual Recording the information
83. Prototype for Lesson Construction Additive Inverse Opposites Absolute Value 1 2 Symbols Simply record keeping! Verbal: Discussion of the concrete Quantity: Concrete display of concept (-5) + (-6) = (-11) -5<5 8 – 6 = 8 + (-6) V. Faulkner and DPI Task Force adapted from Griffin
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86. The Number Line Source: http://www.learningwave.com/chapters/integers/numline.html
Counting Principles 1-1 Correspondence: one word tag to each object Stable Order: order of word tags invariant across sets Cardinality: final word tag = quantity of set Abstraction: objects of any kind can be collected and counted Order-Irrelevance: items in set can be tagged in any sequence Recall the basics of counting. Counting is keeping track of quantity.
This is a critical concept that we tend to lose in our procedural instruction. Yet it is critical. Consider what it means to add. We can only add things that have a defined commonality. In life we can talk about this concept and say, I am going to add all of the people and see how many I have, or I can say I am going to add all of the things with legs. So now I can include the tables and chairs. With mathematics we are tied to the idea of the UNIT when we add. We need to have LIKE UNITS in order to add things. Remember the 1/6 + 1/6 situation. The issue here is that our unit is 6ths and we are seeing “How many 6ths”. The addition is driven by the like units. If we had 1/6 + 1/5 we need to change the form so that we can get like units. Again the unit drives the addition. Format discussion like teacher in the Sims article “Look Who’s Talking” in the Trainer Notes (page 123 bottom of blue box). Engage students as this teacher engages her students. Make sure participants see that the issue of Same Unit is critical when we consider quantity.
Note to trainer—Have a participant read this slide.
This is a visual representation linking magnitude to number lines. We have discussed this slide in a previous unit. Further information is provided in Unit 5 Trainer Notes Slide 18. Questions that will instill quantity are listed below: How many? Who has more or less? What did you roll? How many spaces did you move? How high are you ? How far did you move around the dial? What was the question we asked in trashketball? (Which team scored the highest number of correct free throws? Which team had the most opportunity for improvement?)
When young students are told a story and asked to retell a story there is often difficulty. This issue relates to math when a student is asked to pattern colors, example red, blue, blue, red, blue, blue, etc. and ordinal counting. With respect to visual spatial issues, a student will not always be able to complete a pattern. Students are also unable to reverse patterns. 1, 2, 3, 4, 5, 6, etc. Looking at q quantity and naming it with a number.
See Trainer notes Unit 5 Slide 11 Dialogue.
The Preliminary item - This item is generally mastered around the age of 3 years. It is included in this test as a warm-up item, to orient children to the nature of the test and to give them a successful experience at the start. Level 0 items - These items assess children's ability to count and to quantify small sets, when concrete objects are available and can be touched and manipulated. Teacher says out loud everything that is not in parentheses. Everything in parentheses are directions for the teacher. Trainer note: Make sure to communicate stopping point in the testing process for this level is 3 questions correct. Stress that In order to have an Item that has an a and b, both a and b must be correct to be counted in the score.
Level 1 items - There are two classes of items at this level. Items 2, 3, 4, 5, 6 and 9 assess children's knowledge of the number sequence. Items 1, 7, and 8 assess children's ability to handle simple arithmetic problems. To solve items at this level, children need to rely on something like a &quot;mental counting line&quot; inside their heads.
Level 2 items - There are two classes of items at this level as well: those that assess knowledge of the number sequence and those that assess knowledge of arithmetic. The primary distinction between items at level 1 and level 2 is that items at this level require children to deal with double-digit numbers (i.e., tens and ones) and/or depend on the use of two mental number lines for successful solution.
Level 3 items - Like the earlier levels, there are two classes of items at this level: those that assess knowledge of the number sequence and those that assess knowledge of arithmetic. The primary distinction between items at level 2 and level 3 is that items at this level require children to deal with triple-digit numbers and/or to solve more complex problems involving double-digit numbers (e.g. addition and subtraction problems that require re-grouping). Have participants in pairs give the NKT to each other, taking turns administering and being the test taker. Assign each participant to give the Number Knowledge Test between Day 3 and 4. Give them information for on how to score the test. Bring back the results to Day 4 meeting. Guidelines from the Number Worlds Website. http://clarku.edu/numberworlds
See Trainer Notes Unit 5 Slide 18 Dialogue.
Trainer responsible to read and review this article. “Subitizing” is in the Trainer Notes Refer to Subitizing: What is it and why should we teach it. Doug Clements Make available for teachers to read on their own. Discuss how subitizing is the ability to ‘see suddenly’ the amount. Subitizing is a pre-skill for students to be able to see number as composed of other numbers.
If we only present numbers through manipulatives we are not supporting students’ ability to subitize well. They then live in “ones world’ and do not see how numbers can be composed. If 7 is presented like this, the tendency is to count each dot.
If we present 7 as a 3 and a 4, we begin to encourage students to see numbers as compositions of other numbers. This is a critical element of quantity. Recommend that elementary teachers, particularly early elementary teachers read and re-read the Clements article.
Notice how the picture land supports students in developing their ability to see number quickly, or to subitize. The repeated presentation of number through the standard arrangements in dice, cards and dominos is an excellent way to support kids in seeing number without counting.
Take for instance—a lesson on percentages. Later on in this Foundations training, we will build on student’s fundamental understanding of quantity and magnitude to explore, develop and understand percentages. We believe that this is a more powerful way for us to communicate percentages, ratio, division, fractions and rate than through procedural and oftentimes disconnected methods emphasizing multiplication and “moving it over two”.
Algebra is so concrete there are many examples of algebra in everyday life. Take your run of the mill speeding ticket, this can be turned into an exciting and motivating lesson for young high school about to get there license. Algebra can help them find the answer to that all important question of “where is the best place to speed”? For instance in the symbolic section here we have a state that calculates how many miles over the 65 MPH speed limit you were (s-65) multiplies that number by four and then charges you a $10.00 constant fee for court cost. How much would a speeding ticket cost in this state if you were clocked at 80 MPH. So algebra allows us to create a line that will tell us our exact cost for every possible speed.
This lesson maps Hands On Equations which is an excellent program that teaches algebraic equations using manipulatives. Consider how most of us were taught algebra particularly solving equations—it is taught in a highly symbolic fashion that is completely removed from quantity and magnitude. Let’s give it a try – Note to trainer: Each participant will need 1 Hands On Equation Packet.
Hands-on-Equations shows us how even solving abstract algebraic equations has a fundamental quantity component that students can HOLD ON TO. This is a very Physical presentation of algebra which is NOT a gimmick this IS the real math that is going on in Equations. Note also that this approach fits in nicely with the research not only of Griffin but also of Cecil Mercer whose focus is special education. He has found consistently that the struggling learner shows improvement when they are given direct instruction in moving from the concrete to the REPRESENTATIONAL to the abstract. Presenter may chose to click on the website and show one or two of the videos to see the language that students use with this approach to algebraic equations.
These are the types of equations that students will be able to solve readily after just a few days to weeks of work with Hands-on-equations. They learn to solve equations in a way that connects to what they already understand both the physical and the idea of reacting to stimuli sort of like they do with a video game, for example. The engagement is really amazing.
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As you know, another one of the Components of Number Sense is Equality. This mode of teaching equations not only emphasizes quantity but also the Equality behind equations.
At this stage of Hands On Equations, students solve by trial and error. Set up a blue pawn on the your right hand side of the balance and a number cube of 3 on the left hand side. (Participants should view the equation with the pawn(s) and number cubes as shown above. Your number cubes should face out, but participant number cubes should face up so they can be seen more easily. Model your thinking. Hmmm—if my system is balanced, then the blue pawn must be equal in value to three. Also, if I try the value of 1 for the blue pawn in the second example, it will create an unbalanced system, since 1 + 1 + 2 = 4 and not 8 + 1. I need to try some other values, higher values. Let me try 2 + 2 + 2 = 6 and 8 +2 = 10, so I must need a higher value for my pawn. Let me try 6. Yes! It works since 6 + 6 + 2 = 8 + 6 (Both sides of my system has a value of 14, so my scale balances. Let the students know that the pawn has a special name “x”. A check of the equation is part of the process so if x =6, then 14= 14 with a check above the equals sign.
Lesson centers around equations, variables, and unknowns on both sides of a setup---but still at the intuitive stage. The word variable is not used at all. Students acquire concepts through work with the materials. Show participants how to set up the equation physically. Try x = 3 When I try this solution, I have 7 on one side of my balance but 9 on the other side. Let me try 4. When x = 4, I have 8 on once side and 11 on the other side of my balance. My numbers on each side of the equation are now further apart than when I tried 3, so let me try 2. When x = 2, I have 6 on one side of the equation and 7 on the other side. Hmm—my numbers on both sides are getting closer together. Let me try x = 1. Yes---that works to balance my system. Set up the check by setting up the equation physically again and substituting in 1 for the blue pawn. The check is 5 = 5 so x does equal 1.
A legal move is when the same value is removed from both sides simultaneously. Show how when you physically set up 4x + 2 = 3 x + 9, you can remove 1 blue pawn from both sides, then another blue pawn from both sides and finally a third blue pawn from both sides. You now have (1 blue pawn) X+ 2 = 9. Students will readily see the solution for x is 7. Set up the original equations and check. 30=30 so x does indeed equal 7. We have a balanced system when x = 7.
Have participants try this equation by removing pawns from both sides. 2x + 2 = 10 X = 4 works since 4 + 4 + 2 = 10 Students do not remove number cube values from both sides at this point.
Note what happens in algebra. Up until now, we have almost always maintained a value. In algebra, you actually change values as long as you maintain equality, this is okay. Here is where the “soul of mathematics” the equals sign becomes the most important consideration/concept. Note that if I add 4 to both sides, I am changing my value on both sides. Why is that okay now? It is okay because our purpose has changed. We are no longer just simplifying numbers or writing the same value in different forms, now we are using the power of an equation to chop away at things until we find the hidden value of a variable.
Participants can now be shown about subtracting the same number-cube value from each side of a balance setup. This removal of the same value will leave the setup in balance. Four can be removed from both sides of the balance. Some participants may want to decompose 10 into 4 + 6. Our equation now reads 2x = 6 so students will readily see that x = 3. Checking with the original equations 10=10, so x is equal to 3.
Show participants how to remove the pawns first. Three blue pawns (3x) can be removed from both sides Try removing one from one side and two from the other side and ask if our system remains in balance. Click---t6 show our new equation. The value of 2 can be removed from both sides. This one is a little more difficult to decompose 14 since there are not enough of the correctly numbered cubes, but some participants may try 10 and 4 and then take 2 away from 4 to get 2. Adding back in the 10, we have a value of 12.
Have participants try this equation. They should set up the equation and keep the system balanced by first removing 2 blue pawns from both sides and then the value of 5 from both sides. 13 may be decomposed into 5 + 8 or 10 + 3 to help in the solution process. X = 4 in the check process.
Participants take away pawns as a part of the set-up process. 5 blue pawns are placed on the balance and then 3 are removed. Participants use legal moves to solve the equation. 1 pawn from each side and a value of 2 from each side. Click to show each stage of the symbol with the hands on.
Participants should set up the equation on the balance. Three blue pawns and a 4 number-cube on left side and five pawns on the right side, then remove three pawns from the right side of the balance to show the subtraction, then place a 9 number-cube on the right hand side. Allow participants to use legal moves to solve for x.
Participants learn that what is on the inside of the parentheses is doubled as the number outside the parentheses are distributed over each value. The set-up is pawn plus 3 two times. Have students display the doubled portion in two rows on the mat. On the teacher set-up the doubling in seen next to each other. Subtract one x from each side and six from each side, the value of x is 2. 10 = 10 so it checks.
Students can begin to transfer their concrete, hands-on experience in solving algebraic linear equations to a representation with pencil and paper.
Have participants use the concrete to model the equation, represent the equation with paper and pencil and show their solution. Click on each step of the solution above and on the next slide.
Click to show the set up of the equation and the subsequent steps. Demonstrate that the balance now illustrates x + x = 8. Discuss that at this stage students will use their intuitive knowledge as they did in the first lessons to go to 2 x = 8 to x = 4. Most students will not be thinking of these equations in terms of procedures at this point.
Remember Quantity and Magnitude have many dimensions
Note Trainer: Make sure you are comfortable with the Mayer reading for the later parts of this section. Review both these slides and the reading to develop your comfort level.
Student need to understand fraction from different angles. It is important that we develop students ability to use benchmark fractions to understand quantity at the rational number level. The magnitude model of the number line is an excellent way to do this. Play from minutes 18:00 to 24:40
Explain that this is an adaptation of the problem used by Mayer in his article.
Ask the participants what type of knowledge is used in translating. Depends on both semantic (knowledge of facts and objects—knowing butter comes in sticks and there are 100 cents in a dollar) and linguistic knowledge (language used to present the problem) i.e. Per stick means each stick.
Problem solving must select the relevant information and organize it into a coherent mental representation such as a mental number line with Food Lion at 65 and Lowes at two steps to the right. Involves schematic knowledge. (Seeing the problem as a total cost= Unit cost x number of units)
Strategic knowledge is needed. Breaking a solution plan into its component parts. Monitoring and reflecting as you work through the plan is a part of planning.
Mayer’s stage of Planning. While we are planning we need to make sure that we are training kids to think about these types of issues as they approach problems. When you ask kids to organize a set of problems by similarity, students in the U.S. will organize them by their surface similarity---car problems, food problems, recipe problem whereas students from higher performing countries are far more likely to organize the problems by their structural similarity---in other words—problems that can be solved by algebraic slope would be organized together, and ratio problems would be organized together. What we tend to seem to do in the classrooms in the US, is that we work through each problem trying to get an answer for that problem rather than using physical quantity to connect to mathematical structure. Consider our work so far. We now know that the money problem fits in with the basketball percentage problem Structurally. Before our work, we may have organized all of the basketball problems together and not noticed which ones were related to 100 as the unit whole like a money problem. Just as Sharon Griffin found that quantity is more than just the quantity itself, but rather deeply attaching it to the structure of the number line and magnitude, we need to make sure that whenever we are using quantity (real-world problems) that we are at the same time connecting this to the underlying mathematical structure.
Procedural knowledge is needed. Symbols represent the quantity. Strategy or component error can be determined. Sense-making comes into play here.
Documentation Mayer article
One last note on subtraction. As you do word problems and discuss the math remember that there are 3 basic types of things that can happen in life that we can model and solve utilizing the operation of subtraction. This is NOT a key word strategy. It is designed to help students understand the underlying structure of the mathematics. As you will learn through foundations training, key words don’t work. Teaching underlying structure and teaching students how to represent these structures to make sense of the mathematics does work and develops students sense-making skills.
Repeated Addition: I am going to get 3 dollars each day I dog sit for my neighbors and they will be out of town for 4 days. Array/Row-Column: Area Questions such as My garden is 4 feet by 3 feet, how many square feet is it? Or questions such as, I need to set up the chairs in 4 rows of 3 chairs, how many chairs do I need? Counting Principle: I have 3 types of cones and 4 flavors of ice cream, how many unique one-scoop cones can I make? Also works for others with same structure like 3 tops and for pairs of shorts, how many different outfits, etc. These are the three types of things that happen in the world that we can model and then solve using multiplication. Students should be explicitly taught these structures and then use them to understand word problems and to model the problems at the representational level.
These are examples of assignment sentences – the meaning here can be picked up form a sentence without contextualized interpretation. Consider that key word strategies assume that every math word problem is made up of only assignment sentences. Often times teachers seem to convince themselves that key word strategies work by creating problem sets that are so controlled that they only contain assignment sentences. In the beginning stages of problem solving key words may work. This is an opportunity to begin to teach children structure that will last a lifetime; not key words that are sentencing them to “walking the plank” they will fall off if key words are their problem solving strategy.
Here is another place where the “plank” ends! Just as our elementary students struggled with comparison subtraction problems middle school and upper elementary faced with relational sentences stumble and fall into the sea of misunderstanding
Have participants translate these relational sentences. They have already seen this same type of problem with the students and principals in Unit 2, Slide 50. Extensive notes on this slide are in Unit 2 Trainer notes that will match the dialogue you may need to have again in this slide. What you can expect is the following equation 8 R= D because we read left to right—we translate left to right. You may also get R=D+8 In order to translate the sentence into R=D+8 the translator is not thinking proportionally. Proportional reasoning is not additive but rather multiplicative. Let’s consider the R=D+8, is it an accurate representation of the relational sentence. If there are 8 raccoons how many deer are there? 8= 0+8. Does this match our intuitive feeling about quantity? Another common misrepresentation of relational sentences is to understand that it is a proportional sentence and that the operation of multiplication is appropriate. The translator, therefore, invokes multiplication but perhaps because of earlier practice using key words and assignment sentences they literally translate from left to right. Incorrectly translate to 8R=D. Does this work? What is this sentence saying it means 8 multiplied by R = D. Why does this not work? Note to Trainer: This is algebra – take time to write / draw out / diagram what the sentence is saying. Make sure to have the participants use a number line for Zelda and Yolanda. Ask the question--Who is older? Mark that on the line. So where does Zelda go? What do we know?
Concrete and Quantity help the students to translate. Representational/Structural and Verbal help the student to Integrate and Plan The abstract symbolic level is where the students execute the plan.
Read through this slide with participants so that they get an idea of what is involved in numeration.
If you are working off the SCOS you may not recognize that there are foundational skills necessary for proper understanding as per the developmental sequence. Look at the standard course of study and find out how often a particular concept fits into your math curriculum. Trainer: access the website and glance at the NCSCOS, and other resources for K-12 math Proposed changes can be found to the new curriculum. Make certain that you show the participants the SCOS links, the Indicators by grade level, Week by Week Essentials, classroom strategies and black line masters that are in the resources to the right on the page.
How do we know what the students do know? When we look at a broad objective such as 1.01 in Grade 7, it is imperative to find out what the students do know in order for that student to master that objective. The student may have a simple misconception that is getting in the way of mathematical success. Once we know what the student does know, we can address what they don’t know. Note to trainer: This is just one example of how NCSCOS fits these probes. See handout folder. These probes will be used in Unit 6 also.
The arithmetic sub-test on the WISC factors in on the verbal section. For example y=x+5. This statement just says that y is the same value as x, just x has five more.
Teacher models counting sequence, emphasizing the new part. 2. Teacher models and test students on the new part of the sequence until students are able to say the new part correctly three times in a row. 3. Tell the students what number they will be ending with. 4. Test students from the beginning through the new part. 5. Frequently repeat in multiple settings in one to two minute sessions. NOTE: When the child is counting DO NOT interject corrections as the child is counting.
Often children get the multiplication facts dealing with 2, 5, 10’s. Why is this? It is because we have taught them to skip count earlier in the SCOS. Consider using basic numeration skills to support students with their multiplication tables. Why would you teach counting in this order? Because you can connect to student’s prior knowledge of skip counting by 10’s. 9’s (using the distributive property) is just one less than 10. 12 is (11+1) therefore 12 x 2 is (11 x 2 + 2) and 12 x 3 is (11 x 3 + 3) and etc. The language piece is very important. Ask the student, “How do you do that?”
Procedurally, how do we figure these problems out? Base Ten---trust it!! Discuss the importance of these questions as an opening activity regarding ten percent. Possible answers to the questions. Connecting to prior math knowledge Pointing out patterns Connecting to benchmark fractions Reinforcing the importance of benchmark fractions Creating opportunity for students to discuss and help make sense of the concept Creating opportunity for students to connect to what they already know Getting smaller Less Less Ask where 10 % falls on a number line. Closer to 0 or closer to 1? (0) Not very good since it is less than ½ of his/her shots It means that for every $10 you pay—you get $1 off or you pay 90 % which is $ 9 for every $10.00 Stress here that what you pay and your discount equal 100%.
Practice, practice, practice, but what are your practicing? Procedurally, how do we do this? How do you teach this without just practicing the procedure of “moving it over one”? How do you get kids to visualize and deeply understand ten percent? What is ten percent? It means, for instance, if someone has money, you get ten dollars our of every 100 dollars. So, if they have $100, you get $10. But what if they have $400, how much do you get? You could teach this by saying, let’s move it over one, but particular for the struggling leaner it is important that you connect through counting/numeration and not just a procedural trick that we know from experience that doesn’t build the conceptual understanding of 10%, nor does it work long term as trick for this type of student. So what you are practicing for the kid for $400, “ So, let’s think what this means, let’s see if he has $400, that mean s we get 10 for the first $100, 10 for the next $100, then 10 for next 100, and ten for the last $100. Have I counted for $400? Okay, so how much do I get? 10, 10,10 and 10 more makes 10, 20, 30, 40. So, I get 10% of $400 and that is 10 for every 100, which is $40.00!” “Remember, that is what ‘per cent’ means. So in this case, I get 10 per cent which means 10 per hundred, which we figured out to be $40 per $400. A link to the hundreds board is appropriate as a visual for students. Example 1: What is 10% of 50. Well 50 is ½ of a hundreds grid. If I had 10% of a hundreds grid, then I would have ten, but I only have ½ of a hundreds grid so I must have 5. Example 2: What is 10 % of 400 Ask participants what 10 % of 100 would be. (10) But I have 400 which is 4 x 100 so I must have 4 x 10 which is 40. Proportional reasoning is stressed here as well in this though process.
Thinking about hundreds grids again could be very helpful as you guide participants through this discussion. 20% of 800 10 % of a hundreds grid is 10, so 20 % of a hundreds grid is 20. I don’t have just one hundreds grid, I have 8, so 8 x 20 = 160. 20 % of 800 is 160 Find 15% of 800 10% + 5% means that I have 10% of 8 100s grids---so I have 80, but then I have another ½ of that amount since 5% is ½ of 10%. I have 80 + 40 which is 120. Also, think about complements. We know that if we have to find 20% of something, the complement to that is finding 80% of that amount. If I go to the mall to buy a pair of pants and they are have a 40% off sale, then I know I will be paying 60% of the original price. For example, the cost of the pants is $30.00. 40 % off would be 3 + 3 + 3 + 3 = 12 off (10% + 10% + 10% + 10%) or we could think of it as 3 + 3 + 3 + 3 + 3 + 3= 18 that we will pay (10% + 10% + 10% + 10% + 10% + 10%). Students will be empowered with these connections to base ten and equality.
Fifty percent of what is important to know as we measure quantity.
Bullet 1 Keep it, change it, switch it. What does this mean???? We teach students to keep the sign of the first integer the same, change the operation sign from subtraction to addition and then change the sign of the second integer to its opposite. Are we teaching the understanding of why this is so or are we just teaching them to memorize the trick. Bullet 2 When good things happens to bad guys, that is bad, right? ( + times a – is a -) When bad things happen to good guys, that is bad, right? (- times a + is -) When a good thing happens to a good guy, that is good. ( + time + is +) When a bad thing happens to a bad guy, that is good. (- times – is +) Bullet 3 When two socks that match are pulled out of drawer---that is good (+). (+ times + = + and – times – is +) When two socks that don’t match are pulled out of drawer, that is bad(-). (- times + = - and + times – is -). What might confuse our students here is that putting socks together implies adding them not multiplying serving to only make things more difficult in terms of understanding. Bullet 4 What does the word “larger” imply? Is -8 larger than 3? No, we need to communicate to students the meaning of absolute value and how we find difference between two absolute values and then use the sign of the number with the greater absolute value.
If yellows are positive and reds are negative, what is the value of the quantity? ( 0 ) What happens when you add two numbers that are opposites, like 5 and -5? In a sense, they &quot;cancel&quot; each other, resulting in zero. In other words, the sum of a number and its additive inverse is zero. What is the absolute value of the red counters? (3) How many units is that from zero? What is the absolute value of the yellow counters? (3) How many units is that from zero? What is the opposite of the red counters? (Yellow counters) What is the value of the red counters. (-3)
This is model for vocabulary that helps students to develop linguistic understanding of mathematical concepts. It’s power is in the use of non-examples. Non-examples should be introduced after students have been exposed and understanding examples of the concept.
There is a difference between a tool, such as a calculator, which does the thinking and supplies the answer and a concrete model that helps students explore concepts. Model Integers on a number line. Students perform operations with integers on number lines. Two color chips can provide a visual that can then be represented as a tool to provide meaning for integers and their operations.
Think of all of the math that can be communicated with this mathematical tool.
If a number is to the left of a number on the number line, it is less than the other number. If it is to the right then it is greater than that number.
Discuss how -3 can be modeled on the number line. How would we add the following expressions on the number line? 2 + 2 -2 + 2 1 + -2 -1 + 2 Subtraction -1 - -2 In order to subtract, I must add the opposite, so, I begin at -1 and then I move to the right 2, since I must reverse directions in order to add the opposite. What about multiplication? It is just repeated addition! -1 x 3 = -3 (I have three groups of -1) Since I can commute numbers, 3 x -1 is also -3. Division is the opposite operation of multiplication so if we begin with -12, how many groups of -3 can we make with -12? (4 groups)
See Slide 89 Dialogue in Trainer Notes.
We talk about distance above and below sea level. (Sea level is at 0) Stocks rise and fall from one day to the next so this rise and fall is measured as gains and loses with respect to particular benchmark---the day before, the month before, etc. Temperature rises and falls, again with respect to a particular benchmark---last year at the same time, etc. Golf is played according to par—over or under. In this case of quantity--the lower score wins the game since you were able to play the course in fewer strokes.
Connecting the Components of Number Sense of Quantity, Magnitude and Numeration therefore is more than just working with computations. A student must be able to not only be accurate but understand the math well enough to use different solution methods and make decisions about which solution method is best for a given situation. Toni Meyer and Elizabeth Murray---- Adding It Up