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Mathematical notes
The purpose of this page is to link to documents to inform educators and other people seeking information to facilitate mathematical literacy. Information is organized in four broad areas:
 Knowledge base of concepts, definitions, & outcomes for mathematical literacy.
 Pedagogical knowledge to develop professionally and to create learning experiences to facilitate mathematical literacy and many teacher tools to assist.
 Development, planning, activities  problems, learning units & packets to develop mathematical literacy listed by mathematical dimensions.
 Number value & Operations (addition, subtraction, multiplication, & division)
 Activities  problems, units, packets ...
 Algebra, Patterns, Ratio, & Proportion
 Activities  problems, units, packets ...
 Geometry & Visual spatial reasoning & representation
 Activities  problems, units, packets ...
 Data Analysis & Probability
 Activities  problems, units, packets ...
 Measurement
 Activities  problems, units, packets ...
 Technology  calculator, programming, and other math related tech
Process dimensions
Content dimensions
Knowledge base & pedagogical tools for mathematical literacy
Knowledge base
The mathematical knowledge base includes concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Includes
concepts, outcomes, & vocabulary by content, processes, and attitudinal (values & dispositions) dimensions by levels.
 Algebra & patterns
 Attitudinal (disposition & value statements)
 Classification
 Communication
 Connections & perspectives
 Data Analysis & Probability
 Geometry
 Measurement & measurement estimation
 Number value
 Operations & computation estimation
 Problem solving
 Reasoning & proof
 Relative Position and Motion
 Variable
Pedagogical knowledge to facilitate develop professionally and to create learning experiences to facilitate mathematical literacy and many teacher tools to assist
Different fields of mathematics map & explanation video (11:05)
Resources for professional development
 Mathematical vocabulary lists by dimensions & levels
 Definitions of mathematics and focus questions to analyze them
 Kinds of numbers  fact sheet
 Books
 Conceptual and procedural knowledge  Illustrated with card trick  magic ...
 Considerations for Planning to Teach: Big ideas, concepts, and generalizations
 General information for professional educators
 Hands on: What is it really?
 History of mathematics in North America
 Independent study: planning guide to help select and refine topics
 Memory: learning basic facts & algorithms  getting learners to slow down & think, making better choices on how to proceed
 Mental math quiz to compare traditional algorithmic solutions to non traditional algorithmic solutions mathematicians are likely to use
 Misunderstood mind  video & information on PBS
 Principled procedures for Math Educators
 Reflecting on mathematical instructional practice  suggestions to reflect beyond a summary of what happened
 Sample outline with suggestions for creating a professional portfolio
 Survey your current mathematical practices and research , then compare your thoughts with what current research suggests .
 Mathematical vocabulary lists to review when planning learning activities
 Visual & Spatial Representations  Modes of visual learning, literacy, & research, suggestioins & activities
Math teacher's planning tool box
General planning
Planning information
 General instructional theory  explanation with examples flexible enough to plan any topic & intended learning
 Overview for planning and creating curriculum documents
 Outline for a year plan and suggestions
 Outline for a sequence or unit plan with suggestions
Examples to prep intended learnings
 Framework with explanations of mapping categories for planning to teach big ideas, concepts or generalizations
 Blank planning map
 Classification sample planning map
 Equality and equal arm balance
 Geometry, Area sample planning map
 Proof and reasoning sample planning map
 Thoughts about math games, competitions, and simulations
 Literature to teach math  Includes  uses of literature to facilitate math literacy, ways to mathematize literature, list of literature books with mathematical themes, literature suggestions by math dimensions, sample lessons using literature books
Suggestions for selecting problems and investigations
 Problems by categories
 Selecting problems >> > sample problems with different formats , characteristics of a good problem , guiding questions to prepare mathematical investigations, tasks, or powerful problem
Sample parts and pieces of sequence plans
 Sequence plan with activities to build to cardinality . Good start with multiple concepts, sequence of concepts, and very good lists of activities for concepts. Need more integration of mathematical dimensions for practice of mathemtics and scoring guides with levels.
 Sequence plan for measurement, data analysis, and proportion Includes beginning ideas for integration of mathematical dimensions and some for mathematical practices, good for proportion, could use editing for clarity and more depth of procedures.
Activity plans
 Card trick  Includes integration of mathematical dimensions, good for proof, could use some editing for clarity card trick demonstration with different instructional strategies [conceptual & procedural].
 Reading and writing numbers with numeral and words  Good activity for skill building and necessary skill, doesn't include integration of mathematical dimensions,
 Estimating chocolate chip  Has potential, but needs a lot more information for integration of mathematical dimensions and extended activities.
Assorted tools
 Characteristic of mathematically talented & development of talent
 Mathematics & Logic adapted Frayer Model
 Manipulative materials list
 Parent letter for using manipulatives and representation with multiplication & division
 Eight different uses of literature for teaching mathematics
 Annotated list of literature and classroom uses in NCTM Journals
 Mathematical terms and their representations with Cuisenaire Rods
 Investigations with Cuisenaire rods & student work sheets
 Comparison or English  Spanish wording in math. Translations of some mathematical ideas . Notice how some are the similar and others are very different. Demonstrates that direct translation is generally not as simple as one thinks unless one is bilingual.
 Math Their Way notes
Assessment
 Assessment introduction and review: [ Suggestion for creating criteria referenced assessments ] [ Normreferenced test information ]
 Letter to parents for student lead conferences and authentic assessment
 Questions from standardized achievement tests . SAT, FAA, MCAT, ...
 PISA 2012 ProblemSolving sample questions
 Curriculm Documents with concepts, outcomes, and assessment levels
Sample assessments
 Student survey  open ended about student's math practice  one page
 Student survey  rating sheet (always, mostly, some, not)  about two page
Curriculum
 Comparison of dimensions and categories of NCTM and CCSS of math
 Concepts, suggested sequences, plans, rubrics, and other stuff may be found linked at the top of the page in mathematics content.
 Standards
 Curriculm Documents with concepts, outcomes, and assessment levels
 Rubrics click the desired mathematics dimensions at the top of the page to locate concepts... assessment... evaluation ...
 Notes on exemplary and promising math programs
Workshops, courses, tutorials
 ESU #1 Algebra Workshop information
 Information for graduate Class or professional development
 Information for preservice Class or professional development
 Videos for professional development
 Technology Help Pages  ILife and Windows Media tutorials and help pages for using/ making: photos, audio, and video >>> directions and tutorials...
 NATM fall conference presentation 2006
 NATM spring conference presentation 2005
 NCTM regional presentation fall 2004
Numbers: values and basic operations
Knowledge base & tools to develop mathematical literacy
Knowledge base
Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathematically literate and misconceptions that must be over come. Particularly important for how learners develop their understanding of numbers, their values, operations, patterns, and relationships are:
Concepts related to number sense:
 Number sense
 Classification  children must know how to classify to determine groups (sets) to count & determine number values. More advanced classification ideas are needed to compare number values.
 Whole numbers
 Addition and subtraction
 Place value
 Multiplication & division
 Fractions
 Decimals
 Percents
Information to facilitate literacy in numbers (values & basic operations) & tools to achieve it
Numbers & their definitions  Fact sheet of natural, integer, rational, irrational and real numbers. Their definitions, examples, & number line visual representation.
Vocabulary lists related to number sense
 Number sense & operations
 Vocabulary for operations by grade levels
 Operations
 Fractions
Numbers: Counting, recognition, cardinality
 Prenumber sense (birth  6/8 years)  beginning development of number sense
 Conservation  development of conservation of number & other reasoning tasks
 Subitizing before counting  development of subitizing before accurate counting
 Counting  overview of the development of pre counting to number sense & hierarchial inclusion
 Skip & choral counting  instructional suggestions with examples
 Skip & choral counting by threefifths  example with worksheets
 Whole Number value (6/8 years & beyond)  development of number value after number conservation & onetoone correspondence to hierarchial inclusion
 How to say, read, & write numbers from billion to billionths  lesson plan & worksheets
 Framework for planning counting and cardinality units
 See also place value development
Assessment of development
 13 Whole number assessment tasks  Scripts, materials, scoring guides, and Summative record sheet for all , K2.
 Counting with synchrony or onetoone correspondence & cardinality [ Synchrony information]  script with materials, problems, suggestions & scoring guide
 Rote counting to 100  script with problems, suggestions & scoring guide
 Writing numerals 19  mnemonic phrases to teach to help students remember how to form numerals and assessment script with scoring guide.
 Rote counting by multiples, 2, 5, 10, & 3  script with materials, problems, suggestions & scoring guide
 Counting back from 10, 20, 100  script with materials, problems, suggestions & scoring guide
 Counting to and from 1 000 .  script with materials, problems, suggestions & scoring guide
Counting to 0nethousand requires memory of the necessary words, but also a conceptual understanding of ten repeated 100's.  Numeral recognition  (120) with worksheet, script, suggestions & scoring guide
 Number recognition  prompt sheets (numerals 110, & 1020 randomly arranged) & directions
 Writing or forming numerals  script with materials, problems, suggestions & scoring guide
 Subitizing or instant recognition of values  script with materials, problems, suggestions & scoring guide
 Assessment for Onetoone correspondence with matching  script with materials, problems, suggestions & scoring guide
 Cardinality matching  subitizing or visual pattern recognition, script with materials, problems, suggestions & scoring guide
 More or less  script with materials, problems, suggestions & scoring guide
 Cardinality match numeral and number word  script with materials, problems, suggestions & scoring guide
 Hierachial inclusion to five  script with materials, problems, suggestions & scoring guide
 Cardinal scoring guide and rubric
 Conservation tasks  six conservation of number tasks with instructions, developmental notes, & possible outcomes
Place value development & assessment
Development
 Place value  development from age 5 on: ten as big number, group by ten, pre place value, unitize, place value relationships, decimals, exponents, place value & decimal worksheets
Assessment
Fractions, ratio, & proportion development & assessment
Development
 Development of fractional number values , instructional ideas, sample activities, worksheets, and assessment. Focus is on the development of fractions as equal parts of a whole, sub groups, & equivalent fractions and decimal fractions. Work sheets include: fraction circles, circle disks, paper folding, area models, measurement models, number lines, hundred charts, challenges, & problem suggestions.
 Fractions, decimals, percent, ratio & proportion  development of fractions, decimals, percent, ratio, proportion, their operations , and connections
 Whole numbers, fractions, rational & irrational numbers, infinity, & number lines  instructional notes
 Fractional value & equivalency  scoring guide suggestions & rubric
Operations
Addition & subtraction  whole number development & assessment
Development
 Addition & subtraction  development of addition and subtraction as operations and ways to assess it. Includes an analysis of the four types of addition & subtraction problems  chart with multiple examples for the three subtypes of all four types
 Memory, learning, and developing skill with basic facts & dot plates
 Dot plates
 Strategies for addition & subtraction  includes instructional suggestions and sequences to help develop people's ablity to solve addition and subtraction problems mentally.
Assessment
 Addition and Subtraction  suggestions, directions with problems, scoring guides, & score sheets to assess the development of + &  from four ways: join, separate, partpartwhole, & combine with scoring guide ideas to understanding and use of algorithms
Multiplication & division of whole number development & assessment
Development
 Multiplication & division  development of understanding and strategies
 Development of multiplication facts
Addition, subtraction, multiplication, & division  fractional numbers development & assessment
 Addition and subtraction sequence
 Story problems with representational starters for addition of fractions with unlike denominators thanks Brant
 Multiplication and division sequence
Division problems represented with squares and rectangles
 Sample problems : 1/2 ÷ 1/2, 1/2 ÷ 1/4 , 1/2 ÷ 3/4, 1/2 ÷ 3/5 (thanks Brant),
 Division problem (5 8/10 ÷ 5) with eight different solutions
 Multiplication of fractions with Cuisenaire Rods explanation how to multiply
Addition, subtraction, multiplication, & division  decimal numbers development & assessment
Activities to facilitate number values, & operations literacy
Number sense  whole numbers
 100's of Instructional ideas and activities for number sense & place value
 Counting and cardinality Lesson plan  review also as a planning framework
 Bean toss directions and Fold book patterns for tossing 110 beans
Number relationships >>> see patterns and algebra
Dot plates & subitization: activities
 Dot plates video and explanation about memory, number value, and basic facts
 Eight dots in one pattern with different interpretations
 Subitize information & resources
Manipulatives for students to use to learn number value
Cards for learners to sort  make a set of cards by combining cards appropriate for a particular learner: numerals, words, ordinal, dot, ten frame with dots. The cards can be sorted into paper bags, a sorting box, or on any flat surface. Different games or activities can be created: sort dots, sort dots and label with the set's cardinality numeral word, sort and label with ordinal word, sort all cards and beat the clock ...
Sample cards to print and cut out :
 Blank cards
 Numerals 0  9
 Number words zero  ten
 Ordinal words first  tenth
 Dot cards:
 Ten frame cards
Dice game  helps students subitize number value and introduction to addition. Roll the dice and flip the values on the die or the sum of the die. Continue to roll until all the numbers (112) are flipped (win) or a value rolled has had all it's possible plays, already flipped. Example: roll 2 & 2. Can flip 4, or 1 & 3.
Die roll  simulate a roll of a die .
Number talks: are good tools to show equality of numbers and different ways to represent the same values. Sample number talk outcomes for 16 .
Number line : is a good mathematical tool to represent mathematical values and operations. However, children who are not familiar with them and confident with their use, are not sure what to count: numbers, lines, spaces and where to start (0, 1, ...).
Therefore, before they’re used as a representation to explain and prove mathematical ideas, learners need experiences to develop their understanding and fluency with their use.
Activities, used when counting, such as numbered polka dots organized in a line can provide good background before introducing a number line.
 Introduction to a number line
 Number line  place values from billions to billionths
 Numbers listed  in order from billions  billionths
 Addon or countback
Place value  whole numbers & decimal numbers
Manipulatives for multiples of 10 and 100
 Blank ten strip
 Page of 10 blank ten frames
 Ten and more (teen numbers) Fold Book
 Ten and more (teen numbers) worksheet
 Blank twenty strip
 Hundred chart
 Zero to 100 chart
 Blank hundreds frame four 10 cm squares.
 Blank hundred chart larger
 Thousand chart
 Hundred chart puzzles and puzzle pieces puzzles
 Number line  place values from billions to billionths
 Numbers listed  in order from billions  billionths
Really big numbers and infinity
 Olber's Paradox  if the universe is infinite in time and space, stars should occupy every point in space and fill the night sky with light.
 How many grains of sand in the Universe? Archimedes' proof in The Sand Reckoner.
Number Theory
Number theory is critical for students to do well in algebra and higher levels of mathematics. However, the sad thing is most teachers are unfamiliar with these ideas, or skip them when they occur in their math text or program, as their view of mathematics is limited to basic facts and operations as calculations of numbers.
The following are examples related to number theory.
 Using primes to find common factors, multiples, LCM and GCF ALL together
 Sample dialogue that explores a learners conjecture about multiples and moves to primes and then to the fundamental theory of arithmetic (FTA) ...
  Factors of first 50 & 100 notes  Factors to 100 chart 
 The Fundamental Theory of Arithmetic
 Fibonacci sequence
 Patterns with a factor & a constant factor to find a function  relates to casting out nines
Fractions, Decimals, & Percents number values
 Fractions  development of fractional number values, instructional ideas, sample activities, workheets, and assessment. Focus is on the development of fractions as equal parts of a whole & equivalency. Work sheets include: fraction circles, circle disks, paper folding, area models, measurement models, number lines, hundred chart, challenges, & problem suggestions.
 Fractions  Worksheet to represent 1/2, 1/4, 1/8, 1/16, 1/32 on a number line
 Fractions  Worksheet to explore the fraction 3/5 with skip or coral counting & its patterns
 Fractional values compared problems
 Using fractions to find the area of nine shapes in a 17 square unit rectangle  The Pharaoh's problem
 Percentage  sample activities and worksheets
 Ratio and Proportion
Problems
 Sequences
 Number line with whole & decimal numbers  place values from billions to billionths
 Numbers for whole & decimals  in order from billions  billionths
 Decimals either end in zero or repeat as nonterminating decimals or rational numbers.
Example: place these numbers on number line 1, 2/3, .5, .3 , 1 1/2, 2, .6 , 1/2, .9 , 1 .1 , 1/3 Misconception .9 doesn’t equal 1. Recognize 1/3 = .3 & 2/3 = .6 . but not .9 = 1.0
Basic Operations (+*/)
Addition and subtraction whole numbers (+ )
 Transitional activities from number sense to addition and subtraction
 Sample subtraction algorithmic solutions
 Addition facts table
 Add on or count back  in comic form
 Bucket problems application of addition, subtraction, problem solving, combinations, reasoning.
Multiplication and division of whole number (* /)
 Multiplication arrays Molly B
 Multiplication area or grid representations > two examples > two digit numbers example > a couple of reasons why this is an important representation related to algebra & for two digit multiplication
 Multiplication facts table
 Representations of 21 / 7
Fractional numbers (+  * /)
Activities for number sense particularly: fractions, decimals, and percents
 Addition and subtraction sequence
 Story problems with representational starters for addition of fractions with unlike denominators thanks Brant
 Multiplication and division sequence
Division problems represented with squares and rectangles
 Sample problems : 1/2 ÷ 1/2, 1/2 ÷ 1/4 , 1/2 ÷ 3/4, 1/2 ÷ 3/5 (thanks Brant),
 Division problem (5 8/10 ÷ 5) with eight different solutions
 Multiplication of fractions with Cuisenaire Rods explanation how to multiply
Decimal numbers (+  * /)
Activities for number sense particularly: fractions, decimals, and percents
Integer values and operations (+  * /)
Addition and subtraction of positive and negative integers might be conceptualized by representing and controling two properties: position on a number line and direction of body.
The number line can have two halves with the same numbers being both positive and negative. You have two choices for each number + or  as a reference to position. If the number has a value of 4, it could be represented as a positive four or negative four.
You can represent those two numbers by standing on either +4 or 4. Secondly when you are standing on a number line you can face one of two direction, since the number line is bidirectional, you might be facing left or right, forward or backward, in a positive direction or negative direction. or if it is thought of as a thermometer, you can walk hotter or colder.
Problems can be posed as :
Find a starting point, say 3 and if you add a +3, face warmer and walk forward 3.
Find a starting point, say 3 and if + 3, then face warmer and walk backward.
Find a starting point, say 3 and if   3, then face colder and walk backward.
Find a starting point, say 3 and if  +3, then face colder and walk forward.
Again both are important and a concrete model can be used to describe a procedural rule, as demonstrated as a plus and a plus are plus, a plus and a minus are a minus, and a minus and a minus are a plus.
Another idea is to record a video with motion. People walking or running forward and backward. They might display signs with forward and backward as they do so. However, it probably will be obvious which direction they moving when it is recorded. When the video is made, then each can be viewed with the movie play forward and backward. A chart of the different results can be made (forward, forward, results in forward; forward, backward, results in backward; backward, forward results in backward; backward, backward results in forward.
After several concrete experience have been experienced explore the this challenge.
How many different ways can two numbers be represented with different a sign and an operation (2 ++3; 2 +3; 2 3; 2 +3) (2 ++3; 2 +3; 2 3; 2 +3).
Algebra Patterns and Functions
Knowledge base & tools to develop mathematical literacy
Knowledge base
Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for how learners develop their understanding for algebra, patterns, and functions are:
Concepts related to:
 Algebra & Patterns
 Ratio & proportion
 Concepts and Outcomes from Science for All Americans (SFAA) & National Assessment for Educational Progress (NAEP)
Information to facilitate literacy in algebra, patterns, & functions & tools to achieve it
 Assessment
 Functions & patterns  Work sheet of a sample of Authentic Assessment for a Bridge pattern
 Functions & patterns  Work sheet of a sample of a Directed Assessment for a Bridge pattern
 Assessment  process proof and conjecture  content zero and properties of zero  addition and subtraction  with teacher reflection
 Scoring guides
 Equality & equal symbol  scoring guide or rubric across levels
 Classification or Sorting  Activities to assess by 1, 2, & 3 properties; use of Venn diagrams to represent intersecting & disjoint groups.
 Conjectures  Articulating, editing, & refining conjectures in a second or third grade class
 Equality and equal sign in equations  Discusses equality, equals symbol, its big ideas, supporting concepts, development, scoring guide, & ideas to facilitate understanding. Includes the use of an equal arm balance to discover equality. With photos and different models made from K'Nex.
 Graphing  Instructional Plan for teaching graphing & solving for two unknowns four ways. (812 Grade)
 Justification & proof with 3 examples: concrete, pictoral, & symbolic levels for odd plus odd is even
 Multiples, common multiples, & FTA  Discussion script includes: least common multiple, the next common multiple, fundamental theory of arithmetic (FTA) see FTA chart in activities
 Primes and composites  Diagram to use to explore the primes of whole numbers 1  40 & invent the Fundamental Theory of Arithmetic (FTA)
 Algebra, patterns, & function vocabulary  lists of vocabulary for primary, middle, & upper levels.
 Statistics timeline and history
 Reflections from ESU #1 workshop
Activities to facilitate algebra, patterns, & functions literacy
 Adding consecutive numbers  Work sheet challenge to find a sshort cut to add 4 consecutive numbers, hint, and verification proof.
 Arithmetic Properties  Chart with properties, conjectures, examples, & non examples of numbers and operations: addition, subtraction, multiplication, division, identity, inverse, zero, commutative, & distributive.
 Calendar math problems  List of challenges & hints to identify dates which are: factors, primes, relatively prime, pi, triangle numbers, square numbers, palindromes, perfect number, reciprocals, reciprocal numbers.
 Equality, relationships, & equal arm balances
 Expressions & formulas  Bus, train boarding and deboarding problems for problem solving, expressions & recursive formulas, problem solving strategies of working backwards& acting out.
 Factors  Planning notes, 4 Work sheets: table to list factors 150, table to sort factors by least to most for numbers 150, table to list factors 1100, table to sort factors by least to most for numbers 1100,
 Fibonacci Sequence explanation Slide show that explains the Fibonacci sequence and how to generate it. for generating the sequence & Fibonacci Numbers in Nature First 12 numbers in the sequence .
 Function exploration  includes definitions with examples, diagrams, & representations including a function machine. Also activity with work sheet and charts to explore a function with a variable factor (3) & constant factor (27) & discover casting out nines
 Fundamental Theory of Arithmetic (FTA)  Chart with primes & prime factorization to 30 and blank to 40 to represent the FTA.
 Graph, the math representations behind the modern graph  Story of Nicole Oresme, Father of the modern graph, or as he would say: the Latitude of Forms
 Graphing Notes  Data sheet for coordinate graph, manipulated, responding, independent, dependent, variables, relationships, extrapolation, interpolation, continuous & categorical data
 The Königsberg bridges problem
 Linear equations  fact sheet for y=mx+b, slope, & y intercept
 Graphing Pictures plots  Work sheet to plot a picture and create your own points to plot. Source Johnnie Ostermeyer
 Growth  plants and the mathematics of growth
 Pattern activities  Investigatons to develop patterns & formulas. Starting with number strips, dot patterns & other growing patterns: v, w, pyramid, prism, tower, triangle, rectangle, log stack, can stack, hand shake, dancers, step pyramid, & disc ...
More ... Even, odd, and super even patterns exploration and investigation activities
 Categorical sorting  Activities descriptions, Venn diagrams, sort by one, two, three properties, disjoint groups and intersecting groups
 Happy numbers 
 Stairways and stacking blocks problems  Work sheet, also known as odd squares & tower problem
 Sixteen Colored Squares in an Eight by Eight square problem how many?
 Growth  plants and the mathematics of growth patterns
 Petals around the Rose  is a simulation to find a strategy to determine the number of petals around the rose as simulated by dice. Direcctions & hints .
 Probability activities for see data analysis & probability
 Magic phone number  Work sheet Magic explained with math, Inverse properties of numbers & operations.
 M&M's what colors in a package ?  Work sheet Count, chart, determine percentage, mean , median, mode of M&M's in a package by color.
 Multiplication of whole numbers, multiplication of binomials (x+1)(x+1), & area patterns
 Necklace perfectly beaded  Work sheet to identify a perfect beaded necklace repeating pattern & create one
 Primes and composites. A study of whole numbers  Chart with prime factors for 2  40 to invent the Fundamental Theory of Arithmetic
 Pythagorean Theorem proof
 Toothpick challenges
 Growing squares challenge  Growing squares challenge slides for the challenge above. Slides step through adding toothpicks to make growing squares, record data in a table, and ind the a pattern of increasing squares and increasing number of toothpicks. v
 Growing squares worksheet  Use toothpicks to find a pattern & formulas to determine the number of toothpicks added to make larger squares. Toothpicks in the perimeter, toothpicks to make the squares in the squares, and the number of squares in the growing squares.
 Toothpicks in a rectangular pattern  Worksheet with hints to write equations for the sums of toothpick patterns in a rectangle and discussion ideas for what is an equation, function, and can they be both.
 Relationships  Pools and sidewalks Worksheet growing squares and perimeter patterns, relationships, & functions .
 Simultaneous Equations proofs
 Stem and leaf plot  Work sheet with problems and Java plotter available on the internet
 Variables. One variable activities  pingpong ball race experiment, collect data, analyze (pingpong ball race, how many stars in a minute).
Ratio & proportion
 Jo Tall and Jo Short  Work sheet ratio & proportion activity, historical research problem used to investigate the develop of reasoning as suggessted by Piaget
 Measuring heights of trees & other objects  clinometer, shadows, scale (Biltmore) stick
 Candle problems  Work sheet with sample rate & proportion problem and solution with follow up problem and questions.
 Sack Lunches  Teacher randomly passes out lunches. What are the odds?  Work sheet with hints & discussion ideas.
 Wages for Pet Care
 Acrobats Logic Problem  Work sheet draw a diagram, proportion & problem solving diagrams
Geometry & Spatial reasoning
Knowledge base & tools for mathematical literacy
Geometry originates from Greek ge or gaia meaning earth or land and metria meaning measuring, or measurement of earth or land. Today geometry’s big ideas include the study of spatial properties and relations of points, lines, surfaces, solids, and higher dimensional quantities. Their shapes, variables or invariables, definitions, & proofs.
Analytic geometry is the combination of geometry and algebra.
Relative position is used to locate objects in space with distance and directions.
Knowledge base
Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for geometry and spatial reasoning are:
Tools to facilitate literacy in geometry & spatial reasoning & tools to achieve it
Background information for learning and teaching.
 Spatial and visual representations, abilities, & literacy  article
 Stages of geometric thinking  Pierre and Dina van Hiele' levels of geometric thinking in a chart
 Developmental sequence of geometric ability
 Longitudinal development of volume at the middle level  short article
 Visual and spatial reasoning  article includes visual thinking, related careers, research, skills, perception, memory, creative, six modes of visual learning (developed by the Polaroid Education Program), & instructional activities to develop visual spatial abilities
 Geometry vocabulary  lists of vocabulary for primary, middle, & upper levels.
Activities to facilitate geometry & visual spatial literacy
 Assorted activity suggestions  No Work sheet or detailed explanation: topics include: Volume, Area, Visualization ...
 Area & perimeter assessment  Work sheet with seven direct assessment questions. It defines a hexaright shape, which is used to assess the depth of understanding of area, perimeter and their relationships.
 Area & perimeter  pools and sidewalks  Work sheet patterns, relationships, & functions with area and perimeter
 Cartography map making, angle measurement, compass, clinometer, measure height, distance, locating objects with distance & angle combinations, contour maps, using triangles to measure
 Measuring heights of trees & other objects  clinometer, shadows, scale (Biltmore) stick
 Earth's Circumference, Eratosthenes, Astronomy, Solstices, Equinoxes, Latitude, Science, and Math  Background information and work sheet with instructions to calculate circumference of the Earth.
 Metatrons Cube and Platonic Solids  Link to external site
Puzzles & challenges, with work sheets
 Area of nine shapes in a 17 square unit rectangle  The Pharaoh's problem
 Biggest body of water challenge  Select two bodies of water, lakes, rivers, sea, gulf, ocean.. and use their measurements and geometry to document reasoning for why you believe one is bigger than another. Decide how to measure: one dimension, length, width, height, or two dimensions  area, three dimensions volume or four? Time? …
 Broken Square Puzzle 4 pieces  Work sheet to make a square & solution
 Broken Tee puzzle 1  Work sheet to make a capital T & solution
 Broken Tee puzzle 2  Work sheet to make another capital T & solution
 Five  five piece puzzles  Classic square and four other shapes, directions, hints, solutions, worksheets and puzzle pieces patterns
 Triangles in the shapes puzzle  Work sheet to find triangles & hints
 Squares & triangles  puzzle, challenge, worksheet to find squares & triangles
 Tangrams  Directions for folding & cutting a square sheet of paper or oak tag to make the Tangrams
 Tangram puzzles  Worksheet to find & record fifteen different geometric shapes made with Tangrams
 Tangrams  Work sheet pattern with Tangram pieces in a square
 Three dimensional puzzles  images and patterns
 Four name word puzzle  Work sheet & puzzle
 Fractional parts of a subdivided square Challenge  Work sheet to find & record halves, fourths, eights, & sixteenths of a square with inscribed shapes
 Pentominoes  Describes Pentominoes shapes, history, and puzzles
 How many squares  Cooperative learning work sheet with puzzle that has 40 squares. Also for team building .
 Pattern block star challenge  Work sheet to find the angles of a trapezoid, triangle, parallelogram, and see if they can combine to make a perfect circle. Find what the Angles and Lengths of the Sides are in this
 Surveilence camera puzzle  Work sheet determine the best placement of cameras to surveil eight streets.
 Printable cube pattern  Work sheet
 Pythagoras theorem  Explanation sheet to justify a ^{2} + b ^{2} = c with 1. Area model & 2. lengths of sides model
 Relative position & topology  Work sheet to identify: in, on, out, for 3d & 2d pictures, & draw the 5 shapes. Age: preschool  first grade
 Tessellation  Work sheet: How to make a tessellation
 Triangles  properties, kinds, angles, sides, and relationships
 Ruler and compass constructions
 Cross sections
Tangram puzzle sets and tables
Data Analysis & Probability
Knowledge base & tools to develop mathematical literacy
Knowledge base
The mathematical knowledge base includes concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come to be literate in geometry and spatial reasoning.
 Data analysis
 Probability
 Concepts 912 from Science for All America s (SFAA) & National Assessment for Educational Progress (NAEP )
 Data analysis & probability vocabulary  lists of vocabulary for primary, middle, & upper levels.

Graphing sequence
 K  3 average, bar graph, concrete real object graph, data, grid, picture graph, table, tally, trial
 4  6 broken line graph, descriptive statistics, equally likely events, extrapolate, frequency, generalization, histogram, inference, interpret, interpolate, line graph, line plot, mean, media, midway, mode, ordered pair, outcome, probability, quadrant, random, range, ratio, sample, scale drawin,g simple event, statistics
 7  9 box plot, compound event, circle graph, error of measurement, extrapolate, mutually exclusive, outlier, quartile, scatter plot, skew, lines stem and leaf plot, whisker
 StemandLeaf Plotter
 Box and whisker plotter
 Statistics timeline and history
Information to facilitate literacy in data analysis & probability & tools to achieve it
 Probability  Development with sample problems, explanation of experimental & theoretical probability (with examples) & Reader's Theater carnival problem to solve.
 Probability Rubric  applicable for all levels
 Probability Instructional map  Maps probability as theoretical & experimental concepts, to dice activities, & instruction with a learning cycle instructional method
 Coordinate Graphing Notes  Data sheet for data analysis, variables, relationships, and graphing
 Graph paper  11 high 9 wide 2 cm squares
 Normal curves  Visual graphs & charts with normal curve, standard deviation, z scores, stanines, percentiles, SAT, ACT, IQ, cummulative percentages, quartiles, percent in stanines
 Frequency distribution  visual examples with charts and graphs; central tendency; percentiles; and correlations
Activities to facilitate data analysis & probability literacy
Data analysis
 Bean toss for combination of addends with two dice & probability
 Find your class's hang time and vertical jump statistics  work sheet range, median, mean, mode, graph
 One variable analysis activities  pingpong ball race, how many stars in a minute, collect data, analyze mean, median. mode.
Probability
 One die & two die sum  Complete lesson plans & Work sheets
 Probability unit  6 activities, combines die, spinner, dice, tiles in socks, coin flip, & cup flip in a unit plan for experimental & theoretical probability.
 Sampling number of chips in chocolate chip cookie  Complete lesson plan in linear format
 Spinners  3 activities with spinners of equal sections, some with the same numbers and others with different numbers. Work sheet with hints and spinner patterns.
 Create your own BINGO card for 09 addition  Work sheet with instructional suggestions
 Creat your own BINGO card for 09 subtraction  Work sheet with instructional suggestions
 Body proportions  Work sheet to compare the proportion of four body parts to arm length.
 Flippping 1, 2, 3, & 4 coins  Work sheet with hints.
 Pick two card game  Work sheet for card game to pick two of six to win includes charts, hints, & tree diagram.
 Sickle Cell investigation  Work sheet with Background information & hints. Use of a Punnett Square to determine different probabilities theoretically & experimentally probability
 M&M® Challenge  activity directions and work sheets for data analysis, mean, median, mode, & graphing of colors of M&M's. Can be used for probability.
 Simulation to count a species in a pond population with capture, tag, release, and recapture  Work sheet with hint
 Simulation to determine populations of five species with capture, release, and recapture using ratio and proportion  Work sheet with hint
 Frogs in a pond problem  Work sheet as assessment of ratio, proportion, and capture release. Also used as a Thinking Puzzles: Concrete and Formal Operational Thought Solutions
 The Case of the Carnival Probability Game  Instructional materials: Sample problem with explanation as experimental & theoretical probability & Reader's Theater script for the carnival problem to solve.
 Three coin problem  Work sheet with hints & discussion ideas
 Car keys sweepstakes fair or fixed?  Work sheet with hints & discussion ideas
 Envelopes of cash or not. What are the odds?  Work sheet with hints & discussion ideas.
 Aces for dinner: Which version gives you the better odds?  Work sheet with hints & discussion ideas.
 Sack lunches. Teacher randomly passes out lunches. What are the odds?  Work sheet with hints & discussion ideas.
 Rubber band stretch simulation  is it proportionally accurate?
 John Snow : Analysis for cholera in London, England.
 Snow's map, demonstrated the spatial clustering of cholera deaths around the Broad Street well , provided strong evidence in support of his theory that cholera was a waterborne disease. Snow used some protoGIS methods to buttress his argument: first he drew Thiessen polygons around the wells, defining straightline leastdistance service areas for each. A large majority of the cholera deaths fell within the Thiessen polygon surrounding the Broad Street pump, amd a large portion of the remaining deaths were on the Broad Street side of the polygon surrouding the badtasting Carnaby Street well . Next, using a pencil and string, Snow redrew the service area polygons to reflect shortest routes along streets to wells. An even larger proportion of the cholera deaths fell within the shortesttraveldistance area around the Broad Street pump.
 Finding Pi with probability . Use a toothpick, nail, or needle & sheet of construction paper. Draw parallel lines that are spaced apart the length of the nail across the width of the paper for the length of the paper. Drop the needle from about a half meter above the paper and record if the toothpick lands on a line or not 50 times. Calculate the ratio and see how close it is to π (pi = 3.141592653589793...)
Ratio and Proportion information
Measurement
Knowledge base & tools to develop mathematical literacy
Knowledge base
The mathematical knowledge base includes concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for measurement include:
 Measurement & estimation concepts & misconceptions by levels
 Measurement vocabulary  lists of vocabulary for primary, middle, & upper levels.
 Measurement outcomes  linear, area, volume, capacity, weight, mass, time, money, temperature, estimation. by levels
Information to facilitate literacy in measurement & tools to achieve it (pedagogy)
 Development of measurement is dependent on child and adolescent development , the ability to conserve and perform conservation tasks , unitizing , visualization of units of measurement, and conceptualization of measurement concepts and relationships between different measures.
 Cartography unit  measuring with map making, angle measurement, compass, clinometer, measure height, measure distance, locating objects with distance and angle combinations, contour maps
 Calendars 1776  2000  Print document. Can you discover the day of the week different people were born?
 A History of Measurement in the United States  Timeline
 History of the metric system in the United States  Essay
 Measurement activities  Lesson plan & Work sheets for linear, volume, mass, & density
 Metric fact sheet  Fact sheet vwith metric units: meter, liter, gram, prefixes on a number line, diagram of dust particle & human hair, metric prefixes with symbols to + &  powers of thirty.
 Metric quiz & crossword puzzle review  Work sheet & word bank with answers
 Outcomes by levels  Developmental list of linear, area, volume, capacity, weight, mass, time, money, temperature, estimation outcomes
 Telling time activities  Procedures and activities to teach time at second grade Includes sample pictures
 Timelines : American Education: people, events, and ideas that shaped American Education, Significant Historical Political Events in Human, Development of Significant Science, Mathematical, & Technological Events, History of media and children' literature, & Statistics timeline and history
Historical weights found around the world suggest traders recognized the need for standard measurements.
Activities to facilitate measurement literacy
 Introductory activities should have learners directly compare objects as big and little. Compare hands, clothes, bowls, eating utensils, toys, crayons, blocks, and more. Then have them directly compare and order three objects as big, medium, & small. Work towards sequencing larger sets of objects.
 Use non standard units to measure all sorts of objects.
 Measurement unit  primary level introduction to measurement with first straw activity to develop a concept of standard units of linear measurement , then similar hands on measurement activities for volume, mass, & temperature.
 Measuremen unit  middle level review of linear, mass, volume, measurement of matter, with an introduction to density with directed inquiry activities & blank learner lab notes
 Metric fact sheet  Fact sheet with metric units: meter, liter, gram, prefixes on a number line, diagram of dust particle & human hair, metric prefixes with symbols to + &  powers of thirty.
 Observation & measurement unit  How do we observe? properties to observe, change, measurement, properties to measure, & measurement procedure work sheet
 Density unit  learning cycle inquiry activities for middle grade to apply measurement: linear, mass, volume, of matter to calculate density of solids, liquids, & gases. Includes 14 activities with lesson plans & lab sheets.
 Cartography map making, angle measurement, compass, clinometer, measure height, measure distance, locating objects with distance & angle combinations, contour maps,
 Measuring heights of trees & other objects  clinometer, shadows, scale (Biltmore) stick
 21 Centimeter card game : make a deck with 45 cards. Draw a slanted straight line (at different degrees on each card) with the following lengths, on cards as listed below:
 4 cards each with a  1 cm line,
 4 cards each with a  2 cm line,
 4 cards each with a  3 cm line,
 4 cards each with a  4 cm line,
 4 cards each with a  5 cm line,
 5 cards each with a  6 cm line,
 5 cards each with a  7cm line,
 5 cards each with a  8 cm line,
 5 cards each with a  9 cm line, &
 5 cards each with a  10cm line.
 Linear & volume 
 Solar system  big distances & volume activities (5E learning cycle) to explore the size and relative position of the solar system aligned to the Common Core Standards, Next Generation Science Standards, National Science Education Standards, & 21st Century Skills: critical thinking, communications and measurement.
 Temperature How does temperature in centigrade Celsius compare to temperature in Fahrenheit? Facts: Water freezes at 32 degrees Fahrenheit and 0 degrees Celsius. Water boils at 212 degrees Fahrenheit and 100 degrees Celsius. Find a thermometer that has both scales and read from one scale to the other. C = (F  32) * 5/9; C = (F  32) / 9/5; F = (C * 9/5) + 32 Verify or prove the formulas.
 Money  Sequence of outcomes to develop with activities to learn to count change:
 Know people use money to buy goods and services.
 Sort money by its appearance.
 Draw pictures of coins and paper money.
 Select coins on their desk (penny, quarters, dimes and nickel) as called.
 Identify and state the values of money (penny = 1 cent , quarter = 25 cents, dime = 10 cents, nickel = 5 cents, ... ).
 Sort money by its value.
 Order money by its value.
 Recognize different coins can have similar values.
 Represent coins and their values: draw a circle for each coin (penny, quarter, dime, nickel) with the value inside: (25, 25, 10, 10, 5).
 Identify a coin, its value, and point to its value represented on a number dot sequence, trail, or line . Select a second coin. Identify the coin, its value, and show how to add and represent its value and sum of both.
 Identify a coin, its value, and point to its value represented on a labeled hundreds chart . Select a second coin. Identify the coin, its value, and show how to add and represent its value and sum of both. Draw an arrow from 1 to the value of the first coin. Circle the value and draw an arrow for the value of the next coin and circle the sum. (arrow math)
 Use arrow math and a labeled hundreds chart chart to solve coin problems with two addends.
 Use arrow math and a labeled hundreds chart chart to solve coin problems with more than two addends.
 Given an assorted collection of coins (that add to less than 100 cents), Use arrow math and a labeled hundreds chart chart to solve coin problems with more than two addends.
 Identify different orders to count change more efficiently. (Start with the largest coins, group similar coins, group by easy numbers (10, 50, 100, & 25).
 Repeat the above procedures on a blank hundreds chart .
 Repeat the above procedures with paper money.
Game play : Deal three cards to each player and play like 21. Player that line lengths add closest to 21 cm, without going over is the winner. After the deal each player can choose to be dealt another card or hold. Player closest to 21 cm, with out going over wins. Winner measures each card with a cm ruler to verify their score.
Alternative version : Play as above and record the lengths of their cards, if they don't go over 21. First player to 121 wins.
Can use the counting money strategy for counting decimal numbers. Write the starting value (1.89), select a count (.01), write it, write the sum (1.90), write the next coin (.10), then the sum (2.00), and continue as before.
Process dimensions of mathematics
Information to plan and facilitate the learning and use of skills in the process dimensions of mathematics.
Problem solving Tools to facilitate mathematical problem solving
 Problem solving oncepts, misconceptions, & outcomes by levels in the mathematical knowledge base
Information to facilitate literacy in problem solving & tools to achieve it
 Development of problem solving  page contents includes:
 Heuristic :
 Strategies for solving problems with sample problems
 Planning & teaching tools pedagogical and curriculum
 Teacher's role in problem solving
 Suggestions to help learners solve problems
 Assessment sample questions to assess learners' mathematical problem solving
 Characteristics of a good problem ideas to consider when writing or selecting problems
 Planning questions to prepare to facilitate problem solving & investigating suggestions to consider when writing or planning to use problems
 Problem Solving Goals and Outcomes to review for the inclusion in appropriate curricular documents and plans
 Problem solving vocabulary to introduce to learners
 Dispositions to review & select to focus on
 Learner's materials
 Learners problem solving guide or checklist to use to as a guide with suggestions to solve problems
 Learners solving checklist  with before, during, and after suggestions
 Learners Solving Attitude Inventory with yes  no questions
 See also communication dimension 
Activities to facilitate problem solving
 Bucket problems application of addition, subtraction, problem solving, combinations, reasoning.
 How many squares? Problem / activity
 Teaching notes for forty squares problem / activity
 Problems to solve by strategy for solving
 Expressions & formulas  Bus, train boarding and deboarding problems for problem solving, expressions & recursive formulas, problem solving strategies of working backwards& acting out
 Sack Lunches  Teacher randomly passes out lunches. What are the odds?  Work sheet with hints & discussion ideas.
 Acrobats Logic Problem  Work sheet draw a diagram, proportion & problem solving diagrams
 Sack Lunches  Teacher randomly passes out lunches. What are the odds?  Work sheet with hints & discussion ideas.
 Wages for Pet Care
 Acrobats Logic Problem  Work sheet draw a diagram, proportion & problem solving diagrams
Reasoning and proof
 Reasoning & proof Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base
Information to facilitate literacy in reasoning & proof & tools to achieve it
 Reasoning and Proof Explanations and development of reasoning, proof, & logic  information on reasoning and proof with examples of different types of proof, teaching suggestions, & an extensive scoring guide
 Reasoning and proof big idea, concept, or generalization instructional planning map
 Development of reasoning from birth to maturity
 Reasoning, conservation, and cognitive development tasks & activities
 Application of proof by elimination & verification with a card trick
 Bucket problems reasoning, proof, conjecture, combinations
 Decision making, critical thinging, and change processes
 Structure for Analyzing and Presenting Arguments
 Logical Reasoning and Reasoning Errors
 Position analysis
 Issue analysis & sample issues analysed for democratic education for all
 Reasoning solutions for SUDOKU
 Imagining
 An Outline of Goals for a Critical Thinking Curriculum and Its Assessment
Activities to facilitate reasoning & proof
 Bucket problems application of addition, subtraction, problem solving, combinations, reasoning.
 How many squares? Problem / activity
 Teaching notes for forty squares problem / activity
 SUDOKU
 Jo Tall and Jo Short  Work sheet ratio & proportion activity, historical research problem used to investigate the develop of reasoning as suggessted by Piaget
 Acrobats Logic Problem  Work sheet draw a diagram, proportion & problem solving diagrams
Representation
Representation Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base
Information to facilitate representation & tools to achieve it
 Examples of mathematical representations with language as nouns and verbs.
 Function exploration  includes definitions with examples, diagrams, & representations including a function machine. Also activity with work sheet and charts to explore a function with a variable factor (3) & constant factor (27) & discover casting out nines
Area representations
Lynne Outhred and Michael Mitchelmore asked learners to illustrate the area of a rectangular shape. They found learners demonstrated four different strategies: incomplete covering, visual covering, concrete covering, and measurement. The drawings below show samples from each.
Representation of relative position of objects
Illustration of Internal Representation
Actual object, real world, or external representation
Representation of external objects with points on an other external object
What does this have to do with mathematical representations?
Activities to facilitate representation
 Representations of 21 / 7
 Area & perimeter  pools and sidewalks  Work sheet patterns, relationships, & functions with area and perimeter
 Growing squares challenge  Growing squares challenge slides for the challenge above. Slides step through adding toothpicks to make growing squares, record data in a table, and ind the a pattern of increasing squares and increasing number of toothpicks. v
 Growing squares worksheet  Use toothpicks to find a pattern & formulas to determine the number of toothpicks added to make larger squares. Toothpicks in the perimeter, toothpicks to make the squares in the squares, and the number of squares in the growing squares.
 Toothpicks in a rectangular pattern  Worksheet with hints to write equations for the sums of toothpick patterns in a rectangle and discussion ideas for what is an equation, function, and can they be both.
 Pattern activities  Investigatons to develop patterns & formulas. Starting with number strips, dot patterns & other growing patterns: v, w, pyramid, prism, tower, triangle, rectangle, log stack, can stack, hand shake, dancers, step pyramid, & disc ...
Communication
Communication Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base
Information to facilitate literacy in communitation & tools to achieve it
 Different ways to communicate problems with sample problems
 Ways to communicate mathematical representations as objects, properties, actions, & operations
 Graphing with line graphs as communication
Activities to facilitate communication
Connections & perspective
Connections & perspective Concepts, misconceptions, & outcomes by levels in the mathematical knowledge base
Information to facilitate literacy in connections & perspective & tools to achieve it
 Mathematical ideas build upon each other.
 Mathematical ideas are connected to other mathematical ideas.
 Mathematical ideas are connected to the world.
Activities to facilitate reasoning & proof
Technology, computer programming, software, 3D printing
 Coding software 
Mozilla free
Google free
 Made with code  projects, mentors, resources ...
Technology
 History of the cell phone Read ... Cell phone mathematics . The story of how Brad Parkinson used creativity and mathematics to enable the use of cell phones.
Math software
 History of graphing software Read : The Graphing Calculator Story by Ron Avitzur, then below the article are links to download the software.
Coding resources
 Spies in the Computer  basic software program and class activity on a 10 x 10 grid
Pedagogical resources
 Viewpoint: Arguments for a LOGO  Pascal Computer Curriculum
 Logo dissertation review by Barbara Elias  Dealing with Logo and the transfer of knowledge from learning one programming program to learning another
3 D printing
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Page Overview
 Mathematical literacy knowledge base of concepts & outcomes
 Pedagogical knowledge to develop professionally and create learning experiences to facilitate mathematical literacy and teacher tools to assist.
Development, planning, activities , problems, learning units & packets to develop mathematical literacy by dimension
Process dimensions:
Content : dimensions
 Number value & Operations (addition, subtraction, multiplication, & division)
 Activities  problems, units, packets ...
 Algebra, Patterns, Ratio, & Proportion
 Activities  problems, units, packets ...
 Geometry & Visual spatial reasoning & representation
 Activities  problems, units, packets ...
 Data Analysis & Probability
 Activities  problems, units, packets ...
 Measurement
 Activities  problems, units, packets ...
 Technology  calculator, programming, & other math related tech ideas
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There are 10 kinds of people in the world. Those who understand binary and those who don't.
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Jeffery Weld asks:
Why is it whenever students don't do well on a national test the news agencies report one or a few particular items that were on the test by stating the test questions, then they believe it is necessary to provide the answers.
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If their listeners need the answers, why should we expect students to be able to answer it correctly?
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The Jibaro Indians of the Amazon rain forest express the number five by the phrase:
"wehe amukei,"
"I have finished one hand,"
and the number ten by
"mai wehe amukei,"
"I have finished both hands."
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"These landmarks must not be treated as objectives to be marked off. The goal is to mathematize not pass landmarks. Enjoy the journey create systems to do mathematics with the knowledge that students have. One should recognize the fallacy of thinking there can be one activity or sequence of activities for all students."
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One of the universal customs that man has successfully established on earth is the
Hindu  Arabic numerals
to record numbers.
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Math is meditation for thinking about the world.
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Geometry is the science of correct reasoning on incorrect figures.
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Galileo was put under house arrest and allowed visitors, but no mathematicians .
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Common sense & knowledge
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Counting in Winnebago
 WI ^{n}
 NO ^{n} BA
 THABTHI ^{n}
 DUBA
 SATO ^{n}
 SHAPE
 PETHO ^{n} BA
 PETHABTHI ^{n}
 SHO ^{n} KA
 GTHEBO ^{n}
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Interesting book
The Power of Logical Thinking: Easy Lessons in the Art of Reasoning... and Hard Facts About Its Absence in Our Lives, Marilyn Vos Savant (1996) ISBN 0312139853 Saint Martin's Press: New York.
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Olber's Paradox
If the universe is infinite in time and space, stars should occupy every point in space and fill the night sky with light.
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Directory links :
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Mathmaticians think about more than math.
Earth will run out of the basic resources, and we cannot predict what will happen after that. We will run out of water, air, soil, rare metals, not to mention oil. Everything will essentially come to an end within fifty years. What will happen after that? I am scared. It may be okay if we find solutions, but if we don’t then everything may come to an end very quickly. Mathematics may help to solve the problem, but if we are not successful, there will not be any mathematics left, I am afraid!
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U.S. Students
Score in the top tier on
the PISA 2012 Math Creative Problem SolvingAssessment
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Measurement is never exact .
It is only as accurate as who and with what is doing the measurement.
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Read ...
The story of how Brad Parkinson used creativity & mathematics to enable the use of cell phones.
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Data is just numbers. They need an explanation or theory to make it data .
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Algebra was greatly advanced by alKhwarizmi’s book:
Hisab aljabr w'almuqabala From which the word algebra came.
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