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 1-10 beans

Number relationships --->>> see patterns and algebra

• Card set of different colored shapes for classification

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:
  • 1 dot
  • 2 dots
  • 3 dots
  • 4 dots
  • 5 dots
  • 6 dots
  • 6 dots with five as anchor
  • 7 dots
  • 7 dots with five as anchor
  • 8 dots
  • 8 dots with five as anchor
  • 9 dots
• Ten frame cards
  • zero dots
  • 1 dot
  • 2 dots
  • 3 dots
  • 4 dots
  • 5 dots
  • 6 dots
  • 7 dots
  • 8 dots
  • 9 dots
  • 10 dots

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 (1-12) 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
• Add-on or count-back

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 (+-*/)

• Number line as representation for Subtraction
• Activities for 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

• Addition and subtraction sequence
• Multiplication and division sequence

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).

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 1-50, table to sort factors by least to most for numbers 1-50, table to list factors 1-100, table to sort factors by least to most for numbers 1-100,
• 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 - ping-pong ball race experiment, collect data, analyze (ping-pong 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

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

• 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

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 - ping-pong 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 0-9 addition - Work sheet with instructional suggestions
• Creat your own BINGO card for 0-9 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 water-borne disease. Snow used some proto-GIS methods to buttress his argument: first he drew Thiessen polygons around the wells, defining straight-line least-distance 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 bad-tasting 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 shortest-travel-distance 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

• See ratio & proportion in algebra, patterns, & functions

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.

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.

• 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.
    • Solar System Size & Scale K-4th
    • Solar System Size & Scale 5th
    • Solar System Size & Scale Middle School
    • Solar System Size & Scale K - 4th Grade Lesson
    • Solar System Size & Scale 5th - 8th Grade Lesson
• 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.

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.