Numbers: values and basic operations

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 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
• 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

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 important representation and 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).