# Numbers, their values and operations

*Page Overview*:

- Knowledge base & tools to develop mathematical literacy
- Kinds of numbers & their definitions
- Development & assessment
- Operations
- Activities
- Number sense - counting, cardinality, number vlaue, < > =
- Place value
- Number theory
- Fractions, decimals, & %,
- Basic Operations

(+ - * /) - Integers

Other development & activities pages:

*Process dimensions*:

*Content dimensions*:

## 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
- 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

*Vocabulary lists related to number sense:*

- Number sense & operations
- Vocabulary for operations by grade levels
- Operations
- Fractions

### Developmental information for numbers sense

Numbers & their definitions - *Fact sheet* of natural, integer, rational, irrational and real numbers. Their definitions, examples, & number line visual representation.

*Numbers: Counting, recognition, cardinality*

*Development*

- 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
- Whole Number value (6/8 years & beyond) - development of number value after number conservation & one-to-one correspondence to hierarchial inclusion
- How to say, read, & write numbers from billion to billionths - lesson plan & worksheets
- Framework for planning counting and cardinality units

*Assessment information*

- 13 Whole number assessment tasks - Scripts, materials, scoring guides, and Summative record sheet for all, K-2.
- Counting with synchrony or one-to-one correspondence & cardinality [
*Synchrony*information] - script with materials, problems, suggestions & scoring guide - Rote counting to 100 - script with problems, suggestions & scoring guide
- Writing numerals 1-9 - 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 0ne-thousand requires memory of the necessary words, but also a conceptual understanding of ten repeated 100's. - Numeral recognition - (1-20) with worksheet, script, suggestions & scoring guide
- Number recognition - prompt sheets (numerals 1-10, & 10-20 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 One-to-one 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

- Counting with synchrony or one-to-one correspondence & cardinality [

#### Place value development & assessment

*Development*

- Place value- development from age 5 on

*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, part-part-whole, & 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 1-10 beans

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

#### Dot plates & electronic dots for number sense, & subitization: activities

- Dot plates video and explanation about memory, number value, and basic facts
- Dots flash one to five
- Dots flash one to ten
- Add-on or count-back
- 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 (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*: Children are often confused with number lines. They are not sure what to count: numbers, lines, spaces and where to start (0, 1, ...) Therefore, their use for children below second grade is questionable.

However, numbers in a line, or numbers on dots organized in a line are appropriate for young learners. See:

- Introduction to a number line.
- Number line - place values from billions to billionths
- Numbers listed - in order from billions - billionths

### 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
- 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
- Percentage - sample actvities 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 (+-*/)

#### 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

### 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).