# Numbers and their operations

*Overview*

- Kinds of numbers & their definitions
- Number sense
- Place value
- Fractions, decimals, & %,
- Basic Operations

(+ - * /) - Integers

## Kinds of Numbers and their definitions

Natural numbers (whole numbers), integers, rationals, real numbers, and complex numbers.

*Natural numbers*known as counting numbers and whole numbers. They are 1, 2, 3, 4, ... and sometimes include 0.*Integers*positive and negative whole numbers. Numbers that can be written without a fraction. They are ... -4, -3, -2, -1, 0, 1, 2, 3, 4, ...*Rational numbers*are numbers that can be represented as a fraction (N/D) of two integers. N is the a numerator and D is the denominator. D is always defined as not zero (0). If D is equal to one (1), then it is an integer. Therefore, every integer is a rational number. The set of all rational numbers is usually represented as**Q**. A decimal number is a rational number if it terminates. That is it has a finite number of digits or repeats a finite sequence of digits (.125125...). These statements hold true for any integer base number (binary, hexadecimal).*Irrational numbers*are numbers such as √2 (1.41421356…, π (3.14159265…. Numbers that do not terminate as a decimal number.*Real numbers*include rational numbers (integers, fractions) and irrational numbers. Real numbers can be represented as a point on a number line. Where points represent equally spaced integers.

*Photo source*: Phrood [Public domain], via Wikimedia Commons

## Development of Number sense

- How children and others learn Prenumber sense (birth - 6/8 years)
- How children and others learn number sense beyond cardinality (6/8 years and beyond)
- Classification concepts & misconceptions by levels for mathematical & science thinking
- Subitize
- Summary of counting concepts outline (3 - 8 years)

## Instructional Activities and Teacher tools

- Number sense concepts & misconceptions
- Whole numbers concepts & misconceptions
- Vocabulary for number sense & operations
- Writing numerals mnemonic phrases to help student remember how to for numerals.

### Assessment - Scoring guides and rubrics for number sense

- Whole number assessment information [Summative record sheet for all catgories]
- Rote counting to 100
- Rote counting by multiples, 2, 5, 10, & 3
- Counting back from 10, 20, 100
- Counting to and from 1 000. 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
- Writing or forming numerals
- Subitizing or instant recognition of values
- Counting with synchrony or one-to-one correspondence & cardinality [
*Synchrony*information] - Assessment for One-to-one correspondence with matching
- Cardinality matching subitizing or visual pattern recognition
- More or less
- Cardinality match numeral and number word
- Hierachial inclusion to five

- Cardinal scoring guide and rubric
- Number recognition worksheet (numerals 1-10, & 10-20 randomly arranged)
- Conservation of number tasks
- Pre-Place Value, number sequence 10 and more to 100 assessment and record sheet

### Activities and Instructional Suggestions for number sense

- 100's of Instructional ideas and activities for number sense
- Counting and cardinality Lesson plan - review also as a planning framework
- Memory, basic facts, and dot plates
- 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 for number value

- Numerals 0 - 9
- Number words 0-9
- Ordinal words 1-10
- Dot cards - 1 dot
- Dot cards - 2 dots
- Dot cards - 3 dots
- Dot cards - 4 dots
- Dot cards - 5 dots
- Dot cards - 6 dots without five as anchor
- Dot cards - 6 dots with five as anchor
- Dot cards - 7 dots with five as anchor
- Dot cards - 8 dots with five as anchor
- Blank cards

Die roll ---

Children confuse numbers, lines, and spaces on number lines making their use questionable for children below second grade.

Introduction to a number line.

## Place value

- Development of place value from age 5 on.
- Place value concepts and misconceptions
- Assessment activities and recording sheets

### 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 smaller.
- Blank hundred chart larger
- Hundreds frame with dots.
- Thousand chart
- Hundred chart puzzles and puzzle pieces puzzles

### 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.

## Fractions, Decimals, & Percents

- Concepts & misconceptions [fractions] [decimals] [percents]
- Fractional values compared problems
- Ratio and Proportion Development with sample problems
- Ratio and Proportion
Problems

- Sequences
- Assessment ideas and activities
- Vocabulary
- Rubric
- Ramblings or maybe not about Whole numbers and number lines and rational and irrational numbers and Infinity

## Basic Operations

- Vocabulary for operations by grade levels
- Concepts & misconceptions for addition and subtraction
- Number line as representation for Subtraction.

### Addition and subtraction whole numbers

- Transitional activities from number sense to addition and subtraction
- Developmental Sequence and ideas to facilitate instruction for addition and subtraction of whole numbers
- Types of addition and subtraction problems - chart
- Research supporting

- Addition and Subtraction - check list with problems
- Addition and subtraction assessment Join, separate, part-part-whole, & combine & Possible levels of calculation guide
- Math strategies for adding and subtracting mentally with instructional suggestions
- Sample subtraction algoritmic solutions
- Add on or count back - slide example
- Bucket problems application of addition, subtraction, problem solving, combinations, reasoning.

### Multiplication and division of whole number

- Concepts and misconceptions
- Development or multiplication and division understanding and strategies
- Instructional notes
- 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 in.
- Unpack standards for multiplication and division.
- Representations of 21 / 7

### Fractional numbers

- Concepts and misconceptions
- 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
- Fractions, ratio, and proportion notes

### Decimal numbers

- Addition and subtraction sequence
- Multiplication and division sequence

### Integers

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).