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Mathematics

Arithmetic

The concepts and processes to be acquired and mastered in arithmetic constitute the building blocks of mathematics, since they are applied in all other branches of this subject.

Understanding and writing numbers

Meaning of operations involving numbers
Operations involving numbers

Number sense is a concept that is developed in early childhood and is refined as students progress through school. In elementary school, it is developed first by looking at natural numbers and then enriched by studying rational numbers.1

At the outset, counting rhymes, counting, constructions, representations, ordering and establishing relationships among numbers are essential in order for students to understand number systems. Using appropriate manipulatives, students first learn about counting groups (grouping) and gradually replace this concept with place value. However, care must be taken not to progress too quickly from one concept to another, as this could affect the way students understand operations or learn new numbers. 

It is in elementary school that students acquire the basic tools for understanding and using fractions. Students must first understand concepts (meaning) before they can understand calculation processes (operations). This can be achieved by allowing students to systematically use concrete materials and pictorial representations when dealing with situations involving fractions.

The table below presents the learning content associated with understanding and writing numbers. The concepts and processes targeted will provide students with increasingly complex tools that will help them develop and use all three mathematics competencies.

Understanding and writing numbers

Student constructs knowledge with teacher guidance.

Student applies knowledge by the end of the school year.

 

Student reinvests knowledge.

Elementary
Cycle
One
Cycle
Two
Cycle
Three
1 2 3 4 5 6
  1. Natural numbers less than . . .
1000 100 000 1 000 000
  1. Counts or recites counting rhymes involving natural numbers
    1. counts forward from a given number
       
    1. counts forward or backward
    1. skip counts (e.g. by twos)
  1. Counts collections (using objects or drawings)
    1. matches the gesture to the corresponding number word; recognizes the cardinal aspect of a number and the conservation of number in various combinations
         
    1. counts from a given number
       
    1. counts a collection by grouping or regrouping
    1. counts a pre-grouped collection
   
  1. Reads and writes any natural number
  1. Represents natural numbers in different ways or associates a number with a set of objects or drawings
    1. emphasis on apparent, accessible groupings using objects, drawings or unstructured materials
      (e.g.  tokens, nesting cubes, groups of ten objects placed inside a bag and ten of these bags placed inside another container)
       
    1. emphasis on exchanging apparent, non-accessible groupings, using structured materials
      (e.g. base ten blocks, number tables)
     
    1. emphasis on place value in non-apparent, non-accessible groupings, using materials for which groupings are symbolic (e.g.  abacus, money)
   
  1. Composes and decomposes a natural number in a variety of ways
      (e.g. 123 = 100 + 23
      123 = 100 + 20 + 3
      123 = 50 + 50 + 20 + 3
      123 = 2 × 50 + 30 − 7
      123 = 2 × 60 + 3)
  1. Identifies equivalent expressions
    (e.g. 52 = 40 + 12, 25 + 27 = 40 + 12, 52 = 104 ÷ 2)
  1. Compares natural numbers
  1. Arranges natural numbers in increasing or decreasing order
  1. Describes number patterns, using his/her own words and appropriate mathematical vocabulary (e.g. even numbers, odd numbers, square numbers, triangular numbers, prime numbers, composite numbers)
  1. Locates natural numbers using different visual aids
    (e.g. hundreds chart, number strip, number line)
  1. Identifies properties of natural numbers

    1. odd or even numbers
       
    1. square, prime or composite numbers
       
  1. Classifies natural numbers in various ways, based on their properties
    (e.g. even numbers, composite numbers)
  1. Approximates a collection, using  objects or drawings
    (e.g. estimate, round up/down to a given value)
  1. Represents the power of a natural number
       
    Vocabulary
    Grouping, digit, number, unit, tens place, hundreds place
    Natural number, even number, odd number
    Is equal to, is bigger than (is greater than); is smaller than (is less than)
    Increasing order, decreasing order
    Number line
    Symbols
    0 to 9, <, >, =, numbers written using digits
       
    Vocabulary
    Base ten, position, place value, thousand, thousands place, ten thousands
    Is not equal to; is greater than; is less than
    Square number, composite number, prime number
    Symbols
    ≠, numbers written using digits
       
    Vocabulary
    Hundred thousands, million
    Exponent, power, squared, cubed
    Parenthesis
    Symbols
    ( ), numbers written using digits, exponential notation
       
  1. Fractions (using objects or drawings)
1 2 3 4 5 6
  1. Identifies fractions related to everyday items (using objects or drawings)
       
  1. Represents a fraction in a variety of ways, based on a whole or a collection of objects
  1. Matches a fraction to part of a whole (congruent or equivalent parts) or part of a group of objects, and vice versa
       
  1. Identifies the different meanings of fractions (sharing, division, ratio)
   
  1. Distinguishes a numerator from a denominator
       
  1. Reads and writes a fraction
       
  1. Compares a fraction to 0, ½ or 1
       
  1. Verifies whether two fractions are equivalent
   
  1. Matches a decimal or percentage to a fraction
     
  1. Orders fractions with the same denominator
       
  1. Orders fractions where one denominator is a multiple of the other(s)
       
  1. Orders fractions with the same numerator
       
  1. Locates fractions on a number line
       
    Vocabulary
    Fraction, half, one third, one quarter
       
    Vocabulary
    Numerator, denominator
    Whole, equivalent part, equivalent fraction
    Symbol
    Fractional notation
       
  1. Decimals up to . . .
1 2 3 4 5 6
 

 hundredths 

 thousandths 

  1. Represents decimals in a variety of ways (using objects or drawings)
   
  1. Identifies equivalent representations (using objects or drawings)
   
  1. Reads and writes numbers written in decimal notation
   
  1. Understands the role of the decimal point
       
  1. Composes and decomposes a decimal written in decimal notation
   
  1. Recognizes equivalent expressions
    (e.g. 12 tenths is equivalent to 1 unit and 2 tenths; 0.5 is equivalent to 0.50)
   
  1. Locates decimals on a number line
    1. between two consecutive natural numbers
   
    1. between two decimals
     
  1. Compares two decimals
   
  1. Approximates
    (e.g. estimates, rounds to a given value, truncates decimal places)
   
  1. Arranges decimals in increasing or decreasing order
   
  1. Matches
    1. a fraction to its decimal
       
    1. a fraction or percentage to its decimal
       
    Vocabulary
    Decimal, tenth, hundredth
    Symbol
    Decimal notation
       
    Vocabulary
    Thousandth
    Symbol
    Decimal notation
       
  1. Integers
1 2 3 4 5 6
  1. Represents integers in a variety of ways (using objects or drawings)
    (e.g. tokens in two different colours, number line, thermometer, football field, elevator, hot air balloon)
     
  1. Reads and writes integers
       
  1. Locates integers on a number line or Cartesian plane
       
  1. Compares integers
       
  1. Arranges integers in increasing or decreasing order
       
    Vocabulary
    Integer
    Negative number, positive number
    Symbols
    Integer notation, +/– calculator key
       

Meaning of operations involving numbers
Operations involving numbers

1.  The set of rational numbers includes the set of integers, which contains the set of natural numbers.

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