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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.

Operations involving numbers

Understanding and writing numbers
Meaning of operations involving numbers

As students gradually develop their number and operation sense, they will be called upon to develop their own processes and adopt conventional ones in order to perform various operations. They will learn to recognize equivalencies between these different processes and to develop certain automatic responses. Using these processes and the properties of operations, they will also learn to estimate results and obtain accurate results using mental and written computation.

The situations presented should involve numerical and non-numerical patterns (e.g. colours, shapes, sounds) to allow students to observe and describe various patterns and series of numbers and operations, such as a sequences of even numbers, multiples of 5 and triangular numbers. These situations will also require students to add terms to a series, state general rules or build models. Thus, students will learn to formulate or deduce definitions, properties and rules.

In all cycles, calculators may be used to good advantage as a calculation, verification and learning tool (e.g. in situations involving patterns, number decomposition, or the order of operations).

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

Operations involving 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. Natural numbers
    (based on the benchmarks for each cycle)
1 2 3 4 5 6
  1. Approximates the result of
    1. an addition or subtraction involving natural numbers
       
    1. any of the four operations involving natural numbers
   
  1. Builds a repertoire of memorized addition and subtraction facts1
    1. Builds a memory of addition facts2 (0 + 0 to 10 + 10) and the corresponding subtraction facts, using objects, drawings, charts or tables
       
    1. Develops various strategies that promote mastery of number facts and relates them to the properties of addition
     
    1. Masters all addition facts (0 + 0 to 10 + 10) and the corresponding subtraction facts
     
  1. Develops processes for mental computation
    1. Uses his/her own processes to determine the sum or difference of two natural numbers
       
    1. Uses his/her own processes to determine the product or quotient of two natural numbers
   
  1. Develops processes for written computation (addition and subtraction)
    1. Uses his/her own processes as well as objects and drawings to determine the sum or difference of two natural numbers less than 1000
       
    1. Uses conventional processes to determine the sum of two natural numbers of up to four digits
       
    1. Uses conventional processes to determine the difference between two natural numbers of up to four digits whose result is greater than 0
       
  1. Determines the missing term in an equation (relationships between operations): a + b = □, a + □ = c, □ + b = c, ab = □, a – □ = c, □ – b = c
       
  1. Builds a repertoire of memorized multiplication and division facts
    1. Builds a memory of multiplication facts (0 × 0 to 10 × 10) and the corresponding division facts, using objects, drawings, charts or tables
       
    1. Develops various strategies that promote mastery of number facts and relate them to the properties of multiplication
     
    1. Masters all multiplication facts (0 × 0 to 10 × 10) and the corresponding division facts
     
  1. Develops processes for written computation (multiplication and division)
    1. Uses his/her own processes as well as materials and drawings to determine the product or quotient of a three-digit natural number and a one-digit natural number, expresses the remainder of a division as a fraction, depending on the context
       
    1. Uses conventional processes to determine the product of a three-digit natural number and a two-digit natural number
       
    1. Uses conventional processes to determine the quotient of a four-digit natural number and a two-digit natural number, expresses the remainder of a division as a decimal that does not go beyond the second decimal place
       
  1. Determines the missing term in an equation (relationships between operations): a × b = □, a × □ = c, □ × b = c, a ÷ b = □, a ÷ □ = c, □ ÷ b = c
   
  1. Decomposes a number into prime factors
     
  1. Calculates the power of a number
       
  1. Determines the divisibility of a number by 2, 3, 4, 5, 6, 8, 9, 10
       
  1. Performs a series of operations in accordance with the order of operations
       
  1. Using his/her own words and mathematical language that is at an appropriate level for the cycle, describes
    1. non-numerical patterns (e.g. series of colours, shapes, sounds, gestures)
       
    1. numerical patterns (e.g. number rhymes, tables and charts)
       
    1. series of numbers and family of operations
  1. Adds new terms to a series when the first three terms or more are given
  1. Uses a calculator and
    1. becomes familiar with its basic functions (+, –, =, 0 to 9 number keys, all clear, clear)
       
    1. becomes familiar with its × and ÷ functions 
       
    1. becomes familiar with memory keys and change of sign keys (+/–)
       
    Vocabulary
    Pattern, series
    Symbols
    Calculator keys
       
  1. Fractions (using objects or diagrams)
1 2 3 4 5 6
  1. Generates a set of equivalent fractions
   
  1. Reduces a fraction to its simplest form (lowest terms)
       
  1. Adds and subtracts fractions when the denominator of one fraction is a multiple of the other fraction(s)
       
  1. Multiplies a natural number by a fraction
       
    Vocabulary
    Irreducible fraction
       
  1. Decimals
1 2 3 4 5 6
  1. Approximates the result of
    1. an addition or a subtraction
   
    1. a multiplication or division
       
  1. Develops processes for mental computation
    1. adds and subtracts decimals
   
    1. performs operations involving decimals (multiplication, division by a natural number)
       
    1. multiplies and divides by 10, 100, 1000
       
  1. Develops processes for written computation
    1. adds and subtracts decimals whose result does not go beyond the second decimal place
       
    1. multiplies decimals whose product does not go beyond the second decimal place
       
    1. divides a decimal by a natural number less than 11
       
    Symbols
    $, ¢
       
  1. Using Numbers
1 2 3 4 5 6
  1. Expresses a decimal as a fraction, and vice versa
       
  1. Expresses a decimal as a percentage, and vice versa
       
  1. Expresses a fraction as a percentage, and vice versa
       
  1. Chooses an appropriate number form for a given context
       
    Vocabulary
    Percentage
    Symbol
    %
       

Understanding and writing numbers
Meaning of operations involving numbers

1.  The development of a repertoire of number facts requires more than mere memorization of tables.
2.  The basic additions (and the corresponding subtractions) and multiplications (and the corresponding divisions) include operations whose terms and factors are less than 11. 

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