• Properties of operations are helpful for carrying out calculations.
  • The identity property: a + 0 = a, a – 0 = a, a × 1 = a,  = a.
  • The commutative property: a + b = b + a, a × b = b × a.
  • The associative property: (a + b) + c = a + (b + c), (a × b) × c = a × (b × c).
  • The distributive property: a × b = (c + d) × b = c × b + d × b.
• The commutative, associative, and identity properties can be applied for any type of number.
• When an expression includes multiple operations, there is a convention that determines the order in which those operations are performed:
  • Do calculations in the brackets first.
  • Then evaluate the exponents and roots (exponentiation).
  • Then multiply and divide in the order that these operations appear from left to right (multiplication/division).
  • Then add and subtract in the order that these operations appear from left to right (addition/subtraction).
• Multi-step problems may involve working with a combination of whole numbers, decimal numbers, and positive fractions.
• Multi-step problems may involve working with a combination of relationships, including ratios, rates, and percents.
• There may be more than one way to solve a multi-step problem.

Note

• This expectation supports most other expectations in the Number strand and is applied throughout the grade. Whether working with numbers or operations, recognizing and applying properties and relationships builds a strong foundation for doing mathematics.
• Problems that involve rational numbers in this grade include whole numbers, integers, positive decimal numbers, and positive fractions.
• Solving problems with more than one operation involves similar processes to solving problems with a single operation. For both types of problems:
  • Identify the actions and quantities in a problem and what is known and unknown.
  • Represent the actions and quantities with a diagram (physically or mentally).
  • Choose the operation(s) that match the actions to write the equation.
  • Solve by using the diagram (counting) or the equation (calculating).
• In multi-operation problems, sometimes known as two-step problems, there is often an ultimate question (asking for the final answer or result being sought), and a hidden question (a step or calculation that must be taken to get to the final result). Identifying both questions is an important part of solving these types of problems.
• The actions in a situation inform the choice of operation. The same operation can describe different situations.
  • Does the situation involve changing (joining, separating), combining or comparing? Then it can be represented with addition or subtraction.
  • Does the situation involve equal groups (or rates), ratio comparisons (scaling), or arrays? Then it can be represented with multiplication or division.
  • Representing a situation with an equation is often helpful in solving it.
• The same situation can be represented with different operations. Each operation has an “inverse” operation – an opposite that “undoes” the other. The inverse operation can be used to rewrite an equation to make it easier to calculate, or to check whether a calculation is true.
  • The inverse of addition is subtraction, and the inverse of subtraction is addition. So, for example,  + ? =  can be rewritten as  − = ?.
  • The inverse of multiplication is division, and the inverse of division is multiplication. So, for example, × ? =  can be rewritten as  ÷  = ?.

• A perfect square can be represented as a square with its area the value of the perfect square and a side length that is the positive square root of that perfect square number. In general, the area (A) of a square is side (s) × side (s), A = s².
• Any integer multiplied by itself produces a square number, or a perfect square, and can be represented as a power with an exponent of 2. For example, 9 is a square number because 3 × 3 = 9 or 3² = 9.
• A square number can be composed of a product of a perfect square and an even number of tens. For example, the square roots for 9, 900, 90 000, 9 000 000 are 3, 30, 300, 3000.

Note

• Negative integers expressed in exponential notation need to have a bracket around them to indicate it is the base of the power. Without the bracket it would not have the same result. For example, −3² = −(3 × 3) = −9 versus (−3)² = (−3 × −3) = 9.

• Multiplying a number by 0.1 is the same as dividing a number by 10. Therefore, it can be visualized by shifting the digit(s) to the right by one place. For example, 500 × 0.1 = 50, 50 × 0.1 = 5 and 5 × 0.1 = 0.5.
• Multiplying a number by 0.01 is the same as dividing a number by 100. Therefore, it  can be visualized by shifting the digit(s) to the right by two places. For example, 500 × 0.01 = 5, 50 × 0.01 = 0.5, and 5 × 0.01 = 0.05.
• Mentally multiplying and dividing whole numbers and decimals by powers of ten builds on the constant 10:1 ratio that exists between place-value columns. For example, 1000 is ten times greater than 100, or 100 is one tenth of 1000. Similarly, one hundredth (0.01) is ten times greater than one thousandth (0.001), or 0.001 is one tenth of 0.01.
• Multiplying a whole number and a decimal number by a positive power of ten can be visualized as shifting the digits to the left by one for each multiplication by 10.
  • For example, since 54.3 ×10⁴ means 5.43 × 10 × 10 × 10 × 10, the digits “543” shift to the left four places to become 543 000. This is true for whole numbers and decimals.
• Dividing a whole number and a decimal number by a power of 10 can be visualized as shifting the digits to the right by one for each division by 10.
  • For example, for 5.43 ÷ 10 ÷ 10 ÷ 10, the digits “543” shift to the right three spaces to become 0.00543.
  • Dividing by 10 is the same as multiplying by 0.1, thus 5.43 ÷ 10 ÷ 10 ÷ 10 is equal to 5.43 × 0.1 × 0.1 × 0.1 = 0.00543.

Note

• Making connections between division by 10 and multiplication by 0.1 can support students in converting a number in scientific notation of the form 5.43 × 10^-3 to a number in standard form.
• Mental math refers to doing calculations in one’s head. Sometimes the numbers or the number of steps in a calculation are too complex to completely hold in one’s head, so jotting down partial calculations and diagrams can be used to complete the calculations.
• Estimation is a useful mental strategy when either an exact answer is not needed or there is insufficient time to work out a calculation.

• When given a context, considerations can support the selection of an appropriate model and operation to solve the problem. For example: 
  • Are the integers representing a quantity or change?
  • Is the situation involving the addition of integers with like signs or different signs?
  • Is the situation involving the comparison of two integers?
  • When modelling the situation, do zero pairs need to be used to carry out the operation?
  • What will the result of the calculation mean in relation to the problem being solved?
• When adding and subtracting integers, it is important to pay close attention to all of the elements of the statement. For example:
  • (+4) + (−3) may be interpreted as combining positive four and negative three.
  • 4 + (−3) may be interpreted as adding negative three to positive four.
  • (−4) − (+3) may be interpreted as determining the difference between negative four and positive three by comparing them.
  • (−4) − 3 may be interpreted as taking away positive three from negative four.
  • 4 − 3 may be interpreted as taking away positive three from positive four.
  • −4 − 3 may be interpreted as taking away positive three from negative four.
• The order that integers are written in an addition statement does not matter because the commutative property holds true (e.g., −5 + 3 = 3 + (−5)). It is important to note that the sign directly in front of the number belongs to the number.
• The order in which integers are written in a subtraction statement does matter because the commutative property does not hold true. For example, (−5) − (+3) = −8 and (+3) − (−5) = +8; the two expressions do not produce the same result.
• Addition and subtraction are inverse operations; therefore, a subtraction expression can be rewritten as an addition expression by adding its opposite (e.g., (−5) − (+3) becomes (−5) + (−3); 2 − (−4) = 2 + (+4)).
• When two positive integers are added together, the result is positive. This can be visualized on a number line as:
  • two vectors moving in a positive direction (right or up);
  • a vector moving in a positive direction from a positive starting position.
• When two negative integers are added together, the result is negative. This can be visualized on a number line as:
  • two vectors moving in a negative direction (left or down);
  • a vector moving in a negative direction from a negative starting position.
• When a positive and a negative integer are added together, the result is negative if the absolute value of the negative integer is greater than the absolute value of the positive integer. This can be visualized on a number line as:
  • one vector moving in a positive direction and the other vector with a greater magnitude moving in a negative direction, the sign of the resultant vector is negative;
  • a vector moving in a negative direction from a positive starting position and the head of the vector is to the left (or below) zero;
  • a vector moving in a positive direction from a negative starting position and the head of the vector is to the left (or below) zero.
• When a positive and a negative integer are added together, the result is positive if the absolute value of the positive integer is greater than the absolute value of the negative integer. This can be visualized on a number line as:
  • one vector moving in a negative direction and the other vector with a greater magnitude moving in a positive direction, the sign of the resultant vector is positive;
  • a vector moving in a positive direction from a negative starting position and the head of the vector is to the right (or above) zero;
  • a vector moving in a negative direction from a positive starting position and the head of the vector is to the right (or above) zero.

Note

• If two integers added together have the same sign, then their magnitudes are added together.
• If two integers added together have different signs, then their magnitude is determined by taking the absolute difference between them.
• Depending on the models and the integers that are involved in a subtraction, zero pairs may need to be introduced in order to act out the situation. For example, if the situation involves taking away a negative amount but only positive amounts are shown, then adding zero pairs will allow for the negative amount to be removed.
• If the situation involves comparing two integers, the two integers can be represented as positions on a number line to determine the distance between the two points (magnitude).
• The order the subtraction statement is written is important in determining the sign. The direction of the sign is based on the movement from the point represented by the integer behind the minus sign (subtrahend) to the point represented by the integer in front of the minus sign (minuend). For example:
  • For 10 − (+2) = +8, the distance between positive 10 and positive 2 is 8; the movement from positive 2 to 10 is in a positive direction.
  • For (+2) − (+10) = −8, the distance between positive 2 and positive 10 is 8; the movement from positive 10 to positive 2 is in a negative direction.
  • For (2) − (−10) = + 12, the distance between positive 2 and negative 10 is 12; the movement from negative 10 to positive 2 is in a positive direction.
• Situations involving addition and subtraction can be modelled using tools such as a number line and integer tiles.
• Change can be represented by a positive or negative integer (e.g., rise of 4 expressed as +4, drop of 4 expressed as −4).
• A quantity relative to zero can be represented by a positive or negative integer (e.g., temperature is 3 degrees, temperature is −5 degrees).
• The integers in a situation may be interpreted as changes or as quantities. For example, if the temperature outside drops 5 degrees and then 3 degrees, this may be expressed as the addition of two drops [(−5) + (−3)] or as a subtraction of 3 degrees (−5 − 3). Both statements result in the same answer (−8), meaning the temperature decreased by 8 degrees.
• Familiar real-world contexts for negative and positive integers (temperature, elevators going up and down, parking garages, sea level, golf scores, plus/minus in hockey, gaining and losing money, walking forward and backwards, debts and surplus) provide an authentic opportunity to understand how integers are used in real life to describe a quantity or change. 

• When adding and subtracting proper and improper fractions with the same denominator, the numerators are added, and the denominator remains the same. When the denominators are the same (e.g., three fourths and nine fourths) they have the same units and so can be added (twelve fourths).
• Strategies to add and subtract fractions with unlike denominators depends on the types of fractions that are given. For example:
  • Mental math can be used to create wholes (ones). For example, for  +  = , knowing that three fourths is composed of one half and one fourth, the two halves are combined to make one, and then one fourth is added on.
  • Equivalent fractions are created so that both fractions have a common denominator (e.g.,  +  can be scaled so that both have a denominator of 6, which results in the equivalent expression  + ).
• One strategy to add and subtract mixed fractions is to decompose the mixed fraction into its whole and fractional parts. The wholes are added or subtracted, and the fractional parts are added or subtracted. If the result of the fractional part is greater than one, it is rewritten as a mixed number and the whole combined with the other wholes. Another strategy to add and subtract mixed fractions is to first rewrite them as improper fractions.

Note

• Fractions are commonly added and subtracted in everyday life, particularly when using imperial units (inches, feet, pounds, cups, teaspoons). Imperial units are commonly used in construction and cooking.
• Only common units can be added or subtracted, whether adding or subtracting whole numbers, decimals, or fractions. Adding fractions with like denominators is the same as adding anything with like units:
  • 3 apples and 2 apples are 5 apples.
  • 3 fourths and 2 fourths are 5 fourths.
• The numerator in a fraction represents the count of unit fractions. The denominator represents what is being counted (the unit). To add or subtract fractions is to change the total count of units. This is why only the numerator is added or subtracted. There are helpful ways to visualize the addition and subtraction of fractions. Drawings, fraction strips, clock models, and rulers in imperial units can be used to generate equivalent fractions and model how these common units can be combined or separated.
• The three types of addition and subtraction situations (see SE B2.1) also apply to fractions.

• The multiplication and division of two fractions can be interpreted based on the different ways fractions are used: as a quotient, as parts of a whole, as a comparison (ratio), and as an operator. 
• Multiplication as scaling is one way to multiply a fraction by a whole number. For example, 2 ×  can be interpreted as doubling two one thirds, which is four one thirds or .
• The multiplication of two proper fractions as operators can be modelled as follows:
  • For  × , the fraction two thirds can be shown as two thirds of a rectangle:

•  as an operator can be shown by taking one half of the two thirds:

• In general, the result of a fraction multiplied by a fraction can be obtained by multiplying the numerators and multiplying the denominators. In this example the product of the denominators are the partitions that were created in the rectangle, and the numerator is the resulting count of these partitions.

• Multiplying a mixed fraction by a mixed fraction can be modelled as the product of the area of a rectangle with its dimensions decomposed into wholes and fractions. For example, the rectangle to model  x has a width of two and one third and a length of three and two fifths. The areas of the four smaller rectangles formed by the decomposition are 2 × 3 = 6, 2 ×  = ,  × 3 = 1, and  ×  = .  The sum of all the areas is 6 +  + 1 +  = 7 +  +  = .

• Division of fractions can be interpreted in two ways:
  • 4 ÷   = ? can be interpreted as “How many one halves are in four?”

• Two one halves make 1, so eight one halves make 4. Therefore, 4 ÷ = 8.

• 4 ÷  = ? can also be interpreted as “If 4 is one half of a number, what is the number?”

• Since 4 is one half of a number, the other one half is also 4. Therefore, 4 ÷  = 8.

• Division of a fraction by its unit fraction (e.g., ÷ ) can be interpreted as how many counts of the unit are in the fraction (i.e., how many one eighths are in five eighths)?" The result is the number of counts (e.g., there are 5 counts of one eighth).
• Dividing a fraction by a fraction with the same denominator (e.g., ÷ ) can be interpreted as “How many divisors are in the dividend?” In the fraction strips below, notice there are three counts of two eighths that are in six eighths. Similar to the division of a fraction by its unit fraction, the result is the count.

• Sometimes the division of a fraction by a fraction with the same denominator has a fractional result. For example, ÷ .

• Notice there are 2 two eighths in five eighths, and then  of another two eighths. 
• Therefore, ÷  = .

• In general, when dividing a fraction by a fraction with the same denominator, the result can be obtained by dividing the numerators and dividing the denominators.
• To divide fractions by fractions that have unlike denominators, a strategy is to create equivalent fractions so that the two fractions have a common denominator and then divide numerators and divide denominators. For example:

   ÷ 

= ÷  

= 

= =

• A fraction divided by a whole number can use the same strategy. For example:

÷ 5

=  ÷  

=

• When division involves mixed numbers, a strategy is to convert them to improper fractions and then multiply accordingly.

Note

• When multiplying a fraction by a fraction using the area of a rectangle, first the rectangle is partitioned horizontally or vertically into the same number of sections as one of the denominators. Next, the region represented by that fraction is shaded to show that fraction of a rectangle. Next, the shaded section of the rectangle is partitioned in the other direction into the same number of sections as the denominator of the second fraction. Now it is possible to identify the portion of the shaded area that is represented by that fraction.
• Any whole number can be written as a fraction with one as its denominator. A whole number divided by a fraction can be used to support students in understanding the two ways division can be interpreted. If context is given, usually only one or the other way is needed. Dividing a whole number by a fraction also supports making connections with division of a fraction as the multiplication of its reciprocal.
• In general, dividing fractions with the same denominator can be determined by dividing the numerators and dividing the denominators.
• Since division is the inverse operation of multiplication, the division statement can be rewritten in terms of multiplication. For example, if  ÷  = n, then  × n = .
  • To solve for n, each side of the equal sign is multiplied by :
  •  ×  × n =   ×
  • therefore n =  ×
  • Now the numerators can be multiplied, and the denominators can be multiplied.
  • n =  
  • The fraction  is the reciprocal of  because the product of these two fractions is 1.
  • Therefore, another strategy to divide two fractions is to multiply the dividend by the reciprocal of the divisor.
• Multiplying fractions follows a developmental progression that may be helpful in structuring tasks for this grade:
  • A proper or improper fraction by a whole number.
  • A whole number by a proper or improper fraction.
  • A unit fraction by a unit fraction.
  • A unit fraction by a proper or improper fraction.
  • A proper or improper fraction by a proper or improper fraction. 
  • A mixed fraction by a proper or improper fraction.
  • A mixed fraction by a mixed fraction.
• Dividing fractions follows a developmental progression that may be helpful in structuring tasks for this grade:
  • A whole number divided by a whole number.
  • A proper or improper fraction divided by a whole number.
  • A whole number divided by a unit fraction.
  • A whole number divided by a proper or improper fraction.
  • A proper or improper fraction divided by a unit fraction.
  • A proper or improper fraction divided by a proper or improper fraction with the same denominators and a result that is a whole number.
  • A proper or improper fraction divided by a proper or improper fraction with the same denominators and a result that is a fractional amount (e.g.,  ÷ ).
  • A proper or improper fraction divided by a proper or improper fraction with unlike denominators.
  • A proper or improper fraction divided by a whole number or mixed fraction.
  • A mixed fraction divided by a whole number.
  • A mixed fraction divided by a mixed fraction.

• Multiplication and division facts for whole numbers can be used for multiplying and dividing integers. The difference is consideration of the sign, which is determined by the numbers being worked with.
• A positive integer multiplied or divided by a positive integer has a result that is positive.
• A positive integer multiplied by a negative integer has a result that is negative. Since multiplication can be understood as repeated equal groups, then the positive integer can represent the number of groups and the negative integer can represent the quantity in each group. For example, 3 × (−4) can be modelled as (−4) + (−4) + (−4) = −12.
• The commutative property holds true for the multiplication of integers, so (+3) × (−4) = (−4) × (+3). Therefore (−4) × (+3) = −12 or (−12).
• Since division is the inverse operation, the rules for the signs with multiplication are the same for division:
  • A positive number divided by a positive number has a result that is positive.
  • A positive number divided by a negative number has a result that is negative.
  • A negative number divided by a positive number has a result that is negative.
  • A negative number divided by a negative number has a result that is positive.

Note

• When two brackets are side by side, it is understood to be multiplication. For example: (−3)(−4).
• Division of two numbers can be indicated using the division symbol or using the division bar. For example, 12 ÷ (−3) = . 
• Multiplication can be understood as repeated equal groups, where the first factor is the number of groups and the second factor is the size of the groups. When the first integer is positive, regardless of the sign of the second integer, this is helpful for visualizing the situation since it links multiplication with the repeated addition of a group.
  • A 3° rise in temperature 4 days in a row can be represented as (+4) × (+3) = (+12).
  • A 3° drop in temperature 4 days in a row can be represented as (+4) × (−3) = (−12).
• It is difficult, although not impossible, to conceive of a negative number of groups. To overcome this, properties and reasoning can help:
  • The commutative property says that (+4) × (−3) is the same as (−3) × (+4), so both must equal (−12).
  • Patterning can be used to determine that (−3) × (−4) must be (+12).
• Understanding division of integers requires a strong understanding of the operation and its relationship to multiplication. Grouping division asks, “How many groups of ___ are in ___? ” Sharing division asks, “How many does each receive if ___ are shared among ___?” Both are helpful for understanding division with integers and their relationship to multiplication, repeated addition, and repeated subtraction:
  • (+20) ÷ (+5) draws on work in earlier grades for an answer of (+4).
  • (−20) ÷ (−5) can mean how many groups of (−5) are in (−20). Since there are 4 groups of (−5) in (−20), (−20) ÷ (−5) = (+4).
  • (−20) ÷ (+5) can mean that (−20) can be shared between 5 groups. Since each group would receive (−4), (−20) ÷ (+5) = (−4).
  • (+20) ÷ (−5) can draw on patterns and the inverse relationship between multiplication and division to rewrite this statement as (−5) × __ = (+20) to see that (+20) ÷ (−5) = (−4).
• There are conventions for expressing multiplication and division in ways that make algebraic expressions clearer:
  • Multiplication may be shown with the multiplication sign: (−3) × (−4).
  • Multiplication may be shown with no multiplication sign: (−3)(−4).
  • Multiplication may be shown with the dot operator: (−3)·(−4).
  • Division may be shown with the division sign (÷).
  • Division may be shown with the fraction bar (―).

• If two quantities change at the same rate, the quantities are proportional. Proportional growth, when plotted on a graph, forms a straight line (i.e., linear growth) because each point changes at a constant rate.
• Proportions involve multiplicative comparisons (ratios) and are written in the form a : b = c : d or expressed using fractional notation as  = . When ratios are represented using fractional notation, they are usually read as 3 out of 4, or 3 to 4, rather than as a fraction, three fourths. Writing ratios using fractional notation is helpful for making comparisons and calculating proportions.
• There are four ways a proportion can be written for it to hold true. For example, 3.7 km for every 5 hours and 7.4 km for every 10 hours can be expressed as:
  •    = or  =  or  =  or  = . 
• Problems involving proportional relationships can be solved in a variety of ways, including using a table of values, a graph, a ratio table, a proportion, and scale factors.

Note

• One strategy when using a proportion to solve for an unknown value is to position that unknown in the upper part of the equation (e.g., ).
• In solving for proportional situations, comparisons can be made within a situation (i.e., the unit rate is constant) and between situations (i.e.., the scaling factor is constant). So, 6 items costing $9 is proportional to 12 items costing $18:

• Scaling a ratio creates other proportional situations. For example, the relationship of 2 blue marbles to 3 red marbles (2 : 3) is in proportion to 6 blue marbles and 9 red marbles (6 : 9). The fractions  and  are equivalent, so the situations are proportional.

• Ratio tables, double number lines, and between-within diagrams are helpful tools to identify and compare proportional relationships, and to solve for unknown values.
• Tables and graphs are helpful for seeing proportional (or non-proportional) relationships. Any of the points marked on the graph is proportional to each of the other points.