• 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) = (a × b) + (a × c).
• The commutative, associative, and identity properties can be applied for any type of number.
• The order of operations property needs to be followed when given a numerical expression that involves multiple operations. Any calculations in the brackets are done first. Multiplication and division are done before addition and subtraction. Multiplication and division are done in the order they appear in the expression from left to right. Addition and subtraction are done in the order they appear in the expression from left to right.
• Multi-step problems may involve working with a combination of whole numbers, decimal numbers, and 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.
• 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-step problems, sometimes known as two-step problems, there is 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 a critical 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 the situation can be represented with addition and subtraction.
  • Does the situation involve equal groups (or rates), ratio comparisons, or arrays? Then the situation can be represented with multiplication and division.
• Representing a situation with an equation is often helpful for solving a problem. Identifying what is known and unknown in a situation informs how an equation is structured.

• There are number patterns that can be used to quickly test whether a number can be evenly divided by another number.
• Divisibility rules can be used to determine factors of numbers.

Note

• Divisibility rules can be applied to all integers; the signs can be ignored.
• Divisibility rules do not apply to decimal numbers that are not whole numbers.

• Percents represent a rate out of 100 (“per cent” means “per hundred”) and are always expressed in relation to a whole. To visually determine the percent of an amount, the whole is subdivided into 100 parts (percent) and described using the percent symbol (%).
• Since 1% is 1 hundredth of an amount, and 10% is 1 tenth, other percents can be calculated by mentally multiplying an amount by tenths and hundredths. For example, 0.01 × 500 = 5 or  of 500 = 5.
• Calculating the percent of a whole number can be determined by decomposing the percent as a multiple of 1%. For example, 3% of 500 can be determined by decomposing 3% as 3 × 1%. Since 1% of 500 is 5, then 3% of 500 is 3 × 1% of 500 = 3 × 5 = 15.

Note

• Multiplying a whole number by a percent is the same as multiplying a whole number by a fraction with a denominator of 100 (e.g., 3% × 500 =  × 500 or 0.03 × 500).
• Dividing an amount by 100 is the same as multiplying it by 0.01. Since 0.01 × 500 is 5 (i.e.,  of 500), then 1% of 500 is also 5.
• Percents can be composed from other percents. Since 1% of 500 is 5, then 3% of 500 is 15. This builds on an understanding of fractions and the meaning of the numerator:  is the same as 3 one hundredths.
• Relationships between fractions, decimals, and percents also provide building blocks for mentally calculating unknown percents. For example:  = 25%;  = 50%;  = 75%. Five percent is half of 10%, and 15% is 10% plus 5%.
• Calculating a percent is a frequently used skill in everyday life (e.g., when determining sales tax, discounts, or gratuities).
• Mental math is not always quicker than paper and pencil strategies, but speed is not its goal. The value of mental math is in its portability and flexibility, since it does not require a calculator or paper and pencil. Practising mental math strategies also deepens understanding of numbers and operations.

• Situations involving addition and subtraction may involve:
  • adding a quantity onto an existing amount or removing a quantity from an existing amount;
  • combining two or more quantities;
  • comparing quantities.
• If an exact answer is not needed, an estimation can be used. The estimation can be made by rounding the numbers and then adding or subtracting.
• Estimation may also be used prior to a calculation so that when the calculation is performed, one can determine if it seems reasonable or not.
• If an exact answer is needed, a variety of strategies can be used, including algorithms.

Note

• There are three types of situations that involve addition and subtraction. A problem may combine several situations with more than one operation to form a multi-step or multi-operation problem (see B2.1). Recognizing the type and structure of a situation provides a helpful starting point for solving problems.
  • Change situations, where one quantity is changed, either by having an amount joined to it or separated from it. Sometimes the result is unknown; sometimes the starting point is unknown; sometimes the change is unknown.
  • Combine situations, where two quantities are combined. Sometimes one part is unknown; sometimes the other part is unknown; sometimes the result is unknown.
  • Compare situations, where two quantities are being compared. Sometimes the greater amount is unknown; sometimes the lesser amount is unknown; sometimes the difference between the two amounts is unknown.
• The most common standard algorithms for addition and subtraction in North America use a compact organizer to decompose and compose numbers based on place value. They begin with the smallest unit – whether it is the unit column, decimal tenths, or decimal hundredths – and use regrouping or trading strategies to carry out the computation. (See Grade 4, B2.4 for a notated subtraction example with decimals, and Grade 3, B2.4 for a notated addition example with whole numbers; the same process applies to decimal hundredths.)
• When carrying out an addition or subtraction algorithm, only common units can be combined or separated. This is particularly noteworthy when using the North American standard algorithms with decimals because, unlike with whole numbers, the smallest unit in a number is not always common (e.g., 90 − 24.7). The expression “line up the decimal” is about making sure that common units are aligned. Using zero as a placeholder is one strategy to align values.

• The type of models (e.g., linear model, area model) and tools (e.g., concrete materials) that are used to represent the addition or subtraction of fractions can vary depending on the context.
• Addition and subtraction of fractions with the same denominator may be modelled using fraction strips partitioned into the units defined by the denominators with the counts of the units (numerators) being combined or compared. The result is based on the counts of the same unit.
  • For example, if adding, 3 one fourths (three fourths) and 2 one fourths (two fourths) are 5 one fourths (five fourths), or .
  • For example, if subtracting, taking 2 one fourths (two fourths) from 7 one fourths (seven fourths) leaves 5 one fourths (five fourths), or . Or, when thinking about the difference, 5 one fourths (five fourths) is 2 one fourths less than 7 one fourths (seven fourths).
• Addition and subtraction of fractions with unlike denominators may be modelled using fraction strips of the same whole that are partitioned differently. When these fractions are combined or compared, the result is based on the counts of one of the denominators or of a unit that both denominators have in common. 
• Hops on a number line may represent adding a fraction on to an existing amount or subtracting a fraction from an existing amount. 

Note

• The three types of addition and subtraction situations (see B2.4) also apply to fractions.
• As with whole numbers and decimals (see B2.4), only common units can be added or subtracted. This is also true for 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.
• When adding and subtracting fractions as parts of a whole, the fractions must be based on the same whole. Thus, avoid using a set model because the tendency is to change the size of the whole.
• 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, so only the numerator is added or subtracted.
• If students are adding and subtracting fractions with unlike denominators, they may need to estimate the sum and difference, depending on the tools they are using. This kind of estimation will support fraction sense.
• Without a context, the addition and subtraction of fractions are assumed to be treating the fractions as parts of a whole. Fractions as parts of a whole are commonly added and subtracted in everyday life (e.g., construction, cooking), particularly when combining or comparing units that are commonly used, such as imperial units (inches, feet, pounds, cups, teaspoons).
• Adding and subtracting fractions as comparisons may also have everyday applications. For example, when adding up test scores – a student got 3 of the 4 possible marks () for question 1 and got 4 of the 5 possible marks () for question 2. For the two questions together, the student got 7 of 9 possible marks (). In this example, the fractions are comparing what a student got compared to what was possible. 

• A number can be decomposed as a product of its factors.
• A prime number can only be expressed as a product of two unique factors, the number itself and 1, for example, 1  = 11 × 1.
• A composite number can be expressed as a product of two or more factors. For example, 8 can be written as a product of the factors 1 × 8, 2 × 4, and 2 × 2 × 2.   
• The number 1 is neither prime, nor composite, since it has only one unique factor: itself. It is called a unit.
• Any whole number can be written as a product of its prime factors. Factor trees can be used to show how a number can be repeatedly decomposed until all of its factors are prime.

Note

• Prime and composite numbers can be visualized using rectangles. Rectangles with areas that are prime numbers have only one possible set of whole number dimensions; rectangles with areas that are composite numbers have more than one. For example, there is only one rectangle with whole number dimensions that has an area of 11 cm² (1 cm × 11 cm), but there are two rectangles that have an area of 4 cm² (1 cm × 4 cm and 2 cm × 2 cm).
• A factor may also be decomposed into other factors.
• The factors of a number can assist with mental calculations. For example, 36 × 4 might be challenging to do mentally, but thinking of this as the product 4 × 3 × 3 × 4 means that the known fact 12 × 12 can be used to determine the product.

• An area model can be used to visualize multiplication with decimals.
  • The two numbers being multiplied can be the dimensions of a rectangle.
  • The dimensions can be decomposed by their place value.
  • The number of smaller rectangles formed will depend on how the dimensions have been decomposed.
  • Known facts can be used to determine each of the smaller areas.
  • The smaller areas are added together resulting in the product.

• There are many different algorithms that can be used for multiplication. Students may use one of these algorithms, or their own, and are not required to know all or more than one method. Standard multiplication algorithms for whole numbers can also be applied to decimal numbers. As with whole numbers, these algorithms add partial products to create a total. For example, with 235 × 0.3, the partial products are formed by multiplying each whole number by three tenths. Note the connection between this and multiplying a whole number by 30% (see B2.3) and by a fraction (see B2.9).

• Another algorithm approach uses factoring and properties of operations. It enables multiplication by tenths to be treated as a whole number calculation, which is then multiplied by a tenth (0.1). For example:
  • 235 × 0.3 can be thought of as 235 × 3 × 0.1.
  • A standard algorithm determines that 235 × 3 equals 705.
  • 705 is then multiplied by one tenth (0.1).
  • One tenth of 705 is 70.5.

• The context of multiplication problems may involve:
  • repeated equal groups, including rates.
  • scale factors – ratio comparisons, rates, and scaling.
  • area and certain other measurement attributes.
  • the number of possible combinations of attributes given two or more sets (see Data, D2.2).
• Connections can be made between the multiplication of a whole number by a decimal number and multiplying a whole number by a percent. For example, 235 × 0.3 is connected to multiplying a whole number by 30% (see B2.3) and by a fraction (see B2.9).
• Multiplication of a whole number by a decimal number between 0 and 1 will result in a product much less than the original number.
• Estimating a product prior to a calculation helps with judging if the calculation is reasonable.

• A strategy to divide whole numbers by decimal numbers is to create an equivalent division statement using whole numbers. For example, 345 ÷ 0.5 will have the same result as 3450 ÷ 5. 
• Often division does not result in whole number amounts. In the absence of a context, remainders can be treated as a leftover quantity, or they can be distributed equally as fractional parts across the groups.
  • When using the standard “long-division” algorithm, the whole number dividend can be expressed as a decimal number by adding zeroes to the right of the decimal point until a terminating decimal number can be determined, or until a decimal number is rounded to an appropriate number of places. For example, 27 ÷ 8 can be expressed as 27.000 ÷ 8 to accommodate an answer of 3.375.
  • A remainder can be expressed as a fraction (e.g., ).

Note

• Multiplication and division are related (see B2.1).
• When dividing by tenths, contexts often use quotative division and ask “How many tenths are in this amount?” It is more difficult to think of division with decimals as partitive, where an amount is shared evenly among a tenth, although it is possible. For example, thinking of 22 ÷ 0.5 partitively means thinking that if 22 is only 5 tenths of the whole, what is the whole?
• The context of a division problem may involve:
  • repeated equal groups, including rates;
  • scale factor – ratio comparisons, rates, and scaling;
  • the area of rectangles;
  • the number of possible combinations of attributes given two or more sets (see Data, D2.2).
• In real-world situations, the context determines how a remainder should be dealt with:
  • Sometimes the remainder is ignored, leaving a smaller amount (e.g., how many boxes of 5 can be made from 17 items?).
  • Sometimes the remainder is rounded up, producing a greater amount (e.g., how many boxes are needed if 17 items are packed in boxes of 5?).
  • Sometimes the remainder is rounded to the nearest whole number, producing an approximation (e.g., if 5 people share 17 items, approximately how many will each receive?).
• Division of a whole number by a decimal number between 0 and 1 will result in a quotient greater than the original whole number.
• Estimating a quotient prior to a calculation helps with judging if the calculation is reasonable.

• A proper fraction can be decomposed as a product of the count and its unit fraction (e.g.,  = 3 ×  or  × 3). 
• The strategies used to multiply a whole number by a proper fraction may depend on the context of the problem.
  • If the situation involves scaling, 5 ×  may be interpreted as "the total number of unit fractions is five times greater". Thus, 5 ×  = 5 × 3 ×  = 15 ×  =  (15 fourths).
  • If the situation involves equal groups, 5 ×  may be interpreted as "five groups of three fourths". Thus, 5 ×  =  +  +  +  +  =  or .
  • If the situation involves area, 5 ×  may be interpreted as "the area of a rectangle with a length of five units is multiplied by its width of three fourths of a unit". The area could be determined by finding the area of a rectangle with dimension 5 by 1 and then subtracting the extra area, which is 5 one fourths. Therefore:
  •   5 × 
    = (5 × 1) − 5 × ()
    = 5 − 
    = 5 − 1
    = 3

Note

• How tools are used to multiply a whole number by a proper fraction can be influenced by the contexts of a problem. For example:
  • A double number line may be used to show multiplication as scaling.
  • Hops on a number line may be used to show multiplication as repeat addition.
  • A grid may be used to show multiplication as area of a rectangle.
• The strategies that are used to multiply a whole number by a proper fraction may depend on the type of numbers given. For example, 8 ×  = 8 × 3 × . Using the associative property, the product of 8 ×  may be multiplied first and then multiplied by 3. This results in 2 × 3 = 6. Another approach is to multiply 8 × 3 first, which results in 24, which is then multiplied by , resulting in 6.

• Multiplication and division are related. The same situation or problem can be represented with a division or a multiplication sentence. For example, the division question 6 ÷  = ? can also be thought of as a multiplication question,  × ? = 6.
• The strategies used to divide a whole number by a proper fraction may depend on the context of the problem.
  • If the situation involves scaling, 24 ÷  may be interpreted as “some scale factor of three fourths gave a result of 24”.
    • Therefore,  × ? = 24
    • 3 ×  × ? = 24 or  × ? = 8
    • Therefore, the quotient is 32 because 32 one fourths is 8.
  • If the situation involves equal groups, 24 ÷  may be interpreted as “How many three fourths are in 24?” Either three fourths is repeatedly added until it has a sum of 24 or it is repeatedly subtracted until the result is zero.
  • If the situation involves area, 24 ÷  may be interpeted as “What is the length of a rectangle that has an area of 24 square units, if its width is three fourths of a unit?” Therefore,  × ? = 24 may be determined by physically manipulating 24 square units so that a rectangle is formed such that one dimension is three fourths of one whole.

Note

• In choosing division situations that divide a whole number by a fraction, consider whether the problem results in a full group or a partial group (remainder). In Grade 6, students should solve problems that result in full groups.

• Multiplication and division are related. The same situation or problem can be represented with a division or a multiplication sentence.
• The strategies used to divide a decimal number by a single digit whole number may depend on the context of the problem and the numbers used.
  • If the situation involves scaling, 2.4 ÷ 8 may be interpreted as “some scale factor of 8 gave a result of 2.4” or “What is the scale factor of 8 to give a result of 2.4?” Therefore, 8 × ? = 2.4. The result of 0.3 could be determined using the multiplication facts for 8 and multiplying it by one tenth.
  • If the situation involves equal groups, 3.24 ÷ 8 may be interpreted as “How much needs to be in each of the 8 groups to have a total of 3.24?” The result of 0.405 could be determined using the standard algorithm.
  • If the situation involves area, 48.16 ÷ 8 may be interpreted as “What is the width of a rectangle that has an area of 48.16 square units, if its length is 8 units?” Therefore, 8 × ? = 48.16. The result of 6.02 could be determined using short division.

Note

• Using the inverse operation of multiplication is helpful for estimating and for checking that a calculation is accurate. For example, 1.935 ÷ 9 = ? can be written as 9 × ? = 1.935, which verifies that the missing factor must be less than 1.

• A ratio describes the multiplicative relationship between two or more quantities.
• Ratios can compare one part to another part of the same whole, or a part to the whole. For example, if there are 25 beads in a bag, of which 10 are red and 15 are blue:
  • The ratio of blue beads to red beads is 15 : 10 or , and this can be interpreted as there are one half times more blue beads than red beads.
  • The ratio of red beads to the total number of beads is 10 : 25 or , and this can be interpeted as 40% of the beads are red.
• Any ratio can be expressed as a percent.
• A rate describes the multiplicative relationship between two quantities expressed with different units. For example, walking 10 km per 2 hours or 5 km per hour.
• Problems involving ratios and rates may require determining an equivalent ratio or rate. An equivalent ratio or rate can be determined by scaling up or down. For example:
  • The ratio of blue marbles to red marbles (10 : 15) can be scaled down to 2 : 3 or scaled up to 20 : 30. In all cases, there are  or approximately 66% as many blue marbles as red marbles.
  • The walking rate 10 km per 2 hours (10 km/2h) can be scaled down to 5 km/h (unit rate) or scaled up to 50 km/10 h. 

Note

• Ratios compare two (or more) different quantities to each other using multiplication or division. This means the comparison is relative rather than absolute. For example, if there are 10 blue marbles and 15 red marbles:
  • An absolute comparison uses addition and subtraction to determine that there are 5 more red marbles than blue.
  • A relative comparison uses proportional thinking to determine that: 
    • for every 2 blue marbles there are 3 red marbles;
    • there are  as many blue marbles as red marbles;
    • there are 1.5 times as many red marbles as blue marbles;
    • 40% of the marbles are blue and 60% of the marbles are red.
• A three-term ratio shows the relationship between three quantities. The multiplicative relationship can differ among the three terms. For example, there are 6 yellow beads, 9 red beads, and 2 white beads in a bag. This situation can be expressed as a ratio of yellow : red : white beads = 6 : 9 : 2. The multiplicative relationship between yellow to white is 6 : 2 or 3 : 1, meaning there are three times more yellow beads than white beads. The multiplicative relationship between yellow and red beads is 6 : 9 or 2 : 3, meaning there are two thirds as many yellow beads as there are red beads.
• Ratio tables can be used for noticing patterns when a ratio or rate is scaled up or down. Ratio tables connect scaling to repeated addition, multiplication and division, and proportional reasoning.