• The commutative property holds true for addition and for multiplication. The order in which the numbers are added or multiplied does not matter; the results will be the same (e.g., 45 + 62 = 62 + 45 and 12 × 6 = 6 × 12).
• The associative property holds true for addition and for multiplication. The pairs of numbers first added or multiplied does not matter; the results will be the same. For example,  (24 + 365) + 15 = 24 + (365 + 15). Similarly, (12 × 3) × 5 = 12 × (3 × 5).
• The distributive property can be used to determine the product of two numbers. For example, to determine 12 × 7 the 12 can be rewritten as 10 and 2 and the sum of their products is determined (i.e., 12 × 7 = (10 + 2) × 7, which is (10 × 7) + (2 × 7)).
• Addition and subtraction are inverse operations. Any subtraction question can be thought of as an addition question (e.g., 154 – 48 = ? is the same as 48 + ? = 154). This inverse relationship can be used to perform and check calculations.
• Multiplication and division are inverse operations. Any division question can be thought of as a multiplication question, unless 0 is involved (e.g., 132 ÷ 11 = ? is the same as ? × 11 = 132) and vice versa. This inverse relationship can be used to perform and check calculations.
• Sometimes a property may be used to check an answer. For example, 12 × 7 may be first determined using the distributive property as (10 × 7) + (2 × 7). The factors could also be decomposed as 2 × 6 × 7 and the associative property applied: 2 × (6 × 7) to verify the results.
• Sometimes the reverse operation may be used to check an answer. For example, 32 ÷ 4 = 8 could be checked by multiplying 4 and 8 to determine if it equals 32.

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.
• When addition is used to solve a subtraction question, this is often referred to as finding the missing addend.
• Addition and subtraction strategies can be used to think about and solve multiplication and division questions (see B2.6 and B2.7).
• The context of a problem may influence how students think about performing the calculations.
• Operation sense involves the ability to represent situations with symbols and numbers. Understanding the meaning of the operations, and the relationships between and amongst them, enables one to choose the operation that most closely represents a situation and most efficiently solves the problem given the tools at hand.
• 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 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 as an equation is often helpful for solving a problem. Identifying what is known and unknown in a situation informs how an equation is structured:
  • For addition and subtraction, is the start, change, or result unknown? Addition determines an unknown result. Subtraction determines an unknown starting amount or the amount of change.
  • For multiplication and division, is the total, the number of groups, or the size of the groups unknown? Multiplication names the unknown total (the product). Division determines either the unknown number of groups (quotative or grouping division) or the unknown size of each group (partitive or sharing division).

• The identity principle states that when multiplying an amount by 1 or dividing an amount by 1, the amount stays the same (e.g., 1 × 5 = 5 and 5 ÷ 1 = 5).
• The facts of 1, 2, 5, and 10 can be used to determine the facts for other numbers. For example:
  • 2 × 12 can be determined by knowing 12 × 2.
  • 7 × 12 can be determined by knowing 7 × 10 and adding two more 7s.
  • 8 × 12 can be determined by knowing (8 × 5) + (8 × 5) + (8 × 2), which is 40 + 40 + 16, or 96.
• Division facts can be determined using multiplication facts (e.g., 24 ÷ 6 can be determined using the multiplication facts for 6).
• When multiplying any number by zero, the result is zero. For example, 5 groups of zero is 0 + 0 + 0 + 0 + 0 = 0. Also, zero groups of anything is nothing.
• Zero divided by any non-zero number is zero. For example,  = n can be rewritten as 5 × n = 0. If 5 represents the number of groups and n represents the number of items in each group, then there must be zero items in each group.
• Any number divided by zero has no meaning and is said to be “undefined”. For example, there is no answer to the question, “How many groups of 0 are in 5?”

Note

• Automatic recall of math facts is an important foundation for doing calculations, both mentally and with paper and pencil. For example, knowing facts up to 12 is important for mentally converting inches to feet, units that are commonly used in everyday life.
• The commutative property of multiplication (e.g., 11 × 12 = 12 × 11) reduces, by almost half, the number of facts to be learned and recalled.
• The distributive property means that a multiplication problem can be split (decomposed) into smaller parts, and the products of those smaller parts can be added together (composed) to get the total. It enables a known fact to be used to find an unknown fact. For example, in building on the facts for 1 to 10:
  • Multiplication by 11 adds one more row to the corresponding 10 fact; there are also interesting patterns in the 11 facts up to × 9 that make them quick to memorize.
  • Multiplication by 12 adds a double of what is being multiplied to the corresponding 10 fact; it can also be thought of as the double of the corresponding × 6 fact.
• The associative property means that the × 12 facts can be decomposed into factors and rearranged to make a mental calculation easier. For example, 5 × 12 can be thought of as 5 × 6 × 2 or double 30.
• Practice is important for moving from understanding to automaticity. Practising with one set of number facts at a time (e.g., the 11 facts) helps build understanding and a more strategic approach to learning the facts.

• The inverse relationship between multiplication and division helps when doing mental math with powers of ten.
• Multiplying a number by 0.1 is the same as dividing a number by 10. Therefore, a shifting of the digit(s) to the right by one place can be visualized. 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, a shifting of the digit(s) to the right by two places can be visualized. For example, 500 × 0.01 = 5; 50 × 0.01 = 0.5; and 5 × 0.01 = 0.05.
• Mental math strategies for addition and subtraction of whole numbers can be used with decimal numbers. The strategies may vary depending on the numbers given. For example:
  • If given 0.12 + 0.15, the like units can be combined to get 0.27.
  • If given 44 − 31.49, 31.49 can be rounded to 31.5, and then 31.5 subtracted from 44 to get 12.5.

Note

• Mental math may or may not be quicker than paper-and-pencil strategies, but speed is not the 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 an understanding of numbers.
• Mental math involves using flexible strategies that build on basic facts, number relationships, and counting.
• As numbers and calculations become too difficult to keep track of mentally, partial quantities are written down and totalled as a separate step.
• When adding and subtracting numbers, the like units are combined. For example, hundreds with hundreds, tens with tens, ones with ones, tenths with tenths, and hundredths with hundredths.
• Estimation can be used to check the reasonableness of calculations and should be continually encouraged when students are doing mathematics.

• Situations involving addition and subtraction may involve:
  • adding a quantity on to an existing amount or removing a quantity from an existing amount;
  • combining two or more quantities;
  • comparing quantities.
• Acting out a situation, by representing it with objects, a drawing or a diagram, can help identify the quantities given in a problem and what quantity needs to be determined.
• Set models can be used to add a quantity on to an existing amount or to remove a quantity from an existing amount.
• Linear models can be used to determine the difference by comparing two quantities.
• Part-whole models can be used to show the relationship between what is known and what is unknown and how addition and subtraction relate to the situation.

Note

• An important part of problem solving is the ability to choose the operation that matches the action in a situation. For additive situations – situations that involve addition or subtraction – there are three “problem types”:
  • Change situations, where one quantity is changed by having an amount either 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 larger amount is unknown; sometimes the smaller amount is unknown; sometimes the difference between the two amounts is unknown.
• A variety of strategies may be used to add or subtract, including algorithms.
• An algorithm describes a process or set of steps to carry out a procedure. A standard algorithm is one that is known and used by a community. Different cultures have different standard algorithms that they use to perform calculations.
• 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 (ones) 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 decimals” is really about making sure that common units are aligned. Using a zero as a placeholder is one strategy to align unit values. Unpacking the compactness and efficiency of the standard algorithm strengthens understanding of place value and the properties of addition and subtraction.

• As with whole numbers and decimal numbers, only common units can be combined or separated. 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.
• Fractions with the same denominator can be added by combining the counts of their unit. For example, 3 one fourths and 2 one fourths are 5 one fourths (i.e., ).
• Fractions with the same denominator can be subtracted by comparing the counts of their unit. For example, 7 one fourths is more than 2 one fourths by 5 one fourths (i.e., ).

Note

• The numerator in a fraction can describe the count of unit fractions (e.g., 4 one thirds is written in standard fractional form as ).
• The type of models and tools that are used to represent the addition or subtraction of fractions with like denominators can vary depending on the context. For example:
  • Hops on a number line may represent adding a fraction on to an existing amount or subtracting a fraction from an existing amount. The existing amounts are positions on a number line.
  • An area model may be used to combine fractional areas or remove fractional areas.

• Numbers multiplied together are called factors, and their result is called a product.
• The multiplication of two two-digit numbers using the distributive property can be modelled as the area of a rectangle:
  • When the dimensions of a rectangle are decomposed, the area is also decomposed.
  • When the two-digit length is decomposed into tens and ones, and the two-digit width is decomposed into tens and ones, the area is subdivided into four areas – tens by tens, ones by tens, tens by ones, and ones by ones.
  • Known facts can be used to determine each of the smaller areas.
  • The smaller areas are added together resulting in the product.
• The area model is a visual model of the standard algorithm showing the sum of the partial products.
• The product can be determined using an area model or the standard algorithm.

Note

• The context of multiplication problems may involve:
  • repeated equal groups;
  • scale factors – ratio comparisons, rates, and scaling;
  • area measures;
  • combinations of attributes given two or more sets (see Data, D2.2).
• The array can be a useful model for showing multiplication and division because it structures repeated groups of equal size into rows and columns. The array makes visual connections to skip counting, the distributive property, the inverse relationship between multiplication and division, and the measurement of area.
• The area model using a rectangle is sometimes referred to as an open array. Even though the area model can be used to represent the multiplication of any two numbers, support students in not confusing it with the actual context of the problem.
• Open arrays show how a multiplication statement can be thought of as the area of a rectangle (b × h). An unknown product is decomposed into partial products (smaller rectangles) with “dimensions” that access known facts and friendly numbers. The partial products are then totalled (see Grade 4, B2.4, for more details). Standard algorithms for multiplication are based on the distributive property.
• The most common standard algorithm for multiplication in North America is a compact and efficient organizer that decomposes factors based on the distributive property. It creates partial products, which are then added together to give the total product. There are variations on how this algorithm is recorded, with some being more compact than others; however, the underlying process is the same. Priority should be given to understanding, especially when an algorithm is first introduced.

• Multiplication and division are inverse operations (see B2.1). 
  • The numbers multiplied together are called factors. The result of a multiplication is called the product.
  • When a multiplication statement is rewritten as a division statement, the product is referred to as the dividend, one of the factors is the divisor, and the other factor is the quotient (result of division).
• Using the area model of a rectangle to solve a division question draws on multiplication as its inverse operation. A rectangle is gradually created by arranging all the square units (dividend) into rows and columns for a given dimension (divisor).
• Determining the quotient using an algorithm requires an understanding of place value, multiplication facts, and subtraction.
• Division does not always result in whole number amounts. For example, 320 ÷ 15 is 21 with a remainder of 5, which can also be expressed as  or one third.
• The context of a problem can influence how the remainder is represented and interpreted. For example:
  • A rope is 320 cm long and is divided into 15 equal sections; how long is each section? (320 ÷ 15= ?). Each section is  centimetres. In this case, measuring  of a centimetre of ribbon is possible, given that it is a linear dimension.
  • A van holds 18 students. There are 45 students. How many vans are needed to transport the students? Dividing 45 by 18 means that 2.5 vans are needed. This requires rounding up to 3 vans.

Note

• The context of a division problem may involve:
  • repeated equal groups;
  • scale factors – ratio comparisons, rates, and scaling (see Grade 3, B2.9);
  • area measures;
  • combinations of attributes.
• Multiplication and division are related and therefore the rectangle area model can be used to show how a division question can be solved using repeated addition or repeated subtraction. The area model using a rectangle is sometimes referred to as an open array. Even though the area model can be used to represent division, support students in not confusing it with the actual context of the problem.
• For each division situation, there are two division types:
  • equal-sharing division (sometimes called partitive division):
    • What is known: the total and number of groups;
    • What is unknown: the size of the groups;
    • The action: a total is shared equally among a given number of groups.
  • equal-grouping division (sometimes called measurement or quotative division):
    • What is known: the total and the size of groups.
    • What is unknown: the number of groups.
    • The action: from a total, equal groups of a given size are measured.
• Note that since area situations use base and height to describe the size and number of groups, and because these dimensions are interchangeable, the two types of division are indistinguishable.
• 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. For example, the answer to 17 ÷ 5 can be written as 3 with 2 remaining, or as 3 and , where the 2 left over are distributed among 5. So, the result is  or 3.4.
• 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 to pack 17 items into 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?).
• There are two common algorithms used for division in North America (with variations on each). In both algorithms the recording scheme is not immediately clear, and both will require direct instruction for students to understand and replicate the procedure. Visual models are very important for building conceptual understanding.
  • The most common division algorithm, sometimes referred to as “long division” or “bring-down division”, decomposes the total using place value. Unlike other algorithms, this algorithm starts at the left and moves to the right. Column by column, it “shares” each place-value amount and trades the remainder for smaller pieces, which it adds to the amount in the next column. The partial quotients are then added together for the full quotient. Note that there are variations in how long division is recorded for this algorithm.

• Another well-known algorithm, sometimes called the “repeated subtraction” or “grouping division” algorithm, uses estimation and “think multiplication” to produce partial products. The partial products can be groups of any size, and are determined by a combination of estimation strategies, known facts, and mental strategies. Unlike other algorithms, the amount to be shared is not decomposed into place-value partitions but is considered as a whole.

• Multiplication and division can describe situations involving repeated equal groups.
• The multiplication of a whole number with a unit fraction such as 4 ×   can be interpreted as 4 groups of one third of a whole and can be determined using repeated addition. For example, 4 ×  = 4 one thirds . 
• Since multiplication and division are inverse operations, the division of a whole number by a unit fraction such as 4 ÷  can be interpreted as “How many one thirds are in 4 wholes?” Since it takes 3 one thirds to make 1 whole, it will take four times as many to make 4 wholes, so 4 ÷  = 12.

Note

• Counting unit fractions, adding unit fractions with like denominators, and multiplying unit fractions all represent the same action of repeating (or iterating) an equal group, in this case a unit fraction. This count is also reflected in the numerator.
• The use of drawings, tools (fraction strips, number lines), and objects can help visualize the role of the unit fraction to solve multiplication and division problems.

• A ratio describes the multiplicative relationship between two quantities.
• Ratios can compare one part to another part of the same whole, or a part to the whole. For example, if there are 12 beads in a bag that has 6 yellow beads and 6 blue beads, then:
  • the ratio of yellow beads to blue beads is 6 to 6 (6 : 6) or 1 to 1 because there is one yellow bead for every blue bead;
  • the ratio of yellow beads to the total number of beads is 6 to 12 (6 : 12) or 1 to 2, and this can be interpreted as one half of the beads in the bag are yellow, or that there are twice as many beads in the bag than the number that are yellow.
• Determining equivalent ratios involves scaling up or down. 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 two thirds () as many blue marbles as red marbles.
• A rate describes the multiplicative relationship between two quantities expressed with different units. For example, 1 dime to 10 cents can be expressed as 1 dime per 10 cents or 2 dimes per 20 cents, or 3 dimes per 30 cents and so on.

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 ones;
  • a relative comparison uses multiplication and division to determine that there are as many blue marbles as red marbles.
• Like ratios, rates make comparisons based on multiplication and division; however, rates compare two related but different measures or quantities. For example, if 12 cookies are eaten by 4 people, the rate is 12 cookies per 4 people. An equivalent rate is 6 cookies per 2 people. A unit rate is 3 cookies per person.
• 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 = 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.
• A ratio table is very helpful 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.
• A ratio or rate relationship can also be described using fractions, decimals, and percents.