## B2. Operations

### Specific Expectations

#### Properties and Relationships

B2.1

use the properties of operations, and the relationships between addition, subtraction, multiplication, and division, to solve problems involving whole numbers, including those requiring more than one operation, and check calculations

- The commutative property holds true for addition and for multiplication. The order of the numbers does not matter; the results will be the same. For example, 4 + 6 = 6 + 4 and 4 × 6 = 6 × 4.
- The associative property holds true for addition and for multiplication. The pairs of numbers that are added first or multiplied first does not matter; the results will be the same. For example, (2 + 3) + 5 = 2 + (3 + 5). Similarly, (2 × 3) × 5 = 2 × (3 × 5).
- The distributive property can be used to determine the product of two numbers. For example, to determine 8 × 7 one can rewrite 8 as 5 and 3 and find the sum of the products for 5 × 7 and 3 × 7 (i.e., 8 × 7 = (5 + 3) × 7 which equals (5 × 7) + (3 × 7), which is 35 + 21, or 56).
- Addition and subtraction are inverse operations. Any subtraction question can be thought of as an addition question (e.g., 54 – 48 = ? is the same as 48 + ? = 54) and vice versa. 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., 16 ÷ 2 = ? is the same as ? × 2 = 16), 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, 4 × 7 may be first determined using the distributive property as (2 × 7) + (2 × 7), and then checked by decomposing (4 × 7) as (2 × 2) × 7 and using the associative property 2 × (2 × 7).
- 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 many 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.
- The four operations are related. Addition and subtraction strategies can be used to think about and solve multiplication and division questions (see
- When addition is used to solve a subtraction question, this is often referred to as finding the missing addend.
- 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 among them, enables one to choose the operation that most closely represents a situation and most efficiently solves the problem given the tools at hand.

#### Math Facts

B2.2

recall and demonstrate multiplication facts for 1 × 1 to 10 × 10, and related division facts

- The identity principle states that when multiplying an amount by 1 or dividing an amount by 1, the amount stays the same (e.g., 5 × 1 = 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 × 7 can be determined by knowing 7 × 2.
- 7 × 3 can be determined by knowing 7 × 2 and then adding one more group of 7.
- 7 × 4 can be determined by knowing 7 × 2 and then doubling.

- Division facts can be determined using multiplication facts (e.g., 24 ÷ 6 can be determined using the multiplication facts for 6).

*Note*

- Having automatic recall of multiplication and division facts is important when carrying out mental or written calculations, and frees up working memory when solving complex problems and tasks.
- The development of the other facts using the facts for 1, 2, 5, and 10 is based on the commutative, distributive, or associative properties and in being able to decompose numbers. For example:
- 2 × 7 can be determined by knowing 7 × 2 (commutative property).
- 7 × 3 can be determined by knowing 7 × 2 and then adding one more group of 7 (decomposing and using the distributive property).
- 7 × 4 can be determined by knowing 7 × 2 and then doubling (decomposition and associative property).
- 7 × 6 can be determined by knowing 7 × 5 and adding one more 7.
- 7 × 9 can be determined by knowing 7 × 10 and taking away 7.

- The array can be used to model multiplication and division because it structures repeated groups of equal size into rows and columns.
- In a multiplication situation, the number of rows and the number of columns for the array are both known.
- In a division situation, the total number of objects is known, as well as either the number of rows or the number of columns. In order to create an array to represent a division situation, the objects are arranged into the rows or columns that are known until all the objects have been distributed evenly.

- A strategic approach to learning multiplication and division facts recognizes that some facts are foundational for learning other facts. Although the precise order might differ and different strategies are certainly possible, researchers tend to suggest learning facts in related clusters. For example:
- Foundational facts: 2-facts, 5-facts, 10-facts.
- Near facts: 3-facts, 6-facts, 9-facts (using 2-facts, 5-facts, and 10-facts and adding or subtracting a group).
- Doubles: 4-facts, 6-facts (doubling strategies); 8-facts (doubling a double).
- Adding or subtracting doubles: 8-facts, 7-facts.

- Practice is important for moving from understanding to automaticity. Focusing on one set of number facts at a time (e.g., the 6-facts) and related facts (5-facts or 3-facts) is a useful strategy for building mastery.

#### Mental Math

B2.3

use mental math strategies to multiply whole numbers by 10, 100, and 1000, divide whole numbers by 10, and add and subtract decimal tenths, and explain the strategies used

- Multiplying a whole number by 10 can be visualized as shifting of the digit(s) to the left by one place. For example, 5 × 10 = 50; 50 × 10 = 500; 500 × 10 = 5000.
- Multiplying a whole number by 100 can be visualized as shifting of the digit(s) to the left by two places. For example, 5 × 100 = 500; 50 × 100 = 5000; 500 × 100 = 50 000.
- Mentally multiplying a whole number by 1000 can be visualized as a shifting of the digit(s) to the left by three places. For example, 5 x 1000 = 5000; 50 × 1000 = 50 000; 500 × 1000 = 500 000.
- Mentally dividing a whole number by 10 can be visualized as a shifting of the digit(s) to the right by one place, since the value of the numbers will be one tenth of what they were. For example, 5000 ÷ 10 = 500, 500 ÷ 10 = 50, 50 ÷ 10 = 5, 5 ÷ 10 = 0.5.
- Mental math strategies for addition and subtraction of whole numbers can be used with decimal numbers.
- To mentally add and subtract decimal numbers, the strategies may vary depending on the numbers given. For example:
- If given 44.9 + 31.9, one could round both numbers to 45 and 32 to make 77 and then remove 0.1 twice from the rounding, to make 76.8.
- If given 34.6 + 42.5, one could first make 1 by combining the 0.5 from both of the numbers, then add it to 34 to make 35. Next add 40 from 42 onto the 35 to make 75. Then add on the remaining numbers 2 and 0.1 to make 77.1.

*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 strategies. These strategies continue to expand and develop through the grades.
- When mentally adding and subtracting decimals – or anything – the unit matters. Only like units are combined. For example, hundreds are combined with hundreds, tens with tens, ones with ones, and tenths with tenths.
- Estimation can be used to check the reasonableness of calculations and should be continually encouraged when students are doing mathematics.

#### Addition and Subtraction

B2.4

represent and solve problems involving the addition and subtraction of whole numbers that add up to no more than 10 000 and of decimal tenths, using appropriate tools and strategies, including algorithms

- 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.

- There are a variety of tools and strategies that can be used to add and subtract numbers, including decimal tenths:
- Acting out a situation, by representing it with objects, a drawing, or a diagram, can help support students in identifying the given quantities in a problem and the unknown quantity.
- Set models can be used to add a quantity to an existing amount or removing a quantity from an existing amount.
- Linear models can be used to determine the difference between two quantities by comparing them visually.
- 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 structures”:
*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.

- The use of drawings and models, including part-whole models, helps with recognizing the actions and quantities involved in a situation. This provides insight into which operation to use and helps in choosing the appropriate equation to represent the situation.
- 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*recompose*numbers based on place value. They begin with the smallest unit – whether it be the ones column or decimal tenths – and use regrouping or trading strategies to carry out the computation. (See Grade 3, B2.4.) - 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 numbers because unlike with whole numbers, the smallest unit in a number is not always common (e.g., 90 − 24.7). In this case, the number 90 can be changed to 90.0 so that the units can more easily be aligned; that is, 0 is used as a placeholder.
- Making explicit the compactness and efficiency of the standard algorithm strengthens understanding of place value and the properties of addition and subtraction.

#### Multiplication and Division

B2.5

represent and solve problems involving the multiplication of two- or three-digit whole numbers by one-digit whole numbers and by 10, 100, and 1000, using appropriate tools, including arrays

- Situations involving multiplication include:
- groups of equal quantity – involves determining the total quantity given the number of equal groups and the size of each group;
- scale factor – involves changing the size of an initial quantity;
- area – involves a multiplication of two linear measures;
- combinations – involves determining the total number of combinations of two or more things.

- A variety of tools and strategies can be used to represent multiplication problems:
- Acting out a situation, by representing it with objects, a drawing, or a diagram, can help to identify the given quantities in a problem and the quantity.
- The array can be used to represent groups of equal quantity.
- A double number line can be used to represent scaling.
- Rectangular grids can be used to represent area measures.
- A tree diagram can be used to represent various combinations.

*Note*

- The numbers that are multiplied together are called factors. The result of a multiplication is called the product.
- Situations involving multiplication include:
- The array can be a model for showing multiplication and division because it structures repeated groups of equal size into rows and columns (see Spatial Sense, E2.5). The array makes visual connections to skip counting, the distributive property, the inverse relationship between multiplication and division, and the measurement of area.
- A double number line can be used to show the comparison between the original amount (one number line) and the scaled amount (another number line).
- A grid showing a rectangle partitioned vertically and horizontally can be used to show the decomposition of two factors and the sum of these parts.

B2.6

represent and solve problems involving the division of two- or three-digit whole numbers by one-digit whole numbers, expressing any remainder as a fraction when appropriate, using appropriate tools, including arrays

- Situations involving division include:
- groups of equal quantity – involves determining either the number of groups or the size of each group;
- scale factor – involves determining either the original quantity or the value that the original value was multiplied by;
- area – involves determining the value of either linear measure;
- combinations – involves determining the number of possible values of one attribute or the other.

- A variety of tools and strategies can be used to represent division problems:
- Acting out a situation, by representing it with objects, a drawing, or a diagram, can help identify the given quantities in a problem and the unknown quantity that needs to be determined.
- The array can be used to represent groups of equal quantity.
- A double number line can be used to represent scaling.
- Rectangular grids can be used to represent area measures.
- A tree diagram can be used to represent various combinations.

*Note*

- Multiplication and division are inverse operations (see B2.1).
- The numbers that are 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).

- Situations involving multiplication and division include:
- When an array is used to represent division, the total quantity is given, and one of the factors (which can be either a row or column of the array). The total quantity is equally divided among these rows or columns.
- When a double number line is used to represent a division in which the original and new quantities are known, one needs to determine the scale factor that is used to go from the original number line to the new one.
- When a rectangle is used to represent division, and the total number of 1-unit squares and one dimension are known, one needs to arrange the unit squares into a rectangle that has the given dimension.
- For each division situation, there are two division types:
- equal-sharing division (also called “partitive division”):
*What is known:*the*total*and*number*of groups.*What is unknown:*the*size*of the groups.*The action:*a total shared equally among a given number of groups.

- equal-sharing division (also called “partitive division”):
- See B1.5 for connections between equal-sharing division and fractions.
- equal-grouping division (also called “measurement division” 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 measured out.

- equal-grouping division (also called “measurement division” or “quotative division”):
- Division does not always result in whole number amounts. The real-life situation determines whether the fraction is rounded up or rounded down or remains a fraction. For example:
- 17 items shared among 5 (i.e., 17 ÷ 5) means each receives 3 items and of another item.
- 17 people needing to go in cars that hold 5 people means that 3 cars are needed for 15 of them, plus another car is needed for the remaining 2, so 4 cars are needed in all.
- Determining how many $5 items can be bought with $17 means that 3 items can be purchased; there is not enough money for the fourth item.

B2.7

represent the relationship between the repeated addition of a unit fraction and the multiplication of that unit fraction by a whole number, using tools, drawings, and standard fractional notation

- The numerator in a fraction describes the count of unit fractions. So, 4 one thirds (four thirds) is written in standard fractional form as .
- There is a relationship between the repeated addition of a unit fraction, the multiplication of that unit fraction, and standard fractional notation:
- 4 one thirds (four thirds) can be represented as = 4 × or .

*Note*

- It is important that students recognize the connection between counting unit fractions (see B1.6), repeated addition and multiplication of unit fractions, and the meaning of the numerator (see B1.4).
- As students come to associate multiplication with the count (the numerator) and division with the unit size (the denominator), they come to understand the standard fractional notation and its connection to the operations of multiplication and division.

B2.8

show simple multiplicative relationships involving whole-number rates, using various tools and drawings

- A rate describes the multiplicative relationship between two quantities expressed with different units (e.g., bananas per dollar; granola bars per child; kilometres per hour).
- A rate can be expressed in words, such as 50 kilometers per hour.
- A rate can be expressed as a division statement, such as 50 km/h.
- There are many applications for rates in real life.

*Note*

- 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, then 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.