## B2. Operations

### Specific Expectations

#### Properties and Relationships

B2.1

use the properties of addition and subtraction, and the relationship between addition and subtraction, to solve problems and check calculations

- When zero is added or subtracted from a quantity, the quantity does not change.
- Adding numbers in any order gives the same result.
- Addition and subtraction are inverse operations, and the same situation can be represented and solved using either operation. Addition can be used to check the answer to a subtraction question, and subtraction can be used to check the answer to an addition question.

*Note*

- Students need to understand the commutative and identity properties, but they do not need to name them in Grade 1. These properties help in developing addition and subtraction facts.
- This expectation supports most other expectations in the Number strand and is applied throughout the grade. Whether working with numbers or with operations, recognizing and applying properties and relationships builds a strong foundation for doing mathematics.
- Part-whole models help with noticing the inverse operations of addition and subtraction (see B2.4).
- The inverse relationship can be used to check that a solution is correct.

#### Math Facts

B2.2

recall and demonstrate addition facts for numbers up to 10, and related subtraction facts

- Understanding the relationships that exist among numbers and among operations provides strategies for learning basic facts.
- Knowing the fact families can help with recalling the math facts (e.g., 4 + 6 = 10, 10 – 4 = 6, and 10 – 6 = 4).
- There are many strategies that can help with developing and understanding the math facts:
- Counting on and counting back supports + 1, + 2, − 1, and − 2 facts.
- The commutative property (e.g., 6 + 4 = 10 and 4 + 6 = 10).
- The identity property (e.g., 6 + 0 = 6 and 6 – 0 = 6).
- Doubles, doubles + 1, and doubles – 1 (e.g., 4 + 5 can be thought of as 4 + 4 plus 1 more; 9 can be thought of as 10 less 1 or double 5 less one).

*Note*

- Addition and subtraction are inverse operations. This means that addition facts can be used to understand and recall subtraction facts (e.g., 5 + 3 = 8, so 8 – 5 = 3 and 8 – 3 = 5).
- Having automatic recall of addition and subtraction facts is useful when carrying out mental and written calculations and frees up working memory when solving complex problems and tasks.

#### Mental Math

B2.3

use mental math strategies, including estimation, to add and subtract whole numbers that add up to no more than 20, and explain the strategies used

- 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.

*Note*

- To do calculations in one’s head involves using flexible strategies that build on known facts, number relationships, and counting strategies. These strategies continue to expand and develop through the grades.
- 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 the relationships between numbers.
- Estimation can be used to check the reasonableness of calculations and should be continually encouraged when students are doing mathematics.
- Number lines, circular number lines, and part-whole models can help students visualize and communicate mental math strategies.

#### Addition and Subtraction

B2.4

use objects, diagrams, and equations to represent, describe, and solve situations involving addition and subtraction of whole numbers that add up to no more than 50

- 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.

- Acting out a situation by representing it with objects, a drawing, or a diagram can support students in identifying the given quantities in a problem and the unknown quantity.
- 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 between two numbers by comparing 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. Addition and subtraction are useful for showing:
- when a quantity
*changes*, either by*joining*another quantity to it or*separating*a quantity from it; - when two quantities (parts) are
*combined*to make one whole quantity; - when two quantities are
*compared*.

- when a quantity
- In addition and subtraction, what is unknown can vary:
- In
*change*situations, sometimes the result is unknown, sometimes the starting point is unknown, and sometimes the change is unknown. - In
*combine*situations, sometimes one part is unknown, sometimes the other part is unknown, and sometimes the total is unknown. - In
*compare*situations, sometimes the larger number is unknown, sometimes the smaller number is unknown, and sometimes the difference is unknown.

- In
- It is important to model the corresponding equation that represents the situation. The unknown may appear anywhere in an equation (e.g., 8 + ? = 19; ? + 11 = 19; or 8 + 11 = ?), and matching the structure of the equation to what is happening in the situation reinforces the meaning of addition and subtraction.
- Sometimes changing a “non-standard” equation (where the unknown is not after the equal sign) into its “standard form” can make it easier to carry out the calculation. Part-whole models make the inverse relationship between addition and subtraction evident and help students develop a flexible understanding of the equal sign. These are important ideas in the development of algebraic reasoning.
- Counting up or counting down are strategies students may use to determine an unknown quantity.

#### Multiplication and Division

B2.5

represent and solve equal-group problems where the total number of items is no more than 10, including problems in which each group is a half, using tools and drawings

- With equal-group problems, a group of a given size is repeated a certain number of times to create a total. Sometimes the size of each group is unknown, sometimes the number of groups is unknown, and sometimes the total is unknown.

*Note*

- For this expectation, students are always given the size of the equal groups and they determine either the number of equal groups or the total needed (not to exceed 10). In B1.6, students solve fair-share problems that find the size of an equal group.
- It is important that students represent equal-group situations using tools and drawings; this enables them to use counting to solve the problem.
- Solving equal-group problems lays a strong foundation for work with skip counting, using doubles as a fact strategy, multiplication and division, and fractions.