• some important properties of addition and subtraction:
  • identity property:
    • 7 + 0 = 7:

• 7 – 0 = 7:

• commutative property of addition:
  • 4 + 2 = 2 + 4 = 6:

• checking calculations using the relationship between addition and subtraction:
  • checking that 24 – 7 is 17 by adding 7 onto 17:

• relationship between addition and multiplication:
  • 30 can be interpreted as:
    • the sum of 5 + 5 + 5 + 5 + 5 + 5
    • six skip counts of 5:

• “six groups of 5”:

• relationship between subtraction and division:
  • the hops on the number line below can be interpreted as:
    • 10 – 2 – 2 – 2 – 2 – 2 (repeated subtraction of two):

• 10 separated into groups of two, which results in five hops:
  • 2 + 2 + 2 + 2 + 2 = 10

• When zero is added or subtracted from a quantity, the quantity does not change.
• Two numbers can be added in any order because either order gives the same result.
• When adding more than two numbers, it does not matter which two numbers are added first.
• 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.
• Repeated addition can be used as a multiplication strategy by adding equal groups of objects to determine the total number of objects.
• Repeated addition can also be used as a division strategy by adding equal groups of objects to reach a given total number of objects.
• Repeated subtraction can be used as a division strategy by removing equal groups of objects from a given total number of objects.
• Repeated addition or repeated subtraction can be used to check answers for multiplication and division calculations when they are not used as the initial strategy to do the multiplication or division calculation.
• The commutative property for addition states that the order in which two numbers are added does not change the total. For example, 5 + 3 is the same as 3 + 5, because 3 can be added onto 5 or 5 can be added onto 3; either way the result is 8. This is particularly helpful when learning math facts (see B2.2).
• The commutative property does not hold true for subtraction. For example, 5 – 4 = 1; however, it is not the same as 4 – 5 = –1. Students in Grade 2 do not need to know that 4 – 5 = –1, only that it has a result that is less than zero. To help students grasp this concept, show them how the scale on a thermometer includes numbers less than zero.
• The associative property states that when adding a group of numbers, the pair of numbers added first does not matter; the result will be the same. For example, in determining the sum of 8 + 7 + 2, 8 and 2 can be added first and then that result can be added to 7. Using this property is particularly helpful when doing mental math (see B2.3) and when looking for ways to “make 10” is useful.

Note

• 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.
• Students need to develop an understanding of the commutative, identity, and associative properties, but they do not need to name them in Grade 2. These properties help to develop automaticity with addition and subtraction facts.
• Support students in making connections between skip counts and repeat addition.

• strategies to develop addition facts:
  • doubles facts:
    • 8 + 8:

• “near doubles", “doubles plus one", or “doubles minus one” facts:
  • 7 + 6:
    • 7 + 7 − 1
    • 6 + 6 + 1
• “making 10” facts:
  • 3 + 8:
    • 2 + 8 + 1 = 11
• strategies to develop subtraction facts:
  • “counting on by ones” and “using 10 as a bridge”:
    • 15 – 7:
      • start at 7 and count up by ones: “8, 9, 10; that’s 3 and 5 more is 8”
  • “back down through 10”:
    • working backwards with 10 as a “bridge”
    • 13 – 5 can be solved by subtracting 3 from 13 to get to 10, and then subtracting 2 from 10 to make 8
  • ”think addition” with sums up to 20
    • 13 – 7:
      • 7 + ? = 13
• using the commutative property of addition:
  • 4 + 9 = 9 + 4 = 13

• The focus in Grade 1 math facts was on recalling and demonstrating addition facts for numbers up to 10, and related subtraction facts. In Grade 2, students will expand their range to include numbers that add up to 20, e.g., 9 + 9 = 18 and related subtraction facts, e.g., 18 – 9 = 9.
• There are many strategies that can help with developing and understanding math facts:
  • Working with fact families, such as 7 + 6 = 13; 6 + 7 = 13; 13 – 7 = 6; 13 – 6 = 7.
  • Using doubles with counting on and counting back; for example, 7 + 9 can be thought of as 7 + 7 plus 2 more; 15 can be thought of as 16 less 1 (double 8 less one).
  • Using the commutative property (e.g., 5 + 8 = 13 and 8 + 5 = 13).
  • Using the identity property (e.g., 6 + 0 = 6 and 6 – 0 = 6).
  • "Making 10" by decomposing numbers in order to make 10. For example, to add 8 and 7, the 7 can be decomposed as 2 and 5, resulting in 8 + 2 + 5.

Note

• Ten is an important anchor for learning basic facts and mental math computations.
• 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 important when carrying out mental or written calculations, and frees up working memory to do complex calculations, problems, and tasks.

• mental math strategies for addition:
  • counting on from the larger number:
    • 3 + 38:
      • start at 38 then count up three: “39, 40, 41”
  • making 10s:
    • 13 + 28:

• using a mental number line:
  • 13 + 28:
    • “I’m going to think about this as 28 + 13, because that will be easier.”
    • start at 28, jump 10, jump 3
    • land on 41

• using compensation:
  • 78 + 15:
    • 80 + 15 = 95 (add 2 to 78)
    • 95 – 2 = 93 (take away the 2 that was added to the 78)
  • 27 + 9: can be represented as 26 + 10 = 36
• mental math strategies for subtraction:
  • counting on by ones:
    • 82 – 77:
      • start at 77 and count up by 1s: ”78, 79, 80, 81, 82” (5 is the result)
  • counting back by ones:
    • 93 – 87:
      • start at 93 and count down by 1s to 87: “92, 91, 90, 89, 88, 87” (6 is the result)
  • subtracting in parts:
    • 26 – 12:
      • First do 26 − 10 = 16
      • then subtract 2 more to get 14 using compensation
    • 78 – 9:
      • 78 – 10 = 68 (add 1 to the 9)
      • 68 + 1 = 69 (take away the 1 that was added to compensate)

• Mental math refers to doing a calculation 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.
• 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. 
• Number lines, circular number lines, and part-whole models can be used to show strategies for doing 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 solution.

Note

• Strategies for doing mental calculations will vary depending on the numbers, facts, and properties that are used. For example:
  • For 18 + 2, simply count on.
  • For 26 + 13, decompose 13 into 10 and 3, add 10 to 26, and then add on 3 more.
  • For 39 + 9, add 10 to 39 and then subtract the extra 1.
• 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.

• situations involving addition and subtraction:
  • changing a quantity by joining another quantity to it (change/join):
    • adding 6 on to 52:

• changing a quantity by separating or removing another quantity from it (change/separate): 
  • taking 6 away from 52:

• combining multiple quantities (parts) to make one whole quantity (part-part-whole):
  • combining 48 and 32 (parts) to make 80 (whole):

• determining the unknown part (?) when given the whole (18) and a part (11):

• comparing two quantities by finding the difference between them (compare):
  • determining the difference between 82 and 37:

• 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 represent adding 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 numbers 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. 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.
• In addition and subtraction situations, 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 order to reinforce the meaning of addition and subtraction, it is important to model the correct equation by matching its structure to the situation and placing the unknown correctly; for example, 8 + ? = 19, or ? + 11 = 19, or 8 + 11 = ?.
• 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 support students in developing a flexible understanding of the equal sign. These are important ideas in the development of algebraic reasoning.

Have students represent and solve a wide variety of problem types and structures. The following are some examples appropriate for Grade 2 (also shown on BLM2: B2.4, Problem Types). Describe situations verbally to ensure that reading is not a barrier, and use contexts that are engaging to the students in the class. [Note that the intent here is not that students memorize or learn about the different problem types, but that they are exposed to different problem structures in order to acquire a more robust understanding of how the operations can be applied in real life.]

For each situation, have students draw a model or act it out using concrete materials. Support students in writing the number sentence to match the situation, and discuss the strategies used to find an answer.

• repeated equal groups in real life:
  • groups of 1: tail on a dog, beak on a blue jay
  • groups of 2: legs on a duck, wheels on a bicycle
  • groups of 3: wheels on a tricycle, sides of a triangle
  • groups of 4: legs on a horse, wheels on a car
  • groups of 5: fingers on a hand, toes on a foot
  • groups of 6: eggs in a half carton
• representing equal groups using tools and drawings:
  • 2s on a hundred chart:

• 5s using tallies:

• 10s on a rekenrek:

• three fourths using paper folding:

• repeats of one half using hops on a number line*:

*Students should not be expected to write fractions or mixed numbers using fraction notation, but teachers may model this convention, making connections to the representations when appropriate.

• Multiplication can describe situations involving repeated groups of equal size.
• Multiplication names the unknown total when the number of groups and the size of the groups are known.

Note

• Multiplication as repeated equal groups is one meaning of multiplication. In later grades, other meanings that students will learn include scaling, combinations, and measures, all of which require a major shift in thinking from addition.
• With addition and subtraction, each number represents distinct and visible objects that can be counted. For example, 7 + 3 can be represented by combining 7 blocks and 3 blocks. However, with multiplication involving repeated equal groups, one number refers to the number of objects in a group, and the other number refers to the number of groups or number of counts of a group. For example, 7 × 2 can be interpreted as 7 groups of 2 blocks or a group of 7 blocks, 2 times.
• Multiplication requires a “double count”. One count keeps track of the number of equal groups. The other count keeps track of the total number of objects. Double counting is evident when people use fingers to keep track of the number of groups as they skip count towards a total.

• problems involving division:
  • determining the number of groups:
    • I have 12 peaches in all. I have 4 peaches in each basket. How many baskets do I have?
  • determining the size of the group:
    • I have 12 peaches packed equally in 3 baskets. How many peaches are in each basket?

• Division, like multiplication, can describe situations involving repeated groups of equal size.
• While multiplication names the unknown total when the number of groups and the size of the groups are known, division names either the unknown number of groups or the unknown size of the groups when the total is known.

Note

• The inverse relationship between multiplication and division means that any situation involving repeated equal groups can be represented with either multiplication or division. While this idea will be formalized in Grade 3 (see B2.1), it is helpful to notice this relationship in Grade 2 as well.
• While it may be important for students to develop an understanding of these operations separately at first, it is also important for students to observe both multiplication and division situations together, to recognize similarities and differences.
• There are two different types of division problems.
  • 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 is shared equally among a given number of groups. Equal-sharing division is also being used to develop an understanding of fractions in B1.6 and B1.7.
  • 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 are measured out. (Students often use repeated addition or subtraction to represent this action.)
• Equal-group situations can be represented with objects, number lines, or drawings, and often the model alone can be used to solve the problem. It is important to model the corresponding equation (addition or subtraction and division) for different situations and to make connections between the actions in a situation, the strategy used to solve it, and the operations themselves.
• Although the number sentences representing both division situations might be the same, the action suggested and the drawing used to represent each of them would be very different. It is important to provide students with experiences representing both types of division situations.