Addition and Subtraction
Conceptual Understanding of Addition
Addition combines quantities to find a total. Word problems involving addition typically fall into these structures:
Join problems
An initial quantity increases. "Maria had 5 apples. She picked 3 more. How many apples does she have now?"
Part-part-whole problems
Two parts combine to make a whole. "There are 4 red blocks and 6 blue blocks. How many blocks are there in all?"
Compare problems with "more"
"Tom has 8 stickers. Sam has 3 more stickers than Tom. How many stickers does Sam have?"
Conceptual Understanding of Subtraction
Subtraction can be understood through multiple models:
Take-away model
Starting with a quantity and removing some. "There were 12 cookies. We ate 5. How many are left?"
Comparison model
Finding the difference between two quantities. "Sara has 15 crayons. Ben has 9 crayons. How many more crayons does Sara have?"
Missing addend model
Finding what must be added to reach a total. "I have 7 stamps. I need 12 stamps. How many more do I need?"
Properties of Addition
Commutative property
The order of addends does not change the sum. a + b = b + a. Example: 4 + 7 = 7 + 4 = 11.
Associative property
The grouping of addends does not change the sum. (a + b) + c = a + (b + c). Example: (3 + 5) + 2 = 3 + (5 + 2) = 10.
Identity property
Adding zero to any number gives that number. a + 0 = a. Example: 9 + 0 = 9.
Mental Math Strategies for Addition and Subtraction
Counting on
Starting from the larger number and counting up. For 8 + 3, start at 8 and count: 9, 10, 11.
Making ten
Decomposing numbers to create a group of ten. For 7 + 5, think: 7 + 3 = 10, then 10 + 2 = 12.
Doubles and near doubles
Using known doubles facts. For 6 + 7, think: 6 + 6 = 12, so 6 + 7 = 13.
Compensation
Adjusting numbers to make calculation easier. For 48 + 29, think: 48 + 30 = 78, then 78 - 1 = 77.
Standard Algorithms
The standard algorithm for multi-digit addition involves adding digits column by column from right to left, regrouping when a column sum exceeds 9.
The standard algorithm for multi-digit subtraction involves subtracting digits column by column from right to left, regrouping (borrowing) when the top digit is smaller than the bottom digit.
Understanding place value is essential for explaining why these algorithms work. When regrouping in addition, 10 ones become 1 ten. When regrouping in subtraction, 1 ten becomes 10 ones.
Multiplication and Division
Conceptual Understanding of Multiplication
Multiplication represents repeated addition of equal groups. The expression 4 × 3 means four groups of three, or 3 + 3 + 3 + 3 = 12.
Multiplication can also be understood through:
Array model
Objects arranged in rows and columns. A 4 × 3 array has 4 rows with 3 objects in each row.
Area model
The product represents the area of a rectangle with given dimensions.
Scaling
Multiplication as resizing or stretching a quantity.
Conceptual Understanding of Division
Division can be understood through two models:
Partitive (sharing) division
Dividing a quantity into a known number of equal groups to find how many in each group. "24 cookies shared equally among 6 children. How many cookies does each child get?"
Measurement (quotitive) division
Determining how many groups of a known size can be made from a quantity. "24 cookies put into bags of 4. How many bags can be made?"
Properties of Multiplication
Commutative property
a × b = b × a. Example: 3 × 7 = 7 × 3 = 21.
Associative property
(a × b) × c = a × (b × c). Example: (2 × 3) × 4 = 2 × (3 × 4) = 24.
Identity property
a × 1 = a. Example: 8 × 1 = 8.
Zero property
a × 0 = 0. Example: 5 × 0 = 0.
Distributive property
a × (b + c) = (a × b) + (a × c). Example: 4 × 13 = 4 × (10 + 3) = 40 + 12 = 52.
Mental Math Strategies for Multiplication
Doubling and halving
Adjusting factors to make calculation easier. For 4 × 15, think: 2 × 30 = 60.
Using known facts
Building on multiplication facts already memorized.
Breaking apart (distributive property)
For 7 × 8, think: (7 × 5) + (7 × 3) = 35 + 21 = 56.
Division with Remainders
When dividing, the quotient may not be a whole number. The remainder is the amount left over after dividing as completely as possible with whole numbers.
For 23 ÷ 5: 5 goes into 23 four times (5 × 4 = 20), with a remainder of 3. We write 23 ÷ 5 = 4 R3.
Interpreting remainders depends on context:
Sometimes we ignore the remainder (full groups only)
Sometimes we round up (need an extra unit)
Sometimes we express the remainder as a fraction or decimal
Order of Operations
When expressions contain multiple operations, follow this order (PEMDAS):
Parentheses (and other grouping symbols)
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)
Worked Example
3 + 4 × 2 = 3 + 8 = 11 (multiply before adding) Example: (3 + 4) × 2 = 7 × 2 = 14 (parentheses first)
