Multiplication and Division

Definitions

x: the multiplication symbol means: "groups of"

factor: the numbers multiplied together

product: the answer to a multiplication equation

equal groups: a set of groups with the same number of object in each

multiples: the products of a particular time table Example: multiples of x3 include: 3, 6, 9, 12, 15 ...

arithmetic pattern: a pattern created by the multiples of a times table

divide: the operation that separates into equal groups (sharing)

dividend: the number to be divided in the equation; the total number of objects

divisor: the number of groups or number in each group you are dividing into

quotient: the answer to a division equation

Multiplication

Strategies for Multiplication

Multiplication: multiple equal groups that create a whole

____________ x _____________ = ____________

# of groups size product

There are many ways to solve multiplication equations before your child has committed them into his/her memory. Here are the different ways:

Repeated Addition: using addition to solve the equation
Skip Counting on a number line: jumping along the number line skip counting by one of the factors. (3x4)
Groups of: creating groups (circles) and using objects like dots or tally marks to represent the number of object in each group.
Arrays: rows times columns
Number Bonds: Total amount in the top with the number of groups attached (like legs) and the size in each part.
Skip Counting on a number line: jumping along the number line skip counting by one of the factors.
Tape Diagram: The bar is the total amount. the groups are partitioned into the bar, then the size goes into each part.

Multiples

Using hundreds boards (charts), students found all the multiples of each times table (x2-x10). We discussed and recorded the arithmetic patterns that are true for each table.

Multiples of 2

- - Counting by 2's
  - EVEN
  - The ones place only has 2, 4, 6, 8, or 0.
  - Always has a half

Multiples of 3

- Counting by 3's
- Even and Odd
- All 3-digit numbers, with the SAME digit 3 times (111, 222 ...)

Multiples of 4

- Counting by 4's
- EVEN
- The ones place only has 2, 4, 6, 8, or 0.
- Double the Double (Ex. 4 x 3 ... double the 3 = 6 ... double the 6 = 12)

Multiples of 5

- Counting by 5's
- Even and Odd
- The ones place only has 5 or 0 ONLY

Multiples of 6

- Counting by 6's
- EVEN
- All multiples of 6 are also multiples of 3
- multiples of 6 and 3 are connected (3 + 3 = 6)

Multiples of 7

- Counting by 7's
- Even and Odd

Multiples of 8

- Counting by 8's
- EVEN
- The ones place only has 2, 4, 6, 8, or 0.
- Double the Double and DOUBLE again! (Ex. 8 x 3 ... double the 3 = 6 ... double the 6 = 12 ... double the 12 = 24)

Multiples of 9

- Counting by 9's
- Even and Odd
- If you add the digits of the products together they will always equal 9 (you may have to do 2-steps).
  - EX: 9 x 3 = 27 ... 2 + 7 = 9
  - EX. of 2-step: 9 x 11 = 99 ... 9 + 9 = 18 ... 1 + 8 = 9
- Ask your child about the "Johnny Trick" ... number 0-9 down and then back up
- Finger Trick: (thumbs in or out ... just start counting from the LEFT hand)

Multiples of 10

- Counting by 9's
- Even and Odd
- If you add the digits of the products together they will always equal 9 (you may have to do 2-steps).
  - EX: 9 x 3 = 27 ... 2 + 7 = 9
  - EX. of 2-step: 9 x 11 = 99 ... 9 + 9 = 18 ... 1 + 8 = 9
- Ask your child about the "Johnny Trick" ... number 0-9 down and then back up
- Finger Trick: (thumbs in or out ... just start counting from the LEFT hand)

INput/Output Charts

- In this example, the Rule is +4. So if 2 is the input (or "goes in") then 6 is the output (or "comes out") because 2 + 4 = 6.
- Students will have to be able to fill in missing inputs or outputs AS WELL AS figure out the rule.
- If you have to figure out an input, you can either do the opposite operation (-4 here) or you can create a missing number equation (__ + 4 = 16)
- Input/Output tables can look like either (horizontal and vertical)

Multiplying MULTIPLES of 10

Multiples of 10

When students are multiplying multiples of ten, they need to remember what all multiples of ten have in common. They all end in zero! 10, 20, 30, 40, etc.
When solving 60 x 4 = ___ students are counting by 60, four times. 30 + 60+ 60+ 60 = ___.
If a student is solving 60 x 4 = ___, they should know it will end in a zero. Because of this, students can break down this problem into an equation they are used to.

If 6 x 4 = 24

Then 60 x 4 = 240

Properties of Multiplication

Zero Property &

Identity Property

Commutative Property

Associative Property

Distributive Property

Division

Strategies for Division

Multiplication: multiple equal groups that create a whole

____________ ÷ _____________ = ____________

Dividend divisor quotient

There are many ways to solve division equations before your child has committed them into his/her memory. Here are the different ways:

- Fact Families: These are made up of 3 numbers and they create 4 equations, 2 multiplication and 2 division (unless it's a double, then there will only be 1 of each).
- Equal Groups: (Fair Share and Size Share) Take the divisor and draw that many circles. Then take the dividend (total), and share that number out equally between the groups. The quotient will be the number of items in each group
- Array: Take the divisor and draw that many rows. Then take the dividend and pass that number out equally on each row, creating columns. The quotient will be the number of items in each row.
- Repeated Subtraction: Start with the dividend and subtract the divisor. Then take that answer and subtract the divisor again. Continue until the answer is zero. Count the number of equations you had to do in order to get to zero.
- Skip Counting: Skip count by the divisor until you get to the dividend. ** You can also do this with a number line. **

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