Multiplication and Division
Strategy Notebook
3rd Grade
MCC3.OA.7
Multiplication Strategies for Fluency of the Basic Facts (Important)
Strategy / Description / ExamplesZero / If I have 0 groups of anything, I always have 0.
Think: The answer is always 0! / 0x5 Think: It’s always 0.
Ones (or the Identity) / If I have 1 group of any number, I just have that number.
Think: 1 times any number is that number. / 1x6 Think: It is just the number, so it is 6!
Twos / Double the number. OR
This is “Doubles” from addition! / 2x8 Think: Double 8, so 8+8 is 16.
Threes / Double plus another. / 3x6 Think: Double 6 is 12, add 6 again, that’s 18.
Fours / Double and double again.
OR
Double and then add the double to itself / 4x7 Think: Double 7 is 14 and double 14 is 28.
OR
Double 7 is 14, and 14 and 14 is 28.
Fives / Count by fives.
OR
Multiply by 10 and halve that. / 5x6 Think: 5, 10, 15, 20, 25, 30 That’s six 5s so the answer is 30.
OR
10 x6 is 60, half of 60 is 30.
Sixes / Triple the number and then double it. OR
Multiply by five and add another. / 6x7 Think: 3x7 is 21, double 21 and get 42.
OR
5x7 is 35, add another 7 and get 42.
Sevens / Multiply by five and add the double. / 7x8 Think: 5x8 is 40, 2x8 is 16, 40+16 is 56.
Eights / Double the number three times. / 8x6 Think: 2x6 is 12, 2x12 is 24, and 2x24 is 48.
Nines / Multiply by 10 and subtract the number. / 9x7 Think: 10x7 is 70, subtract 7 to get 63.
Tens / Add a zero to the number. / 10x6 Think: Put 6 in the tens place and 0 in the ones place to get 60.
Multiplication - Repeated Addition / Skip Counting (Important)
This is often a beginning strategy for students who are just learning multiplication. Connecting repeated addition and skip counting to an array model provides an essential visual model for multiplication.
6 x 15
15 + 15 + 15 + 15 + 15 + 1515 + 15 = 30
30 + 15 = 45
45 + 15 = 60
60 + 15 = 75
75 + 15 = 90 / Students will often think of the multiplication problem as an addition problem.
Multiplication - Making Landmark and Friendly Numbers
Sometimes a multiplication problem can be made easier by changing one of the factors to a “friendly” or “landmark number”.
- Friendly numbers are numbers that end in 0. They are called friendly because once the rule for multiplying 0 is understood, that understanding can be extended to larger numbers that end in 0.
- Landmark numbers are similar to friendly numbers. Some examples are 25, 50, 75, 100.
Students who are comfortable multiplying by multiples of ten will often adjust factors to allow them to take advantage of this strength. Drawing arrays provides a great model for helping students understand this strategy.
9 x 15
9 x 15+1 (group of 15)
10 x 15 = 150
150 – 15 = 135 / With this strategy, notice that not just one, but one group of 15 was added.
Since one extra group of 15 was added, it now must be subtracted.
The initial problem was 9 x 15, but it was changed to 10 x 15, which resulted in an area of 150 squares.
Here are some problems you can use during your number talk time that support “Making Landmark or Friendly Numbers.”
7 x 9 / 4 x 20 / 2 x 49 / 9 x 6 / 9 x 32 x 26 / 5 x 19 / 4 x 78 / 2 x 19 / 4 x 49
Multiplication - Partial Products Strategy – Distributive Property (Important)
This strategy is based on the distributive property. When students break apart factors into addends (through expanded form and distributive property), then they can use the smaller problems to solve more difficult ones. As students invent “partial product” strategies, they can break one or both factors apart. While both factors can be represented with expanded notation, it is sometimes more efficient to keep one number whole.
Example #1:
6 x 9
6 x 96 x (4 + 5)
6 x 4 = 24
6 x 5 = 30
24 + 30 = 54
6 x 9
(3 + 3) x (4 + 5)
3 x 4 = 12
3 x 5 = 15
3 x 4 = 12
3 x 5 = 15
12 + 15 + 12 + 15 = 54 / In this example, the 6 was kept whole and the 9 was thought about as 4+5.
This time both factors are broken apart. This works better with larger numbers.
Example #2:
7 x 12
7 x 127 x (10 + 2)
7 x 10 = 70
7 x 2 = 14
70 + 14 = 84
“Partial Products” Continued……
Here are some problems you can use during your number talk time that support “Partial Product Strategy.”
8 x 7 / 6 x 15 / 8 x 6 / 2 x 36 / 8 x 163 x 26 / 2 x 36 / 2 x 45 / 8 x 56 / 125 x 6
More Multiplication Strategies for Basic Facts
Unknown Facts: Build From Facts You Do Know
Examples: 6 x 7 = ?
Hmm, I know 6 x 6 = 36, so another 6 makes 42.
4 x 9 = ?
I know 2 x 9 is 18, so double this and get 36.
13 x 12 = ?
Let’s see, 13 x 10 = 130 and 13 x 2 = 26, and 130 + 26= 156.
6 x 19 = ?
Well, 6 x 20 = 120 and 120-6 = 114.
Multiplication Rhymes: These are fun ways for your students to remember multiplication facts!
4 x 4
A 4 x 4 is a mean machine
I’m gonna get one, when I’m 16.
4 x 4 = 16
6 x 6
Six cold six packs.
Thirsty-chicks will have for snacks.
6 x 6 = 36
7 x 7
7 x 7 made with lines
Bend em up and down to make 49.
7 x 7 = 49
8 x 8
I 8 (ate) 8 times and fell on the floor.
Couldn’t get up till I was 64
8 x 8 = 64
6 x 4
6 x 4 I have my eggs, I’m at the store.
Oooops!
I dropped them on the floor.
Would have been exactly 24!
6 x 4 = 24
7 x 4
The animals are coming
7 x 4
Better open the gate.
There are 28!
7 x 4 = 28
4 x 8
4 x 8
Clean your plate.
I ate the ‘goo’
Now I’m thirsty-too! (32)
4 x 8 = 32
6 x 7
Happy Birthday to Kevin
He’s 6 x 7
He blew and blew
Cause he’s 42.
6 x 7 = 42
7 x 8
7 packs of gum
Each with 8 sticks
Open up, big mouth,
Can you chew all 56?
6 x 8
Flight 6 x 8! Don’t be late!
Leaving at gate 48.
6 x 8 = 48
Division
Repeated Subtraction or Sharing/Dealing Out (Important)
These are strategies that students often use when they are just learning division. Both of these strategies help students make connections to multiplication as the teacher records multiplication and division sentences to match student thinking.
Repeated subtraction is one of the least efficient strategies, especially when the numbers begin to get bigger. Students may use it with story problem contexts that encourage dealing out. For example:
June has 30 lollipops. She plans to give 5 lollipops to each friend. How many friends does June have?
30 ÷ 530 – 5 = 25
25 – 5 = 20
20 – 5 = 15
15 – 5 = 10
10 – 5 = 5
5 – 5 = 0 / Teacher: You kept taking away groups of 5 until the lollipops were gone. How many groups of 5 did you find?
Student: Six
Teacher: Is there a quicker way to use this strategy? Do you know how many two groups of five would be?
Student: ten
Teacher: Yes. So two times five equals ten. Do you have enough lollipops to do this again?
Student: Yes. We can get two more fives, and then we can do that one more time.
Teacher: 2 x 5 = 10
2 x 5 = 10
2 x 5 = 10
Which gives us: 6 x 5 = 30
The sharing and dealing out strategy is also a beginning division strategy. If a student is not comfortable with division they will share or deal out amounts in small quantities.
In this strategy the student associates the divisor with the number of groups between which the whole is being shared.
June has 30 lollipops. She plans to give 5 lollipops to each friend. How many friends does June have?
5 x 6 = 30 / June draws five boxes to represent the five friends. She then deals out one lollipop to each person. June repeats dealing out one lollipop at a time until all thirty lollipops are gone. She then counts how many lollipops each person received.
Teacher: Each person has six lollipops. We can write this as 5 x 6. I noticed you shared the lollipops one at a time. Do you think there is another way we could have shared them?
Student: Maybe two at a time?
Teacher: Let’s test that idea
Partial Quotient Strategy (Important)
The “partial quotients” strategy uses place value and allows students to build on multiplication facts with friendly numbers.
Look at the problem 96 ÷ 4. Notice, to the right, how the student wrote down facts that were easy for him/her and those were the only facts he/she used. This is a good way to introduce students to this strategy. After a while, students will gain enough number sense that they will realize there may be other multiplication facts to use that might be more efficient.
The partial quotient strategy can work with any division problem.
Division Strategies for Fluency of Facts
Mastery of division facts depends solely on the students’ understanding of relationships between multiplication and division and the mastery of multiplication facts. Fact families should be discussed, modeled, and written.
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