Packets  /  Multiply and Divide Master
Grades 3-5Elementary

Multiply and Divide Master

Master multiplication facts, arrays, multi-digit multiplication, division facts, long division with remainders, and multiplying or dividing by 10, 100, and 1000. Learn the concepts, practice with instant feedback, and beat the 60-second challenge.

1Multiplication as Equal Groups

Key idea: Multiplication is a fast way to add equal groups: 4 times 3 means 4 groups of 3.
  1. Count how many groups there are (this is the first factor).
  2. Count how many items are in each group (this is the second factor).
  3. Add the groups together: 3 + 3 + 3 + 3, or multiply: 4 times 3.
  4. Check your answer by skip-counting: 3, 6, 9, 12.
Worked example
There are 3 bags with 4 apples each. How many apples?
3 groups of 4. Repeated addition: 4 + 4 + 4 = 12. So 3 times 4 = 12 apples.
Worked example
5 times 2 = ?
5 groups of 2: 2 + 2 + 2 + 2 + 2 = 10. So 5 times 2 = 10.
Watch out: 4 times 3 means 4 groups of 3 (12), not 4 + 3 (7). Watch for the multiplication sign!

2Arrays and the Commutative Property

Key idea: An array is a picture of multiplication: rows times columns. Flipping the array (turning rows into columns) does not change the total — this is the commutative property.
  1. Count the rows and the columns of the array.
  2. Write the fact: rows times columns.
  3. Flip it: columns times rows gives the same product.
  4. Check: a 3-by-4 array and a 4-by-3 array both have 12 dots.
Worked example
An array has 3 rows and 4 columns. How many dots?
Rows times columns: 3 times 4 = 12 dots.
Worked example
Show 2 times 7 the other way around.
Commutative property: 2 times 7 = 7 times 2 = 14.
Watch out: Order does not change the answer in multiplication, but describe the picture correctly: 3 rows of 4 is different from 4 rows of 3, even though both equal 12.

3Multiplication Facts and Strategies

Key idea: You do not have to memorize everything at once — use strategies: doubles, 5s and 10s, and the nines trick.
  1. Doubles: 7 times 8 = (7 times 4) doubled = 28 + 28 = 56.
  2. Fives: count by 5s. 6 times 5 = 5, 10, 15, 20, 25, 30.
  3. Tens: multiply by 10 by adding a zero: 7 times 10 = 70.
  4. Nines trick: 9 times n = (10 times n) minus n. So 9 times 7 = 70 minus 7 = 63.
  5. Break apart: 12 times 5 = (10 times 5) + (2 times 5) = 50 + 10 = 60.
Worked example
7 times 8 = ?
Double strategy: 7 times 4 = 28. Double it: 28 + 28 = 56.
Worked example
9 times 6 = ?
Nines trick: 10 times 6 = 60, minus 6 = 54.
Watch out: 7 times 8 is 56, not 54. When you use a strategy, do the final addition carefully!

4Division as Sharing and Grouping

Key idea: Division undoes multiplication and has two meanings: sharing fairly (how many in each group) or grouping (how many groups).
  1. Decide the meaning: are you sharing items among groups, or making groups of a certain size?
  2. Sharing: 12 cookies shared by 3 friends. Deal them out: each friend gets 4.
  3. Grouping: 12 cookies, 4 per bag. Make bags: you get 3 bags.
  4. Both are division: 12 divided by 3 = 4, and 12 divided by 4 = 3.
Worked example
20 stickers shared equally by 5 friends. How many each?
Sharing: 20 divided by 5 = 4 stickers each. Check: 5 times 4 = 20.
Worked example
20 stickers, 5 per page. How many pages?
Grouping: 20 divided by 5 = 4 pages. Same numbers, different story!
Watch out: The answer means something different in each story: '4 stickers each' vs '4 pages'. Always name your units!

5Division Facts and Fact Families

Key idea: Every multiplication fact has a partner division fact. The numbers 3, 8, and 24 make one family: 3 times 8 = 24, 8 times 3 = 24, 24 divided by 8 = 3, 24 divided by 3 = 8.
  1. Think of the multiplication fact you know: 3 times 8 = 24.
  2. To divide 24 by 8, ask: '8 times what makes 24?'
  3. Write all four facts in the family.
  4. Check each fact by covering one number and solving from the other two.
Worked example
56 divided by 7 = ?
Ask: 7 times what makes 56? 7 times 8 = 56, so 56 divided by 7 = 8.
Worked example
Write the fact family of 6, 9, and 54.
6 times 9 = 54; 9 times 6 = 54; 54 divided by 9 = 6; 54 divided by 6 = 9.
Watch out: Division is NOT commutative: 24 divided by 8 is 3, but 8 divided by 24 is not even a whole number. Order matters!

6Multiplying and Dividing by 10, 100, 1000

Key idea: Multiplying by 10, 100, or 1000 shifts digits left (add zeros); dividing by them shifts digits right (drop zeros).
  1. Count the zeros: 10 has 1, 100 has 2, 1000 has 3.
  2. To multiply, attach that many zeros: 25 times 100 = 2,500.
  3. To divide a number ending in zeros, drop that many zeros: 700 divided by 100 = 7.
  4. For two non-zero parts, multiply the parts then add all zeros: 30 times 40 = 3 times 4 with 2 zeros = 1,200.
Worked example
5 times 1,000 = ?
1,000 has 3 zeros. Attach them: 5,000.
Worked example
30 times 40 = ?
3 times 4 = 12. Each number has 1 zero, so add 2 zeros: 1,200.
Watch out: You can only drop zeros when dividing if the number actually ends in zeros. 45 divided by 10 is 4.5, NOT 4.

7Multi-Digit Multiplication

Key idea: Break a big number into tens and ones, multiply each part, then add the partial products.
  1. Split the larger number: 23 = 20 + 3.
  2. Multiply each part by the other factor: 20 times 4 = 80; 3 times 4 = 12.
  3. Add the partial products: 80 + 12 = 92.
  4. For two 2-digit numbers, make all four partial products and add.
Worked example
23 times 4 = ?
20 times 4 = 80. 3 times 4 = 12. 80 + 12 = 92.
Worked example
12 times 15 = ?
12 = 10 + 2. 10 times 15 = 150. 2 times 15 = 30. 150 + 30 = 180.
Watch out: Never forget to carry and add the carried digit! In 14 times 6: 4 times 6 = 24, write 4 carry 2; then 1 times 6 = 6, plus 2 = 8. Answer 84.

8Division with Remainders

Key idea: Sometimes division does not come out even. The leftover is the remainder, and it must always be smaller than the divisor.
  1. Divide: how many full groups fit? 14 divided by 3: 3 fits 4 times (12).
  2. Multiply back: 4 times 3 = 12.
  3. Subtract: 14 minus 12 = 2. That is the remainder.
  4. Check: remainder 2 is less than divisor 3. Answer: 4 r 2.
Worked example
29 divided by 5 = ?
5 fits 5 times: 5 times 5 = 25. 29 minus 25 = 4. Answer: 5 r 4.
Worked example
84 divided by 4 = ?
Long division: 8 divided by 4 = 2, 4 divided by 4 = 1. Answer: 21 exactly, no remainder.
Watch out: If your remainder is bigger than the divisor, your quotient can go higher! A remainder of 5 when dividing by 3 means you missed another full group.

9Multiply and Divide Word Problems

Key idea: Read carefully, decide multiply or divide, estimate first, then solve and check with the opposite operation.
  1. Read the story and name what you need to find.
  2. Clues: 'each' with groups to combine means multiply; sharing fairly or 'per group' means divide.
  3. Estimate first: 5 boxes times 8 crayons is about 5 times 10 = 50.
  4. Solve, then check with the opposite operation: 40 divided by 8 = 5 boxes. Correct!
  5. For two-step problems, solve the first step, then use its answer in the second step.
Worked example
Mia has 5 boxes with 8 crayons each. How many crayons in all?
'Each' with groups to combine: multiply. 5 times 8 = 40 crayons. Check: 40 divided by 8 = 5.
Worked example
A store has 3 boxes of 10 oranges and sells 8. How many are left?
Step 1: 3 times 10 = 30 oranges. Step 2: 30 minus 8 = 22 oranges left.
Watch out: Extra numbers can be a trap. Read the whole problem before grabbing the numbers!
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60-Second Challenge

How many multiply and divide master problems can you solve in 60 seconds?