Operations & Algebraic Thinking
Multiplication and Division within 100
Grade 3
- ✓By the end of this lesson students will be able to understand multiplication as finding the total number of objects in equal groups.
- ✓By the end of this lesson students will be able to understand division as partitioning a set of objects into equal shares or as finding an unknown factor.
- ✓By the end of this lesson students will be able to apply properties of operations (commutative, associative, and distributive) to multiply and divide.
- ✓By the end of this lesson students will be able to solve two-step word problems involving multiplication and division within 100.
- ✓By the end of this lesson students will be able to use the inverse relationship between multiplication and division to solve problems.
Key concepts
Multiplication is a way to find the total number of items when you have a certain number of groups, and each group has the same number of items. For example, if you have 3 groups of 4 apples, you can multiply 3 x 4 to find the total number of apples.
Division is a way to share a total number of items equally among a certain number of groups, or to find out how many equal groups can be made from a total. For example, if you have 12 cookies and want to share them equally among 3 friends, you divide 12 ÷ 3 to find out how many cookies each friend gets.
This property states that you can multiply numbers in any order and the product will be the same. The order of the factors does not change the product.
This property states that when you multiply three or more numbers, the way you group the factors does not change the product. You can change where the parentheses are placed.
This property lets you break apart one of the factors in a multiplication problem into an addition problem, multiply each part, and then add the products. It can make multiplying larger numbers easier.
Multiplication and division are inverse operations, meaning they 'undo' each other. If you know a multiplication fact, you also know related division facts. For example, if 3 x 4 = 12, then 12 ÷ 3 = 4 and 12 ÷ 4 = 3.
These are word problems that require two separate math operations to solve. You need to read carefully to identify the first step and then use that answer to complete the second step.
Key facts to remember
- 1Multiplication is a shortcut for repeated addition.
- 2Division is a shortcut for repeated subtraction, or sharing equally.
- 3The product is the answer to a multiplication problem.
- 4The quotient is the answer to a division problem.
- 5Any number multiplied by 0 is 0.
- 6Any number multiplied by 1 is that number itself.
- 7Any number divided by 1 is that number itself.
- 8Any number (except 0) divided by itself is 1.
Worked examples
Example 1
Ms. Davis has 5 rows of desks in her classroom. Each row has 6 desks. How many desks are there in total? Then, if she adds 1 more row of 6 desks, how many desks will she have?
Answer
There are 36 desks in total.
This is a two-step problem. First, we multiply to find the initial total, then we add to find the new total.
Example 2
A baker made 24 cupcakes. He wants to put them into boxes, with 4 cupcakes in each box. How many boxes will he need? Use the inverse relationship with multiplication to check your answer.
Answer
The baker will need 6 boxes.
Remember that division and multiplication are inverse operations. You can use one to check the other.
Example 3
Use the Distributive Property to solve 7 × 8.
Answer
7 × 8 = 56
You could also break 7 into 5 + 2, or 8 into 4 + 4. The Distributive Property works no matter how you break apart one of the factors, as long as the sum is the original factor.
Common mistakes
- ✗Confusing the operations: Students sometimes multiply when they should divide, or vice versa, especially in word problems.
- ✗Not understanding the meaning of factors and products, or dividends, divisors, and quotients.
- ✗Making calculation errors, especially with basic facts, which can lead to incorrect answers in multi-step problems.
- ✗Forgetting to complete all steps in a two-step word problem, only solving the first part.
- ✗Incorrectly applying properties, such as thinking the Commutative Property applies to division (e.g., 10 ÷ 2 is not the same as 2 ÷ 10).
Exam tips
- ★Read word problems carefully, sometimes twice, to understand what is being asked and to identify the operations needed.
- ★Draw pictures or use manipulatives (like counters) to model equal groups for multiplication and division problems.
- ★Show all your work step-by-step, especially for two-step problems, so you can track your thinking and find any errors.
- ★Check your answers using the inverse operation. If you divided, multiply to check. If you multiplied, you can divide to check.
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