The Multiplication - Division Relationship

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Students will be able to explain why multiplication can help solve division problems and explain their strategies in solving several word problems.

Big Idea

Critical Area 1 for third grade states that students must use a variety of solution strategies in order to learn the relationship between multiplication and division. This lesson takes the students through several strategies.


15 minutes

To begin the mini-lesson today, I ask the students to share what they know about multiplication and division being related. This is a concept we have been working with for the whole school year, but it is still difficult for the children to see the operations as related, rather than separated.  (This is not uncommon but needs to be specifically addressed.)

I put a multiplication problem with a missing factor (8 x f = 24) on the board and ask the students to gather with their math reflection journals. I then have them look at the equation and turn to tell their partner what they think the variable is and why.  I then have them write their evidence in their journals.

Many use known multiplication facts, but some use division, which I was glad of.  I made sure to share those responses with the whole class, as well as other strategies.  

Be sure to pull out the conversation of division and relate the whole lesson to multiplication, in context, repeatedly.  Work to point out to the students that they are naturally thinking of both operations as you confer and model.

Active Engagement

20 minutes

To continue with the idea of relating the operations, I decided to use an equation (5 = d ÷ 3) that is complicated in several ways.  The quotient is first and it is a division problem with the dividend (the whole) unknown. The students will need to make some sense of this prior to choosing their strategy. 

Again, when you have your students share out, be sure to pull out the idea that multiplication was helpful and use the terms factors, product, quotient, dividend, and divisor.

This student and her partners are able to solve for the variable, but are unsure if their strategy and dividend make sense.  In their conversation, they know their number works, but because of the quotient being first, they are reading it as a possible negative number.

Remember that this kind of questioning is an excellent opportunity for a quick mini-lesson. In this context the focus is Math Practice 8: Look for and express regularity in repeated reasoning. 

I work with these boys for nearly 5 minutes.  They begin unsure of how to attack the problem. With prompting, they start to draw out the groups of 3.  Finally, one exclaims, "OH, I GET IT!" as he recognizes the pattern.  Following is the rest of our conversation.

Journal Response

15 minutes

For the next step, I ask the students to discuss at their tables what they tried and what helped them with the last equation.  I then had them place a math prompt in their journals.  

If you journal often, I have found it very helpful to have the prompts pre-printed on mailing labels, so they can just be passed out and students can focus on the work, not the writing of the prompt. 

I choose this prompt because it called for the students to work on explaining how the multiplication and division problems are related.  This is a completely different situation than finding a missing factor or quotient.   

In the resources, you will see a few examples from our session.  One student used a fact triangle and went on to explain that division is the opposite of multiplication.  To an 8 year old, that is appropriate, but I hope to bring him along to the point where he can discuss missing factors and products becoming quotients.

The other student example shows that the student understands that two equations can be created using the multiplication equation given.

Sharing Journals

10 minutes

I am always working on different ways to share out thinking and learning at the end of a lesson. Today, I ask students to trade their journals with another person, but to not talk about them. I ask everyone to look at their peers' journal entries and work to figure out what strategy they used and how they explained their thinking. 

Then, we sit in a circle and try to explain each other's thoughts.  Not only does this push the students to look carefully at another's work and make sense of their thinking, but it pushes them to think about their own understanding as well.