Multiplication Models and Structure

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SWBAT solve for unknown variables using the structure of a multiplication sentence of groups x items in a group = whole amount.

Big Idea

Multiplication sentence structure using models to solve for unknown variables in multiplication.


15 minutes

I say to the students, Today we are moving from addition to multiplication!  This is growing from second grade math to third grade math.  This will be easy and fun, and you are going to think this is SO easy. 

I show the students two groups of five objects such as counters, unifix cubes, paper clips, etc.  I choose an easy amount so they can quickly determine how many items I have in the two groups. I say, I have two groups of 5 counters.  I repeat this with different numbers of groups and objects for each group. 

My experience has been that students do not focus on the factors of a multiplication sentence but rather on the product or answer.  This process of drawing a model requires them to identify each of the factors. 


15 minutes

Next, I draw the Groups Dots Model, with groups as circles and dots to represent the counters inside of the group.  I put an outer circle around the entire drawing to represent the whole amount.  Number sentences are written using the format of groups x dots = whole amount.    

To define the group, I label the group circle with a small letter g and the dot inside with a d. The outer circle is labeled as the whole with a w.  I choose numbers that are not doubles at this point to easily identify the group number and the number of objects in the number sentence. 

As models are drawn, I ask the question, What is the equation that matches this model?  I write the multiplication number sentence under each.  For example, five groups of four items is written as 5 x 4 = 20.  Critically, I do not write the related fact because it would not match the diagram.    

The Common Core Math Practices require students to create models of mathematics.  I teach this type of mental model because it can be used by students with any type of writing instrument. Our state testing allows students to use scratch paper but not the hands on manipulative, and this mental model provides them with a strategy for solving multiplication problems. 

Let's Draw Together

15 minutes

Students record the diagrams in their math journal for reference later in the lesson. 

Every new diagram gets its own number sentence using the format of groups x dots = whole amount using the labels for each part.

It is important to repeat, rather than give one example, so the students have several with different numbers and labels. My examples are created so that they scale from easier to more complex models.  I like to start with one of the factors being two or five.  I then choose number sentences such as, 2 x 3, 2 x 4, 5 x 3, 5 x 2, etc.  

As each model is drawn, I ask students, How is the factor 3 shown in the model? What is represented by the small circles?  Where do you see the quantity for the second factor?

I also avoid doubles at this time to make sure the students have the structure of groups and items within a group, and using doubles can confuse them.  As students demonstrate success, I increase the difficulty of the multiplication factors. 

Important questioning to determine student understanding should include, "What would happen to the model if you switched the order of the factors?" This allows students to compare the differences in models for the related fact.

I also demonstrate leaving out different variables for the students to find after their diagram is drawn.  5 x ____ = 20.  In this example the students would draw five groups, and put one dot in each group until they get to twenty. 

Because the two factors are clearly represented in the mental model, I have found it important to emphasize the product is represented by the entire model of groups and dots.  Circling the entire model and labeling the product connects the number sentence to the model.  

Questioning students about the factors, symbols, and equal sign support the students' conceptual understanding of multiplication.  Questions for this include, "What does_______mean to you?" (e.g. symbol, quantity,  diagram)

Try It Yourself

15 minutes

Give students number sentences with different variables missing.  I have them record their work on blank white paper folded into fourths to divide the workspace neatly.  They write the number sentence and fill in the missing variable and label each part of the diagram.  Students use their math journals for reference, and they also work with a partner. 


Wrap Up

10 minutes

Students share out a diagram on an individual whiteboard with their team at their table and the team has to write the number sentence to match the diagram one student writes.  

Because I want to be sure students understand how a diagram changes, I ask them How does the model change if the order of the factors changes?  Please draw the new diagram.