Combinations Using Multiplication Trees

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SWBAT create a model to find the total number of combinations that can be made from groups of objects.

Big Idea

Using a model helps students develop a conceptual understanding of multiplication.

Warm Up

5 minutes

To begin this lesson, the students begin by playing a multiplication game, Write A Different Fact, used on a regular basis in our classroom to build multiplication fact fluency.  This activity involves giving each table group a factor.  Each student writes a math fact with that factor and solve it.  To stay in the game, the students must write a different fact from everyone else at their table.  This is a rapid game, and the students use whiteboards to write their facts. 


15 minutes

The intent of this lesson is for students to begin to see how different combinations of items can be determined using math trees.  This is a task typically found on our state test each spring, although it isn't in the Common Core Standards.  As the transition is made to the new assessments with Common Core, this task could be used to challenge students and extend and push them to practice beyond basic multiplication skills. 

To model this task, I use straws and pipe cleaners (bendable chenille stems) to create the trees.  I find that using these items in different colors helps students sort the information and track the different components. 

The problem I model involves food because it is something the students can make a connection to in their own lives.  The problem uses choices for lunch including sandwiches, soup, and fruit.  I write out the three choices for the sandwiches - turkey, ham, and peanut butter, the choices for the soup - chicken noodle, tomato, and vegetable, and then I write out the choices for the fruit - banana, apple, grapes, and strawberries.  These choices are based on suggestions made by students when asked in a survey the previous day.

Using the straws and pipe cleaners, I use word cards to connect each of the sandwiches to the three types of soups.  Each soup gets its own color of straw for the sandwiches.  I then continue with different colors using the pipe cleaners for each type of fruit connected to each type of soup.

Once the model is created, I show students the different combinations including: 

  • turkey, chicken noodle, banana 
  • turkey, chicken noodle, apple 
  • turkey, chicken noodle, grapes 
  • turkey, chicken noodle, strawberries
  • turkey, tomato, banana 
  • turkey, tomato, apple  
  • turkey, tomato, grapes
  • turkey, tomato, strawberries
  • turkey, vegetable, banana  
  • turkey, vegetable, apple
  • turkey, vegetable, grapes
  • turkey, vegetable, strawberries


And so on for the remaining sandwich, soup, and fruit options. 

The list continues so that students can see all options, and then I connect it to the multiplication sentence of 3 x 3 x 4 so students can see that using multiplication will determine the number of possible combinations.  

Try It On Your Own

20 minutes

The students are given the task of creating a tree for combinations of two types of ice cream, three sauce toppings, and final toppings.  This will result in a number sentence of 2 x 3 x 2 = 12 options. I chose to make the independent practice less complicated (simple multiplication) because I do not want students to become frustrated with manipulating larger numbers. I want their cognitive load to be on making an equation that represents their model.

For example, the students list chocolate and vanilla ice cream with chocolate sauce, strawberry sauce, and caramel sauce.  Their final toppings included whipped cream and cherries.   The students build the tree using different items including the straws and pipe cleaners.  Other students chose to write out their trees on paper.  


10 minutes

Following the creation of tree diagrams, students work with a partner to check if their diagrams have the same combinations, but unlike common practice of using table partners, they pair up with different person from another table group.  

The students record their tree diagrams in their math journals.  I also provide a model of the tree diagram on the screen for the student with special needs so that they could complete the activity.